T M-E E 
D F T
From Exchange-Correlation Functional Design to
Applied Electronic Structure Calculations
Rickard Armiento
Doctoral Thesis
KTH School of Engineering Sciences
Stockholm, Sweden 2005
TRITA-FYS 2005:48
ISSN 0280-316X
ISRN KTH/FYS/--05:48--SE
ISBN 91-7178-150-1
KTH School of Engineering Sciences
AlbaNova Universitetscentrum
SE-106 91 Stockholm
Sweden
Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till
offentlig granskning för avläggande av teknologie doktorsexamen i teoretisk fysik fredagen den 30 september 2005 klockan 14.00 i Oskar Kleins auditorium, AlbaNova Universitetscentrum, Kungl Tekniska högskolan, Roslagstullsbacken 21, Stockholm.
© Rickard Armiento, september 2005
Elektronisk kopia: revision B
iii
Abstract
The prediction of properties of materials and chemical systems is a key component in
theoretical and technical advances throughout physics, chemistry, and biology. The properties of a matter system are closely related to the configuration of its electrons. Computer
programs based on density functional theory (DFT) can calculate the configuration of the
electrons very accurately. In DFT all the electronic energy present in quantum mechanics is handled exactly, except for one minor part, the exchange-correlation (XC) energy.
The thesis discusses existing approximations of the XC energy and presents a new method
for designing XC functionals—the subsystem functional scheme. Numerous theoretical
results related to functional development in general are presented. An XC functional is
created entirely without the use of empirical data (i.e., from so called first-principles). The
functional has been applied to calculations of lattice constants, bulk moduli, and vacancy
formation energies of aluminum, platinum, and silicon. The work is expected to be generally applicable within the field of computational density functional theory.
Sammanfattning
Att förutsäga egenskaper hos material och kemiska system är en viktig komponent för teoretisk och teknisk utveckling i fysik, kemi och biologi. Ett systems egenskaper styrs till
stor del av dess elektrontillstånd. Datorprogram som baseras på täthetsfunktionalsteori
kan beskriva elektronkonfigurationer mycket noggrant. Täthetsfunktionalsteorin hanterar
all kvantmekanisk energi exakt, förutom ett mindre bidrag, utbytes-korrelationsenergin.
Avhandlingen diskuterar existerande approximationer av utbytes-korrelationsenergin och
presenterar en ny metod för konstruktion av funktionaler som hanterar detta bidrag—
delsystems-funktionalmetoden. Flera teoretiska resultat relaterade till funktionalutveckling
ges. En utbytes-korrelations-funktional har konstruerats helt utan empiriska antaganden
(dvs, från första-princip). Funktionalen har använts för att beräkna gitterkonstant, bulkmodul och vakansenergi för aluminium, platina och kisel. Arbetet förväntas vara generellt
tillämpbart inom området för täthetsfunktionalsteoriberäkningar.
P
This thesis presents research performed at the group of Theory of Materials, Department
of Physics at the Royal Institute of Technology in Stockholm during the period 2000–
2005. The thesis is divided into three parts. The first one gives the background of the
research field. The second part discusses the main scientific results of the thesis. The third
part consists of the publications I have coauthored. The papers provide specific details on
the scientific work. Comments on these papers and details on my contributions are given
in chapter 10.
List of Included Publications
1. Subsystem functionals: Investigating the exchange energy per particle,
R. Armiento and A. E. Mattsson, Phys. Rev. B 66, 165117 (2002).
2. How to Tell an Atom From an Electron Gas: A Semi-Local Index of Density Inhomogeneity, J. P. Perdew, J. Tao, and R. Armiento, Acta Physica et Chimica Debrecina 36,
25 (2003).
3. Alternative separation of exchange and correlation in density-functional theory,
R. Armiento and A. E. Mattsson, Phys. Rev. B 68, 245120 (2003).
4. A functional designed to include surface effects in self-consistent density functional theory,
R. Armiento and A. E. Mattsson, Phys. Rev. B 72, 085108 (2005).
5. PBE and PW91 are not the same,
A. E. Mattsson, R. Armiento, P. A. Schultz, and T. R. Mattsson, to be submitted for
publication.
6. Numerical Integration of functions originating from quantum mechanics,
R. Armiento, Technical report (2003).
v
C
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preface
v
List of Included Publications . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contents
Part I
1
2
v
vii
Background
1
Introduction
3
1.1 Units and Physical Constants . . . . . . . . . . . . . . . . . . . . . . . .
6
Density Functional Theory
7
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
3
iii
iii
The Many-Electron Schrödinger Equation
The Electron Density . . . . . . . . . . .
The Thomas–Fermi Model . . . . . . . .
The First Hohenberg–Kohn Theorem . .
The Constrained Search Formulation . . .
The Second Hohenberg–Kohn Theorem .
v -Representability . . . . . . . . . . . . .
Density Matrix Theory . . . . . . . . . .
The Kohn Sham Scheme
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7
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3.1 The Auxiliary Non-interacting System . . . . . . . . . . . . . . . . . . . 15
3.2 Solving the Orbital Equation . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 The Kohn–Sham Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4
Exchange and Correlation
21
4.1 Decomposing the Exchange-Correlation Energy . . . . . . . . . . . . . . 21
4.2 The Adiabatic Connection . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 The Exchange-Correlation Hole . . . . . . . . . . . . . . . . . . . . . . 23
vii
Contents
viii
4.4
4.5
4.6
4.7
5
Part II
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Locality . . . . . . . . . . . . . . . . . . . . .
The Local Density Approximation, LDA . . . .
The Exchange Refinement Factor . . . . . . . .
The Gradient Expansion Approximation, GEA .
Generalized-Gradient Approximations, GGAs .
GGAs from the Real-space Cutoff Procedure . .
Constraint-based GGAs . . . . . . . . . . . . .
Meta-GGAs . . . . . . . . . . . . . . . . . . .
Empirical Functionals . . . . . . . . . . . . . .
Hybrid Functionals . . . . . . . . . . . . . . .
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The GGA of Perdew and Wang (PW91) . . . . . . . . . . . . .
The GGA of Perdew, Burke, and Ernzerhof (PBE) . . . . . . . .
Revisions of PBE (revPBE, RPBE) . . . . . . . . . . . . . . . .
The Exchange Functionals of Becke (B86, B88) . . . . . . . . .
The Correlation Functional of Lee, Yang, and Parr (LYP) . . . .
The Meta-GGA of Perdew, Kurth, Zupan, and Blaha (PKZB) . .
The Meta-GGA of Tao, Perdew, Staroverov, and Scuseria (TPSS)
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Scienti c Contribution
41
41
42
42
43
43
44
45
47
General Idea . . . . . . . . . . . . . . . . . .
Designing Functionals . . . . . . . . . . . . .
Density Indices . . . . . . . . . . . . . . . .
A Straightforward First Subsystem Functional
A Simple Density Index for Surfaces . . . . .
An Exchange Functional for Surfaces . . . . .
A Correlation Functional for Surfaces . . . . .
Outlook and Improvements . . . . . . . . . .
Definition of the Mathieu Gas Model . . .
Electron Density . . . . . . . . . . . . . . .
Exploring the Parameter Space of the MG .
Investigation of the Kinetic Energy Density
29
30
32
33
35
36
37
37
38
38
41
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The Mathieu Gas Model
8.1
8.2
8.3
8.4
24
25
25
26
29
Subsystem Functionals
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
8
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A Gallery of Functionals
6.1
6.2
6.3
6.4
6.5
6.6
6.7
7
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Functional Development
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
6
The Exchange-Correlation Energy Per Particle
Separation of Exchange and Correlation . . .
The Exchange Energy . . . . . . . . . . . . .
The Correlation Energy . . . . . . . . . . . .
47
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49
50
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54
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58
ix
9
A Local Exchange Expansion
9.1
9.2
9.3
9.4
9.5
9.6
61
The Non-existence of a Local GEA for Exchange . .
Alternative Separation of Exchange and Correlation
Redefining Exchange . . . . . . . . . . . . . . . .
An LDA for Screened Exchange . . . . . . . . . . .
A GEA for Screened Exchange . . . . . . . . . . .
The Screened Airy Gas . . . . . . . . . . . . . . .
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61
62
62
63
64
65
10 Introduction to the Papers
67
Acknowledgments
71
A Units
A.1
A.2
A.3
A.4
73
Hartree Atomic Units . . . . . . .
Rydberg Atomic Units . . . . . . .
SI and cgs Units . . . . . . . . . .
Conversion Between Unit Systems
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73
74
74
74
Bibliography
77
Index
83
Part III Publications
87
Paper 1: Subsystem functionals in density functional theory: Investigating
the exchange energy per particle
89
Paper 2: How to Tell an Atom From an Electron Gas: A Semi-Local Index
of Density Inhomogeneity
109
Paper 3: Alternative separation of exchange and correlation in densityfunctional theory
117
Paper 4: Functional designed to include surface effects in self-consistent
density functional theory
125
Paper 5: PBE and PW91 are not the same
133
Paper 6: Numerical integration of functions originating from quantum mechanics
143
If we wish to understand the nature of reality, we have an inner
hidden advantage: we are ourselves a little portion of the universe
and so carry the answer within us.
Jacques Boivin
Part I
B
1
Chapter 1
I
The whole is greater than the sum of its parts.
The part is greater than its role in the whole.
Tom Atlee
The interplay of theoretical and experimental physics during the last century has led to a
successful model for the composition and interaction of matter on a very small scale. In
1897 Thomson discovered the negatively charged electron. The experiments of Rutherford
and coworkers in 1909 lead to the conclusion that matter consists of separated positively
charged nuclei. Following this, in 1913 Bohr created a successful model for the building blocks of matter as composed by nuclei orbited by electrons subject to certain rules.
During the 1920s Heisenberg and Schrödinger were two key players in the construction
of a mathematical framework that provides a precise mathematical description of the behavior of the particles, quantum mechanics. The scientific progress following the work
of these pioneers and others has resulted in a conceptual view of matter as composed of
subatomic particles, which interact according to the laws of quantum mechanics to form
atoms (cf. Fig. 1.1).
It is often observed how the combination of a large number of small parts gives a
resulting compound system that shows properties not evident from the properties of the
individual parts. This is known as the phenomenon of emergent properties. In the present
context, even though we have detailed knowledge from quantum mechanics about the
physics governing electrons and nuclei, a piece of solid material has properties that are
very much non-obvious and sometimes even outright surprising (e.g., high temperature
superconductivity).
Modern computers provide a seemingly straightforward approach to handle emergent
properties. A brute force computational physics approach would be to simulate a system in
a computer program by using the detailed quantum mechanical mathematical description
of a large number of nuclei, electrons, and their interactions. However, even for a few
dozen atoms this approach results in a computer program which will take much too long
3
4
Chapter 1. Introduction
Figure 1.1. (left) A conceptual sketch of the atomic model: a positively charged nucleus is surrounded by an electronic cloud built up from individual electrons. (right) A conceptual sketch
of a few atoms in a crystalline solid. The solid curve illustrates the idea that some individual electrons may be weakly bound and travel through the material. These are only conceptual sketches,
and not to scale. Real electronic orbitals are usually more complicated than illustrated here.
time to run even for an extremely powerful computer. One can even question whether such
a brute force computational approach is a scientifically legitimate concept. A reasonable
simulation of a system with as few as 1000 electrons would require the computer’s memory
to keep track of more information bits than the number of particles in the universe. The
exponential growth of the required memory with the number of electrons has been referred
to as the Van Vleck catastrophe 1 . Note that a calculation for a full simulation of just a few
grams of carbon would involve more than 1023 electrons. Hence, it is misguided to claim
that the knowledge of the basic laws of quantum mechanics makes all emergent properties
of matter understood.
It is thus obvious that refined mathematical models are needed for all but the most
trivial computational studies of material properties. One such refinement is the density
functional theory (DFT) 2,3 . In DFT the quantum mechanical theory is reformulated to
model the electrons as a compound cloud, an electron gas. The reformulation focuses on
the density of electrons, rather than on individual electrons (cf. Fig. 1.2). The benefit
of the electron gas view is that no matter how many electrons are involved, the density
of electrons remains a three-dimensional quantity (a ‘field’). In contrast, to keep track
of all individual electrons, a quantity of a dimensionality proportional to the number of
electrons is needed. The price paid for the simpler description of DFT is that one loses the
ability to describe the properties of the system that are related to the motion of individual
electrons. For other properties, the DFT picture is as theoretically fundamental as the view
of individual interacting electrons 1,2 .
Energy is a fundamental property in physics. Physical mechanisms induce ‘changes’ in
a system’s state, and all such changes involve some kind of energy transfer. Hence, a way to
describe the system’s energy as a function of its state is also a description of the underlying
physical mechanisms. Such an energy function shows what changes the system is likely to
5
Figure 1.2. A conceptual sketch of the DFT view of a crystalline solid; there are no individual
electrons, but only a three-dimensional density of electrons.
Energy related
quantity
state
the ground state
Figure 1.3. A schematic sketch of how the ground state of a system is found as a stable minimum
of an energy related quantity. What specific energy related quantity is used depends on what
environment the system is placed in.
undergo, and what state the system naturally prefers in an external environment; i.e., its
ground state. The ground state is the state where no change is induced, which means that it
is a stable minimum of an energy-related quantity (see Fig. 1.3). The accurate computation
of the energy of a matter system therefore is of much interest, and is the focus of this thesis.
The DFT reformulation of quantum mechanics can be transformed into a form suitable for computer calculations of a system’s energy 3 . The most difficult quantum mechanical behavior of the interaction of electrons is put into a quantity called the exchangecorrelation energy functional, Exc . This quantity is usually of minor magnitude, but except
for some fundamental assumptions, it turns out to be the only part that has to be approximated relative to a brute force quantum mechanical solution. Thus, all that is ‘lost’ in
a DFT calculation is condensed into the exchange-correlation energy functional. Hence,
increasingly accurate approximations to Exc provide a better and better description of
matter.
The scientific contribution of this thesis is focused on the development and testing of
an approach for the construction of more accurate exchange-correlation functionals. The
main underlying idea is that a system can be split into several regions. In each region
Chapter 1. Introduction
6
General idea of dividing a system
into subsystems
R1
R3
Original Kohn and
Mattsson approach
Edge
R5
Interior
R2
R4
Figure 1.4. An illustration of the idea that a system can be divided into subsystems, where different functionals are used in the different regions R1 , R2 , ....
a different approximation of Exc can be used. Each such approximation can then be
specifically designed for the part of the system it is applied to. This idea is based on
the locality, or ‘near-sightedness’, of a system of electrons 4,5 . Kohn and Mattsson have
suggested the possibility to split a system into specific interior and edge parts 5 . The here
discussed generalized approach is illustrated in Fig. 1.4.
The main scientific contributions presented in this thesis can be summarized as:
• The theoretical development of a scheme for functional development in density
functional theory based on the partitioning of the electron density into regions with
different properties—the subsystem functional scheme.
• The creation of a simple first-principles exchange-correlation energy functional, using the subsystem functional approach. The functional uses a targeted treatment for
electron density ‘surfaces’.
• Discussion and development of density indices as a means for automatic classification of regions of an electron density.
• The development and study of an advanced DFT model system, the Mathieu gas.
• The construction of a ‘local’ gradient expansion approximation.
1.1
Units and Physical Constants
This thesis uses SI units. See Appendix A for more information on unit systems. The
following physical constants are used:
Electron charge
Electron mass
Planck’s constant
Permittivity of free space
Bohr radius
Speed of light
ec
me
h̄
0
a0
c
≈ 1.6022 · 10−19 C
≈ 9.1094 · 10−31 kg
≈ 1.0546 · 10−34 J s
≈ 8.8542 · 10−12 C2 /(N m)
= 4π0 h̄2 /(me e2c ) ≈ 5.2918 · 10−11 m
= 2.99792458 · 108 m/s
Chapter 2
D F T
I am your density! I mean, your destiny.
George McFly in the movie ‘Back to the Future’
This chapter introduces the theoretical framework of density functional theory (DFT).
We start from the Schrödinger equation and rewrite the problem of electron interactions
into its DFT equivalent. In the end, the ground state electronic energy of a system of
interacting electrons is shown to be given by a minimization over electron densities of a
total electronic energy functional. There are many textbooks and other sources treating
DFT, for example Refs 6–9.
2.1
The Many-Electron Schrödinger Equation
Our starting point is the time independent non-relativistic Schrödinger equation that describes a system of matter. It is the eigenvalue equation for the total energy operator, the
Hamiltonian Ĥ. The equation defines all states Ψ of the system and their related energies
E:
ĤΨ = EΨ.
(2.1)
In the usual model of matter, with electrons in the presence of the positively-charged
nuclei, it is common to assume that the Schrödinger equation can be separated into independent electronic and nucleonic parts. This is the Born-Oppenheimer approximation 10 ,
which is valid when the electrons reach equilibrium on a time scale that is short compared
to the time scale on which the nuclei move. The approximation separates the states into
independent states for nuclei Ψn and electrons Ψe , with energies En and Ee . The Hamiltonian is split into corresponding terms, Ĥn and Ĥe . The interaction energy between
nuclei and electrons is placed in the electronic part. The result is
Ψ = Ψn Ψe ,
Ĥ = Ĥn + Ĥe ,
7
(2.2)
Chapter 2. Density Functional Theory
8
Ĥn Ψn = En Ψn ,
Ĥe Ψe = Ee Ψe .
(2.3)
(2.4)
The nucleonic part is uncomplicated to handle. Our concern in the following therefore
is the electronic part, which describes interacting electrons that moves in a static external
potential created by the charged nuclei.
The energy operator of the electronic part Ĥe is conventionally split into a sum of
three contributions: the kinetic energy of the electrons T̂, the internal potential energy (the
repulsion between individual electrons) Û, and the external potential energy (the attraction
between the electrons and nuclei) V̂. It is also common to use F̂ for the total internal
electronic energy, i.e., T̂ + Û:
Ĥe = T̂ + Û + V̂ = F̂ + V̂.
(2.5)
Let the spatial location of electron i be denoted ri ; its spin coordinate σi = ↑ or ↓; the
total number of electrons in the system N ; and the static external potential, which originates
from the nuclei, v(r). We combine position and spin coordinates in one quantity xi =
(ri , σi ). In a wave-function based approach the system’s electronic states are described as
many-electron wave-functions Ψe = Ψe (x1 , x2 , ..., xN ), subject to two conditions; they
must be normalized
ZZ
hΨe |Ψe i =
Z
2
|Ψe | dx1 dx2 ...dxN = 1,
(2.6)
Ψe (..., xi , ..., xj , ...) = −Ψe (..., xj , ..., xi , ...).
(2.7)
...
and antisymmetric
The state Ψe we are interested in is the ground state wave-function Ψ0 of energy E0 . It is
the solution to the electronic part of the Schrödinger equation Eq. (2.4) that has the lowest
energy.
The contributions to the Hamiltonian can be explicitly expressed as
N
h̄2 X 2
∇ ,
2me i=1 i
2 X
N
ec
1
Û =
,
4π0 i<j |ri − rj |
T̂ = −
V̂ =
N
X
i=1
v(ri ).
(2.8)
(2.9)
(2.10)
2.2. The Electron Density
9
The electronic energy Ee can be obtained as the expectation value of the Hamiltonian,
Ee = hΨe |Ĥe |Ψe i = hΨe |T̂ + Û + V̂|Ψe i = T + U + V =

2 X
2 X
ZZ
Z
N
N
h̄
|Ψe |2
ec
... −
+
Ψ∗e ∇2i Ψe +
2me i=1
4π0 i<j |ri − rj |
!
N
X
2
+
|Ψe | v(ri ) dx1 dx2 ...dxN .
(2.11)
i=1
Here T , U and V are introduced as the individual scalar expectation values of the corresponding operators.
The Rayleigh–Ritz variational principle 11,12 offers a way to solve the electron energy
problem to obtain the ground state wave-function Ψ0 and energy E0 . The ground state
electronic energy is found through a search for the many-electron wave-function that minimizes the energy expectation value in Eq. (2.11),
E0 = minhΨ|Ĥe |Ψi,
Ψ
has minimum for Ψ = Ψ0 ,
(2.12)
where the search is constrained by the normalization and anti-symmetric conditions of
Eqs. (2.6) and (2.7). A direct application of the Rayleigh–Ritz variational method involves
a search for the minimizing wave-function in the space of functions of a dimensionality
proportional to the number of electrons in the system. In the following we will instead take
the DFT approach and rewrite the problem to involve a search over only three-dimensional
functions, i.e., electron densities.
2.2
The Electron Density
The electron density n(r) is defined as the number of electrons per volume at the point r
in space. It is a physical quantity—it can (at least in theory) be measured. The integral of
the electron density gives the total number of electrons,
Z
n(r)dr = N.
(2.13)
The relation between n(r) and the many-electron wave-function Ψe is
ZZ
Z
2
n(r) = N
... |Ψe (rσ1 , x2 , ..., xN )| dσ1 dx2 ...dxN .
(2.14)
The expression on the right hand side looks similar to the wave-function normalization
integration Eq. (2.6) but without one of the spatial integrals, and thus one coordinate is
left free. Here we have arbitrarily removed the integration over the first coordinate r1 ,
but it can be replaced by any of the spatial integrals, due to the antisymmetric property
Chapter 2. Density Functional Theory
10
of the wave-function Eq. (2.7). The requirement that the wave-functions are normalized
Eq. (2.6) guarantees that the integral of the electron density is N as in Eq. (2.13).
If one looks at the three terms in the expression for the electronic energy Eq. (2.11),
one sees that the term for the external potential V is easily rewritten in terms of the density,
ZZ
V =
...
Z X
N
|Ψe |2 v(ri ) dx1 dx2 ...dxN =
i=1
Z
N Z
1 X
n(ri )v(ri )dri = n(r)v(r)dr.
=
N i=1
(2.15)
The other two terms of the electronic energy Eq. (2.11) are not as easy to rewrite. In
the kinetic energy term T , the derivative operator between the wave-functions prevents
rewriting the integrand on the form |Ψe |2 as needed to turn the term into an expression of
the electron density. In the term of the internal potential energy U , the particle positions
in the denominator preclude a direct term by term integration.
A functional is an object that acts on a function to produce a scalar. From the way
the potential energy term V was rewritten in Eq. (2.15), it is an explicit potential energy
functional V [n] of the electron density. This and other functionals with the electron density n(r) as arguments are called density functionals. The other terms in the electronic
energy Eq. (2.11) are not on explicit density functional form, but can at least be written
as functionals of the many-electron wave-function Ψe ,
Ee = T [Ψe ] + U [Ψe ] + V [v, n] = F [Ψe ] + V [v, n].
(2.16)
At this point a question central to DFT enters: is it possible to also rewrite the total internal
electronic energy F [Ψe ] as a density functional F [n]? If such a functional exists, it is a
universal functional in that it is independent of the external potential. The same F [n] may
be used in any electronic energy problem. The question of the existence of an F [n] functional
will be considered in the following.
2.3
The Thomas Fermi Model
A rather direct approach to answer the question if there exists some, at least approximative, density functional for the total internal electronic energy F [n] is to see if it can be
constructed from basic physics ideas. Early attempts to create such an approximation
were made by Thomas and Fermi 13–16 . They used some assumptions about the distribution and the interaction between electrons to approximate the kinetic energy. The electron density in each space point is set equal to a number of electrons in a fixed volume,
n(r) = ∆N/∆V . A system of ∆N free non-interacting electrons in an infinite-well model
of volume ∆V then gives an expression for the kinetic energy per volume. The continuity
limit is then taken, ∆V → 0. The result is integrated over the whole space to give the
2.4. The First Hohenberg–Kohn Theorem
11
approximate Thomas–Fermi functional for the total kinetic energy TT F [n],
2 Z
3
h̄
2 2/3
T ≈ TT F [n] = (3π )
n5/3 (r)dr.
5
2me
(2.17)
Furthermore, the electrostatic energy of a classical repulsive gas J[n] can be used as a simplistic
approximation of the internal potential energy U ,
2 ZZ
ec
n(r1 )n(r2 )
1
dr1 dr2 .
(2.18)
U ≈ J[n] =
2 4π0
|r1 − r2 |
The result is the Thomas–Fermi model:
Z
Ee ≈ TT F [n] + J[n] +
n(r)v(r)dr.
(2.19)
The Thomas–Fermi approximation to the internal electronic energy thus is
F [n] ≈ TT F [n] + J[n].
2.4
(2.20)
The First Hohenberg Kohn Theorem
The early efforts to find and use internal electronic energy functionals F [n] by Thomas and
Fermi, and extensions along the same ideas, were all based on ‘reasonable’ approximations.
It is a great conceptual difference between such rather heuristic approaches and the more
rigorous theoretical framework that followed the work of Hohenberg and Kohn 2 . Two
famous theorems proved in the work of Hohenberg and Kohn will be examined in the
following.
The first Hohenberg–Kohn theorem tells us that the ground state electron density n(r)
determines the potential of a system v(r) within an additive constant (which only sets the
absolute energy scale). Since the original proof is enlightening and simple, it will be reproduced here. Assume two different system potentials, va (r) and vb (r). If they differ by
more than an additive constant, they must give rise to two different ground states in the
Schrödinger equation, Ψa and Ψb . Let us assume the states to be non-degenerate and that
they both have the same electronic density n(r). Let Ĥa be the Hamiltonian for the system with potential va (r). Use the Rayleigh–Ritz variational principle and the functional
notation of Eq. (2.16) to get
Ea = hΨa |Ĥa |Ψa i < hΨb |Ĥa |Ψb i = F [Ψb ] + V [va , n],
(2.21)
and in the same way,
Eb < F [Ψa ] + V [vb , n].
(2.22)
If the two equations are added, the F and V terms on the right hand side can be recollected
into E terms,
Ea + E b < E b + Ea .
(2.23)
Chapter 2. Density Functional Theory
12
The last relation is a contradiction. The logical implication is: for systems without degenerate ground states, two different potentials cannot have the same ground state electron
density.
The key point with the proof is that a ground state electron density uniquely determines the corresponding external potential of the system. This means all ground state
properties of the system are also consequently determined, since in theory anything can be
calculated from the external potential. Hence, we arrive at the main conclusion of the first
Kohn–Sham theorem: the electron density determines all ground state properties of a system.
The ground state wave-function is also a ground state property of the system and can
therefore be considered to be a functional of the ground state density Ψ0 [n]. The existence
of the total energy functional Ee [n] and an internal electronic energy functional F [n]
directly follows as
Ee [n] = h Ψ0 [n] |Ĥe | Ψ0 [n] i
(2.24)
and
F [n] = F [Ψ0 [n]].
(2.25)
The notation Ψ0 [n] explicitly points out that the ground state is assumed to be nondegenerate (because the notation does not specify which one of the degenerate Ψ0 the
functional refers to). It is not very hard to reformulate the proof to lift the requirement
of a non-degenerate ground state 17 , roughly by reasoning in terms of ‘any one of the
degenerate ground state wave-functions’.
2.5
The Constrained Search Formulation
After the initial work of Hohenberg and Kohn it was discovered how an explicit but somewhat artificial definition of the internal electronic energy F [n] can be constructed 18–21 :
F [n] = min hΨ|T̂ + Û|Ψi,
Ψ→n
(2.26)
where the minimum is taken over all many-electron wave-functions Ψ with the specified
electron density n. The existence of an explicit definition simplifies the derivation of the
fundamental theorems. This formulation of DFT is called the constrained search formulation. It does not require any assumptions of a non-degenerate ground state.
2.6. The Second Hohenberg–Kohn Theorem
2.6
13
The Second Hohenberg Kohn Theorem
The second Hohenberg–Kohn theorem reworks the Rayleigh–Ritz variational principle into
a DFT variational principle for the total energy combination† F [n] + V [v, n]. The constrained search formalism makes the proof straightforward. The Rayleigh–Ritz variational
principle Eq. (2.12) can be split into two separate minimizations,
E0 = minhΨ|Ĥe |Ψi = min min hΨ|T̂ + Û + V̂|Ψi = min(F [n] + V [v, n]),
Ψ
n
Ψ→n
n
(2.27)
where the notation is as explained in Eq. (2.26). The many-electron problem thus has been
rewritten into what looks like a straightforward minimization in a three-dimensional quantity n(r), yet no approximations relative to a solution of the many-electron Schrödinger
equation Eq. (2.4) have been made. The problem left is ‘only’ that the definition of F [n] in
Eq. (2.26) is very unpractical. It re-introduces a minimization over many-electron wavefunctions that we set out to avoid. Hence, if one were to perform a constrained search in
practice, one would not gain anything over a brute force wave-function based approach.
In conclusion, the results just described provide a formal footing for DFT in that the existence and possible use of a universal internal electronic energy functional F [n] have been
established. But so far we have presented little hint on how to actually obtain it. There is
no obvious way to create a practical ‘approximative constrained search’.
2.7
v -Representability
The original work of Hohenberg and Kohn 2 assumed that the search for the density that
minimizes the energy was only over densities that correspond to existing external potentials. A density that has such a corresponding external potential is called v -representable.
The problem is that there is no known practical way to restrict a search to be over only
v -representable densities.
In the constrained search formulation as presented in Eqs. (2.26) and (2.27) the electron densities are not assumed to be v -representable. The Rayleigh–Ritz variational principle is defined to work for all N -electron antisymmetric wave-functions, so the only requirement on the electron density is that it must correspond to such a wave-function; it
must be N -representable. It has been shown that any ‘reasonable’ electron density fulfills
the N -representability requirement 22 .
† Note the formal difference between F [n] + V [v, n], and the form shown to exist in Eq. (2.24), E [n] =
e
F [n] + V [v(r, [n]), n]. The former has an explicit dependence on the real external potential v(r) of the system,
whereas the latter uses the external potential that corresponds to the inserted density, v(r, [n]). These two
external potentials are the same only when the true ground state electron density is used. It is obvious that we
need to use F [n] + V [v, n], and not Ee [n], in a variational principle: Consider two different electron densities,
n(r) and ñ(r). If n is the exact density and one uses ñ as a trial density one expects the variational principle to
state that E[ñ] > E[n], since all trial densities should give higher energies than the true density does. But in
a different problem ñ may be the exact density, and if one now happens to use n as a trial density, one would
expect E[ñ] < E[n]. A variational principle for F [n] + V [v, n] does not suffer from this fallacy; the explicit
dependence on the real external potential v(r) differentiates between the two cases.
Chapter 2. Density Functional Theory
14
The solution to the v -representability problem presented by the constrained search
formulation means that there is no formal problem with the Hohenberg–Kohn theorems.
The issue of v -representability is nevertheless still relevant in the context of more practical
definitions of the F [n] functional than the one in Eq. (2.26). Formally one would need
to verify the behavior of approximations of F [n] for non v -representable densities (e.g.,
if they approximate the constrained search F [n] for such densities), but this issue has not
been reported to cause practical problems for DFT calculations.
It is still an active field of research to determine the criteria for a density to be v representable.
2.8
Density Matrix Theory
It has been established above that the internal electronic energy F = T + U can be reformulated as a density functional, but it is not obvious how to do so. As a first step, density
matrices can be used to express it as a functional of simpler quantities than the full electronic wave-function Ψe . The relation between the electron density and the many-electron
wave-function in Eq. (2.14) can be generalized into the first order spinless density matrix,
ZZ
Z
0
n1 (r , r) = N
... Ψe (r0 σ1 , x2 , ..., xN )Ψ∗e (rσ1 , x2 , ..., xN )dσ1 dx3 ...dxN . (2.28)
The kinetic energy can now be expressed as 6
2 Z
2
h̄
T [n1 ] = −
∇r n1 (r0 , r) r0 =r dr.
2me
(2.29)
Another possible generalization of the density is the pair density
ZZ
Z
(N − 1)
2
... |Ψe (rσ1 , r0 σ2 , x3 , ..., xN )| dσ1 dσ2 dx3 ...dxN .
n2 (r0 , r) = N
2
(2.30)
The internal potential energy becomes 6
2 ZZ
ec
n2 (r, r0 )
U [n2 ] =
drdr0 .
(2.31)
4π0
|r − r0 |
One may think the hard work involved in the construction of pure density functionals could be avoided if one instead keeps the density matrices and uses a density matrix
minimization principle. The problem with such a minimization is that any trial density
matrix must correspond to an antisymmetric many-electron wave-function Ψe , i.e., the
trial density matrices must be N -representable. It turns out to be very hard to restrict the
search to be over only N -representable density matrices.
Chapter 3
T K–S S
The real voyage of discovery consists not in seeking new
landscapes, but in having new eyes.
Marcel Proust
In the previous chapter we arrived at a general minimization principle for finding the
ground state electronic energy of a system. The scheme was not useful in practice, since
only an abstract definition of the functional for the kinetic and interaction energies of the
electrons F [n] was available. In the present chapter we discuss the elaborate scheme of
Kohn and Sham 3 to compute the dominating part of F [n].
3.1
The Auxiliary Non-interacting System
Soon after the original Hohenberg–Kohn paper on DFT, Kohn and Sham 3 proposed a
method for computing the main contribution to the kinetic energy functional to good
accuracy, the Kohn–Sham method. Their idea was to rewrite the system of many interacting
electrons as a system of non-interacting Kohn–Sham particles. These particles behave as
non-interacting electrons† .
The first step is to divide the internal electronic energy functional F [n] into three parts,
F [n] = Ts [n] + J[n] + Exc [n].
(3.1)
Here Ts [n] is the non-interacting kinetic energy, i.e., the kinetic energy of a system of noninteracting Kohn–Sham particles with particle density n; J[n] is the electrostatic energy
of a classical repulsive gas as it was defined in the section about Thomas–Fermi theory,
† With non-interacting electrons we refer to fictitious particles that do not interact with each other by
Coulomb forces, i.e., the internal potential energy Û = 0. The particles are still regarded as indistinguishable fermions. The indistinguishability of the Kohn–Sham particles is further commented on in relation to
Eq. (3.11).
15
Chapter 3. The Kohn–Sham Scheme
16
Total electronic energy
Non−interacting kinetic energy
Internal energy of classic repulsive gas
Electron−nuclei interaction
Remaining ’difficult’ part
E e = Ts + J + V +E xc
?
Figure 3.1. The different contributions to the energy in the Kohn–Sham scheme.
Eq. (2.18); and Exc [n] is the exchange-correlation energy, which is defined to make the
relation exact;
Exc [n] = F [n] − Ts [n] − J[n].
(3.2)
Hence, Exc [n] is the component of F [n] which takes care of the non-classical part of the
potential and kinetic energy related to electron interactions. The electronic energy is now
divided into four parts, cf. Fig. 3.1.
The DFT variational principle for the ground state electronic energy E0 in Eq. (2.27)
can be expressed in the new quantities,
E0 = min(Ts [n] + J[n] + Exc [n] + V [v, n]).
n
(3.3)
In the language of variational calculus this energy minimization can be rewritten as a
stationary condition† for the electron density
δTs [n] δExc [n] δJ[n] δV [v, n]
+
+
+
= 0.
δn
δn
δn
δn
(3.4)
Now we look at what the above relations correspond to when DFT is applied to the system of the non-interacting Kohn–Sham particles. The DFT variational principle becomes
Es = min(Ts [n] + V [veff , n]),
n
(3.5)
where we use Es as the ground state energy of the system of Kohn–Sham particles and
veff (r) is the potential in which they move. The stationary condition becomes
δTs [n] δV [veff , n]
+
= 0.
δn
δn
† The
(3.6)
way the minimization is expressed in the formalism of variational calculus as a stationary condition
has some parallels to the search of a minimum of an ordinary function. It is well known how the latter leads to
the condition that the derivative should be zero at the point of extremum.
3.2. Solving the Orbital Equation
17
A comparison between the stationary conditions of the interacting and non-interacting
systems, Eqs. (3.4) and (3.6), shows that the same stationary n(r) is described if
δV [veff , n]
δExc [n] δJ[n] δV [v, n]
=
+
+
.
δn
δn
δn
δn
The functional derivatives are evaluated on both sides to give
2 Z
ec
n(r0 )
veff (r) = vxc (r) +
dr0 + v(r),
4π0
|r − r0 |
(3.7)
(3.8)
where the exchange-correlation potential vxc (r) is defined as
vxc (r) =
δExc [n]
.
δn
(3.9)
The definition of veff Eq. (3.8) is inserted into the expression for the V [v, n] functional
Eq. (2.15) to derive a relation between the energies of the two systems. By identifying the
terms in the relation, the result can be written
E0 = Es − J[n] + Exc [n] − V [vxc , n].
(3.10)
In conclusion, it has been established that the non-interacting Kohn–Sham particle system
with veff as given in Eq. (3.8) has the same ground state density as the system of fully interacting electrons. The energies of the two systems are closely related through Eq. (3.10).
An auxiliary view of a system of interacting electrons is therefore promoted—the view of
non-interacting Kohn–Sham particles in an effective potential veff . The potential veff is
formally expressed in Eq. (3.8) as a functional derivative of the unknown, difficult, part of
the energy that corresponds to non-classical electron interactions, the exchange-correlation
energy Exc . The non-interacting auxiliary view is a central result for the Kohn–Sham
scheme. In the following we will explore how to solve the auxiliary problem, and show
that the non-interacting kinetic energy Ts [n] can be calculated with much less effort than
needed in a brute force constrained search.
3.2
Solving the Orbital Equation
The point of the previous section was that one can perform a minimization of the energy
of an auxiliary problem of non-interacting Kohn–Sham particles Eq. (3.5) instead of a
many-electron energy minimization Eq. (3.3). The non-interacting particle problem can
be handled in a very direct way, through the explicit solution of the (in this case) separable
Schrödinger equation. Separation leads to the Kohn–Sham orbital equation, which determines the one-particle Kohn–Sham orbitals φi (r) and the Kohn–Sham orbital energies i ,
2 h̄
−
∇2 φi (r) + veff (r)φi (r) = i φi (r).
(3.11)
2me
Chapter 3. The Kohn–Sham Scheme
18
Actual one-particle wave-functions are constructed as combinations of position dependent
parts and spin functions, ψi (r, σ) = φi (r)χi (σ). The ground state √
wave-function of the
†
many-independent particle system is a Slater determinant Ψ = 1/ N ! detij ψj (ri , σi ).
The many-particle wave function is inserted in the usual expression for the electron density
Eq. (2.14) to give the particle density,
X
n(r) =
|φi (r)|2 ,
(3.12)
i
where the sum is taken over all occupied spin-states i (i.e., two per fully occupied orbital).
For the usual zero temperature non–spin-polarized case the count of the occupied states
starts with the orbitals of lowest energy and progress upwards until all N electrons have
been accounted for.‡ The total energy of the system is
X
Es =
i .
(3.13)
i
Common matrix methods can be used to solve the Kohn–Sham orbital equation.
Equations (3.8)–(3.13) are the Kohn–Sham equations, which are at the heart of any
Kohn–Sham based DFT computer program. These equations cannot be straightforwardly
solved from top down, because veff in Eq. (3.9) requires the unknown electron density.
However, in the previous section is was argued that the existence of a minimization principle over densities Eq. (3.5) means that the correct electron density n(r) fulfills a stationary
condition, Eq. (3.6). Such a stationary n(r) can be found by an iterative scheme which
works towards self-consistency. First, start with a trial density constructed in some way.
Then repeat these steps until self-consistency is achieved:
1. Insert the density in Eq. (3.9) to produce an effective potential.
2. Solve the Kohn–Sham orbital equation Eq. (3.11).
3. Compute a new Kohn–Sham particle density from the Kohn–Sham orbitals through
Eq. (3.12).
The result is an electron density n(r) that is likely to be the stationary n(r) that minimizes
Es in Eq. (3.5). A schematic outline of the procedure is shown in Fig. 3.2.
† As previously noted, we take the Kohn–Sham particles to behave similar to non-interacting but indistinguishable electrons. The many-electron ground state wave-function for indistinguishable electrons is known to
be in Slater determinant form, and thus the same applies to the Kohn–Sham particles. However, with the internal potential energy Û = 0 there is in fact no difference between the Hamiltonians obtained when either
a Slater determinant or just a product wave-function
are inserted. Furthermore, the density for distinguishable
P
‘independent’ particles in orbitals φi is also i |φi |2 . In the present context it therefore turns out not to be an
important distinction whether the Kohn–Sham particles are regarded as indistinguishable or not. Terminology
belonging to both views are present in literature, e.g., compare Refs. 6 and 8.
‡ It has been discussed that there may exist an interacting electron system with a density that cannot be
constructed as the lowest N eigenstates of a system of non-interacting Kohn–Sham particles 20 , but there are no
reports that such densities generate problems in actual DFT calculations. Furthermore, for practical reasons it is
common in computer implementations to occupy the eigenstates according to a Fermi–Dirac distribution for a
small temperature rather than strictly using the lowest eigenstates.
3.3. The Kohn–Sham Orbitals
19
Start with guessed density.
Repeat until self consistency (input density = output density).
1. Construct new effective potential
veff (r) (depends on density)
‘hiding’ the many−electron interactions.
2. Matrix−solve a non−interacting
particle equation
2
2
( h
+ v eff ) φ = E φ .
2me
3. The orbitals give new density.
Figure 3.2. Schematic representation of the self-consistent solution of the Kohn–Sham equations.
3.3
The Kohn Sham Orbitals
It is common to think about bonding between atoms and molecules in terms of the interaction between electrons in electronic orbitals; but there are no such orbitals inherent
to the many-electron system itself. The single-particle orbitals referred to are introduced
as a component of the Hartree–Fock§ picture of electronic structure. The Kohn–Sham
scheme provides an alternative, and in theory exact, orbital theory.
Despite the possibility of regarding the Kohn–Sham method as an exact orbital theory,
it is important to realize that the orbitals originate from a system auxiliary to the manyelectron system. The connection between the interacting and non-interacting systems is
only through the systems having the same particle density. In particular, the auxiliary
system has not been created with any ‘correct’ orbital description of the many-electron
system in mind. Thus one should not anticipate any strict physical significance of the
orbitals. In the same way one should not expect any simple interpretation of the Kohn–
Sham orbital energies i in Eq. (3.11). It has long been believed that the energy of the
highest occupied Kohn–Sham orbital is the negative of the exact many-electron ionization
energy 23,24 , but more recently this claim has been called into question 25–29 .
Even though a simple physical interpretation of the Kohn–Sham orbitals and energies
is missing, it is still quite common to take them as approximations for the Hartree–Fock
orbitals and energies. The results are usually surprisingly good. Still, one should keep
in mind that to comment on DFT’s relative ‘success’ or ‘failure’ based on how well the
Kohn–Sham orbitals reproduce the Hartree–Fock orbital band structure is theoretically
misguided. It is worth pointing out that DFT’s well known ‘failure’ to reproduce band
gap energies in semiconductors may only be a failure of the habit of using Kohn–Sham
orbitals as approximations for Hartree–Fock orbitals.
§ The Hartree–Fock method approximates the solution to the many-electron problem by assuming that the
many-electron wave-function can be written on the form of a Slater determinant of single particle orbitals. The
theory can be made exact by completing the basis in which the wave-function is expressed with Slater determinants of orbitals of successively higher energies; this extension is called configuration interaction. The Hartree–
Fock method is itself an extension of the Hartree method where the many-electron wave-function is assumed to
be a simple product of one-electron orbitals. The Hartree assumption means that the electrons are described as
purely independent non-interacting particles.
Chapter 4
E  C
When you have come to the edge of all light that you know and
are about to drop off into the darkness of the unknown, faith is
knowing one of two things will happen: there will be
something solid to stand on or you will be taught to fly.
Patrick Overton
The DFT core theory has left us with one specific goal: to construct a density functional
for the internal electronic energy F [n] that is as accurate as possible. The previous chapter
gave a method for the calculation of the largest contributions to this functional, the noninteracting kinetic energy Ts [n] and the electrostatic energy of a classical repulsive gas
J[n]. In this chapter we turn to the last part that remains, the exchange-correlation energy
Exc [n]. This functional encompasses all the difficult quantum mechanical behavior of
interacting electrons.
4.1
Decomposing the Exchange-Correlation Energy
In the previous chapter, the exchange-correlation energy was defined as the exact internal
electronic energy of a many-body electron system F [n] minus the contributions that now
can be computed exactly, Ts [n] and J[n],
Exc [n] = F [n] − Ts [n] − J[n] = (T [n] − Ts [n]) + (U [n] − J[n]).
(4.1)
In the last step, the expression is put on a form that shows explicitly how Exc is a sum
of two more or less unrelated parts: the correction to the kinetic energy due to electron
interactions T [n]−Ts [n], and the correction to the electrostatic energy due to non-classical
quantum mechanical interactions U [n] − J[n].
It is clear that Exc in itself is not a ‘local quantity’ as it has no spatial coordinate dependence. It is equally affected by all changes throughout the system. To get an (arguably)
21
Chapter 4. Exchange and Correlation
22
semi-local quantity to work with, it is common to implicitly define the exchange-correlation
energy per particle xc ([n]; r) by
Z
Exc [n] = n(r)xc ([n]; r)dr.
(4.2)
The quantity xc ([n]; r) has a spatial dependence and is expected 4,5 to show some kind of
‘locality’, in the sense of being mostly dependent on the part of the electron density which
is close to r.
The implicit definition of the exchange-correlation energy per particle xc ([n]; r) leaves
us with a freedom of choice. Let f (r) be a function that gives zero when integrated over
r. Given a valid xc ([n]; r), an equally valid alternative can be constructed as xc ([n]; r) +
f (r)/n(r). The freedom of choice for the exchange-correlation energy per particle is important for the subsystem functional approach and is discussed more in chapter 7 and
paper 1 of part III.
4.2
The Adiabatic Connection
To enable the development of approximations for the exchange-correlation energy per
particle xc ([n]; r), we first consider how to formulate it exactly in quantities easier to
handle than the many-electron wave-function Ψe . One approach would be to use the
quantities of the density matrix theory of section 2.8; the first order spinless density matrix
Eq. (2.28) and the pair density Eq. (2.30). However, an alternative approach is pursued
in this section, the trick of coupling constant integration in the adiabatic connection 6,30–32 .
In the next section the results found here will be used to derive a composite expression for
the exchange-correlation energy that involves a new 6-dimensional quantity with a rather
intricate relation to the pair density, the exchange-correlation hole.
For a real system, described by Ĥe with electron density n(r), one can define a scaled
Hamiltonian Ĥλ where the strength of the electronic interactions is scaled down by a factor
0 < λ < 1,
Ĥλ = T̂ + λÛ + V̂λ .
(4.3)
The potential function in the potential energy operator V̂λ is chosen as in Kohn–Sham
theory† to make the system’s density n be the same for all values of λ. Thus, there exists
a continuum of Hamiltonians, ranging from the Kohn–Sham system at λ = 0 to the real
interacting system at λ = 1. For each λ, the scaled Hamiltonian Ĥλ has a corresponding
ground state many-particle wave-function Ψλ .
The many-particle wave-function gives the total internal electronic energy as a normal
expectation value,
Fλ = hΨλ |T̂ + λÛ|Ψλ i.
(4.4)
† The
here given derivation of the adiabatic connection assumes the electronic density to be of a nature that
allows potential functions to be constructed to keep it constant for different coupling strengths, i.e., that the
density is v -representable; see e.g. Ref 6 for more information.
4.3. The Exchange-Correlation Hole
23
The fully interacting and the non-interacting cases are recognized as
F1 [n] = F [n] = T [n] + U [n]
and
F0 [n] = Ts [n].
(4.5)
The definition of the (fully interacting) exchange-correlation energy Eq. (4.1) is now easily
rewritten
Exc = U [n] − J[n] + T [n] − Ts [n] = F1 [n] − F0 [n] − J[n]
Z 1
∂Fλ
=
dλ − J[n].
∂λ
0
(4.6)
(4.7)
The derivative in the last step can be obtained using the Hellman–Feynman theorem of
quantum mechanics. It is found that
∂Fλ
= hΨλ |Û|Ψλ i.
∂λ
(4.8)
The expression for the exchange-correlation energy is simplified by defining the potential
energy of exchange-correlation at coupling strength λ as
λ
Uxc
= hΨλ |Û|Ψλ i − J[n].
Thus we arrive at the adiabatic connection formula
Z 1
λ
Exc =
Uxc
dλ.
(4.9)
(4.10)
0
An interesting observation can be made 33 : the integral in Eq. (4.10) explicitly only involves
the internal potential energy part of the exchange-correlation energy. The kinetic energy
part is therefore generated, in effect, by the λ integration.
4.3
The Exchange-Correlation Hole
The adiabatic connection formula Eq. (4.10) was expressed in the potential energy of
λ
λ
exchange-correlation Uxc
. The quantity Uxc
involves the full many-particle wave-function.
In the following we work towards a more manageable expression by expressing the adiabatic
connection formula in the pair-density. The many-particle wave-function Ψλ is inserted
into the ordinary wave-function expression for the pair density Eq. (2.30) to generate
nλ2 (r0 , r). To further simplify the formulas, define the averaged pair density
Z
n2 (r0 , r) = nλ2 (r0 , r)dλ.
(4.11)
The adiabatic connection for the exchange-correlation energy Eq. (4.10), when expressed
using the averaged pair density, becomes
2 ZZ
ec
n2 (r0 , r)
Exc =
drdr0 − J[n].
(4.12)
4π0
|r − r0 |
Chapter 4. Exchange and Correlation
24
The final step is to define the exchange-correlation hole n̂xc (r0 , r) from
1
n(r)n̂xc (r0 , r) + n(r0 )n(r)
n2 (r0 , r) =
2
(4.13)
to get the expression
Exc
1
=
2
e2c
4π0
ZZ
n(r)n̂xc (r0 , r)
drdr0 .
|r − r0 |
(4.14)
This final expression may not look very useful at first. The definition of n̂xc (r0 , r) is obviously complicated, involving pair densities created from a continuum of exact solutions
to many-particle problems. However, the exchange-correlation hole is a useful tool for
reasoning. The definition of n̂xc (r0 , r) is deliberately chosen to put the expression for
Exc in Eq. (4.14) on the form of a classical Coulomb interaction integral. Hence, the
exchange-correlation energy Exc can be interpreted as the result of a simple electrostatic
interaction between electrons and their corresponding exchange-correlation holes. The
name ‘exchange-correlation hole’ is motivated by the idea that the quantity represents a
‘hole’ created in the electron density as an electron at r ‘pushes away’ other electrons. The
interpretation of the n̂xc quantity as an electron hole is further rationalized by the exact
exchange-correlation hole sum rule
Z
n̂xc (r, r0 )dr0 = −1.
(4.15)
It means that the ‘size’ of the hole equals that of the electron to which the hole belongs.
The definition of n̂xc (r0 , r) may seem so complicated that it never could be used for actual
calculations, but it turns out to be possible to compute numerical values for simple systems
through Monte Carlo techniques 34–38 . In section 5.10 the definition is also used in a very
practical way to motivate hybrid functionals.
Exchange-correlation holes alternative to n̂xc can be defined. Any function nxc that
gives the total exchange-correlation energy when integrated as in Eq. (4.14) is a ‘delocalized’ unconventional exchange-correlation hole nxc . This is the same kind of freedom of
choice as was discussed for the exchange-correlation energy per particle. By integration by
parts or by the addition of a function whose integral is zero in Eq. (4.14) one arrives at
some alternative nxc .
4.4
The Exchange-Correlation Energy Per Particle
We now have the theoretical framework needed for defining the local and conventional
exchange-correlation energy per particle ˆxc ([n]; r). This is the specific choice of xc ([n]; r)
one gets from the definition of the exchange-correlation energy per particle, Eq. (4.2), and
the relation for Exc expressed in n̂xc , Eq. (4.14),
2 Z
1
ec
n̂xc (r, r0 ) 0
ˆxc ([n]; r) =
dr .
(4.16)
2 4π0
|r − r0 |
4.5. Separation of Exchange and Correlation
25
Some authors 1,5 introduce a notation to stress that they work with the uniquely defined
−1
choice of ˆxc ([n]; r)—the inverse radius of the exchange-correlation hole Rxc
([n]; r). It is
defined with no freedom of choice,
Z
n̂xc (r, r0 ) 0
−1
(4.17)
Rxc
([n]; r) = −
dr ,
|r − r0 |
2 1
ec
−1
ˆxc ([n]; r) = −
(4.18)
Rxc
([n]; r).
2 4π0
4.5
Separation of Exchange and Correlation
It is common to divide the exchange-correlation energy Exc into separate exchange energy
Ex and correlation energy Ec parts. Basically, the separation continues the trend to part
quantities that can be explicitly formulated from ‘the rest’. The explicit expression that
defines Ex , and therefore also defines this division, will be given in the next section. Separate exchange x ([n]; r) and correlation energies per particle c ([n]; r) are defined as for the
combined exchange-correlation energy Eq. (4.2),
Z
Ex [n] = n(r)x ([n]; r)dr,
(4.19)
Z
Ec [n] = n(r)c ([n]; r)dr,
(4.20)
where
Exc [n] = Ex [n] + Ec [n].
(4.21)
It should be obvious that one has the same freedom of choice for the separate x and c
parts as for the compound xc (i.e., any function that when integrated gives zero can be
added to the integrals).
4.6
The Exchange Energy
The exchange part Ex is defined through one possible choice of x ; the local and conventional
exchange energy per particle ˆx ([n]; r),
2 Z
ec
n̂x (r, r0 ) 0
1
dr ,
(4.22)
ˆx ([n]; r) =
2 4π0
|r − r0 |
1 |n1 (r, r0 )|2
n̂x (r, r0 ) = −
.
(4.23)
2
n(r)
Here we have also defined the exchange hole n̂x (r, r0 ). The first-order spinless density
matrix n1 (r, r0 ), as defined in Eq. (2.28), takes a particularly simple form with the Kohn–
Sham (Slater determinant) many-particle wave-function,
X
n1 (r, r0 ) =
φi (r)φ∗i (r0 ),
(4.24)
i
Chapter 4. Exchange and Correlation
26
where the sum is taken over all occupied spin-states i (i.e., two per fully occupied orbital).
The exchange hole fulfills the exchange hole sum rule,
Z
n̂x (r, r0 )dr0 = −1.
(4.25)
Furthermore, it follows directly from Eq. (4.23) that the exchange hole is negative definite;
the non-positivity constraint,
n̂x (r, r0 ) ≤ 0,
∀ r, r0 .
(4.26)
The integration Eq. (4.19) of the above definition of x defines the total exchange
energy Ex (and therefore also defines the separation of the exchange-correlation energy
Exc in exchange Ex and correlation Ec parts). The total exchange energy has a very useful
exchange scaling relation 39 that describes its behavior when presented with a density scaled
by a scalar γ ;
Ex [nγ ] = γEx [n] for nγ (r) = γ 3 n(γr).
(4.27)
The definition of the exchange energy can be included in an alternative Kohn–Sham
scheme capable of an exact treatment of exchange 3 in a Hartree–Fock-like procedure.
However, the non-local dependence on orbitals makes the equations significantly harder
to solve. A much more common way of including exact exchange in DFT calculations is
instead to use the exchange expressions above as the exchange part of a regular DFT functional. Since that functional is not really a density functional, the effective potential veff
cannot be obtained as a direct functional derivative. Instead, one typically produces veff
through an indirect procedure, the optimized effective potential (OEP) method 40–42 . Note
that exact exchange methods does not universally improve the total exchange-correlation
energy. Simultaneous approximation of exchange and correlation can be beneficial in that
it enables a cancellation of errors between exchange and correlation that is not possible in
exact exchange calculations.
The exchange part of the exchange-correlation energy should formally be called the
Kohn–Sham exchange and it is not the same as the Hartree–Fock exchange. The definitions both looks like Eq. (4.22), but the Kohn–Sham exchange Eq. (4.22) uses the
Kohn–Sham orbitals which are not the same as the Hartree–Fock orbitals (cf. section 3.3).
Similar to the exchange-correlation hole, exchange holes alternative to n̂x can be defined. Any function nx that gives the total exchange energy when inserted and integrated
in Eqs. (4.19) and (4.22) is an unconventional exchange hole.
4.7
The Correlation Energy
When the exchange part is subtracted from the exchange-correlation energy per particle,
the remaining part is the correlation energy per particle,
2 Z
1
ec
n̂c (r, r0 ) 0
ˆc ([n]; r) =
dr ,
(4.28)
2 4π0
|r − r0 |
n̂c (r, r0 ) = n̂xc (r, r0 ) − n̂x (r, r0 ),
(4.29)
4.7. The Correlation Energy
27
where the correlation hole n̂c (r, r0 ) is defined by the last relation. By comparing the
sum rule for exchange-correlation Eq. (4.15) with the one for exchange Eq. (4.25), the
correlation hole sum rule follows,
Z
n̂c (r, r0 )dr0 = 0.
(4.30)
Similar to the exchange-correlation and separate exchange holes, correlation holes alternative to n̂c can be defined. Any function nc that gives the total correlation energy
when integrated as in Eqs. (4.20) and (4.28) is an unconventional correlation hole.
Chapter 5
F D
It is the mark of an educated mind to rest satisfied with the
degree of precision which the nature of the subject admits and
not to seek exactness where only an approximation is possible.
Aristotle
In previous chapters all the energy contributions to the total many-electron energy have
been discussed. It has been made clear that the most difficult parts have been condensed
into the exchange-correlation energy Exc . A number of definitions and theoretical results
for working with this quantity were presented in the last chapter. In this chapter we turn
to the methods used for creating practical approximations.
5.1
Locality
Approximations of the exchange-correlation energy per particle xc ([n]; r) are often characterized in terms of their ‘locality’. Two forms of locality are present in this context, and
in the literature different conventions are used, so the discussion easily becomes confusing.
The two forms of locality are: 1) The specific conventional choice of exchange-correlation
energy ˆx ([n]; r), as defined in Eq. (4.16), is the ‘local’ choice. 2) The functional x ([n]; r)
can be a more or less local functional of the electron density. The meaning of “local functional” will be further explained in the following. The exchange-correlation energy Exc is
given as an integration of xc ([n]; r) together with the electronic density over the whole
space. The locality of the functional describes to what extent the largest energy contribution in the integration comes from the parts of n(r0 ) where r0 is close to r. If xc is more
or less independent of the distance r − r0 , it is a very non-local functional.
29
Chapter 5. Functional Development
30
To reiterate,
An approximation to the local exchange-correlation energy is a functional that aims to
approximate the specific local choice of the exchange-correlation energy per particle,
ˆx ([n]; r).
A local functional of the density (or a functional on local form) is a functional xc ([n]; r)
that depends on the electronic density only at the local point r. Thus it is a function, rather than a functional, of the electronic density: xc ([n]; r) = xc (n(r)).
The assumption that the functional is on this form produces the local density approximation of section 5.2.
A semi-local functional of the density (or a functional on semi-local form ) is a functional xc ([n]; r) with a dependence on the electronic density n(r0 ) mostly focused
around r0 = r. If the functional is assumed to be on this form, it can be expressed
as a function of the electron density and its derivatives (i.e., the gradient of the electronic density etc.) These ideas lead to the generalized gradient approximations of
section 5.5.
One can also create exchange-correlation functionals that are strictly not density functionals, but rather use quantities with a direct relation to the Kohn–Sham orbitals (cf.
section 5.8). As long as such a functional is a local functional of the Kohn–Sham orbitals,
most of the computational efficiency of the Kohn–Sham scheme remains.
5.2
The Local Density Approximation, LDA
The local density approximation (LDA) is the most straightforward approximation of the
exchange-correlation energy. It was proposed already in the first works on DFT 2,3 . One
arrives at this functional from the assumption that the exchange-correlation energy per
particle is a local functional of the electron density.
A uniform electron gas system has a constant veff . The symmetry of this system requires the electron density to be constant n(r) = nunif . It also follows that the exchangecorrelation energy per particle is constant in space and thus can be expressed as a function
unif
(not a functional) of the uniform density, ˆunif
). To construct the local density apxc (n
proximation, one takes in each space point r the real system’s electron density and inserts
it into the uniform exchange-correlation per particle function,
ˆLDA
ˆunif
xc (n(r)) = xc (n(r)).
(5.1)
A schematic illustration is shown in Fig. 5.1.
It is straightforward to derive the exchange part of LDA. The Kohn–Sham orbitals for
a constant effective potential are plane waves. When these orbitals are inserted into the
definition of the exchange energy per particle Eqs. (4.22)–(4.24), the result is a constant
exchange energy per particle ˆunif
and a uniform electron density nunif . The expression for
x
5.2. The Local Density Approximation, LDA
For each space−point...
31
... LDA uses a uniform system
with the same electron density
εLDA
xc
n(r)
Figure 5.1. The definition of the local density approximation.
ˆunif
can then be rewritten as a function of the density nunif . Finally, the uniform density
x
is replaced with a generic n(r). The result is
2 1/3
3
ec
LDA
ˆx (n(r)) = −
3π 2 n(r)
.
(5.2)
4π 4π0
It is common to express this in the dimensionless radius of the sphere that contains the
charge of one electron,
1/3
1
3
.
(5.3)
rs =
a0
4πn(r)
The result is
ˆLDA
(n(r))
x
3
=−
4π
9π
4
1/3 e2c
4π0 a0
1
.
rs
(5.4)
Exact LDA expressions for the correlation are only known in two limits. The first is
the limit of high density and weak correlations 43–47
e2c
LDA
ˆc (n(r)) =
(c0 ln rs + c1 + c2 rs ln rs + c3 rs + ...) , rs 1. (5.5)
4π0 a0
The coefficients c0 , c2 , c3 , ... depend on the electron spin configuration. For a spinunpolarized electron gas (equal number of spin up and spin down electrons) the constant
c0 was calculated 43 in 1950s; c0 = (1 − ln 2)/π 2 . However, it was not until 1992 that
c1 was put on a form that could be evaluated to arbitrary precision 46 , c1 ≈ −0.046920.
Furthermore 44,45 c2 ≈ 0.0092292; and 47 c3 ≈ −0.010. There are also results available
for a fully spin-polarized gas (where all electrons have the same spin).
The second known limit is that of low density and strong correlation 48–50 ,
e2c
d0
d1
d2
LDA
ˆc (n(r)) =
+ 3/2 + 4 + ... , rs 1.
(5.6)
4π0 a0
rs
rs
rs
It is common to use the knowledge of the form of this series when interpolation expressions
are created, but usually one does not use calculated numerical values for the coefficients
d0 , d1 , d2 , ... (A fit to data for intermediate densities gives values of the coefficients that
can be seen as ‘effective’ power series coefficients).
Chapter 5. Functional Development
32
Data for the correlation of the uniform electron gas for densities between the two
known limits have been accurately computed by Monte Carlo methods 51 . Useful approximations of the correlation energy have been created by parameterization of the Monte
Carlo data in a way that takes the known limits into account. There are three such parameterizations in popular use. Vosko, Wilk, and Nusair 52 presented in 1980 a careful
analysis and parameterization. In 1981 Perdew and Zunger 53 independently created a parameterization in an appendix of a paper on how to correct the self-interaction error in
DFT. Furthermore, in 1992 Perdew and Wang constructed another parameterization 45
based on the ideas of Vosko, Wilk, and Nusair. Note that none of these popular correlation parameterizations use the most accurate value available for c1 in the high density
limit Eq. (5.5). Section 9.2 and paper 3 discuss an alternative parameterization of LDA
that uses an accurate value of c1 .
LDA was constructed as a suitable approximation for systems with a slowly varying
electron density, but it was found remarkably successful for wider use. It is still used as
the main functional for many solid state applications. At least three reasons have been put
forward to explain why LDA is so successful:
1. The formal definition of the conventional exchange-correlation energy Eq. (4.14)
can be used to show that a complete description of the ‘exact’ exchange-correlation
hole is not needed, only its spherical average. It is found that LDA reproduces the
spherical average of the real hole more accurately than it reproduces the real hole
itself 54 .
2. A number of constraints, which it is known that the exact exchange-correlation energy functional must fulfill, are also correctly reproduced by LDA, e.g., the sum rule
Eq. (4.15).
3. LDA is based on a real physical model system, the uniform electron gas. Both exchange and correlation are reproduced exactly when the functional is applied to this
model system. The fact that a physical model is used, and that exchange and correlation are treated in the same way, leads to compatible exchange and correlation. When
exchange and correlation are approximated in this consistent way, their errors tend
to cancel.
LDA has been a great success e.g. for applications in the solid state. However, there
are cases where its accuracy is not sufficient. For example in the description of certain
molecular system and for systems where explicit surfaces are present. In particular, LDA
has a tendency to make chemical bindings much too strong, i.e., LDA overbinds.
5.3
The Exchange Re nement Factor
The electron density is a quantity of dimension 1/length3 . For the exchange part one can
39 , by dimensional analysis and the exchange scaling relation Eq. (4.27), conclude that the
exact x must depend on the bare density precisely as n(r)1/3 , just like LDA does. It is
5.4. The Gradient Expansion Approximation, GEA
33
therefore common in the context of density functional development to define and work
with the exchange refinement factor Fx ,
x (r; [n]) = ˆLDA
(n(r))Fx .
x
(5.7)
The key point of working with Fx instead of x (r; [n]) is that the LDA prefactor takes
care of the known bare density dependence. It then follows that Fx can only depend on
density-scale invariant dimensionless quantities. Note that there is no known simple scaling
relation for the correlation energy, so that part cannot be simplified in the same way.
In the next section we will discuss approximations to x (r; [n]) that use the gradient
and Laplacian of the density. These density derivatives can be expressed on scale invariant
form; the dimensionless gradient
s=
|∇n(r)|
2(3π 2 )1/3 n4/3 (r)
(5.8)
q=
∇2 n(r)
.
4(3π 2 )2/3 n5/3 (r)
(5.9)
and the dimensionless Laplacian
To verify that these definitions are indeed scale invariant, one can insert a density scaled as
in Eq. (4.27) and observe that the scaling factors cancel. This is in contrast to, for example,
the dimensionless rs parameter.
5.4
The Gradient Expansion Approximation, GEA
Already the earliest works of DFT 2,3 presented the idea of extending LDA in the form of
a gradient expansion approximation (GEA). LDA uses only the local value of the electron
density. The idea behind a GEA is to regard LDA as the first term in a power series
expansion of xc in the density’s spatial variation (described by the derivatives of n(r)). The
second-order GEA thus uses LDA plus the term of next lowest order in density variation,
Taking all symmetries into account 2 , this term is of order O(∇2 ) and the GEA is expressed
in s2 and q as
e2c
e2c
LDA
2
Âxc (n(r))s +
B̂xc (n(r))q + ..., (5.10)
ˆxc = ˆxc (n(r)) +
4π0 a0
4π0 a0
where Âxc (n(r)) and B̂xc (n(r)) are dimensionless functions (not functionals) of n(r).
It is also possible to eliminate the term proportional to the Laplacian by an integration
by parts in the integral over xc Eq. (4.2). However, note that then the known and local
choice of ˆxc is transformed into an unknown and non-local xc ,
e2c
LDA
xc = ˆxc (n(r)) +
Axc (n(r))s2 + ....
(5.11)
4π0 a0
Chapter 5. Functional Development
34
The exchange part of the GEA can be simplified by using the insights from the previous
section: an LDA prefactor can be extracted to take care of all bare density dependence. The
coefficients must then be scalar;
ˆx = ˆLDA
(5.12)
(n(r)) 1 + âx s2 + b̂x q + ... ,
x
x = ˆLDA
(5.13)
(n(r)) 1 + ax s2 + ... .
x
For the latter, transformed, expression one finds 55–57 ax = 10/81. However, for the
untransformed expression an explicit calculation of ˆx for a model system in the limit of
slowly varying electron densities shows that the suggested power expansion generally does
not exist on the above form. This is further discussed in chapter 9 and paper 1 of part III.
For the correlation term, it is common to work with the density variation expressed in
the reduced density gradient t instead of s (but the two are interchangeable),
t2 =
|∇n(r)|2
,
16[3/(πa30 n)]1/3 n8/3 (r)
(5.14)
and write the expansion as,
c = ˆLDA
(n(r)) +
c
e2c
4π0 a0
Ac (n(r))t2 + ....
(5.15)
Ma and Brueckner 58 calculated the value of the dimensionless function Ac ≈ 0.0667244
in the n → ∞ limit. Later an explicit expression for Ac was derived 59–61 and numerically
calculated for a number of values of the density.
The gradient coefficient for exchange ax = 10/81 was not straightforward to establish. First Sham performed 62 a calculation based on the correlation methods of Ma and
Brueckner 58 and obtained a value of 7/81. Another calculation of Gross and Dreizler 63
confirmed the same result; but empirical results 64 indicated that the value was too low.
Antoniewicz and Kleinman obtained 10/81 55 , and after some suggestions of Perdew and
Wang 65 , Kleinman and Lee 56 numerically demonstrated that the cause of the confusion
was an order of limits problem between the Yukawa screening factor k̄Y and the wave
vector of the density variation K . The problem is nicely exemplified by a (here slightly
modified) toy model of a possible explicit form by Perdew and Wang: If
3
K2
7
+
ax (K, k̄Y ) =
,
(5.16)
2
81 81 K 2 + k̄Y
one finds that
7
(Sham result),
81
10
lim lim ax (K, k̄Y ) =
(Antoniewicz–Kleinman result).
K→0 k̄Y →0
81
lim lim ax (K, k̄Y ) =
k̄Y →0 K→0
(5.17)
(5.18)
5.5. Generalized-Gradient Approximations, GGAs
35
The plots of Kleinman and Lee 56 indicate that the qualitative behavior of this example is
not far from the truth. Given this, it is evident that the ‘right’ answer is the Antoniewicz–
Kleinman result, because in a true Coulomb system the Yukawa screening factor is identically zero, and hence must always be smaller than the wave vector of the density variation,
that only tends to zero as we approach a slowly varying density.
However, in the successive papers of Antoniewicz, Kleinman and Lee there appear
some comments on whether the gradient coefficient one should use may depend on how
the correlation energy term is obtained—i.e., perhaps the errors in the Sham exchange are
cancelled by errors in the Ma–Brueckner correlation. In 1989 Kleinman and Tamura 66
pointed out several problems with the work of Ma and Brueckner. Among other things
they state: “Thus the e2 dependence of [the Ma–Brueckner correlation GEA coefficient] may be
nothing more than a mathematical curiosity, valid only when the [density gradient], for which
it is the coefficient, is identically zero.” This casts some doubts on the accepted exchange and
correlation coefficients of the GEA, and it is unknown to the present author if this has yet
been fully resolved.
In a truly slowly varying system, GEA should improve on LDA, but outside of its area
of formal validity the GEA is found to be unsatisfactory when applied in computations.
The fact that it often is less accurate than the LDA is somewhat disappointing. However,
GEA has successfully been used in the derivation of modern nonempirical functionals as
the limit of low-density variation. This approach, discussed in the next few sections, has
given very useful functionals.
5.5
Generalized-Gradient Approximations, GGAs
A generalized-gradient approximation (GGA) is abstractly defined as any generic function of
the local value of the density and its squared gradient s2 that is constructed to approximate
the exchange-correlation energy per particle. Hence,
GGA
Exc
Z
=
n(r)GGA
(n(r), s2 )dr.
xc
(5.19)
A GGA is thus not just meant to be a terminated power expansion valid only for low density
gradients s, like GEA, but rather some expression that aims to give a generally applicable
good approximation of the exchange-correlation energy per particle for all values of s.
The GGA’s view of the density is solely through the local value of the density n(r)
and the density gradient s. It should be evident that there may be situations when this
limited view does not discriminate between physically different situations. For example,
certain points in the inter-shell regions of an atom look the same as points where the
electron density decays exponentially (see section 7.3 and paper 2 of part III). In such
cases, the GGA must use some kind of ‘averaged’ interpretation of what the values of n(r)
and s mean. It follows that users who aim for different applications will prefer different
GGAs ‘tailored’ to interpret the values in a context relevant for them. Hence, a wealth of
different GGA expressions exist and there is an ongoing discussion on what makes the ‘best
Chapter 5. Functional Development
36
and most general’ GGA. The author has also contributed to this field by the derivation of
a new functional on GGA form (presented in chapter 7 and paper 4 of part III).
The view of ‘a GGA’ as any expression on the form of Eq. (5.19) is in common use in
literature. However, the term was first introduced in the context of the real-space cutoff
procedure described in next section, so it is not uncommon to find presentations where
the term is used in that more specific sense.
5.6
GGAs from the Real-space Cutoff Procedure
In a series of articles Perdew and coworkers have developed and refined a process of functional development known as the real-space cutoff technique (see Refs. 67–71 and references therein). For electronic densities which are not slowly varying, the GEA is not
well behaved; in particular it violates the sum rule and the non-positivity constraint for
exchange, Eqs. (4.15) and (4.26). The real-space cutoff solution 67,71 is to introduce a
cutoff radius and use step functions in real-space to cut off the exchange hole at some r.
The step functions are chosen to force the expression to satisfy the sum rule as well as the
non-positivity constraint. One argument for this procedure is that the description of the
exchange-correlation hole by GEA is most accurate close to the electron but gets worse
further away 70,72–75 .
The derivation of Perdew, Burke and Wang from 1996 (Ref. 71) gives a clear account
of the method. The GEA exchange hole is written as
1
nGEA
(r, r + R) = − n(r)y(R),
x
2
(5.20)
where the radial behavior of the exchange hole y(R) is some known, but complicated,
function. Two step functions† θ(x) are inserted to remove the properties of the hole that
is the source of complications. The result is the exchange hole of the cutoff GGA,
1
nGGA
(r, r + R) = − n(r)y(R)θ(y(R))θ(Rc (r) − |R|).
x
2
(5.21)
The first step-function enforces the non-positivity constraint Eq. (4.26). The second uses
a cutoff radius Rc chosen to make the expression satisfy the sum rule Eq. (4.25). A similar
technique is employed for the correlation hole: In the expression for the spherical averaged
correlation hole, a step function is appended with a similar radial cutoff chosen to make it
satisfy the correlation sum rule Eq. (4.30).
When the GEA hole is integrated with these cutoffs in place, one gets a numerical GGA
that can be parameterized by an analytical expression. The result is a functional that can
be applied in calculations.
† The
step function is defined as: θ(x) = 0 for x < 0; and θ(x) = 1 for x ≥ 0.
5.7. Constraint-based GGAs
5.7
37
Constraint-based GGAs
The GGA functional of Perdew and Wang 1991 (PW91; Refs. 69,70) uses the real-space
cutoff scheme presented above but also chooses a form which ensures that some exact
conditions are fulfilled. This approach was taken further by Perdew, Burke and Ernzerhof
(PBE) in 1996 (Ref. 76) as they presented an alternative way of deriving a GGA functional.
They derived all the coefficients from exact constraints and used no fitting to real-space
cutoff data at all. The resulting GGA functional has been argued to be very similar to the
one of Perdew and Wang. The similarity has been put forward as an argument for the
generality of these GGAs. Paper 5 of part III raises some issues with this argued universal
similarity.
The uniform gas is a well studied limiting case and therefore provides some of the most
precise constraints used for creating constraint-based GGAs. However, there have been
some arguments about whether imposing a correct uniform gas limit really is relevant for
functionals used in e.g. quantum chemistry (see for example Becke’s admitted wavering on
the issue 77–79 ). One other constraint has also been debated, the Lieb–Oxford lower bound
80 ,
2 Z
ec
Ex ≥ −1.679
n4/3 (r)dr.
(5.22)
4π0
In for example PBE this bound is implemented on a local level. Such an implementation
is a more strict requirement that the regular Lieb–Oxford bound and may be unnecessarily
strict (i.e., it might not be fulfilled by the exact functional). The local Lieb–Oxford lower
bound is given by
x ≤ 2.273 ˆLDA
(n(r)).
(5.23)
x
5.8
Meta-GGAs
To continue the approach of constructing expressions that fulfill more and more exact
constraints, one has to introduce more information about the electron density than is
given by the local values of the electronic density and its gradient s. This leads to the
so called meta-GGAs. The logical extension of the GGA form would be to add further
derivatives of the electron density, the Laplacian q etc. However, functionals that include
these parameters have been seen to be subject to great numerical difficulties when employed
in a self-consistent Kohn–Sham scheme 81,82 . As an alternative, it is common to instead
introduce the non-interacting kinetic energy density (see section 8.4),
τ (r) =
h̄2
2me
X
|∇φi (r)|2 ,
(5.24)
i
with the sum taken over all occupied Kohn–Sham orbitals. An approximation of the
exchange-correlation energy per particle that is dependent on the kinetic energy density is
strictly not a density functional, but rather a local functional of the Kohn–Sham orbitals.
Chapter 5. Functional Development
38
Along the lines of the general definition of a GGA, Eq. (5.19), one commonly use
an abstract definition of a meta-GGA as a function of the local value of the density, its
squared gradient s2 , its Laplacian q , and the kinetic energy density that is constructed to
approximate the exchange-correlation energy per particle (but possibly one may also allow
for other semi-local parameters). Hence,
Z
mGGA
(5.25)
Exc
= n(r)mGGA
(n(r), s2 , q, τ )dr.
xc
5.9
Empirical Functionals
An alternative to the real-space cutoff scheme and/or satisfaction of exact constraint is
the more pragmatic approach of empirical functionals. One of the earliest examples of
an empirical functional is the Xα approximation of Slater 83 . Among others, Becke and
coworkers 84–86 have had a key role in the development of the empirical approach. Data
are first produced for real systems, usually atomic or molecular. Useful data come from
e.g. computer calculations for simple systems using very time-consuming methods that are
more accurate than DFT, and from experiments. In any case, accurate data must somehow
be produced outside of DFT. The data are then parameterized in the density n(r), its
derivatives (e.g., s and q ), and possibly other available parameters (e.g., τ ).
The empirical approach is commonly criticized for the risk that the functionals are too
strongly influenced by the systems used for fitting. The resulting functionals may be very
accurate for some classes of systems, but lack general applicability.
5.10
Hybrid Functionals
The idea of hybrid functionals grew out of the attempts to use DFT functionals as a computationally cheap way of correcting Hartree–Fock calculations for correlation effects. Becke
formalized the approach 33 in an early hybrid theory that is interesting in itself. Start from
the adiabatic connection formula Eq. (4.10) derived in section 4.2,
Z
Exc =
1
λ
Uxc
dλ.
(5.26)
0
This integral can be approximated using the mean-value theorem of integration as
Exc ≈
1 0
1
1
1
(U + Uxc
) = (Ex + Uxc
),
2 xc
2
(5.27)
0
where, in the last step, Becke argues 33 that Uxc
just is Ex as defined in Eq. (4.22). The
1
quantity Uxc is the exchange-correlation potential energy of the fully interacting real system. An approximation for the latter can be constructed the same way LDA was constructed,
Z
1
LDA
Uxc
≈ Uxc
= uxc (n(r))dr.
(5.28)
5.10. Hybrid Functionals
39
The LDA-like functional uxc (n(r)) is derived as an LDA approximation of the potential
energy part of the exchange-correlation energy, i.e., U [n] − J[n], cf. Eq. (4.1). Becke obtains 33 an expression to use for uxc from the parameterization of regular LDA correlation
1
by Perdew and Wang 45 . It was later shown 87 how an approximation of Uxc
can be created
from any exchange-correlation functional. For a generic density functional approximation
(DFA) one finds
DFA
[nγ ] ∂Exc
1
DFA
DFA
,
(5.29)
Uxc
≈ Uxc
= 2Exc
[n] −
∂γ
γ=1
where nγ is the scaled density as defined in Eq. (4.27).
Becke’s hybrid theory can be viewed both as an correlation correction to the Hartree–
Fock scheme, and as a method for incorporating exact exchange into DFT calculations.
Becke called it “a true hybrid of its components” and named the two-point adiabatic integration “half-and-half theory”.
The half-and-half theory was followed by another three-parameter hybrid formula of
Becke 88 that arguably is less connected to formal theory, but was more successful and
constitutes the basis for several hybrid functionals in use (e.g., B3LYP 89,90 ),
LDA
Exc = a0 (Ex − ExLDA ) + Exc
+ ax (ExGGA − ExLDA ) + ac (EcGGA − EcLDA ). (5.30)
Here a0 , ax , and ac are empirical parameters. The use of scaling parameters in the last
two terms, which represent the GGA’s correction of LDA, was motivated by Becke with
the argument that a GGA partly includes a correction of the failure of LDA to produce
exact exchange in the λ = 0 limit. Since the formula manually corrects this problem the
GGA’s corrections must be scaled down.
However, it was remarked by Levy et. Al. 87 , that the tree-parameter hybrid formula
seems to be a step away from the formal adiabatic connection approach since it apparently
drops the λ-derivative in Eq. (5.29). The empirical parameters may be able to correct
for this fallacy. Furthermore, Perdew, Ernzerhof and Burke 91 looked at the formula with
ax = ac = 1 and discussed its motivation starting from a simple model for the hybrid
coupling-constant dependence:
λ
DFA
Uxc
= Exc,λ
+ (Ex − ExDFA )(1 − λ)k−1 ,
(5.31)
with k an unknown integer. They found that this model led to a theoretical motivation
for choosing the value a0 ≈ 0.25.
To implement hybrid functionals in computer code it is quite common to use Hartree–
Fock exchange to approximate the exact Kohn–Sham exchange used in the derivation of
the Hybrid theory. It is possible that this approximation is somewhat compensated for in
the fit of empirical parameters.
Chapter 6
A G  F
What is your substance, whereof are you made,
That millions of strange shadows on you tend?
William Shakespeare
The previous chapter presented a number of general techniques for the development of
exchange-correlation functionals. In this chapter we go through the most commonly
known functionals developed with these techniques.
6.1
The GGA of Perdew and Wang (PW91)
The GGA of Perdew and Wang 69,70 from 1991 (PW91) is a nonempirical functional based
on fitting to a numerical GGA produced by the real-space cutoff procedure described in
section 5.6. When the dimensionless gradient s → 0, i.e. in the limit of slowly varying
and high density limits, the PW91 parameterization is chosen to reproduce a second-order
GEA, Eq. (5.11), with Shams ax and the Ac (n(r)) of Rasolt and Geldart (cf. section 5.4).
PW91 improves on LDA for most chemical systems, and for certain properties of materials.
For systems with electronic surfaces, such as vacancy systems, PW91 is inferior to LDA
92 . PW91 does not describe the correct uniform scaling to the high density limit. It often
gives spurious wiggles in the exchange-correlation potential for small and large s.
6.2
The GGA of Perdew, Burke, and Ernzerhof (PBE)
The GGA of Perdew, Burke, and Ernzerhof 76 from 1996 (PBE) is a nonempirical functional with parameters derived to satisfy a specific set of exact constraints. This approach
was discussed in section 5.8. PBE does not reproduce a second-order GEA for slowly
41
42
Chapter 6. A Gallery of Functionals
varying densities. Instead it provides a better description of the linear response limit† .
PBE reduces to the LDA for slowly varying densities. It does not uphold a scaling limit
that PW91 upholds (the nonuniform scaling of Ex in limits where the reduced gradient
s → ∞). The PBE authors argue that this constraint is energetically unimportant.
The PBE functional turns out to be very similar to PW91. In fact, PBE and PW91 are
often argued to be roughly equivalent for applications; but paper 5 in part III raises some
issues with the similarity between the functionals. As for PW91, PBE’s results for vacancy
formation energies are inferior to LDA (see papers 4 and 5 of part III). PBE does not have
the spurious wiggles in the exchange-correlation potential found for PW91, and therefore
is more suitable for e.g. pseudopotentials.
6.3
Revisions of PBE (revPBE, RPBE)
Zhang and Yang 93 remarked that enforcing the local Lieb–Oxford bound in the construction of the PBE exchange functional may be too strict. They proceeded by constructing
a functional revPBE that entirely ignored the bound and instead turned one of the PBE
parameters into an empirical value by fitting it to total atomic energies from helium to
argon. They argued that since revPBE still fulfilled the regular Lieb–Oxford bound for
all their test systems (atoms and molecules), this could be a general feature of the functional. The work presented data of improved atomization energies for small molecules.
Furthermore, Hammer, Hansen, and Nørskov found that revPBE also improved upon
PBE for chemisorption energetics of atoms and molecules on transition-metal surfaces 94 .
They also presented a further revised revPBE functional (RPBE) that reintroduced the
local Lieb–Oxford bound.
However, it has been seen that RPBE and revPBE do not always improve on PBE
95 . For example, some material properties are in larger disagreement with experimental
results compared to PBE. This leads us back to one of the points of section 5.5; the way
a GGA interprets the information it is given can be more or less tailored towards certain
applications.
6.4
The Exchange Functionals of Becke (B86, B88)
Becke presented an empirical exchange functional in 1986 (B86). It proposes an analytical
form based on the GEA, but damps the s-dependence to avoid the problems related to
the divergent behavior of the GEA. It contains two empirical parameters determined by
fitting to Hartree–Fock exchange energies of 20 atomic systems. Various improvements
to the analytical form were later presented by Becke and other authors. The exchange
functional of PBE is in fact based on the B86 expression, but determines the parameters
non-empirically.
† The linear response limit means wiggles of small amplitude on a uniform electron gas; the GGA form is too
restricted to simultaneously get both this limit right and reproduce a specified second-order GEA.
6.5. The Correlation Functional of Lee, Yang, and Parr (LYP)
43
In 1988 Becke presented an improved exchange functional (B88) that has become
popular, in particular for applications in quantum chemistry. The goal was to reproduce
a correct asymptotic behavior for the exchange energy per particle outside a finite system.
It leaves one parameter to be determined empirically. Becke fitted its value using Hartree–
Fock exchange energies of six noble-gas atoms.
6.5
The Correlation Functional of Lee, Yang, and Parr (LYP)
Colle and Salvetti 96 presented a formula for the correlation energy in 1975. The formula
was essentially based on a theoretical analysis that started from the Hartree–Fock secondorder density matrix rescaled with a correlation factor. Four empirical parameters were
determined by a fit to exact data for the helium atom. The formula was found to give
good correlation energies for atoms and molecules. Lee, Yang and Parr reworked the Colle–
Salvetti formula into a density functional (LYP). The LYP functional has been used very
successfully in quantum chemistry together with the B88 functional (BLYP), in particular
in the hybrid scheme called B3LYP 89,90 . BLYP and B3LYP are among the most popular
functionals for quantum chemistry, but they perform badly for more electron-gas like
applications, like e.g. solid-state systems 95 .
One of the major criticisms raised against LYP is that it does not reproduce LDA in
the limit of slowly varying densities. It therefore is not surprising that it performs badly
for more electron-gas like systems, e.g. solids 95 . Another issue is that LYP becomes zero
for a fully spin-polarized system, which is not correct for a multi-electron system.
6.6
The Meta-GGA of Perdew, Kurth, Zupan, and Blaha
(PKZB)
Perdew, Kurth, Zupan, and Blaha 97 presented in 1999 a meta-GGA (PKZB) that built on
PBE but added one more input parameter to the GGA form, the kinetic energy density.
PKZB thus is a meta-GGA as discussed in section 5.8. The extra parameter makes it
possible to satisfy more exact constraints. Among other features, the PKZB functional
reproduces both a fourth-order GEA, and a specified linear response function up to forth
order in the wave-vector. The correlation part of PKZB is based on a self-correlation
correction to PBE’s correlation. The PKZB functional contains one empirical parameter
determined by fitting to atomization energies of 20 small molecules (The magnitude of
this parameter was also argued from surface exchange energies of slowly varying densities.)
PKZB improves on PBE for several applications, e.g., surface and atomization energies 95,97,98 . However, it also gives poor equilibrium bond lengths and hydrogen-bonded
complexes 98,99 .
Chapter 6. A Gallery of Functionals
44
6.7
The Meta-GGA of Tao, Perdew, Staroverov, and
Scuseria (TPSS)
Tao, Perdew, Staroverov, and Scuseria presented an improved meta-GGA 100,101 (TPSS) in
2003. Similar to PKZB, TPSS adds the kinetic energy density as a parameter to the GGA
form. The construction of TPSS starts from PKZB and, among other improvements, eliminates the need for an empirical parameter. Extensive tests have been performed 100,102,103 ,
and the TPSS authors conclude that the tests indicate a general, but moderate, improvement of PBE 102 .
Part II
S C
45
Chapter 7
S F
Great acts are made up of small deeds.
Lao Tsu
This chapter presents the subsystem functional approach to functional development. More
details on the material discussed here are given in paper 1 of part III.
7.1
General Idea
The subsystem functional approach is based on the idea of locality (near-sightedness) of
the electron gas 4,5 . The near-sightedness is explained as the observation that an electron is
mainly influenced by those other electrons that are closest. Thus, the electron’s behavior
should be governed by local or semi-local properties of the electron gas.
We start from the implicit definition of the exchange-correlation energy per particle
Eq. (4.2),
Z
Exc =
(7.1)
n(r)xc (r)dr.
This integration over all space may be decomposed into integrations over several separate
spatial regions R1 , R2 , ...;
Z
Z
Z
Exc =
n(r)xc (r)dr +
n(r)xc (r)dr + ... +
n(r)xc (r)dr.
(7.2)
R1
R2
RN
This general idea was illustrated in chapter 1 in Fig. 1.4.
Approximations to xc that can be applied in a partial system like in Eq. (7.2) are
subsystem functionals. Obviously, a subsystem functional may not be based on the assumption that it will be used in the whole system. Rather, it must give a valid approximation
of the integrated value of some exact xc when integrated over only a part of a system. We
47
Chapter 7. Subsystem Functionals
48
have previously discussed that the implicit definition of xc leaves a freedom of choice.
All integrations over parts in Eq. (7.2) must approximate integrated values of one and the
same choice of xc . This is required for the contributions from the different parts to sum
up to the correct total exchange-correlation energy. The straightforward way to enforce
this is if all subsystem functionals applied to a system are taken to approximate the conventional exchange-correlation energy per particle ˆxc . Paper 1 of part III discusses this in
some more detail.
The reason why the above discussion is about a partition in real space, as opposed to
k -space, is the view of a near-sighted electron gas. One could create a partition in k -space
by performing a Fourier-transform of Eq. (7.1) and then partitioning the integral, but
this approach has not been formally investigated. To further pursue the idea one needs to
make a careful examination of what concept of locality is used and discuss for what kinds
of systems the k-space approach would be useful.
The subsystem functional scheme has similarities to the divide and conquer scheme of
Yang 104,105 , but the two approaches are not identical. The latter divides the entire Kohn–
Sham iteration to be over separate subregions. The subsystem functional scheme leaves
the Kohn–Sham scheme unmodified, and the subdivision of a system only occurs within
the exchange-correlation functional.
7.2
Designing Functionals
The functionals presented in chapter 6 use different, unknown, choices of the exchangecorrelation energy per particle. For example, they are derived using GEA power series
integrated by parts and empirical coefficients. While they approximate the correct total
exchange-correlation energy when integrated, their specific local values of xc cannot be
seen as an approximation to the local conventional choice ˆxc . The lack of a consistent
choice of ˆxc in different functionals means that they cannot be combined into a subsystem
functional scheme. Basically, the functionals have been derived on the assumption that
they will be used throughout the space of integration.
To discuss the development of functionals that work in a subsystem functional setting
we have to start from the local density approximation, which approximates the conventional exchange-correlation energy per particle ˆxc . Much of papers 1 and 3 of part III
deal with how to go beyond LDA in the form of a GEA of a local exchange energy per
particle and turn it into a functional for slowly varying electron densities. A local GEA is
derived in paper 3, where also a redistribution of exchange and correlation is performed;
a requirement for the GEA to exist (see section 9.2).
To create subsystem functionals for systems where the electron density is not slowly
varying one can use model systems. The exchange functional for electronic surfaces that is
designed in paper 4 is one example of such use of model systems. The functional is based
on a model where the effective potential is linear. It will be discussed more in section 7.4
and forward.
7.3. Density Indices
7.3
49
Density Indices
We will now discuss the non-trivial problem of performing the partitioning of a system into
subsystems. One approach is to have a computational scientist manually part the system
into subregions. In this case the partitioning would be based on the physical insight of the
system that the scientist has.
A more automatic approach is to build into the functional a mechanism for deducing
how to partition the system. An automatic separation into parts can be created using one
or more density indices. A density index is a functional of the electron density, which for
each space point gives a value between 0 or 1 that describes to what extent the density in
this point can be said to be of a specific type. For example, an index can tell whether the
density in a space point is on an ‘electronic surface’ as opposed to e.g. in the interior of a
system. Another example would be to determine to what extent points of the density are
atom-like.
Let one subsystem functional be the generic functional that is to be used where no
other model is suitable, ˆ(0)
xc . This generic functional can, for example, be ordinary LDA.
(N )
(2)
ˆxc ... ˆxc .
Then imagine a series of subsystem functionals based on different models ˆ(1)
xc , For each of these functionals one has an index, I (1) , I (2) , ..., I (N ) . A straightforward way
to construct an interpolating subsystem functional (ISF) is,
ˆISF
(0)
xc = Xˆ
xc +
N
X
I (n) (n)
ˆ ,
N xc
n=1
(7.3)
where
X=
N
X
I (n)
1−
N
n=1
!
.
(7.4)
However, for functionals that are based on an asymptotic behavior one has to be careful.
The indices must be designed to interpolate in a way that preserves the correct limiting
behavior.
Paper 2 in part III deals with the construction of a density index that describes how
atom-like the density is. It is seen how an elaborate construction involving electron density
and kinetic energy density derivatives is needed to get all parts of the intershell regions of
an atom correctly classified. However, we note that for an actual DFT calculation it may
not be absolutely necessary to use an index with this precision. Even if the density is
interpreted incorrectly in ‘a few points’ in the intershell regions, it may be sufficient to
be right in the major part of the system to reach good accuracy. Furthermore, the index
constructed in paper 2 classifies a point of the density using only information available in
that specific spatial point, i.e., electron density and kinetic energy values and derivatives.
An index that uses more than just the local information might reach the same precision
without elaborate kinetic energy derivatives.
50
7.4
Chapter 7. Subsystem Functionals
A Straightforward First Subsystem Functional
We will now demonstrate the subsystem functional scheme by the construction of a ‘first’
simple subsystem functional. The approach is the one of paper 4 in part III (which in the
following is referred to as AM05), but some additional details are given.
The construction starts from the interpolation formula presented in the previous section. Ordinary LDA is used for the base functional ˆ(0)
xc . One other functional is used
along with LDA, a functional to specifically treat electronic surfaces. An electronic surface
is a region where the electron density rapidly decreases, e.g., outside a surface system or
inside a vacancy. Roughly, one can think of electronic surfaces in terms of the classical
turning points of a system’s most energetic electrons.
When only two functionals are involved in a subsystem functional scheme, the interpolation formula Eq. (7.3) reduces to
ˆDFA
= Xˆ
LDA
+ (1 − X)ˆ
surf
xc
xc
xc .
(7.5)
To complete this functional we thus need an interpolation index X and an exchangecorrelation functional for surface systems surf
xc . These components will be addressed in
the following.
7.5
A Simple Density Index for Surfaces
The dimensionless gradient s diverges outside an electronic surface. The reason is that
the electron density n appears in the denominator of the definition of s, Eq. (5.8), and in
this limit n → 0. An index I that increases towards 1 the more ‘surface like’ the electron
density is can thus be created as
αs2
I=
.
(7.6)
1 + αs2
In the interpolation formula Eq. (7.5) we then use X = (1 − I). The scalar parameter α is
a surface position parameter. When the index interprets the electron density, this parameter
adjusts the overall inward-outward position of the electronic surface. To use the index one
has to provide the parameter α or determine it in some way. Below we will use a fitting
procedure to obtain a useful value of α.
7.6
An Exchange Functional for Surfaces
In AM05 an exchange functional is constructed to target surface regions of the electron density. One starts from the Airy gas model system 5 . The Airy gas is a model of
Kohn–Sham particles in a linear potential, veff (r) = Lz . It models an electronic surface
where the classical turning point of the most energetic Kohn–Sham particles is at z = 0.
The parameter L sets an overall length scale. It is used to rescale the exchange energy
per particle ˆx and the density n(r) into dimensionless and scale-independent quantities;
−1/3
ˆAiry
ˆx (r; [n]), and n0 = L−1 n(r). By solving the Kohn–Sham orbital equation
x,0 = L
7.6. An Exchange Functional for Surfaces
51
Eq. (3.11), and then inserting the orbitals in the usual expressions for the exchange energy
Eqs. (4.23)–(4.24) and electron density Eq. (2.14), one arrives at 5
√ 0
Z ∞
Z
Z
√
−1 ∞ 0 ∞
Airy
0 g( χ∆ζ, χ ∆ζ)
dχ
dχ
dζ
ˆx,0 =
πn0 −∞
∆ζ 3
0
0
0
0
×Ai(ζ + χ)Ai(ζ + χ)Ai(ζ + χ )Ai(ζ 0 + χ0 ),
(7.7)
where ζ = L1/3 z, ∆ζ = |ζ − ζ 0 |, and
g(η, η 0 ) = ηη 0
Z
0
∞
J1 (ηt)J1 (η 0 t)
√
dt.
t 1 + t2
In AM05 the density is given on explicit form
1
Ai(ζ)Ai0 (ζ)
2
n0 =
ζ 2 Ai2 (ζ) − ζAi0 (ζ) −
.
3π
2
Taking derivatives of the density expression directly gives
2
02
dn0 1
, dn0 = ζAi (ζ) − Ai (ζ) ,
s=
4/3 dζ dζ
2π
2(3π 2 )1/3 n0
q=
1
5/3
4(3π 2 )2/3 n0
d 2 n0
,
dζ 2
d 2 n0
Ai2 (ζ)
=
.
2
dζ
2π
(7.8)
(7.9)
(7.10)
(7.11)
A functional based on the Airy gas model should relate a real system’s electron density
in a given spatial point to that of an Airy gas for which the density behavior semi-locally is
as similar to the real system as possible. There is more than one possible implementation
of this. The most straightforward approach is to take the exchange energy from an Airy
gas model that has the same local value of the electron density n and density gradient ∇n
as the real system. The approach has similarities to the construction of LDA in section
5.2. AM05 presents a parameterization of the Airy gas exchange energy: the Local Airy
Approximation (LAA). In the following we give some details of the construction of the
parameterization that is not given in AM05.
The real system’s local value of the electron density in a given spatial point is automatically reproduced by the Airy gas model if the right length scale is chosen; L = n/n0 . The
Airy exchange energy ˆAiry
thus can be separated into a prefactor n1/3 and a dimensionless
x
and density-scale invariant function. This separation is just a special case of the general
separation of the exchange energy per particle into LDA and a refinement factor, as previously discussed in section 5.3. The dimensionless Airy refinement factor FxAiry (s) is a
function of the dimensionless gradient s and is defined by
ˆAiry
(r; [n]) = ˆLDA
(n(r))FxAiry (s).
x
x
(7.12)
It can be expressed in the rescaled dimensionless Airy quantities as
FxAiry (s) =
L1/3 ˆAiry
ˆAiry
x,0
x,0
=
.
ˆLDA
(n(r))
ˆLDA
(n0 )
x
x
(7.13)
Chapter 7. Subsystem Functionals
52
A parameterization of FxAiry (s) is needed to use the Airy gas in density functional
theory computations. One such parameterization is already available, the Local Airy Gas
106 :
FxLAG (s) = 1 + aβ saα /(1 + aγ saα )aδ ,
aα = 2.626712, aβ = 0.041106, aγ = 0.092070, aδ = 0.657946.
(7.14)
(7.15)
However, the subsystem functional we are constructing needs a high-accuracy expression
for use in electronic surface regions. The LAG parameterization was constructed as a
universally acceptable expression for all parts of a system. It is not safe to assume that
this parameterization is accurate enough for our purposes, i.e., especially in the region far
outside the surface. Because of this, an improved parameterization is derived.
The derivation starts with the asymptotic behavior far outside the surface, which is
the key difference between the Airy parameterization constructed here and the one already
available (LAG). The paper of Kohn and Mattsson on the Airy gas 5 gave the asymptotic
behavior of the Airy gas exchange energy per particle as ˆAiry
x,0 → −1/(2ζ). The quantity ζ
can be transformed into an expression in s by inserting asymptotic expressions for the Airy
functions into Eq. (7.10) (carefully including a sufficient number of terms) and inverting.
The procedure results in a function ζ̃(s) that approaches the regular ζ in the s → ∞ limit,
ζ̃(s) =
3
W
2
s3/2
√
2 6
2/3
Airy
x,0 → −
,
1
,
2ζ̃
(7.16)
where W (x) is the Lambert W -function 107 ; the solution w to x = wew . To describe the
(n0 ) must also be expressed as a
asymptotic behavior of FxAiry (s), the LDA factor ˆLDA
x
function of s that is correct in the s → ∞ limit. An expression for this LDA factor is
given by inserting asymptotic expansions of the Airy functions in Eq. (7.9) and then let
ζ → ζ̃(s). The result is
ñ0 (s) =
ζ̃(s)3/2
,
3π 2 s3
FxAiry (s) → −
1
ˆLDA
(ñ0 (s))2ζ̃(s)
x
.
(7.17)
This expression for FxAiry (s) is formally valid in the s → ∞ limit, but it is observed to be
fairly useful even for finite s. To improve it for low s, it should be made to approach the
LDA, i.e., one wants Fx (s) → 1 in the s → 0 limit. The actual behavior of the s → ∞
asymptotic expression
in the s → 0 limit is found by expanding it around s = 0. The
p
leading term is 2/3 4π/(3s1/2 ). However, if one makes the change ζ̃(s) → ζ̃(s)1/2 ,
then the leading term turns into a constant. Thus the asymptotic s → ∞ and the LDA
s → 0 limits can be fulfilled simultaneously by creating a new “effective” interpolated
ζ -coordinate. The following definition of the effective coordinate does a good job in describing the transition,
1/4
˜
ζ̃(s) = C 4 ζ̃(s)2 + ζ̃(s)4
,
C = (4/3)1/3 2π/3.
(7.18)
7.7. A Correlation Functional for Surfaces
53
The scalar C is chosen to make Fx (s) approach 1 (rather than some other constant value).
The new interpolated refinement factor
Fxb (s) = −
1
(7.19)
˜
ˆLDA
(ñ0 (s))2ζ̃(s)
x
still deviates slightly from actual computed values for intermediate values of s. This can
be improved if the expression is pushed slightly more towards LDA in a way difficult to
accomplish by further adjusting Eq. (7.18). The last step therefore is to interpolate the
above expression towards LDA (despite the fact that it already does approach LDA). The
final expression becomes
FxLAA (s) = (cs2 + 1)/(cs2 /Fxb (s) + 1),
c = 0.7168,
(7.20)
where c is obtained through a least-squares fit to the exact Airy exchange data obtained
from Eq. (7.7). The LAA parameterization makes a small improvement to LAG in the
region of intermediate s, but the improvement becomes significant for larger s (i.e. outside
the electronic surface; see Fig. 1 in AM05).
7.7
A Correlation Functional for Surfaces
The preferred way of creating a correlation functional that matches the Airy gas exchange
functional would be to parameterize exact correlation energy per particle data for the Airy
gas model. Such data should be possible to compute by e.g. Monte Carlo methods. However, no correlation data for the Airy gas are yet available to parameterize. Therefore the
correlation functional that is matched with the Airy exchange functional in AM05 is created by a fitting procedure that instead involves jellium surface energies.
The jellium surface model is a model system with a uniform background of positive
charge n̄ for z ≤ 0 and 0 for z > 0 108 . The value of n̄ is commonly expressed in
the dimensionless radius of the sphere that contains the charge of one electron rs as defined in Eq. (5.3). The jellium surface energy of a density functional approximation (DFA)
DFA
xc (r; [n]) is given by
Z
DFA
σxc = n(z) DFA
ˆLDA
(7.21)
xc (r; [n]) − xc (n̄) dz.
An LDA correlation adjusted with a multiplicative factor γ is used for the surface
correlation functional; ˆsurf
= γˆ
LDA
. The multiplicative factor provides an adjustment
c
c
of the LDA correlation energy that scales reasonable with the area of the electronic surface.
It is believed that the most accurate jellium surface energies are given by the improved
random-phase approximation scheme presented by Yan et al. 109 (RPA+). The RPA+ values
are cited as integers in the unit erg/cm2 , and therefore we assume that the absolute errors
are roughly equal throughout all the values (meaning σxc for smaller rs have smaller relative
errors due to their greater magnitude). Hence,
is reasonable to let the least squares fit
P it AM05
RPA+ 2
minimize an unweighted least squares sum rs |σxc
− σxc
| . The fit in AM05 uses
Chapter 7. Subsystem Functionals
54
the RPA+ values for rs = 2.0, 2.07, 2.3, 2.66, 3.0, 3.28, and 4.0 to simultaneously fit the
surface position α in Eq. (7.6) and the LDA correlation factor γ ,
αLAA = 2.804,
γLAA = 0.8098.
(7.22)
This completes the functional,
(n(r)) + (1 − X)ˆ
LDA
(n)FxLAA ,
x (r; [n]) = Xˆ
LDA
x
x
c (r; [n]) = Xˆ
LDA
(n(r)) + (1 − X)γˆ
LDA
(n).
c
c
7.8
(7.23)
Outlook and Improvements
The simple functional constructed in the previous sections has been tested for a few solid
state systems and performs well (see the test results in AM05 for details). Still, there are
several future directions open for improving our currently rather crude procedure:
• One should develop a less rudimentary density index that does a better job in distinguishing between interior and surface regions.
• A better correlation functional for surfaces would most likely improve the results.
• LDA has been used for the interior region. A better approximation for near-uniform
electron gas system, e.g. a gradient corrected functional, would probably improve
the results further.
• Subsystem functionals for other types of systems can be derived and incorporated
into the scheme. For example, a subsystem functional tailored for atomic intershell
regions of the electron density may improve the exchange-correlation energy for such
regions.
Naturally, the author hopes to see future development along one or more of these suggested
improvements of the scheme.
Chapter 8
T M G M
What distinguishes a mathematical model from, say, a poem, a
song, a portrait or any other kind of “model,” is that the
mathematical model is an image or picture of reality painted
with logical symbols instead of with words, sounds or
watercolors.
John Casti
Much of paper 1 in part III of this thesis deals with the numerical study of a specific model
system, the Mathieu gas. This chapter introduces the model and discusses its usefulness as
a DFT model system.
8.1
De nition of the Mathieu Gas Model
The Mathieu gas (MG) can be viewed as a family of electron densities parameterized by
two dimensionless scalar parameters, λ̄ and p̄. The electron densities are obtained from a
system of Kohn–Sham particles moving in an effective potential
veff (r) = µλ̄(1 − cos(2p̄z̄)).
(8.1)
Here µ is the chemical potential of the system and z̄ = kF,u z = (2mµ/h̄2 )1/2 z is the
z coordinate scaled with the Fermi wave vector of a uniform electron gas. By solving the
corresponding non-interacting electron system for specific values of λ̄ and p̄, the Kohn–
Sham orbitals are obtained, and consequently gives an electron density.
8.2
Electron Density
Solving the MG effective potential system for the Kohn–Sham orbitals is significantly easier than it would be for a general system. As the effective potential only depends on the z
55
Chapter 8. The Mathieu Gas Model
56
coordinate, the Kohn–Sham orbital equation can be separated into three one-dimensional
equations. The Fermi surface of a one-dimensional system is only a point, which greatly
simplifies the integration over occupied states. In a non-separable three-dimensional system, the treatment of the Fermi surface is not straightforward.
With constant x and y dimensions the Kohn–Sham orbitals take the form
φν (x, y, z) = √
1
ei(k1 x+k2 y) ϕη (z),
L1 L2
(8.2)
where ν specifies k1 , k2 and η ; L1 L2 is the x, y area of the system and will approach
infinity; ki Li = 2πmi (i = 1, 2, mi integer); and finally ϕη (z) is the one-dimensional
z -direction Kohn–Sham orbital. This orbital is determined by the following Kohn–Sham
equation;
h̄2 d2
+
v
(z)
ϕη (z) = η ϕη (z).
(8.3)
−
eff
2me dz 2
Inserting the MG veff gives the Mathieu differential equation, for which the solutions are
known (see Ref. 110 for definitions of the Mathieu function symbols, seη , ceη , aη and bη )
√
(1/√L) ceη (p̄z̄, −λ̄/(2p̄2 )) if η > 0
ϕη (z) =
,
(8.4)
(1/ L) seη (p̄z̄, −λ̄/(2p̄2 )) if η < 0
µ(p̄2 aη + λ̄) if η > 0
η =
(8.5)
,
µ(p̄2 bη + λ̄) if η < 0
Z η̃
3
kF,u
η
2
p̄
L|ϕη (z)| 1 −
dη,
n(r) =
(8.6)
4π 2
µ
−η̃
where η̃ is the largest possible η that fulfils η ≤ µ.
However, numerical calculations based on these formulas require computer routines
for the Mathieu functions ce and se. Such routines are produced by going back to the
Mathieu differential equation, Eq. (8.3), and solve it by standard matrix methods. Once
the Kohn–Sham orbitals are known, the conventional exchange energy per particle and
other quantities can be obtained by direct numerical calculation. The data in Figs. 7–12
of paper 1 in part III were essentially produced by this method. Details on how to compute
the Mathieu functions and how to perform the integrations above are presented in paper 1.
The energy expression of the MG model, Eq. (8.5), shows a rudimentary energy-band
structure. The parameter η indexes the band structure, much like the wave vector in an
extended Brillouin zone-scheme.
8.3
Exploring the Parameter Space of the MG
The MG model spans a wide variety of systems over the range of possible λ̄ and p̄. We
have found it useful to investigate some specific limits in the MG. These limits of the MG
constitutes model systems on their own.
8.3. Exploring the Parameter Space of the MG
57
The Limit of Slowly Varying Densities
From the construction of the MG family of densities it follows that the limit of slowly
varying densities is found as λ̄, p̄ → 0. However, the two-dimensionality of this limit
makes it challenging to analyze the evaluated numerical data in a consistent way. The data
were therefore plotted versus a new parameter α that index the energy structure of the MG
as a function of λ̄ and p̄;
µ − η1
+ |η1 |,
(8.7)
α=
η2 − η1
where, if µ is inside a z -dimension energy band, η1 is the lowest energy in this band. If
µ is not inside an energy band, η1 is the lowest energy in the band which contains the
z -dimension energy state with highest energy ≤ µ. Furthermore, η2 is the lowest possible
energy of all z -dimension energy states within bands that only contain energies > µ. By
construction η1 and η2 are integer.
The parameter α describes the position of the chemical potential relative to the lower
band edges, that is, the lowest energies of the energy bands in the z dimensional energy
band structure. The parameter α differs from η in that it indexes values of the chemical
potential both within and between the energy bands in the z dimension, making it useful
throughout the parameter space of the MG.
The Free Electron Gas Limit
When λ̄ → 0, the MG effective potential, Eq. (8.1), approaches a constant potential. This
makes the solutions of the MG differential equation approach the plane wave solutions to
a free electron (FE) gas,
1
ϕη (z) = √ exp(iη p̄z̄),
L3
η = µη 2 p̄2 .
(8.8)
(8.9)
Hence, in this limit the MG model describes a weakly perturbed uniform gas. For some
finite but low λ̄ a crystal-like system is described. This view was used in paper 2 to create
a model of sodium and calcium crystals.
In the FE limit the α parameter reduces to
1
1/p̄2 + N (N + 1)
, N=
.
(8.10)
αF E =
2N + 1
p̄
The Harmonic Oscillator Limit
In the limit λ̄/p̄2 → ∞ the MG effective potential approaches an harmonic oscillator
(HO) potential. The energy structure in this limit becomes
q
n = µ 2λ̄p̄2 (2n + 1) .
(8.11)
Chapter 8. The Mathieu Gas Model
58
p
The relation describes equally spaced energy levels (with spacing µ 2λ̄p̄2 ), much like a
typical text book HO system. The corresponding Kohn–Sham orbitals are:
ϕn (z) =
!1/2
p
q
kF,u ( 2λ̄p̄2 )1/2
√ n
Hn (( 2λ̄p̄2 )1/2 z̄)
π2 n!
q
× exp (−[( 2λ̄p̄2 )1/2 z̄]2 /2),
(8.12)
where Hn (x) are Hermite polynomials 110 and n = 0, 1, 2, . . .. The α parameter reduces
to
1
1
αHO = p
− .
(8.13)
2
2
2 2λ̄p̄
One of the primary features of the HO model system is the discrete z energy spectrum.
The model can be said to mimic an atomic-like system, as it effectively is of finite size in
the z direction.
As a part of the thesis work a computer program was written specifically for this limit.
In contrast to Mathieu functions, Hermite polynomials can be computed without having
to resort to solving differential equations. Comparing MG data in the HO limit and data
for the pure HO thus gave an extra check on the numerical procedure.
8.4
Investigation of the Kinetic Energy Density
In the following we use the MG model system to study the power expansion of the noninteracting kinetic energy density. Although the power expansion of this quantity is already
well known, the procedure provides a test of our numerical methods. The study also
serves as a simplified example of the methods of the investigation of the exchange energy
per particle presented in paper 1 of part III.
The kinetic energy density τ (r) is a localized version of the total kinetic energy of
the non-interacting Kohn–Sham system Ts defined in Eq. (3.1). Similar to the implicit
definition of the exchange-correlation energy per particle Eq. (4.2), one defines the kinetic
energy density implicitly as
Z
Ts [n] =
τ (r)dr.
(8.14)
The conventional definition of τ (r) for a spin unpolarized system is
τ (r) =
h̄2
2me
X
|∇φν (r)|2 ,
(8.15)
ν
where the sum is taken over all occupied orbitals. It is known 111–113 that the second order
8.4. Investigation of the Kinetic Energy Density
59
gradient expansion of this quantity is
5
20
τexp (r) = τLDA 1 + s2 + q ,
27
9
2 3
h̄
τLDA (r) = (3π 2 )2/3
n(r)5/3 .
5
2me
(8.16)
(8.17)
The Kohn–Sham orbitals corresponding to the MG family of densities can be inserted
into Eq. (8.15) to compute numerical values of the kinetic energy density. The computed
values in the limit of slowly varying densities are then expected to behave accordingly to
Eq. (8.16). We can verify this expected behavior by evaluating curves for a fixed λ̄/p̄2 = 0.8
and plot them versus 1/α in the limit 1/α → 0; i.e., the limit of slowly varying electron
densities.
Given Eq. (8.16), the following limits are expected for the MG:
τ (r)
20
2
(8.18)
for s = 0, 1/α → 0 :
− 1 /q → ,
τLDA
9
τ (r)
5
for q = 0, 1/α → 0 :
(8.19)
− 1 /s2 → .
τLDA
27
To evaluate these limits numerically using MG densities, values of τ (r) must be computed for some space point r for a series of values of α. The first limit requires the use
of space points where s2 = 0. This requirement is fulfilled at the minimum point of the
effective potential, i.e., z = 0, due to the symmetries of the system. The second limit
requires space points where q = 0. A search was implemented in the computer program
to numerically find a point where q = 0 for every value of α.
Data for the two limits were computed and are plotted in Figs. 8.1 and 8.2. The limits
as predicted by Eqs. (8.18) and (8.19) are correctly reproduced when 1/α → 0. Apart
from this expected result, it is interesting to note the behavior of the KE density at higher
1/α. One can compare the KE figures to the figures of other DFT quantities (as plotted
in paper 1 of part III). These quantities are strongly influenced at values of α where the
chemical potential enters a new z dimension energy band (i.e., where α is an integer, and
thus 1/α = 1/2 and 1/α = 1/3, etc). A similar correspondence between the energy
structure and the behavior of the plotted curves is seen also for the KE density, but less
pronounced.
Chapter 8. The Mathieu Gas Model
60
2.27
MG−tau, s2=0
KE expansion coeff: 20/9
2.26
(τ/τLDA−1)/q
2.25
2.24
2.23
2.22
2.21
2.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1/α
Figure 8.1. The quantity (τ (r)/τLDA − 1)/q vs 1/α for λ̄/p̄2 = 0.8 and s2 = 0. In the limit
of slowly varying densities, 1/α → 0, this quantity approaches the Laplacian coefficient in the
kinetic energy density power expansion, Eq. (8.16), as is expected.
0.6
MG−tau, q=0
KE expansion coeff: 5/27
0.55
0.5
(τ/τLDA−1)/s2
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1/α
Figure 8.2. The quantity (τ (r)/τLDA − 1)/s2 vs 1/α for λ̄/p̄2 = 0.8 and q = 0. In the limit
of slowly varying densities, 1/α → 0, this quantity approaches the gradient coefficient in the
kinetic energy density power expansion, Eq. (8.16), as is expected.
Chapter 9
A L E E
I seldom end up where I wanted to go,
but almost always end up where I need to be.
Douglas Adams
Much of papers 1 and 3 of part III deal with the expansion of the local exchange energy
per particle in the electron density variation. Such an expansion is expected to be useful for
treating slowly varying electron densities beyond LDA in a subsystem functional scheme.
The most important findings of these papers in this context are outlined here.
9.1
The Non-existence of a Local GEA for Exchange
Section 5.4 discussed the gradient expansion approximation (GEA). It was explained that
an expansion of the exchange energy per particle in the limit of slowly varying densities,
taking symmetries into account, leads to the following expression:
ˆx = ˆLDA
(n(r)) 1 + âx s2 + b̂x q + ... .
(9.1)
x
Paper 1 of part III uses the Mathieu gas model system to evaluate the conventional exchange energy per particle in the limit of slowly varying Mathieu gas densities. In doing
so, it is explicitly demonstrated that a general expansion on this form cannot exist! The
conventional exchange energy per particle is thus a non-analytical function of s2 and q
in the limit of slowly varying densities. The paper gives three suggestions for how to deal
with this issue:
1. One can take into account the fact that the conventional exchange energy per particle is non-analytical in the limit of slowly varying electron densities, and create an
expansion on a form alternative to the traditional GEA of Eq. (9.1).
61
Chapter 9. A Local Exchange Expansion
62
2. One can utilize the freedom of choice in the exchange energy per particle (discussed
in section 4.1) to transform the expression into one that is analytical and wellbehaved in the limit of slowly varying electron densities. The motivation behind
this idea is that an expansion on the GEA form is proved to exist for the non-local
exchange expression that has been integrated by parts to fully remove the Laplacian
term (or, at least it is widely believed that this expansion exists; the coefficient of
the gradient term has been confirmed by several works to be 10/81; see section 5.4).
However, since a functional based on the transformed exchange will use a specific
non-local choice of the exchange energy per particle, all subsystem functionals used
together with such a functional must also approximate that same specific choice.
3. The separation of exchange-correlation into exchange and correlation is arbitrary
in DFT. The DFT exchange is defined as an imitation of the exchange expression
of Hartree–Fock theory. But since this definition causes trouble, it is reasonable to
reexamine it.
9.2
Alternative Separation of Exchange and Correlation
Paper 3 of part III shows that the gradient expansion form of Eq. (9.1) is reestablished
when the Coulomb interaction in the definition of exchange Eq. (4.22) is screened. Motivated by this observation, it was suggested that the exchange part can be redefined to
include a finite screening of the Coulomb interaction. The correlation part, defined as
exchange-correlation minus exchange, is then also redefined correspondingly; the exchange
and correlation sum is left unmodified.
The view here, that the definition of the exchange part can be redefined to include
screening, is fundamentally different from the view present in most other works that employ screened exchange. The most common use of screening is as a temporary means to
help analytical manipulation of the exchange expressions. In that case it is always with the
intent of taking the limit of zero screening in the end. Other works discuss screening as
an approximation of the conventional exchange. The procedure suggested here is more
similar to recent works that discuss splitting the exchange into a short and a long-range
contribution 114–119 .
9.3
Rede ning Exchange
To define the screened exchange energy per particle, we take the unscreened expression
Eq. (4.22) and insert a Yukawa screening factor with a Yukawa wave vector kY ,
2 Z
1
ec
n̂x (r, r0 ) −kY |r−r0 | 0
e
dr .
(9.2)
ˆ(x+Y) ([n]; r) =
2 4π0
|r − r0 |
Similar to how regular exchange and correlation was defined, a correlation part corresponding to the screened exchange part is defined from the requirement that the parts sum up
9.4. An LDA for Screened Exchange
63
to the correct exchange-correlation,
ˆxc ([n]; r) = ˆ(c−Y) ([n]; r) + ˆ(x+Y) ([n]; r).
(9.3)
An analogous way of viewing the redefinition of exchange is as a redistribution of a
positive term from correlation into exchange:
2 Z
0
ec
|n1 (r, r0 )|2 1 − e−kY |r−r | 0
1
dr ,
ˆY =
(9.4)
4 4π0
n(r)
|r − r0 |
ˆ(x+Y) = ˆx + ˆY , ˆ(c−Y) = ˆc − ˆY .
(9.5)
The main point here is that the form of ˆY is chosen specifically to make the exchange part
well-behaved in the limit of slowly varying densities.
Arbitrarily screening the exchange does not in itself guarantee a well-behaved exchange
energy per particle in the limit of slowly varying densities. The screening parameter is
chosen to be a function of spatial coordinate (or rather, a function of the electronic density,
which varies with spatial coordinate) and must approach the following expression in the
limit of slowly varying densities:
kY = pF k̄Y ,
(9.6)
where k̄Y is a dimensionless non-zero positive scalar constant that canpbe freely chosen and
pF is the position-dependent Fermi momentum, pF = (2me /h̄2 )1/2 µ − veff (r). In the
limit of slowly varying densities, pF → (3π 2 n)1/3 .
9.4
An LDA for Screened Exchange
Paper 3 of part III derives an LDA for screened exchange. The method is basically the same
as for the derivation of the regular LDA in section 5.2, but uses the screened exchange
expressions. The result is
ˆLDA
ˆLDA
(n(r))I0 (k̄Y ),
x
(x+Y) (n(r)) = where I0 (k̄Y ) is a complicated function of k̄Y ,
2
4
1
2
2
2
24 − 4k̄Y
− 32k̄Y arctan( ) + k̄Y
(12 + k̄Y
) ln( 2 + 1) .
I0 (k̄Y ) =
24
k̄Y
k̄Y
(9.7)
(9.8)
If the screened LDA exchange expression is subtracted from the total exchange-correlation
energy per particle for the uniform electron gas, the remainder can be parameterized as
a function of the electron density. The result is a parameterization of the modified correlation that is compatible with the screened exchange LDA. Such an parameterization
was done in paper 3 of part III to produce two screened LDA expressions, YLDA1 and
YLDA2. The specifics of their construction shows that the parameterization is at least no
more complicated than for regular correlation. Hence, the modified correlation does not
Chapter 9. A Local Exchange Expansion
64
in itself complicate functional development. In fact, since the screening has eliminated an
artificial complication in exchange that is not present in exchange-correlation, the modified correlation may even be more well-behaved than regular correlation.
It is possible to take benefit of parameterizations of the modified correlation, such as
YLDA1 and YLDA2, even when a traditional correlation is needed. A parameterization
of the modified correlation ˆLDA
(c−Y) (n(r)) can be turned into a regular LDA correlation
LDA
parameterization ˆc (n(r)) and vice versa;
ˆLDA
(n(r)) = ˆLDA
ˆLDA
ˆLDA
(n(r))
(9.9)
c
x
(x+Y) (n(r)) + (c−Y) (n(r)) − and conversely,
ˆLDA
ˆLDA
(n(r)) + ˆLDA
(n(r)) − ˆLDA
x
c
(c−Y) (n(r)) = (x+Y) (n(r)).
(9.10)
The latter relation makes it possible to use a regular LDA parameterization in a screened
exchange scheme. However, in that case one does not make use of the properties of the
modified correlation, and thus is limited by the accuracy of the parameterization of the
regular correlation.
9.5
A GEA for Screened Exchange
Paper 3 of part III also derives a GEA for the local screened exchange. It was discussed
in section 5.4 how several works that dealt with the non-local GEA derived an incorrect
coefficient due to the (sometimes covert) use of screening. One starts from an intermediate
step in one of these works 63 where no non-local transformations have yet been made, and
then makes sure to keep the screening finite throughout the derivation. The end result is
2
ˆx = ˆLDA
(9.11)
(x+Y) (n(r)) 1 + â(x+Y) s + b̂(x+Y) q ,
where
8
â(x+Y) =
27
b̂(x+Y) =
3 1 IB (k̄Y ) 1 IC (k̄Y )
−
+
4 3 I0 (k̄Y )
2 I0 (k̄Y )
,
8 IB (k̄Y ) 4
− ,
27 I0 (k̄Y )
9
(9.12)
(9.13)
and
IB =
2
2
2
2
40 + 12k̄Y
− 6k̄Y (4 + k̄Y
) arctan(2/k̄Y ) − (4 + k̄Y
) ln(4/k̄Y
+ 1)
, (9.14)
2
16 + 4k̄Y
IC =
2
2
2
2
k̄Y (4 + k̄Y
) arctan(2/k̄Y ) − 4 − 2k̄Y
− 2(k̄Y
− 4)/(k̄Y
+ 4)
.
2
8 + 2k̄Y
(9.15)
In the paper, these expressions were also confirmed numerically using a screened Mathieu
gas model.
9.6. The Screened Airy Gas
65
0
−0.05
ε(x+Y) (hartree)
−0.1
−0.15
−0.2
kY = 2.0
kY = 1.0
kY = 0.5
kY = 0.1
Unscreened
−0.25
−0.3
−2
−1
0
ζ
1
2
Figure 9.1. The screened Airy gas for different screening parameters k̄Y .
9.6
The Screened Airy Gas
If screened exchange is used to derive the exchange expression for the Airy gas of section 7.6, one arrives at
√ 0
Z
Z
Z ∞
√
−1 ∞ 0 ∞
Airy
0 gY ( χ∆ζ, χ ∆ζ, k̄Y ∆ζ)
ˆ(x+Y),0 =
dζ
dχ
dχ
πn0 −∞
∆ζ 3
0
0
0
0
0
×Ai(ζ + χ)Ai(ζ + χ)Ai(ζ + χ )Ai(ζ + χ0 ),
(9.16)
where
gY (η, η 0 , χ) = ηη 0
Z
0
∞
J1 (ηt)J1 (η 0 t) −χ√1+t2
√
e
dt.
t 1 + t2
(9.17)
The same computational procedure as for the regular Airy gas yields the data shown
in Fig. 9.1. Some preliminary work have been done for producing a parameterization of
the screened Airy exchange.
Chapter 10
I   P
Perfection is achieved, not when there is nothing more to add,
but when there is nothing left to take away.
Antoine de Saint Exupéry
Paper 1: Subsystem functionals in density-functional theory: Investigating the
exchange energy per particle.
The paper presents the subsystem functional approach and examines properties of a suggested form of a subsystem functional for subsystems with slowly varying electron densities. A main result, relevant also outside the context of subsystem functionals, is that the
expansion of the local exchange energy per particle is ill defined. The fact that the expansion is ill defined was demonstrated through explicit computation for model systems. The
paper goes into much detail of the Mathieu gas model system, which is necessary to make
a careful data analysis. In an appendix, the paper gives many details on the construction
of the computer program used to generate the data.
I wrote the computer programs and performed the calculations. My coauthor and I
did the data analysis and theory discussions together. I wrote the first draft of the paper,
and then my coauthor and I completed it jointly.
Paper 2: How to Tell an Atom From an Electron Gas: A Semi-Local Index of
Density Inhomogeneity.
The paper discusses the construction of indices to categorize regions of the electron density. Such indices are necessary in a subsystem functional approach for specifying the interpolation between functionals used within a system. The paper discusses the problem
of distinguishing regions of the density pertaining to atoms from slowly varying gas-like
regions. A main result is that to avoid any confusion between the two classes of density
67
68
Chapter 10. Introduction to the Papers
regions, a rather complicated expression is needed that involves higher order derivatives of
the electron and kinetic energy densities.
I took part in discussions of the ideas and results. I wrote the computer program for
performing the tests of the indices in the Mathieu gas. I wrote the part of the paper that
is about the Mathieu gas tests.
Paper 3: Alternative separation of exchange and correlation in
density-functional theory.
The paper presents a method to create an exchange functional for partial regions where the
electron density is slowly varying. The part of the exchange energy that causes the expansion of the local exchange energy per particle to be ill-defined is separated out and instead
added to the correlation energy. The new ‘revised’ exchange quantity is demonstrated to
be numerically well behaved. Its second order GEA is derived, which provides a functional for slowly varying electron densities. Furthermore, a local density approximation is
constructed based on the revised exchange and correlation.
I wrote the computer program, performed the calculations and created the figures. My
coauthor and I did the data analysis and theory discussions together. I wrote the first draft
of the paper, and then my coauthor and I completed it jointly.
Paper 4: Functional designed to include surface effects in self-consistent
density functional theory
This paper constructs a functional using the subsystem functional scheme. The functional
automatically partitions the electron density into surface and interior regions and applies
suitable approximations in either part. Successful test results of the functional in electronic
structure calculations of aluminum, platinum and silicon are presented.
I implemented the functional in the pseudopotential generation program, which involved extending the software with routines for numerical functional derivatives. I implemented the functional in the DFT program. Calculations of the bulk test results and the
jellium XC surface energies were performed by me. Preparation for the vacancy tests were
done jointly. My coauthor and I did the data analysis and theory discussions together. I
wrote the first draft of the paper, and then my coauthor and I completed it jointly.
Paper 5: PBE and PW91 are not the same
The paper is a comment on an unexpected feature seen in the test data of Paper 4. It is
common practice to regard the PBE functional as basically equivalent to the PW91 functional, but with a simpler derivation. However, we discovered that for metal vacancies and
jellium surface energies, the two functionals perform more differently than expected. We
present a model that relates the difference in vacancy formation energies to the difference
in jellium formation energies.
I did the main work on the model relating jellium and vacancy results and created the
figures. I wrote the first draft of the paper, and then the coauthors completed it jointly.
69
Paper 6: Numerical integration of functions originating from quantum
mechanics
This paper is a technical report on an algorithm for parallel integration used for some of
the data presented in Paper 1. All work and the writing of the report were made by me.
A
At the all-you-can-eat buffet, the only obstacle is yourself.
Scott Adams, in the comic Dilbert
Thanks to all who have helped making this thesis a reality!
The supervisor of my research projects, Ann Mattsson; you have guided me in so many
ways. Your aid and supervision have always led me back on track whenever I have been lost
or confused. Your inexhaustible enthusiasm has been a true source of inspiration for me;
not only in my work, but also for life in general. You have been an excellent supervisor,
but I also see you as a close friend. There are few topics that we have not discussed during
the six years we have known each other, and your insights have enlightened me countless
times. If more people had your attitude to life, the world would be a far better place.
Thank you!
My supervisor at KTH, professor Göran Grimvall, have given me great help with various formal matters throughout my time as a graduate student and provided me with a
very supportive environment in which I have performed my work. We have also had a fair
share of interesting discussions on a wide range of topics. To assist teaching in his course
on thermodynamics was a very educational experience for me. I am sure Göran’s confident
appearance has inspired me to be prepared for whatever next step I take in life.
A special ‘thank you’ goes to my good friends Marios Nikolaou and Tore Ersmark. Our
many lunches and discussions are probably what have kept me sane (?) all these years.
Keep up the good work, and good luck to you both.
Thanks to all the people at the department who over time have contributed to the
friendly atmosphere, and with whom I have had the delight to interact. Mattias Forsblom:
lets make ‘combi’ a lifestyle, Nils Sandberg: gives talks so relaxed you feel like you are
having tea and freshly baked scones in his living room, Blanka Magyari-Köpe: the office
got so silent when you left ;-), keep up that intense energy, Martin Lindén: thinks “cheap
outdoor sports equipment!” when he hears Los Angeles, Sara Bergkvist: sweet on the
outside, but on the inside she is orchestrating evil plans keeping me a computer admin
slave ;-), Jurij Smakov: thanks for an entertaining time in Austin, Håkan Snellman: thank
you for being my mentor back when I was a undergraduate student; as I see it, you are
partly responsible for me being here in the first place, Gunnar Benediktsson: a final ‘Salve!’
71
72
Acknowledgments
to my office neighbor, Tommy Ohlsson: thanks for bringing some spirit to the workplace
by Friday coffee, wall posters, etc. Anders Vestergren: you are a truly entertaining person
and a joy to be around. And then of course, in no particular order, Mattias Blennow,
Tomas Hällgren, Martin Hallnäs, Helena Magnusson, Kristin Persson, Mathias Ekman, Jakob
Wohlert, Gunnar Sigurðsson, Olle Edholm, Jack Lidmar, Mats Wallin, Patrik Henelius, Edwin
Langmann, Göran Lindblad, Bo Cartling, Erik Aurell, Anders Rosengren, Jouko Mickelsson,
John Rundgren, Bengt Nagel, Clas Blomberg, Askell Kjerulf ; and surely other people I have
left out (sorry).
Also, a big ‘thank you’ to the whole Mattsson family for making me feel unreservedly
welcomed into your family life and activities during my many visits to Albuquerque.
Thanks to Thomas who during several interesting preparing-dinner discussions have made
me realize the importance of being knowledgeable in the world around us; world economics, social and political issues etc. Thanks to Carolina for a ski coaching that borders
to the surreal in getting an absolute beginner up to speed. Thanks to Simon for demonstrating that I am not infallible when it comes to having my behind kicked in computer games,
and for hours of fun chicken racing warhogs. Also, thanks to all the friendly people at Sandia National Laboratories in Albuquerque; in particular Peter Schultz who introduced me
to an amazing tuna food dish in Taos with a taste that still lingers in my head.
Thanks to the open source community in general and Linus Torvalds in particular; your
efforts have made my life much simpler, both as a system administrator and as a researcher.
A huge ‘thank you’ goes to my family, Solveig, Michele, Alex and Maria, thanks for
being there and checking up on me while I have been engulfed in my work.
Finally, my warmest heartily thanks go to my love Maria Tengner; thank you for your
constant support. You are the light that brightens my reality. I love you.
Rickard Armiento,
Stockholm, 30 Aug 2005
Appendix A
U
The form of some equations depends on the choice of units in which they are expressed.
This thesis uses SI units, but in the papers the use of unit systems varies. To avoid confusion
because of the differing practices, a brief summary of the relevant unit systems follows.
A.1
Hartree Atomic Units
The bohr unit is introduced as a length based on quantities common for calculations on
atomic scales. The hartree is then defined as the Coulomb repulsion between two electrons
separated by one bohr;
a0 =
4π0 h̄2
= 1 bohr,
me e2c
e2c
= 1 hartree.
4π0 a0
(A.1)
When speaking of hartree atomic units, one usually takes the hartree unit to be dimensionless (i.e., 1 hartree = 1) and additionally sets
1
h̄2
= .
2me
2
(A.2)
Some presentations stop here, because this is enough to get rid of the most common prefactors of quantum mechanical equations, simplifying them significantly. However, it is
not unusual also to make a set of other quantities dimensionless and equal to 1. One sets
ec = 1 ⇔
1
= 1,
4π0
me = 1 ⇔ h̄ = 1.
(A.3)
This practice is consistent with Hartree’s own use of this unit system 120 in 1927. A common alternative notation for expressing the use of full hartree units is
h̄ = me = ec = 1,
73
(A.4)
Chapter A. Units
74
where 1/(4π0 ) = 1 is assumed, i.e., one starts from the cgs-esu system (see below). Numerical values of quantities of other dimensions than mass, charge, energy and length are
then usually marked as given in “a.u.”, designating atomic units. Giving the values with
no unit at all is also formally correct.
A.2
Rydberg Atomic Units
The rydberg atomic units are based on similar ideas as the hartree atomic units but define
the rydberg as the electron energy of the hydrogen atom
1 me e4c
= 1 rydberg.
2 (4π0 )2 h̄2
(A.5)
It is found that 1 hartree = 2 rydberg. Within the rydberg atomic units one takes the
rydberg to be dimensionless (i.e., 1 rydberg = 1) and also sets
h̄2
= 1.
2me
(A.6)
As for hartree atomic units, some presentations stop here; but it is also common to set
e2c = 2 ⇔
1
= 1,
4π0
me =
1
⇔ h̄ = 1.
2
(A.7)
A common alternative notation for expressing the use of full rydberg units is
h̄ = 1, e2c = 2, me = 1/2.
A.3
(A.8)
SI and cgs Units
The cgs and SI systems of units are based on similar ideas within dimensions of mass, time
and length but they differ significantly in the area of electromagnetism. There are at least
two different conventions for the cgs system in this area, cgs-emu and cgs-esu. In cgs-esu
the charge unit has been chosen to simplify equations involving interactions between static
electric charges by fixing the constant in Coulomb’s law to one, giving 0 = 1/4π . In cgsemu the conventions are chosen to simplify equations involving moving charges by fixing
the permeability of vacuum µ0 = 1/(0 c2 ) = 1 thus giving 0 = 1/c2 .
A.4
Conversion Between Unit Systems
To convert a mathematical formula from SI or cgs to atomic units, one sets the physical
constants to their respective dimensionless numerical values in the atomic unit system.
The same procedure is used for converting from SI to cgs.
A.4. Conversion Between Unit Systems
75
To convert from atomic units to SI or cgs one identifies the unit that the mathematical
formula is supposed to have in SI or cgs. Then one combines the dimensionless quantities
of the atomic unit system into a factor that 1) is equal to the value 1 in the atomic unit
system, and 2) has the unit the formula is supposed to have in SI or cgs units. The expression is then multiplied with this factor. The same procedure is used for converting from
cgs to SI. The factor to use in the latter conversion is, of course, dependent on the kind of
cgs system one is working with, which, if unknown, must be determined for example by
observing the appearance of a Coulomb factor in an equation where it is known that such
a factor should appear.
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I
E0 , 8
Ec , 25
Ex , 25
Ee , 7
Exc [n], 16
GGA
, 35
Exc
mGGA
, 38
Exc
F [n], 12
Fx , 33
J[n], 11
N, 8
−1
Rxc
([n]; r), 25
Ts [n], 15
TT F [n], 11
λ
Uxc
, 23
V [v, n], 10
Ψ0 , 8
Ψe , 7
i , 17
c ([n]; r), 25
x ([n]; r), 25
xc ([n]; r), 22
GGA
(n(r), s2 ), 35
xc
mGGA
xc
(n(r), s2 , q, τ ), 38
LDA
ˆc (n(r)), 31
ˆLDA
(n(r)), 31
x
LDA
ˆxc (n(r)), 30
ˆc ([n]; r), 26
ˆx ([n]; r), 25
ˆxc ([n]; r), 24
F̂, 8
Ĥe , 7
T̂, 8
Û, 8
V̂, 8
φi (r), 17
σi , 8
σxc , 53
τ (r), 58
ri , 8
xi , 8
a0 , 6
kY , 62
n(r), 9
n1 (r, r0 ), 14, 25
n2 (r, r0 ), 14
nγ (r), 26
pF , 63
q , 33
rs , 31
s, 33
t, 34
v(r), 8
veff (r), 16
vxc (r), 17
nc , 27
nx , 26
nxc , 24
k̄Y , 63
ˆ(c−Y) ([n]; r), 63
ˆ(x+Y) ([n]; r), 62
n̂c (r, r0 ), 27
n̂x (r, r0 ), 25
n̂xc (r0 , r), 24
83
84
adiabatic connection, 22
Airy gas, 50
anti-symmetry condition, 8
atom, 3
averaged pair density, 23
B3LYP, 43
B86, 42
B88, 43
BLYP, 43
Born-Oppenheimer approximation, 7
compatible exchange and correlation, 32
configuration interaction, 19
constrained search formulation, 12
conventional exchange energy per particle,
25
conventional exchange-correlation energy
per particle, 24
correlation energy, 25
correlation energy per particle, 25, 26
correlation hole, 27
correlation hole sum rule, 27
coupling constant integration, 22
density functional, 10
density index, 49
density matrix, 14
density-scale invariance, 33
DFT variational principle, 13
divide and conquer, 48
Index
exchange scaling relation, 26
exchange-correlation energy, 16
exchange-correlation energy functional, 5
exchange-correlation energy per particle, 22
exchange-correlation hole, 24
exchange-correlation hole sum rule, 24
exchange-correlation potential, 17
expectation value, 9
external potential, 8
external potential energy, 8
first order spinless density matrix, 14
functional, 10
functional on local form, 30
functional on semi-local form, 30
generalized-gradient approximation, 35
gradient expansion approximation, 33
ground state wave-function, 8
half-and-half theory, 39
Hamiltonian, 7
Hartree method, 19
Hartree–Fock method, 19
Hohenberg–Kohn theorem
first, 11
second, 13
hybrid functional, 38
internal electronic energy, 8
internal potential energy, 8
inverse radius of the exchange-correlation
hole, 25
effective potential, 17
electron, 3
jellium surface energy, 53
electron density, 9
jellium surface model, 53
electron gas, 4
electrostatic energy of a classical repulsive kinetic energy, 8
gas, 11
kinetic energy density, 58
empirical functional, 38
Kohn–Sham equations, 18
energy, 4
Kohn–Sham method, 15
exchange energy, 25
Kohn–Sham orbital energies, 17
exchange energy per particle, 25
Kohn–Sham orbital equation, 17
exchange hole, 25
Kohn–Sham orbitals, 17
exchange hole sum rule, 26
Kohn–Sham particles, 15
85
Lieb–Oxford lower bound, 37
linear response limit, 42
Local Airy Gas, 52
local density approximation, 30
local exchange energy per particle, 25
local exchange-correlation energy
per particle, 24
local functional of the density, 30
local functional of the Kohn–Sham orbitals,
30
local Lieb–Oxford lower bound, 37
LYP, 43
many-electron wave-function, 8
Mathieu gas, 55
meta-GGAs, 37
N -representable, 13
near-sightedness, 47
non-interacting kinetic energy, 15
non-interacting kinetic energy density, 58
non-local functional, 29
non-positivity constraint, 26
normalization condition, 8
nucleus, 3
numerical GGA, 36
pair density, 14
PBE, 37, 41
PKZB, 43
position-dependent Fermi momentum, 63
potential energy functional, 10
potential energy of exchange-correlation, 23
PW91, 37, 41
quantum mechanics, 3
random-phase approximation scheme, 53
Rayleigh–Ritz variational principle, 9
real-space cutoff, 36
reduced density gradient, 34
refinement factor, 33
revPBE, 42
RPA+, 53
semi-local functional of the density, 30
Slater determinant, 18
spatial location, 8
spin coordinate, 8
spin function, 18
state, 7
stationary condition, 16
subatomic particle, 3
subsystem functional, 47
surface position parameter, 50
Thomas–Fermi functional, 11
three-parameter hybrid formula, 39
time independent non-relativistic Schrödinger
equation, 7
TPSS, 44
unconventional correlation hole, 27
unconventional exchange hole, 26
unconventional exchange-correlation hole,
24
uniform electron gas system, 30
universal functional, 10
Xα approximation, 38
Yukawa wave vector, 62
Part III
P
87
1
Paper 1
Subsystem functionals in density functional theory:
Investigating the exchange energy per particle
R. Armiento and A. E. Mattsson,
Phys. Rev. B 66, 165117 (2002).
PHYSICAL REVIEW B 66, 165117 共2002兲
Subsystem functionals in density-functional theory: Investigating the exchange energy per particle
R. Armiento*
Department of Physics, Royal Institute of Technology, Stockholm Center for Physics, Astronomy and Biotechnology,
SE-106 91 Stockholm, Sweden
A. E. Mattsson†
Surface and Interface Sciences Department MS 1415, Sandia National Laboratories, Albuquerque, New Mexico 87185-1415
共Received 7 June 2002; published 31 October 2002兲
A viable way of extending the successful use of density-functional theory into studies of even more complex
systems than are addressed today has been suggested by Kohn and Mattsson 关W. Kohn and A. E. Mattsson,
Phys. Rev. Lett. 81, 3487 共1998兲; A. E. Mattsson and W. Kohn, J. Chem. Phys. 115, 3441 共2001兲兴, and is
further developed in this work. The scheme consists of dividing a system into subsystems and applying
different approximations for the unknown 共but general兲 exchange-correlation energy functional to the different
subsystems. We discuss a basic requirement on approximative functionals used in this scheme; they must all
adhere to a single explicit choice of the exchange-correlation energy per particle. From a numerical study of a
model system with a cosine effective potential, the Mathieu gas, and one of its limiting cases, the harmonic
oscillator model, we show that the conventional definition of the exchange energy per particle cannot be
described by an analytical series expansion in the limit of slowly varying densities. This indicates that the
conventional definition is not suitable in the context of subsystem functionals. We suggest alternative definitions and approaches to subsystem functionals for slowly varying densities and discuss the implications of our
findings on the future of functional development.
DOI: 10.1103/PhysRevB.66.165117
PACS number共s兲: 71.15.Mb, 31.15.Ew
I. INTRODUCTION
In density-functional theory1 共DFT兲 the total electron energy E e is written as a formally exact functional of a given
共arbitrary兲 ground-state electron density. The total electron
energy for a system with an external potential v (r) is then
found as the minimum of E e , occurring for the true groundstate electron density n(r) of the system. The Kohn-Sham
共KS兲 formulation2 of DFT casts the search for this minimum
into a self-consistency calculation of a problem of noninteracting electrons moving in an effective potential v eff(r). The
effective potential has been constructed to make the freeelectron density of the resulting free-electron orbitals, the KS
electron orbitals ␺ ␯ (r), give the sought n(r). In a spin unpolarized system,
n 共 r兲 ⫽2
兺␯ 兩 ␺ ␯共 r兲 兩 2
共1兲
共where the sum is taken over all occupied orbitals兲.
Within KS DFT the total electron energy functional E e is
divided into classical contributions and a remaining part, the
exchange correlation energy E xc . In order to decompose E xc
into local contributions, the exchange correlation energy per
particle ⑀ xc is defined as a density functional which gives the
total exchange correlation energy as
E xc ⫽
冕
n 共 r兲 ⑀ xc 共 r; 关 n 兴 兲 dr.
共2兲
This implicit definition of ⑀ xc is not unambiguous. All transformations preserving the value of the total integral yield
possible choices of ⑀ xc . Equivalently expressed, two correct
⑀ xc are equal apart from an additive function that, multiplied
0163-1829/2002/66共16兲/165117共17兲/$20.00
with n(r), integrates to zero over the whole system. This is
an important property that we explore in this paper.
A suitable approximation of some choice of ⑀ xc (r; 关 n 兴 ) is
needed to use KS DFT in calculations. One such approximative functional put forward in the earliest works of DFT was
the local-density approximation2 共LDA兲. It was aimed at systems with very slowly varying electron densities, but was
remarkably successful for wider use. LDA sets ⑀ xc in every
space point r, with density n(r), equal to E xc per electron of
a system with a constant v eff 共a uniform electron gas兲 chosen
such that the density of the uniform system equals n(r). In
this way LDA uses as input only the local value of the denLDA
„n(r)…. Newer functionals,
sity and can be written as ⑀ xc
generalized gradient approximations 共GGA’s兲, use, apart
from the local value of the density, also the first-order denGGA
sity derivative 共the gradient兲: ⑀ xc
„n(r), 兩 ⵜn(r) 兩 …. Further
functional development such as meta-GGA’s, use additional
parameters not always trivially related to the density, e.g.,
kinetic energy densities.
The successively refined approximations of ⑀ xc (r; 关 n 兴 )
described above all take the slowly varying density as their
starting points. The aim has been to create a single universal
functional useful for all kinds of systems, but the resulting
functionals tend to fail in the parts of the system where the
density is far from homogeneous, e.g., at surfaces.3–5 In contrast to this practice of developing universal functionals,
Kohn and Mattsson6 共KM兲 worked towards a functional specifically designed to handle the edge part of a system. They
suggested that this functional could be used together with
another functional taking care of the interior region of the
system. A more generalized idea of using different functionals in different regions of a system is illustrated in Fig. 1.
66 165117-1
©2002 The American Physical Society
PHYSICAL REVIEW B 66, 165117 共2002兲
R. ARMIENTO AND A. E. MATTSSON
FIG. 1. The generalized idea of dividing a system into subsystems, applying different functionals to the different parts. The
left figure refers to the approach presented by Kohn and Mattsson in
Ref. 6.
Functionals used in this way must all adhere to a single
explicit choice of the exchange-correlation energy per particle. This is an important requirement that is discussed in
this paper.
KM introduced the edge electron gas as a suitable starting
point for a functional to use in the edgelike part of a system.
The simplest possible model of the edge electron gas, the
Airy gas, has a linear effective potential and features wave
functions transitioning from oscillatory to vanishing. A functional based on the Airy gas does not relate the density in the
edge subsystem to a slowly varying density, but is instead
based on other assumptions valid only in an appropriate region near an edge. Within this region of validity an Airy gas
based functional should outperform functionals based on the
homogeneous electron gas, but may not be a suitable approximation in the bulk part or interior of a system.
In a related effort Vitos et al. have developed a functional,
the local Airy gas 共LAG兲.7 Roughly, it corresponds to using
the Airy gas exchange energy per particle and the LDA correlation energy per particle in the edge region, while using
LDA exchange and correlation energies per particle in the
interior region. LAG gives mixed results for two reasons.
First, the LDA correlation functional used in the edge region
is not compatible8 with the Airy gas exchange functional.
Second, the use of LDA in the interior region is, in many
cases, inadequate. An Airy gas based correlation functional
and an improved interior region functional are needed to
improve on the LAG.
The uniform electron gas and the edge electron gas are
not the only interesting starting points for functionals. Other
alternatives should be used to develop functionals for a large
variety of subsystem classes. Such functionals can either be
carefully combined by computational scientists targeting
some specific system, or be composed into more general
functionals applicable to a general set of problems, such as
systems with electronic edges, which was the aim of the
original work of KM.6 Functionals derived from alternative
starting points have already been created, for example for
Luttinger liquid systems.9
In addition to the general discussions about the use of
functionals in subsystems, this work also addresses the development of a functional suitable for the interior region of a
system, where the density is slowly varying. We determine if
a specific 共the conventional兲 choice of the exchange energy
per particle can be expressed as a power expansion in the
density variation. The investigation is based on the Mathieu
gas 共MG兲 model, a noninteracting electron system that models the KS orbitals of an effective potential with a cosine in
one of the three dimensions. The MG is presented in detail,
as its properties are important for the interpretation and discussion of our results. It shows a rudimentary energy-band
structure and its parameter space range from the free-electron
共FE兲 gas to a harmonic oscillator 共HO兲 system. From numerical calculations of the MG we show that the conventional choice of the exchange energy per particle has a
nonanalytical behavior in the limit of slowly varying densities, and thus this choice cannot be described by an ordinary
共analytical兲 expansion. The behavior indicates that the conventional definition of the exchange energy per particle is not
a good choice for the derivation of subsystem functionals.
Our results also raise concerns for the inclusion of Laplacian
terms in functionals outside the scheme of subsystem functionals. The discovered nonanalyticity is argued from extensive numerical data for the MG. This presented data might
also be useful outside of our present work for derivation and
testing of exchange functionals.
In Sec. II, we explain and explore the basic requirement
that suitable subsystem functionals in a divided system
scheme must all adhere to a single explicit choice of the
exchange-correlation energy per particle. This is explicitly
discussed in the context of the exchange energy per particle
in a slowly varying system. In Sec. III, the MG is thoroughly
presented and its HO limit is recognized as a valuable model
system in its own right. In Sec. IV the computed density,
density Laplacian, and exchange energy per particle are analyzed in terms of deviations from their uniform electron gas
values, and finite-size oscillations present in the HO-like part
of the MG parameter space are investigated. The deviations
from the uniform gas values for the density and the Laplacian are shown to behave as expected, but the computed deviations from the uniform electron gas value for the exchange energy per particle imply that the conventional
definition of the exchange energy per particle must be modeled by an nonanalytical function of the Laplacian. In Sec. V
the numerical precision of our data is validated. Finally, in
Sec. VI, our findings are summarized and discussed, with
comments on the future development of subsystem
functionals.
II. EXCHANGE ENERGIES PER PARTICLE
The basic idea explored in this work is to divide the integration over the whole system in Eq. 共2兲 into suitable parts
and apply different approximations of the exchangecorrelation energy per particle, ⑀ xc (r, 关 n 兴 ) to each part. Approximations of ⑀ xc (r, 关 n 兴 ), which can be applied to such a
divided system, are referred to as subsystem functionals. In
this section we will discuss requirements a subsystem functional must satisfy.
At this point we are only concerned with the exchange
contribution to the exchange-correlation energy per particle.
The exchange and correlation terms are separated in the
usual way
165117-2
⑀ xc ⫽ ⑀ x ⫹ ⑀ c .
共3兲
SUBSYSTEM FUNCTIONALS IN DENSITY-FUNCTIONAL . . .
PHYSICAL REVIEW B 66, 165117 共2002兲
The freedom of choice of ⑀ xc , as explained in connection to
Eq. 共2兲, also makes ⑀ x nonunique. Similarly as for ⑀ xc , all
choices of ⑀ x must integrate, multiplied with the electron
density, to the same value 共the total exchange energy E x ;
Ref. 10 presents several definitions of E x and discusses how
be the conthey relate to different choices of ⑀ x ). Let ⑀ irxh
x
ventional choice of ⑀ x , which was also used for the Airy
gas.6 There exists an exact relation11 between this exchange
energy per particle and the KS orbitals. Using the first-order
spinless density matrix ␳ 1 (r;r⬘ ) and the inverse radius of the
exchange hole6 共irxh兲, R ⫺1
x , the relation is expressed in cgs
units as
been used in the derivation of modern nonempirical GGAs as
the limit of low-density variation, and has led to very useful
functionals.14,15
In addition to the dimensionless gradient term, there is
another term that should be included in a general expansion.
This term is proportional to the dimensionless Laplacian,
2 ⫺1
⑀ irxh
x ⫽⫺e R x 共 r 兲 /2,
R ⫺1
x 共 r 兲 ⫽⫺
冕
兩 r⫺r⬘ 兩
dr⬘ ,
冕
V
共6兲
兺␯ ␺ ␯共 r兲 ␺ *␯ 共 r⬘ 兲 ,
共7兲
where n x (r;r⬘ ) is the conventional exchange hole density
and e is the electronic charge.
⑀ LDA
„n 共 r兲 …⫽⫺e 2
x
3
关 3 ␲ 2 n 共 r兲兴 1/3.
4␲
共8兲
An improvement to LDA exchange, proposed in the earliest works on DFT,2 was to use gradient expansions. The
traditional gradient approximation approach results in the
second-order gradient expansion approximation 共GEA兲,
冉
⑀ GEA
„n 共 r兲 , 兩 ⵜn 共 r兲 兩 …⫽ ⑀ LDA
„n 共 r兲 … 1⫹
x
x
冊
10 2
s ,
81
共9兲
where s is the dimensionless gradient,
s⫽
兩 ⵜn 共 r兲 兩
2 共 3 ␲ 2 兲 1/3n 4/3共 r兲
.
共10兲
The correct coefficient, 10/81, of the dimensionless gradient
s was finally established by Kleinman and Lee12 in 1988. In
a truly slowly varying system, the GEA performs well, but
outside of its area of formal validity the GEA is found to be
unsatisfactory when applied in computations. Often it is less
accurate than the LDA.13 However, GEA has successfully
4 共 3 ␲ 2 兲 2/3n 5/3共 r兲
共11兲
.
n 4/3
冉
ⵜ 2n
n
⫺
5/3
冊
1 兩 ⵜn 兩 2
dV⫺
3 n 8/3
冖
n ⫺1/3
S
⳵n
dS⫽0, 共12兲
⳵␰
where ⳵ n/ ⳵␰ is the derivative of the density in the direction
of the outward pointing normal to the surface S enclosing the
volume V. Equation 共12兲, showing one choice of a function
integrating to zero, can be added to the exchange part of Eq.
共2兲. Adding the integrand of Eq. 共12兲, multiplied by a factor
proportional to b, to the GEA, Eq. 共9兲, the expansion of all
possible analytical exchange energies per particle becomes
冋 冉
A. Systems with slowly varying densities
For slowly varying densities, the exchange part of LDA is
the most straightforward approximation of ⑀ irxh
x (r; 关 n 兴 ). The
LDA expression is obtained by inserting KS orbitals for a
constant effective potential 共plane waves兲 in Eqs. 共4兲–共7兲,
giving a constant ⑀ irxh
x , which is parametrized in the uniform
electron density to give the familiar expression
ⵜ 2 n 共 r兲
In the following it is explained why this term can be neglected in GEA and why it is not appropriate to neglect it in
the present context of different functionals in different parts
of a system.
By Green’s formula
共5兲
1 兩 ␳ 1 共 r;r⬘ 兲 兩 2
,
2
n 共 r兲
n x 共 r;r⬘ 兲 ⫽⫺
␳ 1 共 r;r⬘ 兲 ⫽2
n x 共 r;r⬘ 兲
共4兲
q⫽
⑀ x 共 r; 关 n 兴 兲 ⫽ ⑀ LDA
„n 共 r兲 … 1⫹
x
冊
册
10 b 2
⫺ s ⫹bq⫹••• ,
81 3
共13兲
where the surface term always vanishes in practical calculations. In a finite system the integration surface is placed far
outside the system, where the normal derivative of the density is very small. Furthermore, the integrands at opposite
sides of the surface cancel due to the opposite sign of the
directional derivatives of the density. In a periodic system the
integrands on opposite sides of the cell also cancel, since
their normals are in opposite directions. Finally, in a divided
system, any surface element on the surfaces enclosing the
different parts of the system have another surface element
with opposite sign that can cancel if the constant b is the
same for the different functionals used. Hence, as long as the
same functional is used in the whole system, the value of b
can be arbitrary. It is traditionally set to zero, motivating that
GGAs need only depend on the gradient and not on the Laplacian. In a divided system, however, all subsystem functionals used must have the same value of b. Unfortunately, an
explicit definition of the exchange energy per particle resulting in b⫽0 is not known. In the choice between searching
for such a definition or establishing the value of b that corresponds to the definition in Eqs. 共4兲–共7兲 we here choose the
latter.
Turning to our choice of exchange energy per particle, the
expansion takes the form
165117-3
LDA
⑀ irxh
„n 共 r兲 …共 1⫹a irxhs 2 ⫹b irxhq⫹••• 兲 ,
x 共 r; 关 n 兴 兲 ⫽ ⑀ x
共14兲
PHYSICAL REVIEW B 66, 165117 共2002兲
R. ARMIENTO AND A. E. MATTSSON
where the gradient coefficient a irxh is expected to be 10/81
⫺b irxh/3, and the Laplacian coefficient b irxh is to be determined. Since the gradient coefficient is fully determined by
the Laplacian coefficient we will only be concerned with the
Laplacian coefficient.
B. General systems
Although only slowly varying systems are explicitly examined in this work, we comment on the extension of subsystem functionals to general systems. Above we discussed
the requirement that all subsystem exchange functionals applied to one slowly varying system must have the same value
of the Laplacian coefficient b. The same arguments can be
repeated for all terms in the Taylor expansion, leading to the
conclusion that different subsystem exchange functionals applied to a general system must all be based on the same
explicit definition of the exchange energy per particle. This
point was illustrated by assuming the exchange energy per
particle to be analytic. However, it is obvious that analyticity
is not required. Hence, to be a subsystem functional, a full
exchange-correlation functional must be based on a specific
set of definitions. When the integration in Eq. 共2兲 is divided
into integrations over subsystems, new nonvanishing terms
must not be introduced.
III. MATHIEU GAS
The development of exchange-correlation energy functionals has predominately been guided by studies of one
model system, the uniform electron gas. For example, the
Monte Carlo calculation by Ceperly and Alder16 of the total
energy of uniform gases with different densities is the foundation of most correlation functionals in use today, and the
exchange energy of the uniform electron gas is the basis for
the LDA exchange energy functional.2 Other model systems,
like the Airy gas6 and the exponential model,17 have been
studied to expand the understanding of strongly inhomogeneous systems such as surfaces. Sahni and co-workers used
model systems, like the step, linear, and finite-linear potential
models, in studies of surfaces.18
One motivation for using model systems is the unified
development of exchange and correlation functionals. LDA
performs so well since the LDA exchange and correlation
functionals are ‘‘compatible.’’8 The error in the LDA exchange is counterbalanced by the error in the LDA correlation, as the combination gives the energy in the uniform
electron gas. This is in contrast to how GGA’s are usually
developed, where the exchange and correlation functionals
are constructed separately, as accurately as possible, and
little attention is paid to the combined quantity. It is well
known that even though the separate GGA exchange and
correlation energies for the jellium surface are much more
accurate that the LDA quantities, the combined quantity is
actually more accurate in LDA than in GGA 共Ref. 19兲 关this
is, however, not true20 for the PKZB meta-GGA 共Ref. 21兲兴.
By creating functionals from model systems it is possible to
obtain compatible exchange and correlation.
Our aim is to go beyond LDA, basing our study on a
model system suitable for interior regions, containing the
FIG. 2. The effective potential of the Mathieu Gas 共MG兲. The
dot marks a minimum point, i.e., one of the points where the dimensionless gradient vanishes. For amplitudes 2␭ much larger than
the chemical potential ␮ , the MG approaches the harmonic oscillator 共HO兲 model, whose effective potential is shown as a fat broken
line. The opposite limit is the free-electron 共FE兲 gas. The limiting
case between the HO domain and the FE domain arises
when 2␭⫽ ␮ .
slowly varying limit where LDA is appropriate. We seek information about the exchange functional from exploration of
yet another model system, the Mathieu Gas 共MG兲. The MG
is the two-parameter model in which the KS effective potential is described by 共Fig. 2兲
v eff共 z 兲 ⫽␭⫺␭ cos共 pz 兲 .
共15兲
where ␭ is the amplitude, and p is the wave vector of the
effective potential. Since we are mainly interested in the Laplacian coefficient b irxh in Eq. 共14兲, we have chosen z⫽0 to
be at a local minimum in the symmetric effective potential.
The dimensionless gradient in Eq. 共10兲 is always zero at this
point, thus eliminating the gradient term.
The dimensionless parameters of this family of potentials
2
⫽2m ␮ /ប 2 is the
are ¯␭ ⫽␭/ ␮ and p̄⫽ p/(2k F,u ), where k F,u
Fermi wave vector of a uniform electron gas with chemical
potential ␮ . In this work k F,u is considered to be independent of position.
A system similar to the MG has recently been studied by
Nekovee et al.22 using Monte Carlo methods, but with emphasis on strongly inhomogeneous densities. As early as
1952 Slater studied a potential with cosines in all three
directions.23 Some of his results are relevant in our context
and will be repeated here.
A. Exact solution of the MG
Following the general method outlined in Ref. 6,
␺ ␯ 共 x,y,z 兲 ⫽
1
A 1/2
e i(k 1 x⫹k 2 y) ␸ ␩ 共 z 兲
共16兲
is inserted into the KS equations2 关 ␯ ⬅(k 1 ,k 2 , ␩ ); k i L i
⫽2 ␲ m i (i⫽1,2, m i integer兲, and A⬅L 1 L 2 the crosssectional area兴. The solutions to the resulting equation for
␸ ␩ (z),
165117-4
PHYSICAL REVIEW B 66, 165117 共2002兲
SUBSYSTEM FUNCTIONALS IN DENSITY-FUNCTIONAL . . .
冉
⫺
冊
ប2 d2
⫹ v eff共 z 兲 ⫺ ⑀ ␩ ␸ ␩ 共 z 兲 ⫽0,
2m dz 2
共17兲
with v eff(z) from Eq. 共15兲, can be written in terms of
Mathieu functions, F ␩ (x). These functions are described in
Ref. 24. We use the Bloch, or Floquet, form:
␸ ␩共 z 兲 ⫽
⫽
1
冑L 3
F ␩ 共 p̄z̄ 兲
⬁
1
冑L 3
exp共 i ␩ p̄z̄ 兲
兺 c 2k␩ exp共 i2kp̄z̄ 兲 ,
k⫽⫺⬁
共18兲
where ␩ p̄k F,u L 3 ⫽2 ␲ m 3 (m 3 integer兲, L 3 the z length of the
system, z̄⫽k F,u z, and the parameter ␩ is the characteristic
␩
are determined from
exponent. The coefficients c 2k
␩
⫺
共 2k⫹␩ 兲 2 c 2k
¯␭
2p̄
2
冉 冊
␩
␩
⫹c 2k⫹2
兲 ⫽a ␩ ,
共 c 2k⫺2
¯␭
2p̄
2
␩
c 2k
,
共19兲
⬁
␩ 2
兺 k⫽⫺⬁
兩 c 2k
兩 ⫽1.
2
¯
These equations
and are normalized with
also give the eigenvalues a„␩ ,␭ /(2 p̄ )… used in the energy.
The energy of an eigenstate of the MG is
⑀ ␯⫽
ប2 2 2
共 k ⫹k 兲 ⫹ ⑀ ␩ ⭐ ␮ ,
2m 1 2
共20兲
where
冉 冊
¯␭
⑀␩
¯ ⫹p̄ 2 a ␩ ,
⫽␭
.
␮
2p̄ 2
FIG. 3. The parameter space of the MG. Parameters in the
shaded areas correspond to a chemical potential in one of the bands,
while parameters in the light areas correspond to a chemical potential in the free-electron continuum between bands. For combinations of parameters on the full lines the chemical potential is at a
band edge. Thick lines correspond to the bottom of bands, while
thin lines correspond to the top of bands. For the sake of clarity
lines near the origin are not shown. The short-dashed line is the
dividing line between the HO domain and the FE domain 共see text兲
and corresponds to a chemical potential at the maximum of the
effective potential 共Fig. 2兲. For combinations of parameters on a
quadratic line the energy-band structure is constant 共see Fig. 4 and
text兲 apart from scaling. From right to left the long-dashed quadratic lines correspond to ¯␭ /p̄ 2 ⫽0.2, 0.4, 0.8, 20, 40, and 100.
共21兲
Equation 共19兲 can be written in an infinite symmetric matrix form. Matrix theory gives that all values of
¯ /(2p̄ 2 )… are real and bounded from below. The same
a„␩ ,␭
system of equations is recovered while shifting ␩ by an even
¯ /(2 p̄ 2 )… also has a ⫾ ␩ symmetry.
integer. The values a„␩ ,␭
The index ␩ have infinite range, ⫺⬁⬍ ␩ ⬍⬁, and with each
value one energy and one wave function are associated. This
is the extended Brillouin-zone scheme. An alternative is to
set ␩ ⫽even integer⫹ ␨ , ⫺1⬍ ␨ ⭐1, and associate an infinite
number of different wave functions and energies with each
value of ␨ . This is the reduced Brillouin-zone scheme. Note,
in the extended scheme, that ␩ ⫽integer will seemingly produce two solutions as the ⫾ ␩ symmetry coincides with the
even-integer shift symmetry. The issue is resolved by noting
that one of the solutions is associated with the ␩ ⫽
⫺ 兩 integer兩 and the other with ␩ ⫽ 兩 integer兩 . This is further
discussed in association with the energy-band structure of the
MG.
Both the Mathieu functions 共in their real forms, see Appendix B兲 and a( ␩ ,Q) are available in numerical computer
software 共e.g., MATHEMATICA兲, making it easy to reproduce
most of Slaters results.23
B. Parameter space
The parameter space of the MG contains two well studied
limiting cases; the weakly perturbed periodic potential 关the
free-electron 共FE兲 gas兴 and the harmonic oscillator 共HO兲.
The two dimensionless parameters of the MG are ¯␭ and p̄,
but in discussions of certain properties there are dimensionless combinations that work better, most notably the combi¯ p̄ 2 , in the HO limit, and ¯␭ / p̄ 2 , when discussing
nations 冑2␭
the energy-band structure. In order to emphasize the two
dimensionality of the parameter space we do not introduce
new notations for these combinations. In the next sections
the different combinations and their meaning are discussed.
We have chosen to use a parameter space spanned by p̄ and
冑2␭¯ p̄ 2 as is shown in Fig. 3.
1. Periodic potential and p̄
The parameter p̄ describes the periodicity of the potential.
The vector 2p̄k F,u ẑ 共where ẑ is a unit vector in the z direction兲 is the reciprocal-lattice vector. All k-space vectors,
(k 1 ,k 2 , ␩ p̄k F,u ), with a magnitude of the z component being
a multiple of p̄k F,u 共i.e., with integer ␩ ) lie on Bragg planes.
For a detailed discussion of the weak periodic potential see
165117-5
PHYSICAL REVIEW B 66, 165117 共2002兲
R. ARMIENTO AND A. E. MATTSSON
Ref. 25. In the parameter space shown in Fig. 3, lines with
constant p̄ are parallel to the vertical axis.
2. FE gas limit and ¯␭
As ¯␭ →0, the system of equations in Eq. 共19兲 decouples
and
␸ ␩共 z 兲 ⫽
1
冑L 3
exp共 i ␩ p̄z̄ 兲 ,
⑀␩
⫽ ␩ 2 p̄ 2 .
␮
共22兲
共23兲
By substituting k 3 ⫽ ␩ p̄k F,u , the plane waves of the uniform
electron gas are recognized.
Lines with constant ¯␭ are straight and start at the origin,
like the short-dashed line ¯␭ ⫽1/2, in the parameter space
shown in Fig 3. The horizontal axis, ¯␭ ⫽0, is the FE gas 共or
uniform electron gas兲 limit.
3. HO and
冑2␭¯ p̄ 2
For ¯␭ ⫽␭/ ␮ →⬁ 共see dashed line in Fig. 2兲 the occupied
energy levels are well described by a harmonic oscillator.
The cosine potential can be expanded around z⫽0 to lowest
order,
␭p 2 2
z ,
v eff共 z 兲 ⫽
2
共24兲
¯ Õp̄ 2
C. Energy-band structure and ␭
Due to the uniform character of the effective potential in
the x and y directions, the MG has a continuous energy spectrum. 关Only the case where the linear dimensions, L i (i
⫽1,2, and 3), of the system are infinite, i.e., k space is
dense, is considered.兴 The density of states at the chemical
potential only depends on the energy-band structure in the z
direction in k space, that is, on the structure of ⑀ ␩ , since for
any ⑀ ␩ ⭐ ␮ , there is always a free-electron energy addition
that brings the total energy to the chemical potential according to Eq. 共20兲. However, the MG does exhibit a rudimentary
band structure due to the Bragg planes in the z direction of k
space. The characteristic exponent ␩ plays the role of a dimensionless scaled wave vector. Energies in the first band
are given by 0⬍ 兩 ␩ 兩 ⬍1, energies in the second band by 1
⬍ 兩 ␩ 兩 ⬍2, and so on. Note, however, that there are never any
band gaps. The chemical potential can be placed in the freeelectron continuum between two bands. In Sec. IV it is
shown that this band structure influences the quantities calculated for the MG.
Recall that k F,u is not the magnitude of the Fermi wave
vector of a MG system with chemical potential ␮ , but that of
the Fermi wave vector of a uniform electron gas with chemical potential ␮ . The Fermi surface for the general MG system is determined by the k vectors fulfilling ⑀ ␯ ⫽ ␮ in Eq.
共20兲.
The energy in Eq. 共21兲 can be scaled in two ways, each
appropriate for one of the limiting cases:
冉 冊
¯␭ /p̄ 2 →0
¯␭
1 ⑀ ␩ ¯␭
⫽ 2 ⫹a ␩ , 2 ——→ ␩ 2
2 ␮
p̄
p̄
2p̄
and
giving the HO model.
The discrete energy levels in the z direction in k space of
¯ p̄ 2 ,
this system are proportional to 冑2␭
⑀n
¯ p̄ 2 共 2n⫹1 兲 .
⫽ 冑2␭
␮
The KS orbitals are
␸ n共 z 兲 ⫽
冉
¯ p̄ 2 兲 1/2
k F,u 共 冑2␭
冑␲ 2 n n!
冊
1/2
¯ p̄ 2 兲 1/2z̄…
H n „共 冑2␭
¯ p̄ 2 兲 1/2z̄ 兴 2 /2…,
⫻exp„⫺ 关共 冑2␭
1
⑀␩
冑2␭¯ p̄ 2
␮
⫽
共25兲
共26兲
24
where H n (x) are Hermite polynomials and n⫽0, 1, 2, . . . .
The vertical axis in the parameter space in Fig. 3 is the HO
¯ p̄ 2 are parallel to the horilimit and lines with constant 冑2␭
zontal axis.
4. Curvature and ¯␭ p̄ 2
The dimensionless Laplacian q in Eq. 共11兲 of the minimum 共black dot in Fig. 2兲 is, to first order, proportional to the
curvature there. The 共dimensionless兲 curvature is proportional to ¯␭ p̄ 2 , as is seen from Eq. 共24兲.
共27兲
冉 冊 冉 冊 冉 冊
¯␭
2 p̄ 2
1/2
⫹
p̄ 2
¯
2␭
1/2
a ␩,
¯␭
2 p̄ 2
¯␭ /p̄ 2 →⬁
——→ 共 2n⫹1 兲 ,
共28兲
where n is the integer nearest below 兩 ␩ 兩 .
The FE gas limit is obtained when ¯␭ / p̄ 2 →0. For FE like
spectra, scaling according to Eq. 共27兲 is appropriate. The HO
limit is when ¯␭ / p̄ 2 →⬁ and, for HO-like spectra, scaling according to Eq. 共28兲 is used. In Fig. 4 we show four scaled
energy-band structures.
Apart from scaling, the energy spectra are the same for
parameters related by constant ¯␭ / p̄ 2 关see Eqs. 共27兲 and 共28兲兴.
¯ p̄ 2 ),
In Fig. 3 共the parameter space spanned by p̄ and 冑2␭
2
¯
long-dashed lines represent ␭ /p̄ ⫽0.2, 0.4, 0.8, 20, 40, and
100. The x axis corresponds to the FE gas limit, ¯␭ / p̄ 2 ⫽0,
and the y axis represents the HO model, ¯␭ / p̄ 2 →⬁. Note that
¯␭ /p̄ 2 is independent of the chemical potential ␮ . Fixing the
chemical potential in the energy-band structure selects a specific point on a line with constant ¯␭ / p̄ 2 , and thereby sets the
scale of the energy-band structure.
In Fig. 3 the full lines show choices of parameters for
which the chemical potential is placed on an energy level/on
165117-6
PHYSICAL REVIEW B 66, 165117 共2002兲
SUBSYSTEM FUNCTIONALS IN DENSITY-FUNCTIONAL . . .
where, if ␮ is inside a z-dimension energy band, ⑀ ␩ 1 is the
lowest energy in this band. If ␮ is not inside an energy band,
⑀ ␩ 1 is the lowest energy in the band which contains the
z-dimension energy state with highest energy ⭐ ␮ . Furthermore, ⑀ ␩ 2 is the lowest possible energy of all z-dimension
energy states within bands that only contain energies ⬎ ␮ .
By construction ␩ 1 and ␩ 2 are integer.
The parameter ␣ describes the position of the chemical
potential relative to the lower band edges, that is, the lowest
energies of the energy bands in the z dimensional energy
band structure. The parameter ␣ differs from ␩ in that it
indexes values of the chemical potential both within and between the energy bands in the z dimension, making it useful
throughout the parameter space of the MG. Integer ␣ 共lower
band edges兲 are shown as thick lines in Fig. 3.
In the pure HO model 兩 ␩ 1 兩 approaches the index of the
highest discrete energy level with energy ⭐ ␮ . Thus it is easy
to retrieve the 共integer兲 value of this highest index by truncating the ␣ parameter. Furthermore, for the HO model and
the FE limit it is straightforward to express the ␣ parameter
in ¯␭ and p̄ 共where b x c is the highest integer ⭐x):
␣ HO ⫽
␣ FE ⫽
FIG. 4. The energy band structure of selected MG models: 共a兲
¯␭ /p̄ 2 ⫽0, the FE limit, 共b兲 ¯␭ /p̄ 2 ⫽0.8, 共c兲 ¯␭ /p̄ 2 ⫽20, and 共d兲 ¯␭ /p̄ 2
→⬁, the HO limit. The reduced index ␨ (⫺1⬍ ␨ ⭐1) is related to
␩ (⫺⬁⬍ ␩ ⬍⬁) by ␩ ⫽even integer⫹ ␨ .
a band edge. The energy levels of the HO broaden into energy bands as the potential becomes weaker and thereby allows for tunneling between neighboring wells. The shortdashed line with ¯␭ ⫽1/2 marks where the chemical potential
is equal to the maximum of the effective potential 共see Fig.
2兲. This line separates HO-like and FE-like systems.
Within a fixed energy structure 共where ¯␭ /p̄ 2 is constant兲 a
FE-like state is always reached when the chemical potential
is raised well above the effective potential 共i.e., going towards the origin on a line with a constant ¯␭ /p̄ 2 and passing
the short-dashed ¯␭ ⫽1/2 line兲. This is seen in Fig. 4共c兲.
The slowly varying limit is at the origin. In this work
paths with constant ¯␭ /p̄ 2 are followed towards the origin, but
any path towards the origin is equally valid.
The position of the chemical potential relative to the different energy levels ⑀ ␩ is important, and a parameter for this
property is needed. We choose the definition
␣⫽
␮ ⫺ ⑀ ␩1
⑀ ␩2⫺ ⑀ ␩1
⫹ 兩 ␩ 1兩 ,
1
¯ p̄ 2
2 冑2␭
1
⫺ ,
2
1/p̄ 2 ⫹N 共 N⫹1 兲
,
2N⫹1
N⫽
共30兲
bc
1
p̄
.
共31兲
A similar explicit expression can not be constructed for the
general MG case. After inserting Eq. 共21兲 in Eq. 共29兲 the
expression cannot be further simplified. In addition, when
using Eq. 共21兲 for energies of band edges 共i.e., integer ␩ , as
is the case here兲 extra care must be taken not to confuse the
lowest energy in a band with the highest energy in the band
below, corresponding to the two different signs of the integer
␩ . For noninteger ␩ both signs give identical energies.
IV. DENSITY, DENSITY LAPLACIAN AND IRXH
EXCHANGE ENERGY PER PARTICLE IN THE MG
In this section we will use the framework of the MG
developed above to examine a number of DFT quantities.
The primary purpose of this study is to investigate the proposed exchange energy per particle expansion of Eq. 共14兲.
The presentation will be kept on a detailed part by part level,
which is needed to show the true origin of the odd behavior
that is found. A higher level summary and discussion of the
results is deferred to Sec. VI.
Infinite systems are considered; L 1 ,L 2 ,L 3 →⬁, and the k
vectors, k 1 ,k 2 , and ␩ , are continuous variables. The FE limit
is solved by inserting the plane wave KS orbitals, Eq. 共22兲
and Eq. 共16兲, into the definition of the density, Eq. 共1兲, and
the definition of the exchange energy per particle, Eqs. 共4兲–
共7兲. The well known results are
共29兲
165117-7
n u 共 r兲 ⫽
3
k F,u
3␲2
,
共32兲
PHYSICAL REVIEW B 66, 165117 共2002兲
R. ARMIENTO AND A. E. MATTSSON
irxh
⑀ x,u
共 r兲 ⫽⫺e 2
3k F,u
.
4␲
共33兲
Using Eqs. 共1兲 and 共4兲–共7兲 we calculate the densities
n m (r) and n h (r) and the exchange energies per particle
irxh
irxh
⑀ x,m (r) and ⑀ x,h
(r) for the MG and the HO, respectively.
From the calculated densities, density Laplacians and gradients are obtained numerically. Details on numerical methods
and calculational schemes are presented in the appendixes.
A. Analyzing the results: Expanding around the uniform
electron gas
For clarity parameters directly related to the MG are used
in the analysis and, unless otherwise stated, the z⫽0 point is
considered. Instead of relating the calculated exchange energy per particle, ⑀ irxh
x , to the LDA values as in Eq. 共14兲 共i.e.,
relate it to the exchange energy of a uniform electron gas
with the same density兲, it is related to the exchange energy of
a uniform electron gas with the same chemical potential.
With a curvature on the potential not only the exchange
energy per particle but also the density and the Laplacian
deviate from the uniform electron gas values. To lowest order
n m 共 0 兲 ⫽n u 共 1⫹a 1¯␭ p̄ 2 兲 ,
共34兲
q m 共 0 兲 ⫽a 2¯␭ p̄ 2 ,
共35兲
irxh
irxh
⑀ x,m
共 0 兲 ⫽ ⑀ x,u
共 1⫹a 3¯␭ p̄ 2 兲 ,
共36兲
irxh
are given in Eqs. 共32兲 and 共33兲. From Eq.
where n u and ⑀ x,u
共8兲 it then follows that
b irxh⫽
a 3 ⫺a 1 /3
.
a2
共37兲
FIG. 5. The density deviations in the minimum point of the MG
¯ p̄ 2 ). The quantity is constructed to
共cf. Fig. 2兲, 关 n m(0)/n u⫺1 兴 /(␭
give the first Taylor coefficient in an expansion of the MG density
in the parameter ¯␭ p̄ 2 , when approaching the limit ¯␭ p̄ 2 ⫽0 关cf. Eq.
共34兲兴. The line dividing the HO and FE domains in the parameter
space is also shown. An oscillatory behavior that is connected to the
energy-band structure is visible in the HO domain 共cf. Fig. 6兲.
From the data in the FE-like domain the expansion of Eq.
共34兲 is confirmed with a 1 ⫽⫺1/2 共Fig. 7兲.
Obtaining a 1 in the HO model
The independent HO expressions 关Eqs. 共25兲, 共26兲, and
Appendix C兴 are used to compare the behavior of the HO
model with the behavior in the HO-like domain of the MG.
The MG model should approach the HO model when ¯␭ / p̄ 2
→⬁, because the effective potential approaches a harmonic
The prefactors a 1 ,a 2 , and a 3 remain to be determined.
B. Determination of the coefficient of density deviation, a 1
We first examine the quantity
¯ 2
关 n m 共 0 兲 /n u ⫺1 兴 ␭ p̄ →0
——→ a 1 .
¯␭ p̄ 2
共38兲
Figure 5 shows this density deviation of the MG, at the minimum point, from a uniform electron gas with the same
chemical potential scaled with the curvature. In Fig. 6 the
same data are shown as a contour plot with the energy-band
structure in Fig. 3 superimposed. A dependence of the density deviation on the energy-band structure is evident.
A dramatic change happens in the behavior along the line
where the chemical potential is at the potential maximum,
¯␭ ⫽1/2, that is, at the line dividing the HO-like and the FElike domains. This change occurs where the chemical potential rises above the most distinct discrete energy level and
enters a more continuous energy-band structure, once again
illustrating the importance of the energy-band structure for
the properties of the system.
FIG. 6. The density deviations of the MG superimposed by the
energy-band structure. The lighter contour lines are the same quantity as shown in Fig. 5. The darker contour lines reproduce the band
edges in the MG energy-band structure, as shown in Fig. 3. A dependence of the density deviations on the energy structure is
evident.
165117-8
SUBSYSTEM FUNCTIONALS IN DENSITY-FUNCTIONAL . . .
PHYSICAL REVIEW B 66, 165117 共2002兲
FIG. 7. Density deviations vs 1/␣ for the curves through the
parameter space of the MG with constant ¯␭ /p̄ 2 ⫽0.2, 0.4, 0.8, 20,
40, and 100 共shown in legends兲, corresponding to the long-dashed
lines in Fig. 3. The lighter lines with ¯␭ /p̄ 2 ⫽0.2, 0.4, and 0.8 show
density deviations in the maximum point z⫽ ␲ /p, while the other
curves show the density deviations in the minimum point z⫽0. The
light oscillatory curve shows the density deviations for the HO
model, corresponding to the limit ¯␭ /p̄ 2 →⬁. The parameter ␣ is
related to the energy-band structure and is defined in Eq. 共29兲. The
slowly varying limit is approached as 1/␣ →0. In that limit we find
a 1 ⫽⫺0.5 关cf. Eq. 共38兲兴.
FIG. 8. The black line is the density deviation for the HO model
of a system with a low temperature k B T⫽0.05␮ . The light line is
the density deviation for the HO model at k B T⫽0. In the slowly
varying limit we find a 1 ⫽⫺0.5 at nonzero temperature, which
agrees with the value extracted in Fig. 7.
oscillator potential. Furthermore, in this limit, the MG energy spectrum approach the energy spectrum of the HO system. Hence the MG density in the HO-like limit should approach the pure HO density. This is confirmed in Fig. 7.
However, using the limiting procedure in Eq. 共38兲, convergence to a single value of a 1 is not obtained. The convergence is prevented by heavy oscillations, a situation similar
to sin(1/x) in the limit x→0, with a range of limiting values.
The sum in the expression for the density, Eq. 共A1兲, can
be evaluated explicitly at z⫽0, yielding
¯ p̄ 2 兲 3/2
n h 共 0 兲 ⫽n u 冑␲ 共 冑2␭
⫻
¯ p̄ 2 ⫺4N e ⫹1 兲 N e 共 2N e 兲 !
共 3/冑2␭
4 Ne
共 N e! 兲2
.
共39兲
N e is the number of discrete energy levels with even index n
and energy ⑀ n ⭐ ␮ . Examining Fig. 7, a periodic behavior
with ⌬ ␣ ⫽2 is seen, where maxima and minima of the oscillations in the density coincide with integer values of ␣ ,
indicating a strong relationship between the oscillations and
the energy-band structure. The limit ¯␭ p̄ 2 →0 is therefore
taken separately for each point with a fixed relative position
to two consecutive even ␣ . By defining a number 0⭐ ␣ e
⬍2 as the smallest number to subtract from ␣ to obtain an
even integer 共i.e., ␣ e is the distance in ␣ from the chemical
potential, ␮ , to the highest even energy level ⭐ ␮ ), N e can
be expressed as
N e⫽
␣⫺␣e
⫹1,
2
共40兲
which is inserted into Eq. 共39兲. Using the explicit expression
for ␣ for the HO, Eq. 共30兲, and keeping ␣ e constant, a Taylor
¯ p̄ 2 gives as coefficient for the term
expansion of n h in 冑2␭
2
¯
proportional to ␭ p̄ ,
5
a 1 共 ␣ e 兲 ⫽⫺ ⫹6 ␣ e ⫺3 ␣ 2e .
2
共41兲
This is a parametrization, in ␣ e , of the range of possible
limiting values of a 1 .
Averaging a 1 ( ␣ e ) over 0⭐ ␣ e ⬍2 gives ⫺1/2, i.e., the
same value of a 1 as extracted from the FE domain of the
MG. Oscillations in the HO model are thus superimposed on
a curve converging to the same value of a 1 as in the FE
domain.
When a low temperature is introduced by adding the usual
temperature factors26 into the KS-orbital system and numerically recalculating the density, a 1 converges to ⫺1/2, as is
seen in Fig. 8. This motivates taking averages over ␣ e in the
zero-temperature HO model, or equivalently, averaging over
the position of the chemical potential in the energy-band
structure, as a way of extracting information valid in more
realistic cases.
To summarize, the density of the MG model behaves differently in the FE-like and HO-like regions of the parameter
space. In the first region the chemical potential is in a FElike energy structure. The density is well behaved, and converges to a 1 ⫽⫺1/2. In the second region the chemical potential is in a HO-like discrete z-dimension energy structure.
The density oscillates heavily with the system parameters.
Curves with ¯␭ /p̄ 2 constant, starting from the HO-like region
and approaching the slowly varying limit 共by going in the
limit ¯␭ p̄ 2 →0) eventually reach the FE-like region where the
oscillations damp out. In the case of the pure HO system,
165117-9
PHYSICAL REVIEW B 66, 165117 共2002兲
R. ARMIENTO AND A. E. MATTSSON
q h共 0 兲 ⫽
¯ p̄ 2 ⫺12N o ⫺3 兲共 2N 2o ⫹N o 兲 共 2N o 兲 !
4c q 共 5/冑2␭
15
4 No
共 N o! 兲2
⫺
¯ p̄ 2 ⫺12N e ⫺1 兲共 N 2e ⫺N e 兲 共 2N e 兲 !
8c q 共 5/冑2␭
15
4 Ne
共 N e! 兲2
⫺
¯ p̄ 2 ⫺4N e ⫹1 兲 N e 共 2N e 兲 !
2c q 共 3/冑2␭
,
3
4 Ne
共 N e! 兲2
c q⫽
¯ p̄ 2 ) vs 1/␣ for the same paFIG. 9. Laplacian deviations q/(␭
rameters as in Fig. 7. In the slowly varying limit we find a 2
⫽⫺1.5 关cf. Eq. 共42兲兴.
however, the chemical potential is stuck between the endless
number of purely discrete energy levels, leaving the oscillations undamped.
The oscillations present in the HO model 共and throughout
the HO-like domain of the MG兲 are a technical issue at zero
temperature and uninteresting when drawing conclusions
about more realistic systems. When introducing a temperature into the HO model, or, equivalently, averaging over the
position of the chemical potential, the limiting value of a 1
⫽⫺1/2 is recovered. Note that no artificial finite size is imposed in our calculations, like using periodic boundary conditions or hard walls. The oscillations emerge naturally from
the discrete energy levels in the HO limit and are present
also in the non-numerical treatments. Hence, when using
such a simplistic model as the HO to test proposed gradient
expansions or for fitting of parameters, some method similar
to our ␣ averages or temperature additions must be used to
quench the oscillations and obtain results valid for general
systems.
C. Determination of the coefficient of Laplacian deviation, a 2
Next, we examine
¯␭ p̄ 2 →0
q m共 0 兲
——→ a 2 ,
¯␭ p̄ 2
冉 冊
nu
n h共 0 兲
5/3
3 冑␲
¯ p̄ 2 兲 5/2.
共 冑2␭
4
共43兲
共44兲
N e is the number of discrete energy levels with even index n,
and N o is the number of discrete energy levels with odd
index m, such that their energies ⑀ n and ⑀ m ⭐ ␮ .
In analogy to ␣ e above, we introduce a parameter 0
⭐ ␣ o ⬍2 as the smallest number that gives an odd integer
when it is subtracted from ␣ . We get
N o⫽
␣⫺␣o 1
⫹ .
2
2
共45兲
The relation between ␣ o and ␣ e is ( 兵 x 其 denotes the decimal
part of x)
␣ o ⫽2
再 冎
␣ e ⫹1
.
2
共46兲
Thus, if ␣ e is constant, ␣ o must also be constant. This relation is based on the equal spacing of the HO energy levels
and thus is only valid in the pure HO model.
Using N o ( ␣ o ) and N e ( ␣ e ) and keeping ␣ e and ␣ o con¯ p̄ 2 gives the
stant, a Taylor expansion of Eq. 共43兲 in 冑2␭
coefficient for the term proportional to ¯␭ p̄ 2 as
a 2 共 ␣ e 兲 ⫽⫺3 共 1⫺ 兩 ␣ e ⫺1 兩 兲 ,
共47兲
where we have eliminated ␣ o by observing that ␣ e and ␣ o
fulfill 1⫹(1⫺ ␣ o ) 2 ⫺(1⫺ ␣ e ) 2 ⫽2(1⫺ 兩 ␣ e ⫺1 兩 ) in the interval of their definition. Averaging a 2 ( ␣ e ) over 0⭐ ␣ e ⬍2
gives ⫺3/2, i.e., the same as the value of a 2 in the FE-like
domain of the MG.
D. Divergence of the coefficient of exchange energy
per particle deviation, a 3
共42兲
When examining
irxh
irxh
⫺1 兴
关 ⑀ x,m
共 0 兲 / ⑀ x,u
where a 2 ⫽⫺3/2 in the FE-like part of parameter space
共Fig. 9兲.
Obtaining a 2 in the HO model
In the HO model, the Laplacian of the density has an
oscillatory behavior similar to that of the density, as seen in
Fig. 9. For z⫽0, the Laplacian, Eq. 共11兲, for the HO model
becomes
¯␭ p̄ 2
¯␭ p̄ 2 →0
——→ a 3 ,
共48兲
as in Fig. 10, no convergence to a value a 3 in the limit
irxh
¯␭ p̄ 2 →0 is observed. This indicates that ⑀ x,m
(0) does not
2
¯
have an analytical expansion in ␭ p̄ , as was assumed in Eq.
共36兲. In Fig. 10 the same expression but with the LDA exchange energy per particle is also shown. As expected the
LDA limiting value is a 1 /3⫽⫺1/6, which is obtained by
inserting Eq. 共34兲 into Eq. 共8兲.
165117-10
PHYSICAL REVIEW B 66, 165117 共2002兲
SUBSYSTEM FUNCTIONALS IN DENSITY-FUNCTIONAL . . .
Since the limiting procedure of low curvature at the maximum point is appropriate only for chemical potentials ␮
⬎2␭, or ¯␭ ⬍1/2, data outside the FE-like part of the parameter space of the MG are not investigated 共Fig. 3兲.
The three quantities to consider thus are
FIG. 10. Deviations from the uniform electron gas exchange
irxh
¯ 2
energy per particle, ( ⑀ irxh
x / ⑀ x,u ⫺1)/(␭ p̄ ), for the same parameters
as in Figs. 7 and 9. In the slowly varying limit this expression is
expected to approach the a 3 coefficient in Eq. 共48兲, but all irxh
curves are diverging and no value can be extracted. For comparison
the same expression for the LDA exchange energy per particle,
¯ 2
¯ 2
( ⑀ LDA
/ ⑀ irxh
x
x,u ⫺1)/(␭ p̄ ) for ␭ /p̄ ⫽0.8, is shown.
a 3 in the HO model
In the HO model, not only the characteristic energy structure related oscillations are present but also the divergence
seen in the FE-like domain of the MG 共Fig. 10兲. Since both
the maxima and the minima diverge in the ¯␭ p̄ 2 →0 limit, the
averaging technique used previously would not cure the divergence. Nor will the behavior be canceled by the other
coefficients when composing b irxh according to Eq. 共37兲.
The divergence of the a 3 coefficient does not imply that
irxh
⑀ x,h
itself diverges. In fact, ⑀ irxh
converges to the FE-limit of
x
Eq. 共33兲 in both the MG and the pure HO. This indicates that
⑀ irxh
is not analytical at all points, which we will discuss in a
x
later section. The divergence in the limit ¯␭ p̄ 2 →0 with ¯␭ /p̄ 2
constant, seems to be of logarithmic kind 共rather than, for
example, x y with y being a fractional number兲. It could be
possible to create a local expansion of such a nonanalytical
function, but not as a regular power expansion as Eq. 共14兲. A
suitable expansion needs one or more nonanalytical terms
that tend to zero in the slowly varying limit, like qlog兩q兩.
E. Analyzing data at the maximum of the potential
The fact that the gradient term in the expansion in Eq.
共14兲 is zero at the minimum of the potential at z⫽0 was used
above, thus giving direct access to the value of b irxh. This is
also the case at the maximum of the potential at z⫽ ␲ /p,
which allows us to analyze the results in terms of negative
curvature.
We must, however, compare with the correct uniform
electron gas, having a chemical potential ␮ max⫽ ␮ ⫺2␭.
Thus k F,u in Eqs. 共32兲 and 共33兲 should be replaced by
¯ , and the negative dimensionless cur(k F,u ) max⫽k F,u 冑1⫺2␭
¯ p̄ 2 ) max⫽⫺␭
¯ p̄ 2 (1
vature must be rescaled according to (␭
⫺2
¯
⫺2␭ ) .
¯ 兲 3 兴 ⫺1
n m 共 z⫽ ␲ / p 兲 / 关 n u 共 冑1⫺2␭
,
2
⫺2
¯ p̄ 共 1⫺2␭
¯兲
⫺␭
共49兲
q m 共 z⫽ ␲ / p 兲
,
¯
¯ 兲 ⫺2
⫺␭ p̄ 2 共 1⫺2␭
共50兲
irxh
irxh冑
¯ 兲 ⫺1
⑀ x,m
1⫺2␭
共 z⫽ ␲ / p 兲 / 共 ⑀ x,u
.
¯ p̄ 2 共 1⫺2␭
¯ 兲 ⫺2
⫺␭
共51兲
In Figs. 7, 9, and 10 the data for the maximum points are
drawn as light lines. No major differences are seen between
darker and lighter lines, confirming the symmetry between
positive and negative curvature in the density and the Laplacian, and implying this symmetry for the inverse radius of
the exchange hole definition of the exchange energy per particle, Eqs. 共4兲–共7兲, at low curvature.
V. COMMENTS ON NUMERICAL RESULTS
Since we only have numerical proof that b irxh is not well
defined, indicating nonanalyticity of the exchange energy per
particle of Eqs. 共4兲–共7兲, the accuracy of our results needs to
be considered. As seen in Fig. 10, LDA has converged well
before the irxh curves are in doubt numerically, which is one
indication that the divergence of the irxh curves is not due to
numerical errors. We base an estimate of the accuracy of our
calculations in the FE-like domain of the MG on an independent numerical inspection which will be explained in this
section. The estimated errors are presented in Table I.
Not only the prefactor 10/81 in Eq. 共9兲 is known but also
prefactors for higher-order terms.27 While remembering that
these factors are valid only as an expansion of the exchange
energy itself, that is, for the expansion integrated together
with the density according to Eq. 共2兲, we use this as an independent check of the accuracy of our numerical calculations of the exchange energy per particle.
The fourth-order expansion is according to Svendsen and
von Barth 共SvB兲,
冉
LDA
⑀ SvB
1⫹
x ⫽⑀x
冊
10 2 146 2 73 2
s ⫹
q ⫺
s q⫹0s 4 .
81
2025
405
共52兲
The higher-order prefactors 73/405 and 0 are not exact but
the possible errors in these prefactors does not influence the
results since s and q are very small in the FE-like domain of
the MG. For values in the HO-like domain, s and q can be
very large and a comparison with the SvB expression is not
adequate.
LDA
LDA
and ⑀ irxh
are compared over a
In Fig. 11 ⑀ SvB
x /⑀x
x /⑀x
half period in the spatial coordinate for one representative set
165117-11
PHYSICAL REVIEW B 66, 165117 共2002兲
R. ARMIENTO AND A. E. MATTSSON
TABLE I. Error estimates for selected points in Fig. 10. The right part of the table refers to Fig. 3 for the
location of the point in the parameter space and to Fig. 11 for the error estimates. The difference ⌬ between
LDA
the value of ⑀ irxh
in z̄⫽0 and z̄ p̄⫽ ␲ /2 is included in the table as a measure of the scale on the y axis
x /⑀x
LDA
in Fig. 11. By adding ␦ ( ⑀ irxh
) to the calculated data, the same total exchange energy is obtained as with
x /⑀x
irxh ¯ 2
the SvB expansion, Eq. 共52兲; see Sec. V and Fig. 11. This corresponds to adding ␦ ( ⑀ irxh
x / ⑀ x,u )/␭ p̄ to the
points in Fig. 10. The third column shows errors for points on the data curves for minima, while the fourth
column shows errors for points on the data curves for the maxima.
¯␭ /p̄ 2
1/␣
0.2
0.2
0.8
0.8
0.8
20
20
100
0.596
0.089
0.582
0.097
0.062
0.075
0.044
0.080
␦
冉 冊
⑀irxh
x 共0兲
⑀ irxh
x,u
冉 冉 冊冊冒
⑀ irxh
x
¯ p̄ 2
/␭
⫺0.0002
⫺0.0116
0.0007
⫺0.0012
0.0113
0.0007
0.0024
0.0155
␦
␲
p
共 ⑀ irxh
x,u 兲 max
共 ¯␭ p̄ 2 兲 max
0.0002
0.0115
⫺0.0003
0.0012
⫺0.0112
N/A
N/A
N/A
of values of ¯␭ and p̄. It is obvious that these two quantities
can only be compared via the integrated values according to
the exchange part of Eq. 共2兲.
The errors in our data points are estimated by comparing
the different integrated values, making the following assumptions: 共i兲 The numerical errors in the calculation of the
density are negligible, compared with the errors made in the
calculation of the exchange energy per particle, since the
density calculation is much less complex 关compare Eqs. 共B2兲
and 共B3兲兴. The density is also well behaved as seen in Fig. 7.
p̄
⌬
0.553
0.089
0.494
0.096
0.062
0.071
0.043
0.061
⫺2.179⫻10⫺3
8.842⫻10⫺6
⫺8.035⫻10⫺3
4.692⫻10⫺5
10.356⫻10⫺6
5.091⫻10⫺4
7.656⫻10⫺5
5.262⫻10⫺3
␦
冉 冊
⑀irxh
x
⑀ LDA
x
⫺4⫻10⫺6
⫺1.45⫻10⫺7
3.5⫻10⫺5
⫺8⫻10⫺8
1.35⫻10⫺7
3⫻10⫺7
1.6⫻10⫺7
2.2⫻10⫺5
This implies that the value of the total exchange energy
based on the SvB expansion in Eq. 共52兲 can be considered an
exact reference value as long as s and q are small. 共ii兲 Statistical errors, due to limited internal numerical precision in
the computer, are negligible compared with systematic errors. This is based on the smoothness of the curve joining
consecutive points in Fig. 11. If there was a statistical error,
the points would be scattered in a band of a width corresponding to the statistical error. 共iii兲 The systematic error is
the same over the entire interval shown in Fig. 11. We have
found no reason why the systematic error should have a dependence on position. The full line in Fig. 11 was created by
LDA
curve
adding a uniform systematic error to the ⑀ irxh
x /⑀x
chosen to make this curve give the same value of the total
LDA
curve.
exchange energy as obtained from the ⑀ SvB
x /⑀x
As a further indication that the discovered behavior is
correct we note that the two model systems, the MG and the
HO, have been treated separately 共see Appendixes B and C兲
and the divergence is present in both models.
VI. DISCUSSION AND CONCLUSIONS
FIG. 11. Exchange functionals based on different sets of definitions can only be compared via the total exchange energy given by
the exchange part of Eq. 共2兲. This is evident in the figure where the
SvB exchange energy per particle from Eq. 共52兲 is shown together
with the irxh exchange energy per particle in Eqs. 共4兲–共7兲 over a
¯ p̄ 2 ⫽0.0049 and p̄
half period in the spatial coordinate for 冑2␭
⫽0.0621. In order to obtain the same total exchange energy from
the SvB and the irxh exchange energy per particle a uniform correction of 1.35⫻10⫺7 is needed for the irxh. This is shown with the
full line. The exchange energy obtained from the SvB expansion in
Eq. 共52兲 can be considered exact because of the small parameters
used in this work.
In the first part of this work we discussed a way, via
subsystem functionals, of extending the successful use of
DFT to more complex systems than are addressed today. The
basic idea of subsystem functionals is to apply different
functionals to different parts of a system. This puts the additional constraint on the functionals that they all must adhere
to a single explicit choice of the exchange-correlation energy
per particle. A limited subsystemlike scheme has already
been implemented and tested.7
To make the scheme of subsystem functionals competitive
with current multipurpose functionals, a subsystem functional more accurate than LDA for the slowly varying interior part of a system is needed. We address the derivation of
such a functional in the second part of this work by examin-
165117-12
PHYSICAL REVIEW B 66, 165117 共2002兲
SUBSYSTEM FUNCTIONALS IN DENSITY-FUNCTIONAL . . .
LDA
FIG. 12. The quantity ( ⑀ irxh
⫺1)/q vs 1/␣ for the same
x /⑀x
parameters as in Figs. 7, 9, and 10, summarizing the data presented
in these plots. In the limit of slowly varying densities, 1/␣ →0, this
quantity is expected to approach the Laplacian coefficient of the
conventional 共irxh兲 exchange energy per particle, b irxh, but the divergence found in Fig. 10 prevents convergence and thus no such
coefficient exist. We thus conclude, in Sec. VI, that the irxh exchange energy per particle can not be expanded in the density variation as suggested in Eq. 共14兲, which indicates that it is not a good
choice when deriving subsystem functionals, which need to adhere
to an explicit choice used throughout the whole system.
ing the conventional definition of the exchange energy per
particle, Eqs. 共4兲–共7兲, for two specific model systems, the
MG and the HO. We arrive at the general result that an
expansion of this exchange energy per particle in the density
variation must contain a nonanalytical function of the dimensionless Laplacian 共if such an expansion exists at all兲. Our
numerical results, presented in Figs. 7, 9, and 10, can be
summarized as in Fig. 12.
Any attempt to model the exchange energy per particle
defined by Eqs. 共4兲–共7兲 with an analytical expression will be
futile, in the sense that it will be unable to reproduce the
nonanalytic behavior found in the slowly varying limit of the
MG. This issue needs to be considered also outside the context of subsystem functionals, particularly when Laplacian
terms are included in GGA-type functionals. The general
scheme of subsystem functionals is unaffected by the
nonanalyticity, but it makes the construction of a subsystem
functional for systems with slowly varying densities less
straightforward. Most importantly it indicates that the conventional 共irxh兲 definition of the exchange energy is not a
good choice for the derivation of subsystem functionals.
The established nonanalytical behavior is consistent
throughout the wide variety of systems encompassed by the
MG model. The MG includes both the finite system of the
HO and the extended system of the weakly perturbed periodic potential, two very dissimilar systems. A functional
based on the results for the MG can potentially become a
true multipurpose functional useful for atoms, molecules,
and bulk systems.
Nonanalytical behavior and improper coefficients have
appeared in previous work28 regarding the same exchange
energy per particle, but only in such a way that it is unknown
whether the difficulties found were caused by problems with
the exchange energy per particle or due to other issues 共such
as in which order the limits have been taken兲. In contrast, our
results show how the unscreened, zero-temperature expressions themselves raise difficulties.
We suspect the long Coulomb tails to be responsible for
the nonanalytic behavior of the exchange energy per particle.
The nonanalyticity should disappear if screening is introduced. This can be done by using a Yukawa potential in
place of the Coulomb potential in Eq. 共5兲. Another way of
taking the screening into account is to perform a full
random-phase approximation 共RPA兲 calculation.
In conclusion, we have found that for the creation of an
expansion for subsystem functionals of the exchange energy
per particle in the density variation, i.e., to go beyond the
LDA exchange in a subsystem, there are two options. Either
the nonanalytical function of the dimensionless Laplacian
must be found and included in a density functional based on
the irxh exchange energy per particle, Eqs. 共4兲–共7兲, or an
alternative definition of the exchange energy per particle
must be chosen. Alternative definitions have been
suggested10 and we plan to continue our investigation by
examining if any of these can give an exchange energy per
particle that can be expanded in a Taylor series. Note, however, that most 共if not all兲 of the exact conditions that are
used when constructing an exchange functional in the traditional way are based on the definition in Eqs. 共4兲–共7兲. New
similar conditions need to be constructed if another definition is used. Some such conditions on alternative definitions
have already been derived.29 As a final remark we note that
the origin of the division of the exchange-correlation energy
into an exchange and a correlation part is based on the
Hartree-Fock method that treats exchange only. In DFT this
division is artificial. An alternative way to proceed could be
to either divide the exchange-correlation energy in another
way or to not divide it at all.
ACKNOWLEDGMENTS
We thank Walter Kohn, John Perdew, Ulf von Barth, Stefan Kurth, and Thomas Mattsson for fruitful discussions. We
also want to thank Saul A. Teukolsky who kindly proposed a
good method for calculating the Mathieu functions. Parts of
the calculations where done on the IBM SP computer at PDC
in Stockholm. Financial support from the Göran Gustafsson
Foundation is gratefully acknowledged. This work was partly
funded by the project ATOMICS at the Swedish research
council SSF. Sandia is a multiprogram laboratory operated
by Sandia Corporation, a Lockheed Martin Company, for the
United States Department of Energy under Contract No. DEAC04-94AL85000.
APPENDIX A: GENERAL COMPUTATIONAL FORMULAS
The density and the inverse radius of the exchange hole,
defined in Eqs. 共1兲 and 共5兲 respectively, are computed according to the formulas in Ref. 6 where the x and y dimensions in both real and reciprocal space are integrated out. For
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PHYSICAL REVIEW B 66, 165117 共2002兲
R. ARMIENTO AND A. E. MATTSSON
completeness these formulas are restated here, in a more general form,
n 共 z 兲 ⫽2
兺␩ 兩 ␸ ␩兩 2 w ␩ ,
w ␩⫽
m
2␲ប2
共␮⫺⑀␩兲
of the tail integral. The approximation is created by composing a new integrand from the asymptotic behaviors of the
integrated functions,
共A1兲
t→⬁
J 1 共 rt 兲 ——→ ⫺
and
⑀ irxh
x 共 r 兲 ⫽⫺
e2
2␲n共 z 兲
冕
dz ⬘
冕
⬁
t 冑1⫹t
⬘
⫻ ␸ ␩ ⬘ 共 z ⬘ 兲共 ⌬z 兲 ⫺3 g 共 k ␩ ⌬z,k ␩ ⬘ ⌬z 兲 ,
g 共 r,r ⬘ 兲 ⫽rr ⬘
1
␸ ␩ 共 z 兲 ␸ ␩* 共 z ⬘ 兲 ␸ ␩*⬘ 共 z 兲
兺␩ 兺
␩
J 1 共 rt 兲 J 1 共 r ⬘ t 兲
t 冑1⫹t 2
0
共A2兲
共A3兲
dt,
To calculate numerical values of g(r,r ⬘ ) a method for
calculating Bessel functions, J 1 (x) is needed. We use the
algorithm described in Ref. 30, as implemented in Ref. 31,
but extended with coefficients for higher accuracy.
The g(r,r ⬘ ) function has a long oscillating tail, which is
handled by separating it into two terms:
g 共 r,r ⬘ 兲
rr ⬘
⫽
冕
J 1 共 rt 兲 J 1 共 r ⬘ t 兲
⬁
t
0
⫻
冉
2
1
t 冑t 2 ⫹1
⫺
1
t2
冊
dt⫹
冕
⬁
0
J 1 共 rt 兲 J 1 共 r ⬘ t 兲
⬁
J 1 共 rt 兲 J 1 共 r ⬘ t 兲
t2
0
⫽
2
3r ⬘ ␲
冋
冉 冊
r ⬘2
r
2
⫺ 共 r 2 ⫺r ⬘ 2 兲 K
冉 冊册
r ⬘2
r
2
0
J 21 共 rt 兲
t2
dt⫽
4r
.
3␲
t→⬁
——→ ⫺
1
2t 4
共A7兲
共A8兲
,
1
共 4 ␲ ⑀ 冑rr ⬘ 兲 4
共A9兲
.
g 共 r,r ⬘ 兲 ——→
冉
冊
1 1
⫺
r ⬘2,
2 ␲r
共A10兲
but this expression has a relative error of as large as 10⫺4 at
the highest values of r needed in our calculation 共about
1000). Since the calculations required a higher precision the
expression is not used.
APPENDIX B:
COMPUTATIONAL FORMULAS FOR THE MG
.
The special case r⫽r ⬘ gives
⬁
t
2
冊
Details on the method used for the numerical integration are
found in Appendix B4.
The speed of the calculation is increased with a lookup
table for g(r,r ⬘ ). Bicubic interpolation is used, with three
million lookup points for values of r and r ⬘ ranging from 0
to 1200. The points are distributed with a nonlinear transformation to increase accuracy for very small r,r ⬘ and when r is
almost equal to r ⬘ . There is a limiting expression for g(r,r ⬘ )
for large values of r and r ⬘ ⬍r that could have been useful
for the construction of the lookup table:
共A4兲
dt.
共A5兲
冕
t 0⫽
dt
共 r 2 ⫹r ⬘ 2 兲 E
1
r→⬁
The first part can be integrated for r⬎r ⬘ , giving 关with K(z)
and E(z) as the complete elliptic integrals of the first and
second kind24兴
冕
⫺
冉
2
␲
cos rt⫺3 ,
␲ rt
4
but leaving out the cosine as it only superimposes oscillations and is ⭐1. When integrating this expression from t 0 to
infinity it gives an approximation of the tail integral, which is
solved for t 0 to give a value for where to end the integration
over t:
where k ␩ ⫽ 关 2m( ␮ ⫺ ⑀ ␩ )/ប 2 兴 1/2; ⌬z⫽ 兩 z⫺z ⬘ 兩 ; and the sums
in Eqs. 共A1兲 and 共A2兲 should be taken over all ␩ of occupied
orbitals in the zero-temperature ground state.
Calculation of g„r,r ⬘ …
2
冑
There is a simple relation between the form of Mathieu
functions used here, the F ␩ (z) of Eq. 共18兲, and the real even
and odd forms of the Mathieu functions,24 ce and se, which
are commonly found in numerical software. Although we
compute F ␩ (z) directly, this relationship is useful for making
independent verifications:
冉
共A6兲
The complete elliptic integrals are calculated using the
implementations of Ref. 31, modified for higher accuracy.
Numerical integration is still needed for the second integral
in Eq. 共A4兲, but the oscillations of this integrand decay much
faster than the oscillations in the original integrand, and
hence are easier to handle.
The infinite interval of integration is treated by introducing an error bound ⑀ and setting it equal to an approximation
F ␩ 共 z 兲 ⫽ce ␩ z,⫺
冊 冉
冊
1 ¯␭
1 ¯␭
⫹ise ␩ z,⫺
.
2 p̄ 2
2 p̄ 2
共B1兲
It was shown in Sec. III A that ␩ enumerates the solutions
of different energies, giving a rudimentary band structure.
When L 3 of Eq. 共18兲 approaches infinity, ␩ can take any
value from ⫺ ˜␩ to ˜␩ , where ˜␩ is the positive number enumerating the state with largest energy ⑀ ˜␩ ⭐ ␮ . The energy ⑀ ␩
is a continuous function of ␩ except at integers, and can be
integrated numerically if formulas that exclude the discon-
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PHYSICAL REVIEW B 66, 165117 共2002兲
SUBSYSTEM FUNCTIONALS IN DENSITY-FUNCTIONAL . . .
tinuous points are used. Besides the practical issues, the discontinuities of ⑀ ␩ have no influence on the values of the
integrals, as they only occur at a finite number of single
points.
The KS orbitals in Eq. 共18兲 can be used to express the
density and the irxh exchange energy per particle as
n m 共 z 兲 ⫽n u p̄
irxh
irxh
⑀ x,m
2p̄ 2
共 z 兲 ⫽ ⑀ x,u
3
2
冕
˜␩
0
nu
n m 共 z̄ 兲
冉 冊
兩 F ␩ 共 p̄z̄ 兲 兩 2 1⫺
冕
⬁
⫺⬁
dz̄ ⬘
⑀␩
d␩,
␮
冕 冕
˜␩
0
d␩
˜␩
0
2. Integrations over the Mathieu index ␩
One solution to the Mathieu matrix equations produces
values for all Mathieu functions with ␩ ⫽even number
⫾ 兩 ␨ 兩 . Because of this, but also as a way to handle the discontinuities of ⑀ ␩ when ␩ is integer, the integrations over ␩
are parted up 共using ˜␨ as the reduced index of ˜␩ ):
冕
共B2兲
˜␩
0
f 共 ␩ 兲d␩⫽
冕
A
i⫽0
0
⫹
d␩⬘
冉兺
冕 冉兺
兩˜␨ 兩
1
B
f 共 2i⫹ ␨ 兲 ⫹
兺 f 共 2i⫺ ␨ 兲
i⫽1
C
兩˜␨ 兩 i⫽0
D
f 共 2i⫹ ␨ 兲 ⫹
冊
d␨
兺 f 共 2i⫺ ␨ 兲
i⫽1
冊
d␨.
共B5兲
⫻Re关 F ␩ 共 p̄z̄ 兲 F ␩* 共 p̄z̄ ⬘ 兲兴 Re关 F ␩*⬘ 共 p̄z̄ 兲 F ␩ ⬘ 共 p̄z̄ ⬘ 兲兴
⫻ 共 ⌬z̄ 兲 ⫺3 g 共 k̄ ␩ ⌬z̄,k̄ ␩⬘ ⌬z̄ 兲 ,
共B3兲
k̄ ␩ ⫽ 冑1⫺ ⑀ ␩ / ␮ ,
共B4兲
where ⌬z̄⫽ 兩 z̄⫺z̄ ⬘ 兩 ⫽k F ⌬z, and we have used F ⫺ ␩ (z)
⫽F ␩* (z).
Important issues with the computation of these formulas
will be treated in the subsections below.
1. Mathieu functions
The algorithm for computing Mathieu functions presented
here has similarities with the one presented in Ref. 32, but
the code was developed by us. In Sec. III A a Fourier expanded Floquet solution was inserted into the Mathieu differential equation, giving a matrix eigenvalue equation describing the solutions, Eq. 共19兲. To solve this equation
numerically the matrix must be truncated at some finite size
2K⫹1. We based the choice of K for a given ␩ on the
numerical testing performed in Ref. 32. The eigenvalue problem is solved by regular numerical methods, using the algorithms from Ref. 31. 共We are aware that these implementations are not as efficient and optimized as more specialized
routines.兲
The index ␩ can be parted into a sum of two terms, an
even integer and a reduced index ⫺1⬍ ␨ ⭐1, as discussed in
Sec. III A. Solutions with the same ␨ , but with different even
integer parts, show up as solutions with different eigenvalues
¯ /(2p̄ 2 )… to the same matrix problem. Solutions for
a„␩ ,␭
␩
␩
→c ⫺2k
.
negative ␩ are obtained from the relabeling c 2k
Hence one single solution of the matrix eigenvalue problem
produces values for all ␩ ⫽even number ⫾ 兩 ␨ 兩 .
The routines for the Mathieu functions are also used to
determine ˜␩ from a known chemical potential ␮ using the
bisection method. Guesses of ˜␩ are refined until an energy as
close to ␮ as possible is obtained. There are more efficient
ways of determining ˜␩ from ␮ , but since this is only done
once per computed data point, the time lost by using bisection is negligible.
For a given ˜␩ , values for A, B, C, and D must be carefully
chosen to make the right-hand expression constitute the
whole interval 0 to ˜␩ . Details on the method used for the
numerical integration are found in Sec. B4.
3. Infinite integration over z̄ ⬘
The integrand over z̄ ⬘ in Eq. 共B3兲 is the expression for the
exchange hole divided by a positive distance and thus always
has the same sign. Furthermore, the doubly infinite integration over all z̄ ⬘ is split at z̄, transformed and re-added into
one integration from 0 to infinity, giving slightly more complicated arguments in the Mathieu functions.
To handle the infinite interval of integration it is possible
to extract a limiting behavior for the z̄ ⬘ integral for the uniform electron gas 共the same cannot be done for the MG兲,
giving a result proportional to 1/z̄ ⬘ 3 . This result, and numerical experiments throughout the parameter space of the MG,
indicate that this is an upper limit on how slowly the oscillations in the integrand can decay. In the HO-like area of the
MG the oscillations die out much more quickly. When approaching the FE limit the decay of the oscillations approaches the result found for the uniform electron gas. Based
on this, our method to handle the z integration is to fit a
function of the form const/z̄ ⬘ 3 to the behavior of the last part
of the integrand. As the integrand decays like this fitted function or more quickly, and has a constant sign, two approximate values of the total integral appear. The first has the
additional const /z̄ ⬘ 3 tail added, and the second totally disregards any tail contributions. These two values for the integral
are approximations of an upper and lower bound on the real
value of the integral. The integration of the z̄ ⬘ integral is
halted when these upper and lower bounds are closer than the
accuracy goal set for the integration.
4. Method of numerical integration
An integration algorithm suitable for parallel computers is
needed, as the multiple levels of integration in the expressions are very time consuming for certain choices of parameters. There are many nonparallel integration routines available, such as the QUADPACK 共Ref. 33兲 routine ‘‘dqag.’’ The
‘‘dqag’’ routine is intended for integration of oscillatory in-
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PHYSICAL REVIEW B 66, 165117 共2002兲
R. ARMIENTO AND A. E. MATTSSON
tegrands, like those encountered in this work. It handles them
adaptively in the sense that it spends most of the time on the
difficult parts of the integrand. For parallel computers there
are only a few commonly available similar adaptive integration routines, as distributing an equal load to each computer
node is difficult.
However, for the integrations encountered in this work the
gain of a proper adaptive integration method is limited, as
the integrands usually are smooth but heavily oscillating,
with a frequency not varying much throughout the interval of
integration. This motivates the choice of a more basic algorithm refining the entire interval of integration at once, which
makes a parallel implementation easier. The algorithm presented here has been developed by us and used in most of the
calculations.
As all finite ranges can be substituted into the range from
0 to 1, only this case will be treated. Ordinary integral substitution using a function, x⫽w(x ⬘ ), fulfilling w(0)⫽0 and
w(1)⫽1, gives
冕
1
f 共 x 兲 dx⫽
0
冕
1
0
f „w 共 x 兲 …w ⬘ 共 x 兲 dx.
共B6兲
We seek an explicit expression for w(x) whose right derivatives, to any order, equals zero as x→⫹0, and whose left
derivatives, to any order, equals zero at x→1. A function
fulfilling these requirements is
w共 x 兲⫽
冕
x
0
2
2
ce ⫺1/(z⫺z ) dz, w ⬘ 共 x 兲 ⫽ce ⫺1/(x⫺x ) ,
c⫽
冉冕
1
2
e ⫺1/(z⫺z ) dz
0
冊
⫺1
,
共B7兲
共B8兲
where c is chosen to meet the requirement w(1)⫽1.
The integration of the combination f „w(x)…w ⬘ (x) can
now be seen as an integration of one period of a periodic
function, as the function values and all derivatives match at
x→⫹0 and x→1. For such integrands ordinary trapezoid
integration converges very rapidly, since error terms cancel.
The argument assumes that f „w(x)…w ⬘ (x) approaches zero
in these limits, which is true unless f (x) is too divergent; similar assumptions are also made for the derivatives
of f (x).
The combination of this substitution and the trapezoid integration can be recast on a form similar to the one used for
Gaussian quadrature formulas 共by also using the requirements limx→⫹0 w ⬘ (0)⫽0 and limx→1 w ⬘ (1)⫽0, the two outermost terms have been disregarded兲:
冕
1
0
1/h⫺1
f 共 x 兲 dx⬇h
x n ⫽w 共 hn 兲 ,
兺
vn f 共 xn兲,
共B9兲
v n ⫽w ⬘ 共 hn 兲 ,
共B10兲
n⫽1
where h is a chosen step length. For each step length the
values of v n and x n can be pre-calculated with some other,
simple, numerical integration algorithm during the program
initialization. For the implementation we note that the two
quantities should be stored intermixed in one array to ensure
good use of the cache memory of the computer.
The integration is performed by iteration, reducing h in
each step, until the relative difference between the results
from two consecutive steps is less than some error bound ⑀ .
A major benefit inherited from the trapezoid integration is
that if h is reduced with a factor of 2 in each step, the previous computed approximation for the next iteration can be
reused. This halves the number of function evaluations
needed.
Despite the fact that Eq. 共B9兲 formally does not include
the end points of the interval 共i.e., it is formally open兲, the
nature of the function w(x) brings x 1 and x n⫺1 extremely
close to 0 and 1 共i.e., for practical purposes the formula is to
be regarded as closed兲. In case the end points of the interval
must be avoided, the interval of integration can be shrunk
minimally and open trapezoid integration used on these
small parts.
For the integrals in this work the described integration
algorithm shows both a rapid convergence and a very stable
behavior. In tricky situations, where the integrand is not entirely smooth, the algorithm results in a trapezoid integration
of a nonperiodic function, and thus converges 共although
slowly兲. However, for the cases where the integrand is well
behaved and smooth 共as it should be兲, the convergence is
much more rapid, imitating the behavior seen with usual
trapezoid integration of whole periods of periodic functions.
For the nonparallel case the results and speed of the described integration method for integrals relevant for this
work were compared with the QUADPACK 共Ref. 33兲 routine
‘‘dqag.’’ That routine seems to be significantly slower, requiring on the average more evaluations of the integrand.
APPENDIX C:
COMPUTATIONAL FORMULAS FOR THE HO
The HO formulas obtained by combining Eqs. 共26兲 and
共A1兲–共A3兲 look roughly similar to the MG formulas but are
computable with less elaborate numerical methods. The KS
orbitals are enumerated by the discrete index of the Hermite
polynomials, making the ␩ , ␩ ⬘ sums of Eqs. 共A1兲 and 共A2兲
range from 0 to N⫺1. The number of occupied orbitals, N,
is related to our input parameters ¯␭ , p̄ by
N⫽
b冑
c
1
1
⫹ ,
¯ p̄ 2 2
2 2␭
共C1兲
where b x c means the highest integer ⭐x.
The speed of the calculations is increased by using an
explicit expression for the Hermite polynomials in z⫽0.
Furthermore, the function g(r,r ⬘ ) is treated as in Appendix
A, but without a lookup table, i.e., the function values are
computed directly when needed.
All integrations in the HO model are performed by a
straightforward implementation of adaptive Gaussian integration. The reason for not using the algorithm described in
Appendix is that the HO model calculations were finished
before the need for a parallelized integration algorithm for
the MG case was discovered. This adds to the independence
165117-16
SUBSYSTEM FUNCTIONALS IN DENSITY-FUNCTIONAL . . .
PHYSICAL REVIEW B 66, 165117 共2002兲
of the two models, and makes the observation that computed
values for the MG model approach values for the HO model
an additional verification of our numerical methods.
As a result, the dimensionless gradient s in Eq. 共10兲, and
Laplacian q in Eq. 共11兲, take the forms
s⫽
APPENDIX D: CALCULATIONAL FORMULAS
FOR THE GRADIENT AND LAPLACIAN
The density is calculated on a fully dimensionless form.
For example, for the MG:
n̄ m 共 z̄ 兲 ⫽
n m 共 z̄ 兲
.
nu
共D1兲
Electronic address: [email protected]
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冏 冏
1
d 2 n̄ m 共 z̄ 兲
5/3
4n̄ m
共 z̄ 兲
dz̄ 2
1
dn̄ m 共 z̄ 兲
dz̄
,
共D2兲
.
共D3兲
The quantities s and q can thus be easily computed by taking
numerical derivatives of the routines that compute the
density.
18
*Electronic address: [email protected]
†
q⫽
4/3
2n̄ m
共 z̄ 兲
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10, 2221 共1974兲.
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165117-17
2
Paper 2
How to Tell an Atom From an Electron Gas:
A Semi-Local Index of Density Inhomogeneity
J. P. Perdew, J. Tao, and R. Armiento,
Acta Physica et Chimica Debrecina 36, 25 (2003).
How to Tell an Atom From an Electron Gas: A Semi-Local Index of Density
Inhomogeneity
John P. Perdew and Jianmin Tao
Department of Physics and Quantum Theory Group,
Tulane University, New Orleans, Louisiana 70118, USA
Rickard Armiento
Department of Physics, Royal Institute of Technology,
AlbaNova University Center, SE-106 91 Stockholm, Sweden
(Dated: July 8, 2003)
From a global perspective, the density of an atom is strongly inhomogeneous and not at all like
the density of a uniform or nearly-uniform electron gas. But, from the semi-local or myopic
perspective of standard density functional approximations to the exchange-correlation energy,
it is not so easy to tell an atom from an electron gas. We address the following problem: Given
the ground-state electron density n and orbital kinetic energy density τ in the neighborhood of
a point r, can we construct an “inhomogeneity index” w(r) which approaches zero for weaklyinhomogeneous densities and unity for strongly-inhomogeneous ones? The solution requires not
only the usual local ingredients of a meta-generalized gradient approximation (n,∇n,∇ 2 n,τ ),
but also ∇τ and ∇2 τ . The inhomogeneity index is displayed for atoms, and for model densities
of metal surfaces and bulk metals. Scaling behavior and a possible application to functional
interpolation are discussed.
I.
INTRODUCTION
How can we tell an atom from a uniform electron
gas, or from an electron gas of slowly-varying or nearlyuniform density? From a global perspective, the answer
is trivial: The atom has a few electrons strongly confined
to a small region of space, while the electron gas has an
infinite number of electrons distributed smoothly over
all space. But from the local or semi-local perspective
of density functional theory, which looks at the electron
density n and perhaps the Kohn-Sham orbital kinetic
energy density τ only in each small volume element, the
answer is not so simple.
Some of the most successful density functionals for the
exchange-correlation energy of a many-electron system
transfer information from the slowly-varying electron gas
to the densities of real atoms, molecules and solids. This
is a major achievement, since most of the density of an
atom is very different from that of a slowly-varying electron gas. To show this, we shall construct an “inhomogeneity index” w(r) which vanishes for a uniform density but approaches unity where the density is strongly
inhomogeneous. No single semi-local inhomogeneity parameter suffices. A composite index (rather like a stock
market index) is needed; it should approach unity when
any one of its many inhomogeneity parameters is large.
As we will see, the construction of an adequate inhomogeneity index from the behavior of the electron density n
and orbital kinetic energy density τ in the neighborhood
of the point r is a subtle problem. It requires using all of
the local ingredients of modern density functionals, and
more.
The local density approximation (LDA) for exchange [1–3] has evolved over the years into the modern Kohn-Sham [3] density functional theory, the cornerstone of most electronic structure calculations in both
condensed matter physics and quantum chemistry. In
LDA, the exchange energy Ex and potential vx (r) for a
ground-state electron-density n(r) are approximated as
Z
ExLDA = d3 r n(r)unif
(1)
x (n(r)),
vxLDA (r) = vxunif (n(r)),
(2)
where
unif
x (n) = −
3
3
(3π 2 n)1/3 = − kF
4π
4π
(3)
is the exchange energy per electron of an electron gas of
uniform density n (in atomic units, where ~ = m = e2 =
1). kF in Eq. (3) is the Fermi wave-vector: n = kF3 /3π 2 .
The exchange potential is
1
1
2 1/3
vxunif (n) = ∂ nunif
= − kF . (4)
x (n) = − (3π n)
π
π
LDA is exact for a uniform or slowly-varying density;
it assumes that each volume element d3 r is like a volume element of a uniform gas at the local density n(r).
2
Equations (1)– (4) are also known as Gáspár-Kohn-Sham
exchange. Slater [4] also pioneered the use of Eq. (4), but
with a coefficient that was not quite right for a slowlyvarying density [1, 3] or for an atom of large atomic number [2].
In modern density functional theory, this idea is extended to include correlation, and the list of local ingredients is expanded. For example, a generalized gradient
approximation (GGA) [5] uses the spin densities (n↑ ,n↓ )
and their gradients (∇n↑ , ∇n↓ ). The meta-GGA for exchange and correlation [6–12] is
Z
MGGA
Exc
= d3 r n(r)
×xc (n↑ , n↓ , ∇n↑ , ∇n↓ , ∇2 n↑ , ∇2 n↓ , τ↑ , τ↓ ),(5)
where
We begin by defining the iso-orbital indicator [14]
X = τW /τ
(0 ≤ X ≤ 1),
(9)
where
τW = |∇n|2 /8n
(10)
is the von Weizsäcker kinetic energy density. For any
one- or two-electron ground-state density, or for any region of space in which one orbital shape dominates both
n and τ , X → 1 [15]. For any slowly-varying density,
we can replace τ by its local-density or Thomas-Fermi
approximation
3
3 2
τ unif =
(3π 2 )2/3 n5/3 = n
kF ,
(11)
10
10
so that
τσ (r) =
occup
X
α
1
|∇ψασ (r)|2
2
(6)
is the orbital kinetic energy density for electrons of spin
σ. The ψασ (r) are the Kohn-Sham orbitals that produce
the density
n(r) =
X
σ
nσ (r) =
X occup
X
σ
|ψασ (r)|2 ,
(7)
α
and are themselves nonlocal functionals [3] of the density
n(r). In the rest of this work, we shall restrict our attention to spin-unpolarized densities (n↑ = n↓ = n/2 and
τ↑ = τ↓ = τ /2).
As we will see in section 2, an adequate inhomogeneity
index requires not only the ingredients n, ∇n, ∇2 n, and
τ , but also ∇τ and ∇2 τ . The last two ingredients are
not currently included in the meta-GGA form of Eq. (5),
but suggest a symmetry between n and τ and arise in the
density matrix expansion [13].
5
p, where p = (|∇n|/2kF n)2
3
(12)
is close to zero.
Our first guess for an inhomogeneity index is then
wX ≡ X 2 ,
(13)
which is close to zero in a slowly-varying electron gas and
equal to one in any one- or two-electron ground state.
Figures 1–3 show wX as a function of r (distance from
the nucleus) for the Hartree-Fock densities [16] of the
atoms Be, Ar, and Zn. Although wX is close to one
near the nucleus (r → 0) and in the density tail (r →
∞), it can be close to zero over large regions, especially
intershell regions, of an atom, although the density is in
fact strongly inhomogeneous in those regions. Thus wX
is inadequate as an inhomogeneity index.
The most important single inhomogeneity parameter
for the exchange energy is probably the Becke parameter [17, 18]
Q=
II. THE MENAGERIE OF DENSITY
INHOMOGENEITY PARAMETERS, WITH
RESULTS FOR ATOMS
5
10
τ
p + q + (1 − unif ),
3
3
τ
(14)
∇2 n
(2kF )2 n
(15)
where
We seek an inhomogeneity index w(r) defined at each
point r of a many-electron system, and bounded in the
range
0 ≤ w ≤ 1.
X→
(8)
We want w(r) to be close to unity for strongly inhomogeneous densities like those of atoms, and close to zero
for weakly inhomogeneous densities like those of slowlyvarying or nearly-constant electron gases. (Note that
nearly-constant densities need not be slowly-varying;
consider a uniform density perturbed by a density wave
of small amplitude but large wave-vector.) We will construct w to be of order ∇4 in the slowly-varying limit.
q=
is the reduced Laplacian and p of Eq. (12) is the square of
the reduced gradient of the density. (p and q tell us how
fast n varies on the scale of the n-dependent local Fermi
wavelength 2π/kF . For further discussion, see Ref. [19].)
The spherically-averaged exchange hole density for any
spin-unpolarized density has the short-range behavior
hnx (r, r + u)isph.avg. = −
n u2 unif
+ τ
[1 − Q]
2
3
+O(u4 ),
(16)
where u is distance from the electron at r. Note that
1−
5 p
τ
=1−
.
τ unif
3X
(17)
3
In a weakly inhomogeneous region of space, Q2 and
the squares of the three individual terms in Q of Eq. (14)
should be much less than 1. So we define
2 2 5
τ 2
10
Y2 =
(18)
p +
q + 1 − unif ,
3
3
τ
and propose our second guess for an inhomogeneity index
wXY ≡
X2 + Y 2
.
1+Y2
(19)
Eq. (19) still makes wXY = 1 in any iso-orbital region
(X = 1), and it makes wXY closer to one than is wX
over much more of the density of an atom (Figs. 1–3).
But there are still “outer intershell” regions of an atom
where wXY ≈ 0. These are regions of space in which
4πr2 n(r) increases with r. In these regions, the usual
meta-GGA parameters of Eq. (5) cannot recognize the
strong inhomogeneity, and thus cannot tell an atom from
an electron gas; indeed, p and q are small there and τ ≈
τ unif , yet these regions are not electron-gas-like.
For example, consider the outer intershell region of the
Be atom, where n = n1s + n2s is dominated by n2s , but
n2s maximizes so that τ = |∇n1s |2 /8n1s + |∇n2s |2 /8n2s
is dominated by n1s . In this region, X of Eq. (9) is
necessarily small. What tells us that this is a region
of strong inhomogeneity? In this region, τ is decaying
rapidly, with a length scale characteristic of n1s , so it is
the derivatives of τ that are needed to tell this atomic
region from an electron gas.
The previous paragraph suggests that we need for τ
the analogs of the dimensionless derivatives of Eqs. (12)
and (15):
2
∇2 τ
|∇τ |
,
(20)
,
qτ =
pτ =
2kτ τ
(2kτ )2 τ
where kτ is defined by
τ=
3 2
k
10 τ
kτ3
3π 2
.
(21)
(pτ and qτ tell us how fast τ varies on the scale of the
τ -dependent Fermi wavelength 2π/kτ .) By analogy to
Y 2 of Eq. (18), we define
2 2
10
5
pτ
+
qτ ,
(22)
Z2 =
3
3
and propose our final inhomogeneity index
wXY Z ≡
X2 + Y 2 + Z2
.
1 + Y 2 + Z2
atom, including the “difficult” intershell regions. wXY Z
dips below one in the valence-shell region of the atom,
suggesting that this region is slightly more homogeneous
than the rest of the atom, as one might expect. The dip in
the valence-shell region seems shallowest for s electrons,
deeper for p electrons, and still deeper for d electrons,
reflecting the increasing orbital overlap from s to p and
p to d shells.
(23)
wXY Z is “balanced” between or symmetric in the n and
τ variables. It is small where the reduced gradient and
Laplacian of n and τ are small in the same sense, and
τ ≈ τ unif , and X 2 1. These conditions are easy to
satisfy in an electron gas, but not in an atom. Figures
1–3 show that wXY Z is close to unity over most of an
III. RESULTS FOR MODEL DENSITIES OF
METAL SURFACES AND BULK METALS
In section 2, we applied our inhomogeneity indices to
atoms, which have a discrete spectrum of Kohn-Sham orbital energies and are strongly-inhomogeneous throughout space. Here we will turn to systems that have a continuous spectrum, and can be either strongly or weakly
inhomogeneous.
In the infinite barrier model (IBM) [20] of a jellium
surface, the non-interacting or Kohn-Sham electrons are
confined to the half-space z > 0 by the effective potential veff (z) = 0 (for z > 0) and +∞ (for z < 0). The
density n(z) then vanishes for z ≤ 0, and tends to a
constant n̄ = k̄F3 /3π 2 as z → ∞, with a first Friedel
peak at 2k̄F z = 6 and smaller Friedel oscillations for
larger z. Figure 4 shows that the surface region is one
of strong inhomogeneity, while the bulk region is one of
weak inhomogeneity, as expected. Note however that
the IBM surface is more inhomogeneous than the selfconsistent [21, 22] jellium surface (Fig. 5) or the surface
of a real free-electron-like metal.
We now turn to the Mathieu Gas (MG) model system,
which is defined by the effective potential
veff (r) =
1 2
k̄ λ̄[1 − cos(2k̄F p̄z)]
2 F
(24)
applied to non-interacting electrons of initially uniform
density n̄ = k̄F3 /3π 2 (i.e., n(r) → n̄ as λ̄ → 0).
Its main properties are determined by the dimensionless
parameters λ̄ and p̄. The inhomogeneity indices are independent of the overall scale, which is set by k̄F . Reference
23 gives more details on the MG model system, the role of
its parameters, and the calculation of the Kohn-Sham orbitals. Here, we simulate bulk Na and Ca by making the
MG effective potential reproduce the corresponding pseudopotential’s first non-zero Fourier term for a direction
perpendicular to a lattice plane. From tabulated coefficients [24] we get the following parameter values (using
bcc monovalent Na with rs = 3.93, and fcc divalent Ca
with rs = 3.27)
Bulk Na model: λ̄ = 0.17, p̄ = 1.140,
Bulk Ca model: λ̄ = 0.087, p̄ = 0.880.
(25)
(26)
For these parameters, the separable MG energy band
structure along the kz direction shows some resemblance
to bulk Na and Ca, with the Fermi level of Na just below,
4
1
1
wXYZ
wXY
0.8
0.8
Be atom
IBM surface (jellium)
0.6
w
0.6
w
wX
wXY Z
0.4
0.4
0.2
wX
wXY
2
4
0.2
0
0
6
8
10
12
14
2k̄F z
0
0
0.5
1
1.5
r (bohr)
2
2.5
3
FIG. 1: The inhomogeneity indices wX (Eq. (13)), wXY
(Eq. (19)), and wXY Z (Eq. (23)) for the Hartree-Fock
density of the Be atom. The 2s valence orbital has
hr−1 i−1 = 1.91 bohr, close to the outer maximum of
4πr2 n(r).
FIG. 4: The inhomogeneity indices wX (Eq. (13)), wXY
(Eq. (19)), and wXY Z (Eq. (23)) for the jellium surface
in the infinite barrier model. The electron density vanishes for z < 0, and approaches a constant n̄ = k̄F3 /3π 2 as
z → ∞, with a first Friedel peak at 2k̄F z = 6. The neutralizing uniform positive background fills the half space
2k̄F z > 3π/4 ≈ 2.36.
1
0.8
wXYZ
wXY
0.4
wX
w
0.6
Ar atom
0.2
0
0
0.5
1
1.5
r (bohr)
2
2.5
3
FIG. 2: Same as Fig. 1, but for the Ar atom. The 2p
valence orbital has hr −1 i−1 = 1.23 bohr, close to the
outer maximum of 4πr 2 n(r).
and that of Ca just above, the first band gap. Figures 6
and 7 show the inhomogeneity indices in the MG bulk
Na and Ca models over half a period of veff , i.e., from
z = 0 to z = π/(2k̄F p̄). As expected for these electrongas-like systems, all the indices are close to zero over the
whole range. The indices are significantly lower for the
Ca model than for Na. This is explained by the observation that, for MG systems, placing the Fermi level successively higher in the energy band structure describes a
path towards the limit of slowly-varying densities (p̄ → 0,
λ̄ → 0) [23]. Hence, the Ca bulk model is expected to be
closer to the slowly-varying limit than the Na model.
For a density wave of relative amplitude A and wavevector k superposed on a uniform density n̄ = k̄F3 /3π 2 ,
i.e., n(r) = n̄[1 + A cos(kz)], we note that p ∝ A2 (k/k̄F )2
1
1
0.8
0.8
Self-consistent surface (jellium)
rs = 2
w
0.6
Zn atom
w
0.6
0.4
wXYZ
0.2
wXY
wX
0.4
wXY Z
wX
wXY
0
0.2
0
2
4
6
8
10
12
14
2k̄F z
0
0
0.5
1
1.5
r (bohr)
2
2.5
3
FIG. 3: Same as Fig. 1, but for the Zn atom. The 3d
valence orbital has hr −1 i−1 = 0.65 bohr, close to the
outer maximum of 4πr 2 n(r).
FIG. 5: Same as Fig. 4 but for the self-consistent jellium surface with bulk density parameter rs = 2 =
(3/4πn)1/3 . The neutralizing uniform positive background fills the half space 2k̄F z > 3π/4 ≈ 2.36, as in
Fig. 4.
5
0.1
0.25
0.2
0.08
W
XYZ
0.06
0.15
w
bulk Ca
w
W
bulk Na
XY
0.04
0.1
0.02
0.05
W
XYZ
W
X
0
0
0.1
0.2
0.3
z/[period length]
0.4
0.5
FIG. 6: The inhomogeneity indices for the Na bulk model
density, obtained from the system described by the MG
effective potential veff in Eqs. (24) and (25). The plot
ranges over half a period of the system, from the density
maximum at z = 0 (at the veff minimum) to the density
minimum (at the veff maximum).
and q ∝ A(k/k̄F )2 . For |A| 1, we shall have p |q|
and wX wXY , as can be seen in the right half of Fig.
4 or 5 and in Figs. 6 and 7. Either |A| 1 or k/k̄F 1
can make p and |q| small, although only k/k̄F 1 is the
limit of slowly-varying densities.
In the slowly-varying limit, we can use the secondorder gradient expansion [25]
20
5
(27)
τ → τ unif 1 + p + q
27
9
to express
X 2 → 2.78p2 ,
W
X
WXY
0
0
0.1
0.2
0.3
z/[period length]
0.4
0.5
FIG. 7: The inhomogeneity indices for the Ca bulk
model. The plot is similar to Fig. 6, but uses a veff with
parameters from Eq. (26).
is easy to see that all three of our inhomogeneity indices
scale:
w(r) → w(γr),
(32)
i.e., the system does not become any more or less inhomogeneous under uniform density scaling.
Under the transformation of Eq. (31), the exchange
energy has a simple scaling [26]:
Ex [n] → Ex [nγ ] = γEx [n].
(33)
If we write
Ex =
Z
d3 r n(r)x (r),
(34)
then
(28)
x (r) → γx (γr).
Y → 2.81p + 0.82pq + 16.05q ,
(29)
Z 2 → 35.15p2 + 41.15pq + 30.86q 2,
(30)
If we know x (r) in both the strongly-inhomogeneous (SI)
and weakly-inhomogeneous (WI) limits, we might make
an interpolation
2
2
2
where p and q are both small. (For the densities of Figs. 6
and 7, Z 2 is not well- represented by Eq. (30). The expansions (27)–(30) have not been used in any of our figures.)
IV. SCALING, FUNCTIONAL
INTERPOLATION, AND OTHER DISCUSSION
Consider a uniform density scaling
n(r) → nγ (r) = γ 3 n(γr),
(31)
γ is a positive
parameter. The number of electrons
Rwhere
R
d3 r nγ (r) = d3 r n(r) is unchanged, but the density
is uniformly compressed (γ > 1) or expanded (γ < 1). It
WI
x (r) = w(r)SI
x (r) + [1 − w(r)]x (r),
(35)
(36)
which preserves the scaling behavior of Eq. (35).
A very accurate non-empirical meta-GGA for
Exc [n] [12] can be constructed using just the local
ingredients n(r), ∇n(r), and τ (r), without the other
ingredients ∇2 n, ∇τ , and ∇2 τ needed to complete our
inhomogeneity index wXY Z of Eq. (23). A possible
explanation is as follows: The only parts of an atom
where wXY of Eq. (19) is small are the “outer intershell
regions”, in which p of Eq. (12) and q of Eq. (15) are
small, as are X of Eq. (9) and (1 − τ /τ unif ) of Eq. (14).
In these regions, the exchange energy densities predicted
by LDA, by the non-empirical PBE GGA [5], and by the
non-empirical TPSS meta-GGA [12] will all agree closely
6
with one another, and the short-range behavior of the
exact exchange hole (Eq. (16)) will be LDA-like. It is
very possible then that LDA, GGA and meta-GGA are
all correct in these regions, even though these regions
are decidedly not electron-gas-like. This suggests using
wXY in place of wXY Z in the interpolation of Eq. (36).
Fortunately, inhomogeneity effects can be weak even
when the inhomogeneity is not. For example, the GáspárKohn-Sham LDA of Eq. (1) for the exchange energy, applied to atoms, never makes an error of more than about
14 %, and usually much less. The relative error seems to
be very small for an atom of large atomic number [2]. Our
inhomogeneity index shows what a remarkable achievement that really is.
Finally, we can define a global inhomogeneity index
Z
.Z
w̄P = d3 r n(r)P w(r)
d3 r n(r)P ,
(37)
where P = 4/3 would be the natural choice for a discussion of the exchange energy.
ACKNOWLEDGMENTS
J.P.P and J.T acknowledge support from the U.S. National Science Foundation under grant DMR-01-35678,
and discussions with S. Kümmel. R.A acknowledges support from the project ATOMICS at the Swedish research
council SSF and from the Göran Gustafsson Foundation.
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3
Paper 3
Alternative separation of exchange and correlation in density-functional theory
R. Armiento and A. E. Mattsson,
Phys. Rev. B 68, 245120 (2003).
PHYSICAL REVIEW B 68, 245120 共2003兲
Alternative separation of exchange and correlation in density-functional theory
R. Armiento*
Department of Physics, Royal Institute of Technology, AlbaNova University Center, SE-106 91 Stockholm, Sweden
A. E. Mattsson†
Computational Materials and Molecular Biology, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA
共Received 15 August 2003; published 30 December 2003兲
It has recently been shown that local values of the conventional exchange energy per particle cannot be
described by an analytic expansion in the density variation. Yet, it is known that the total exchange-correlation
共XC兲 energy per particle does not show any corresponding nonanalyticity. Indeed, the nonanalyticity is here
shown to be an effect of the separation into conventional exchange and correlation. We construct an alternative
separation in which the exchange part is made well behaved by screening its long-ranged contributions, and the
correlation part is adjusted accordingly. This alternative separation is as valid as the conventional one, and
introduces no new approximations to the total XC energy. We demonstrate functional development based on
this approach by creating and deploying a local-density-approximation-type XC functional. Hence, this work
includes both the theory and the practical calculations needed to provide a starting point for an alternative
approach towards improved approximations of the total XC energy.
DOI: 10.1103/PhysRevB.68.245120
PACS number共s兲: 71.15.Mb, 31.15.Ew
Kohn-Sham 共KS兲 density-functional theory1 共DFT兲 is a
successful scheme for electron energy calculations. The long
term goal is chemical accuracy for chemical and material
properties without the need of a careful problem analysis
prior to the calculation. This would enable computerized optimization of chemicals, materials, and compounds to an extent that is not possible today. The accuracy of the KS-DFT
scheme is limited by the approximation for the exchangecorrelation 共XC兲 energy functional. Development towards
improved generic XC functionals has been slow compared to
the progress of algorithms and computer hardware. Almost
40 years of research have passed since the local-density approximation 共LDA兲 was suggested. Even if LDA is not generally accurate enough for applications in molecular systems,
it is still in use in calculations of properties of certain metallic and semiconductor systems. This is not for being ‘‘faster’’
than other functionals, but because it still often delivers the
most accurate results in such applications. Progress made in
functional developments have either 共i兲 sacrificed generality,
defining functionals working good only for certain systems
but decreasing accuracy for others, or 共ii兲 improved the separate exchange and correlation parts of the XC energy without
much improvement of the combined quantity. It is fair to
conclude that current approaches have not yet taken us a
significant step forward towards generic XC functionals. The
present work identifies an inherent problem with the current
approach and supplies the starting point of an alternative
path for approximations of the total XC energy.
KS-DFT is based on a total electron energy functional
E e 关 n(r) 兴 that is minimized by the true ground-state electron
density n(r) of a system. The minimization is done by selfconsistently refining an effective potential v eff(r) of a system
of noninteracting electrons, to make that system’s electron
orbitals ␺ ␯ (r) give n(r) as their 共noninteracting兲 electron
density. The XC energy functional E xc 关 n(r) 兴 is the part of
E e that remains when all more easily treated parts have been
accounted for 共i.e., the potential energy, the kinetic energy of
0163-1829/2003/68共24兲/245120共5兲/$20.00
a system of noninteracting electrons, and the internal potential energy of a classical repulsive gas兲. E xc is decomposed
into a local quantity by defining the XC energy per particle
⑀ xc from the requirement:
E xc 关 n 共 r兲兴 ⫽
冕
n 共 r兲 ⑀ xc 共 r; 关 n 兴 兲 dr.
共1兲
An approximation for ⑀ xc (r; 关 n 兴 ) is referred to as a ‘‘DFT
functional.’’ It is common to further separate this quantity as
⑀ xc ⫽ ⑀ x ⫹ ⑀ c where the separation is defined from the requirement that ⑀ x should give the exchange energy E x when integrated in Eq. 共1兲. The quantity E x can be implicitly defined
through the conventional choice2 of the exchange energy per
particle ⑀ irxh
x . In rydberg atomic units 共a.u.兲, for a spinunpolarized system
⑀ irxh
x ⫽⫺2
冕
1
n 共 r兲 兩 r⫺r⬘ 兩
冏兺
␯
冏
2
␺ ␯ 共 r兲 ␺ ␯* 共 r⬘ 兲 dr⬘ .
共2兲
Recent work3 shows that local values of ⑀ irxh
cannot be
x
described by an analytic expansion in the density variation.
Yet, it is known that the total XC energy density does not
show any corresponding nonanalyticity. Hence, this is not a
problem inherent to the underlying physics, but artificially
created. In the following we present a solution to this problem by separating the XC energy in an alternative way and
show this solution to hold for systems of generic effective
potentials. Finally the ideas are placed in the context of functional development through the construction of a LDA-type
functional. We perform benchmark calculations using an
implementation of this functional. Taken together, these parts
provide a complete starting point for an alternative approach
towards XC functionals that avoids the deficiency of the traditional separation in exchange and correlation.
68 245120-1
©2003 The American Physical Society
PHYSICAL REVIEW B 68, 245120 共2003兲
R. ARMIENTO AND A. E. MATTSSON
If the long-range Coulomb potential is responsible for the
nonanalytical behavior of ⑀ irxh
x , then the insertion of a traditional screening factor of Yukawa type, e ⫺k Y兩 r⫺r⬘ 兩 , into the
integration of Eq. 共2兲, should give a well-behaved quantity
irxh
⑀ (x⫹Y)
. This introduces k Y as the Yukawa wave vector,
which effectively is an inverse screening length for the Coulomb potential that may be dependent on r. A corresponding
irxh
irxh
is defined by the relation ⑀ (x⫹Y)
correlationlike term ⑀ (c⫺Y)
irxh
irxh
⫹ ⑀ (c⫺Y) ⫽ ⑀ xc . This can be seen as moving a term from
correlation to exchange,
⑀ Yirxh⫽2
冕
1⫺e ⫺k Y兩 r⫺r⬘ 兩
n 共 r兲 兩 r⫺r⬘ 兩
irxh
irxh
⑀ (x⫹Y)
⫽ ⑀ irxh
x ⫹⑀Y ,
冏兺
␯
冏
2
␺ ␯ 共 r兲 ␺ ␯* 共 r⬘ 兲 dr⬘ ,
irxh
irxh
⑀ (c⫺Y)
⫽ ⑀ irxh
c ⫺⑀Y
共3兲
irxh
LDA
irxh
irxh
⑀ (x⫹Y)
⫽ ⑀ (x⫹Y)
s 2 ⫹b (x⫹Y)
q⫹ . . . 兴 ,
关 1⫹a (x⫹Y)
共4兲
and is an alternative way of partitioning ⑀ xc without introducing any new approximations. Screened exchange has
been used previously. In the Hartree-Fock scheme, exchange
is known to have singularities originating from the separation in exchange and correlation. Screening the Hartree-Fock
exchange has been shown to remove these singularities.4 In
DFT, several recent functionals and schemes have been constructed based on screened exchange expressions.5 However,
in these works the long-range part has either been thrown
away or handled with another approximative scheme. The
present approach is fundamentally different in that the
screening of the exchange is compensated for by redefining
irxh
constant. This alternative
correlation to keep the total ⑀ xc
separation provides as good a starting point for functional
development as the commonly used separation into unirxh
screened exchange, ⑀ irxh
x , and conventional correlation, ⑀ c .
In Eq. 共3兲 the limit k Y→0 approaches the conventional
partitioning between exchange and correlation 共i.e., ⑀ Y
→0). In the following we use a scaled k Y , k̄ Y⫽k Y /p F with
p F ⫽ 冑␮ ⫺ v eff(r), where ␮ is the chemical potential. Our
aim is now to show that this alternative separation removes
the found problem for exchange, while not introducing any
change in the combined XC energy.
irxh
,
The term of lowest order in density variation of ⑀ (x⫹Y)
i.e., LDA for the exchangelike term, is obtained from insertirxh
ing the KS orbitals for the uniform electron gas into ⑀ (x⫹Y)
关Eq. 共4兲兴. Substituting p F → 关 3 ␲ 2 n(r) 兴 1/3 gives
LDA
⑀ (x⫹Y)
„n 共 r兲 …⫽⫺ 关 3/共 2 ␲ 兲兴关 3 ␲ 2 n 共 r兲兴 1/3I 0 共 k̄ Y兲 ,
⫽ ␮ ¯␭ 关 1⫺cos(2冑␮ p̄z) 兴 . The limit of slowly varying densities is found as ¯␭ , p̄→0. To simplify the analysis of numerical data in this two-dimensional limit, the parameters are
combined in a nontrivial way into a new parameter6 ␣ , with
the slowly varying limit 1/␣ →0. The MG family of densities was also used when demonstrating the nonanalytical bein Ref. 3. We use the computer program in that
havior of ⑀ irxh
x
reference, modified for Yukawa screening, to calculate
irxh
⑀ (x⫹Y)
for 1/␣ →0 in specific r points, for several specific
k̄ Y . The results are investigated based on the expansion of
irxh
⑀ (x⫹Y)
in density variation,
共5兲
s⫽
兩 “n 共 r兲 兩
2 共 3 ␲ 2 兲 1/3n 4/3共 r兲
q⫽
irxh
⑀ (x⫹Y)
⫽⫺
4 共 3 ␲ 2 兲 2/3n 5/3共 r兲
共8兲
,
冉
冊
“ 2 p F2
共 ⵜ p F2 兲 2
1 p F4
I
⫹
I
⫹
I C ⫹••• ,
0
B
n 2␲3
18␲ 3
24␲ 3 p F2
共9兲
2
2
⫺6k̄ Y共 4⫹k̄ Y
I B ⫽ 关 40⫹12k̄ Y
兲 arctan共 2/k̄ Y兲
2
2
2
⫺ 共 4⫹k̄ Y
⫹1 兲兴 / 共 16⫹4k̄ Y
兲 ln共 4/k̄ Y
兲,
共10兲
2
2
I C ⫽ 关 k̄ Y共 4⫹k̄ Y
兲 arctan共 2/k̄ Y兲 ⫺4⫺2k̄ Y
2
⫺2 共 k̄ 2 Y⫺4 兲 / 共 k̄ 2 Y⫹4 兲兴 / 共 8⫹2k̄ Y
兲.
共11兲
Using the expansion of the density in p F from Ref. 8, Eq. 共9兲
can be recast into the form of Eq. 共7兲, with general coefficients as functions of k̄ Y ,
irxh
a (x⫹Y)
共 k̄ Y兲 ⫽
冉
冊
8 3 1 IB 1 IC
⫺
⫹
,
27 4 3 I 0 2 I 0
共12兲
8 IB 4
⫺ .
27 I 0 9
共13兲
共6兲
irxh
b (x⫹Y)
共 k̄ Y兲 ⫽
irxh
⑀ (x⫹Y)
For each r point with density n(r), the value of
for a
uniform electron gas with the same density is used. In the
limit k̄ Y→0, this approaches regular LDA exchange.
irxh
using the Mathieu gas 共MG兲
We numerically study ⑀ (x⫹Y)
family of electron densities. These densities are parametrized
by two dimensionless quantities ¯␭ and p̄, and are obtained
from a noninteracting system of electrons moving in v eff(r)
ⵜ 2 n 共 r兲
Figure 1 confirms this expansion for k̄ Y⬎0 with the dimenirxh
irxh
and b (x⫹Y)
being functions of the
sionless scalars a (x⫹Y)
value of k̄ Y . The behavior is consistent for all investigated
values of ¯␭ / p̄ 2 , i.e., convergence is independent of the path
through the two-dimensional MG parameter space. However,
for k̄ Y⫽0 the expansion of Eq. 共7兲 is not confirmed 共this was
a major point of Ref. 3兲.
A derivation of the convergence points for curves with
k̄ Y⬎0 in Fig. 1 for systems of generic v eff(r) follows. We
start from an expansion of the exchange energy per particle
in p F from Refs. 7 and 8 with all spatial integrations done,
2
I 0 共 k̄ Y兲 ⫽ 关 24⫺4k̄ Y
⫺32k̄ Yarctan共 2/k̄ Y兲
2
2
2
⫹k̄ Y
⫹1 兲兴 /24.
兲 ln共 4/k̄ Y
共 12⫹k̄ Y
,
共7兲
The values extracted from the numerical data from the MG
family of densities 共see Fig. 1兲 are in excellent agreement
with these derived coefficients. This shows that our numerical data illustrate the behavior of a general system. When the
generalized expansion approximation 共GEA兲 gradient coefficient was established,8 –10 there was an order of limits prob-
245120-2
PHYSICAL REVIEW B 68, 245120 共2003兲
ALTERNATIVE SEPARATION OF EXCHANGE AND . . .
merical accuracy shows that the alternative separation indeed
provides an alternative approach to conventional functional
development; 共iii兲 it provides a starting point for further reirxh
irxh
fined approximations of the ⑀ (x⫹Y)
and ⑀ (c⫺Y)
parts.
LDA
The expression for ⑀ (x⫹Y)
关Eq. 共5兲兴 has one free parameter
k̄ Y for which a natural choice is a scaled Thomas-Fermi
where ␥
wave vector k̄ TF⫽k TF/ p F⫽ 冑4r s /( ␲ ␥ ),
⫽(9 ␲ /4) 1/3 and r s ⫽ ␥ / 关 3 ␲ 2 n(r) 兴 1/3 共a.u.兲 is a r dependent
density parameter. A generalized choice is
a
⫽ 冑ar s .
k̄ Y
共14兲
The Yukawa exchangelike term, Eq. 共5兲, is expanded around
r s ⫽0 and ⬁, giving
r s →0
LDA
⑀ (x⫹Y)
→ ⫺
冉
冋
3 ␥ 1 2 ␲ 冑a 1
1
⫺
⫹a ln 2⫺ ln a
2␲ rs
3 冑r s
2
1
1
⫺ ln r s ⫹
2
2
r s →⬁
LDA
⑀ (x⫹Y)
→ ⫺
册冊
共15兲
,
冉
冊
3␥ 4 1
8 1
⫺
.
2 ␲ 9a r s2 15a 2 r s3
共16兲
The expansions for the total XC energy of a uniform electron
gas are known:11–13
r s →0
unif
⑀ xc
→ ⫺ 共 3 ␥ 兲 / 共 2 ␲ r s 兲 ⫹c 0 ln r s ⫺c 1 ⫹c 2 r s lnr s , 共17兲
irxh
LDA
FIG. 1. Effective 共a兲 Laplacian coefficient ( ⑀ (x⫹Y)
/ ⑀ (x⫹Y)
irxh
LDA
⫺1)/q, 共b兲 gradient coefficient ( ⑀ (x⫹Y)
/ ⑀ (x⫹Y)
⫺1)/s 2 , for space
points r where 共a兲 s⫽0 共density maxima; effective potential
minima兲, 共b兲 q is close to zero, for different values of ¯␭ /p̄ 2 and k̄ Y .
irxh
irxh
The quantities are expected to approach 共a兲 b (x⫹Y)
, 共b兲 a (x⫹Y)
, in
Eq. 共7兲 in the limit of slowly varying densities 1/␣ →0. All curves
where k̄ Y⬎0 show convergent trends towards values predicted by
Eqs. 共12兲 and 共13兲 共shown in legend and marked on the y axes兲. The
oscillating behavior was explained in Ref. 3, and is not important in
this context. Due to involved numerics, explicit divergence for k̄ Y
⫽0 can only be demonstrated in 共a兲, but the values in 共b兲 are consistent with an expected divergence towards ⫹⬁. The similarity of
convergence values for k̄ Y⫽0.5 and 1.0 in 共b兲 is coincidental.
lem between the limit k̄ Y→0 and the limit of slowly varying
electron densities. In contrast, our calculations show that an
expansion involving both the gradient and the Laplacian, Eq.
共7兲, cannot describe the conventional exchange energy per
particle regardless of the order of the limits. The solution is
instead to use the alternative separation given by Eq. 共4兲,
keeping k Y⬎0.
The alternative separation needs to be substantiated to be
useful. In the following we show how to create a LDA-type
functional by approximating both the exchangelike and correlationlike terms. The reasons this derivation is important
are that 共i兲 it shows how functional development using the
alternative separation use very similar methods to conventional functional development; 共ii兲 when deployed, its nu-
r s →⬁
unif
⑀ xc
→ ⫺ 共 3 ␥ 兲 / 共 2 ␲ r s 兲 ⫺d 0 /r s ⫹d 1 /r s3/2 ,
共18兲
Setting a
where c 0 ⫺c 4 , d 0 , and d 1 are scalars.
⫽c 0 4 ␲ /(3 ␥ ) makes the leading logarithmic term compatible with Eq. 共15兲. It is now easy to produce a suitable exLDA
,
pression to model ⑀ (c⫺Y)
14
LDA,1
⑀ (c⫺Y)
⫽
冑
b 1 r s ⫹b 2
r s3/2⫹b 3 r s ⫹b 4
冑r s
.
共19兲
Of the four free parameters, b 1 –b 4 , two are fixed by eliminating the 1/冑r s in the low r s limit 共Eq. 15兲, and by rendering the total constant term equal to c 1 . The remaining two
parameters are determined by a least-squares fit, minimizing
LDA
LDA
共 r s 兲 ⫹ ⑀ (c⫺Y)
共 r s 兲 ⫺ ␬ 共 r s 兲兴 /⌬ ␬ 共 r s 兲 兩 2 ,
兺r 兩 关 ⑀ (x⫹Y)
共20兲
s
where ␬ (r s ) and ⌬ ␬ (r s ) are the Ceperley-Alder15 共CA兲 data
and errors, respectively. This gives Yukawa LDA1 共YLDA1兲,
composed by Eqs. 共5兲, 共14兲, and 共19兲 with parameters: a
b 2 ⫽⫺7.576 97,
b3
⫽0.135 718,
b 1 ⫽⫺1.714 78,
⫽5.134 52, b 4 ⫽10.7168. In Table I it is compared with the
CA data and other XC parametrizations currently in use.11 In
the fitting, YLDA1 uses one fitting parameter less than the
other parametrizations but still performs at least as well as
Perdew-Zunger correlation 共PZ兲 and approximately as well
as Vosko-Wilk-Nusair correlation 共VWN兲.
245120-3
PHYSICAL REVIEW B 68, 245120 共2003兲
R. ARMIENTO AND A. E. MATTSSON
TABLE I. 共a兲 Correlation from original CA data 共in mRy兲 and
from different parametrizations of this data, compared to ⑀ xc
⫺ ⑀ irxh
for the YLDA’s. 共b兲 Differences between the values in 共a兲,
x
and the CA data, scaled with the errors in the CA data. An absolute
value ⭐1 means that the parametrization is within the error bars of
the CA data and can be considered exact.
共a兲
rs
CA
PZ
VWN
PW
YLDA1
YLDA2
1 120
119.3
120.0
119.5
120.5
120.3
2 90.2
90.18
89.57
89.52
89.70
90.05
5 56.3
56.68
56.27
56.43
56.21
56.43
10 37.22
37.137
37.089
37.145
37.044
37.104
20 23.00
22.995
23.095
23.060
23.094
23.091
50 11.40
11.332
11.407
11.385
11.421
11.377
100
6.379
6.3429
6.3693
6.3820
6.3695
6.3829
共b兲
rs
PZ
1
2
5
10
20
50
100
⫺0.31
⫺0.07
3.48
⫺1.58
⫺0.11
⫺6.55
⫺7.15
VWN
0.47
⫺1.61
⫺0.62
⫺2.54
3.24
0.96
⫺1.88
PW
⫺0.02
⫺1.73
1.03
⫺1.43
2.06
⫺1.21
0.66
YLDA1
YLDA2
0.94
⫺1.27
⫺1.18
⫺3.44
3.20
2.36
⫺1.83
0.76
⫺0.40
1.01
⫺2.23
3.08
⫺2.01
0.84
An improved YLDA is given by the additional requirements of an independent r s ln rs term and a zero coefficient
for 冑r s in the small r s limit. This is achieved through extending k̄ Y in Eq. 共14兲 to
ab
⫽ 冑ar s ⫹br s3/2
k̄ Y
共21兲
LDA
part,
and adding two parameters to the ⑀ (c⫺Y)
*Electronic address: [email protected]
†
Electronic address: [email protected]
1
P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 共1964兲; W.
Kohn and L.J. Sham, ibid. 140, A1133 共1964兲.
2
Implications of Eq. 共1兲 allowing more than one definition of ⑀ x
were thoroughly discussed in Ref. 3
3
R. Armiento and A.E. Mattsson, Phys. Rev. B 66, 165117 共2002兲.
4
G. Aissing and H.J. Monkhorst, Int. J. Quantum Chem. Symp. 27,
81 共1993兲; H.J. Monkhorst, Phys. Rev. B 20, 1504 共1979兲.
5
A. Seidl, A. Görling, P. Vogl, J.A. Majewski, and M. Levy, Phys.
Rev. B 53, 3764 共1996兲; T. Leininger, H. Stoll, H. Werner, and
A. Savin, Chem. Phys. Lett. 275, 151 共1997兲; J. Heyd, G.E.
Scuseria, and M. Ernzerhof, J. Chem. Phys. 118, 8207 共2003兲.
6
The definition is ␣ ⫽( ␮ ⫺ ⑀ ␩ 1 )/( ⑀ ␩ 2 ⫺ ⑀ ␩ 1 )⫹ 兩 ␩ 1 兩 , where, if ␮ is
inside a z-dimension energy band, ⑀ ␩ 1 is the lowest energy in
this band. If ␮ is not inside an energy band, ⑀ ␩ 1 is the lowest
energy in the band which contains the z-dimension energy state
with highest energy ⭐ ␮ . Furthermore, ⑀ ␩ 2 is the lowest possible
LDA,2
⑀ (c⫺Y)
⫽
冑
e 1 r s ⫹e 2 r s ⫹e 3
2
r s ⫹e 4 r s3/2⫹e 5 r s ⫹e 6
冑r s
.
共22兲
Hence four parameters are fitted to the CA data. This gives
YLDA2 共Ref. 16兲 with a⫽0.135 718, b⫽0.042 605 5, e 1
⫽⫺1.819 42,
e 2 ⫽2.741 22,
e 3 ⫽⫺14.4288,
e4
⫽0.537 230, e 5 ⫽1.281 84, e 6 ⫽20.4080. The performance
of YLDA2 is comparable with the Perdew-Wang correlation
共PW兲 共Table I兲.
To make sure that there is no major difference between
the YLDA’s and the other LDA XC functionals we have
calculated the surface energy of jellium surfaces using selfconsistent densities obtained by the PW correlation. Ranging
over surface systems with constant bulk r s ⫽2, 2.07, 2.30,
2.66, 3, 3.28, 4, 5, and 6, we find no systematic differences.
They all differ from each other in the order of 0.1%, with a
total error in the order of a few percent.17 Furthermore, selfconsistent calculations for bulk silicon18 give a lattice constant of 5.38 Å, and a bulk modulus between 95.2 and
95.6 GPa, regardless of parameterization; i.e., PZ, VWN,
PW, YLDA1, YLDA2 give essentially equal values.
In this paper we have 共i兲 established that the lack of anain the MG
lytical behavior in the slowly varying limit of ⑀ irxh
x
model is caused by the long rangedness of the Coulomb
potential; 共ii兲 shown that this is a general artifact of the conventional definition of ⑀ irxh
x , and is not restricted to limits
taken through MG densities; 共iii兲 shown that an analytical
behavior can be obtained by using a nonconventional separation of exchange and correlation within ⑀ xc ; 共iv兲 derived
and implemented a LDA-type functional based on this alternative separation. This LDA-type functional provides a starting point for further approximate functionals.
We thank Walter Kohn for inspiring discussions. This
work was partly funded by the Göran Gustafsson Foundation
and the project ATOMICS at the Swedish research council
SSF. Sandia is a multiprogram laboratory operated by Sandia
Corporation, a Lockheed Martin Company, for the United
States Department of Energy under Contract No. DE-AC0494AL85000.
energy of all z-dimension energy states within bands that only
contain energies ⬎ ␮ . By construction ␩ 1 and ␩ 2 are integer.
Details on this parameter are found in Ref. 3.
7
The exchange energy per particle expanded in p F is derived in
Ref. 8. The derivation uses an implicit Yukawa screening, but
takes the limit k̄ Y→0 in the end result. Clarifications are found
in Ref. 10.
8
E.K.U. Gross and R.M. Dreizler, Z. Phys. A 302, 103 共1981兲.
9
L. J. Sham, Computational Methods in Band Structure 共Plenum
Press, New York, 1971兲, p. 458; P.R. Antoniewicz and L. Kleinman, Phys. Rev. B 31, 6779 共1985兲; L. Kleinman and S. Lee,
ibid. 37, 4634 共1988兲.
10
J. P. Perdew and Y. Wang, Mathematics Applied to Science 共Academic Press, Boston, 1988兲, pp. 187–209.
11
J.P. Perdew and Y. Wang, Phys. Rev. B 45, 13 244 共1992兲; J.P.
Perdew and A. Zunger, ibid. 23, 5048 共1981兲; S.H. Vosko, L.
Wilk, and M. Nusair, Can. J. Phys. 58, 1200 共1980兲.
12
M. Gell-Mann and K.A. Brueckner, Phys. Rev. 106, 364 共1957兲.
245120-4
PHYSICAL REVIEW B 68, 245120 共2003兲
ALTERNATIVE SEPARATION OF EXCHANGE AND . . .
13
14
G.G. Hoffman, Phys. Rev. B 45, 8730 共1992兲.
Since none of the correlation functionals in use today 共Ref. 11兲
use the proper value of c 1 关found as late as 1992 共Ref. 13兲兴, we
here give: c 0 ⫽2 (1⫺ln 2)/␲2, c 1 ⫽ 关 22⫹32 ln 2⫺24 ln22
⫹9␨(3)兴/6␲ 2 ⫺1/2⫺(ln 2)/3⫺c 0 关 ln(4/( ␲ ␥ )⫺1/2⫹ 具 R 典 兴 , where
␨ (x)
is
the
Riemann
Zeta
function,
具R典
⬁
⬁
⫽ 兰 ⫺⬁
R 2 (u)ln R(u)du/兰⫺⬁
R2(u)du
and
R(u)⫽1
⫺u arctan(1/u). Numerical values to six relevant digits are c 0
⫽0.062 181 4, and c 1 ⫽0.093 840 6 共a.u.兲.
D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45, 566 共1980兲.
The c 2 coefficient of YLDA2 is 0. 00151 共a.u.兲, which is closer to
the exact value than PW 共Ref. 11兲.
17
S. Kurth, J.P. Perdew, and P. Blaha, Int. J. Quantum Chem. 75,
889 共1999兲.
18
For the Si calculations we used the software SOCORRO, developed
at Sandia National Laboratories. Norm-conserving Don Hamann
LDA pseudopotential was used; D.R. Hamann, Phys. Rev. B 40,
2980 共1989兲.
15
16
245120-5
4
Paper 4
Functional designed to include surface effects in self-consistent density functional theory
R. Armiento and A. E. Mattsson,
Phys. Rev. B 72, 085108 (2005).
PHYSICAL REVIEW B 72, 085108 共2005兲
Functional designed to include surface effects in self-consistent density functional theory
R. Armiento1,* and A. E. Mattsson2,†
1Department
of Physics, Royal Institute of Technology, AlbaNova University Center, SE-106 91 Stockholm, Sweden
2Computational Materials and Molecular Biology MS 1110, Sandia National Laboratories, Albuquerque,
New Mexico 87185-1110, USA
共Received 25 May 2005; published 4 August 2005兲
We design a density-functional-theory 共DFT兲 exchange-correlation functional that enables an accurate treatment of systems with electronic surfaces. Surface-specific approximations for both exchange and correlation
energies are developed. A subsystem functional approach is then used: an interpolation index combines the
surface functional with a functional for interior regions. When the local density approximation is used in the
interior, the result is a straightforward functional for use in self-consistent DFT. The functional is validated for
two metals 共Al, Pt兲 and one semiconductor 共Si兲 by calculations of 共i兲 established bulk properties 共lattice
constants and bulk moduli兲 and 共ii兲 a property where surface effects exist 共the vacancy formation energy兲.
Good and coherent results indicate that this functional may serve well as a universal first choice for solid-state
systems and that yet improved functionals can be constructed by this approach.
DOI: 10.1103/PhysRevB.72.085108
PACS number共s兲: 71.15.Mb, 31.15.Ew
I. INTRODUCTION
Kohn-Sham 共KS兲 density functional theory1 共DFT兲 is a
method for electronic structure calculations of unparalleled
versatility throughout physics, chemistry, and biology. In
principle, it accounts for all many-body effects of the
Schrödinger equation, limited in practice only by the approximation to the universal exchange-correlation 共XC兲
functional. In this paper we present an improved XC functional, created with a methodology entirely from first principles, that incorporates a sophisticated treatment of electronic surfaces—i.e., strongly inhomogeneous electron
densities. This directly addresses a weakness of currently
popular functionals.2–4 The result is a systematic improvement of bulk properties of solid state systems and a qualitative improvement for systems with strong surface effects.
The XC functional suggested in the early works on the
theoretical foundation of DFT,1 the local density approximation 共LDA兲, was derived from the properties of a uniform
electron gas, but has shown surprisingly wide applicability
for real systems. For solid-state calculations the LDA is still
often the method of choice. The next level in functional development, the generalized gradient approximations
共GGA’s兲, in many cases significantly improves upon the
LDA. The GGA functionals popular for solid-state
applications5,6 are constructed to fulfill constraints that have
been derived for the true XC functional. However, the resulting functionals improve results in an inconsistent way 共see,
e.g., Ref. 4兲. Even worse, these functionals often are less
accurate than the LDA for properties involving strong surfaces effects, such as the generalized surfaces of metal
monovacancies. Recent work has explained this as a systematic underestimation of the surface-intrinsic energy contribution that, for simple surface geometries, can be estimated by
a posteriori procedure.2,3 A recently developed meta-GGA
functional by Tao, Perdew, Staroverov, and Scuseria7 共TPSS兲
is able to fulfill yet more constraints of the exact XC func1098-0121/2005/72共8兲/085108共5兲/$23.00
tional by allowing for a more complicated electron density
dependence 共i.e., through the kinetic energy density of the
KS quasiparticle wave functions兲 than the present work does.
However, it appears that TPSS does not fully rectify the surface energy problems found for the GGA’s. We repeated the
post-correction scheme in Ref. 3 for TPSS, using published
TPSS jellium XC surface energies,7 and from this a remaining surface error is predicted.
The present work follows an alternate route to functional
development from the traditional path described above. The
LDA’s use of the uniform electron gas model system leads to
physically consistent approximations 共e.g., compatible exchange and correlation that compose the XC functional兲. Our
subsystem functional approach,8 aims to preserve this propitious property of the LDA through the use of region-specific
functionals derived from other model systems. A first effort
in this direction was made with the local airy gas9 共LAG兲. It
extends the LDA by an exchange surface treatment derived
from the edge electron gas model system,10 but keeps the
LDA correlation. This first step is completed with the optimized, compatible, correlation introduced here. It is in this
sequence of functional development, the LDA, LAG, and
then our functional, that the contribution of the present work
is most clear.
The XC energy functional Exc关n兴 operates on the groundstate electron density n共r兲. It is usually decomposed into the
XC energy per particle ⑀xc,
Exc关n共r兲兴 =
冕
n共r兲⑀xc共r;关n兴兲dr.
共1兲
Exchange and correlation parts are treated separately, with
⑀xc = ⑀x + ⑀c. We put special emphasis on the conventional,
local, inverse radius of the exchange hole10 definition of the
exchange energy per particle, ⑀ˆ x. This is in contrast to expressions based on transformations of Eq. 共1兲 that arbitrarily
delocalize ⑀x and therefore cannot directly be combined with
085108-1
©2005 The American Physical Society
PHYSICAL REVIEW B 72, 085108 共2005兲
R. ARMIENTO AND A. E. MATTSSON
each other within the same system.8 The LDA is local in this
sense, while common GGA functionals5,6 are not. The LDA
exchange term is, in Rydberg atomic units,
⑀ˆ LDA
„n共r兲… = − 3/共2␲兲关3␲2n共r兲兴1/3 .
x
共2兲
II. FUNCTIONAL CONSTRUCTION
Kohn and Mattsson10 put forward the Airy electron gas as
a suitable model for electronic surfaces. The Airy gas is a
model of electrons in a linear potential, veff共r兲 = Lz. L sets an
overall length scale and ⑀ˆ x and n共r兲 can be rescaled by
Airy
⑀ˆ x,0
= L−1/3⑀ˆ x共r ; 关n兴兲 and n0 = L−1n共r兲. Parametrizations are
Airy
constructed from the exact ⑀ˆ x,0
and n0 expressed11 in Airy
functions Ai,
Airy
⑀ˆ x,0
=
−1
␲n0
冕 ⬘冕 冕
⬁
d␨
−⬁
⬁
d␹
⬁
0
0
d␹⬘
g共冑␹⌬␨, 冑␹⬘⌬␨兲
⌬␨3
⫻Ai共␨ + ␹兲Ai共␨⬘ + ␹兲Ai共␨ + ␹⬘兲Ai共␨⬘ + ␹⬘兲, 共3兲
n0 = 关␨2Ai2共␨兲 − ␨Ai⬘2共␨兲 − Ai共␨兲Ai⬘共␨兲/2兴/共3␲兲,
共4兲
dn0/d␨ = 关␨Ai2共␨兲 − Ai⬘2共␨兲兴/共2␲兲,
共5兲
where ␨ = L z, ⌬␨ = 兩␨ − ␨⬘兩, and
1/3
g共␩, ␩⬘兲 = ␩␩⬘
冕
⬁
J1共␩t兲J1共␩⬘t兲
t冑1 + t2
0
dt.
共6兲
The LAG functional of Vitos et al.9 uses the ⑀c of
the Perdew-Wang 共PW兲 LDA 共Ref. 12兲 combined
FLAG
from an Airy gas corresponding to a gewith ⑀ˆ x = ⑀ˆ LDA
x
x
neric system’s density n共r兲 and scaled gradient
s = 兩 ⵜ n共r兲兩 / 关2共3␲2兲1/3n4/3共r兲兴. The refinement factor is
共s兲
FLAG
x
a␣
= 1 + a␤s /共1 + a␥
s a␣兲 a␦ ,
共7兲
where a␣ = 2.626 712, a␤ = 0.041 106, a␥ = 0.092 070, and a␦
= 0.657 946. Fx depends only on s since n共r兲 just sets a global scale of the model via L. However, far outside the elecdoes not reproduce the right limiting
tronic surface, FLAG
x
behavior. We have derived an improved parametrization by
using 共i兲 the leading behavior of the exchange energy far
Airy
→ −1 / 共2␨兲, 共ii兲 asymptotic expanoutside the surface,10 ⑀ˆ x,0
sions of the Airy functions in Eqs. 共4兲 and 共5兲, and 共iii兲 an
interpolation that ensures the expression approaches the
LDA appropriately in the slowly varying limit,
共s兲 = 共cs2 + 1兲/共cs2/Fbx + 1兲,
FLAA
x
˜
共ñ0共s兲兲2˜␨共s兲兴,
Fbx = − 1/关⑀ˆ LDA
x
˜˜
␨共s兲 = 兵关共4/3兲1/32␲/3兴4˜␨共s兲2 + ˜␨共s兲4其1/4 ,
冋 冉 冊册
3/2
˜␨共s兲 = 3 W s
冑
2
2 6
2/3
,
ñ0共s兲 =
˜␨共s兲3/2
,
3 ␲ 2s 3
共8兲
FIG. 1. Parametrizations of Airy exchange ⑀ˆ Airy
vs scaled spatial
x
coordinate ␨. The solid black line is the true Airy exchange from
Eq. 共3兲. The inset shows the difference between the parametrizations and the true exchange. Far outside the edge, the LAA is more
accurate than the LAG due to the former’s proper limiting behavior.
least-squares fit to the true Airy gas exchange. Figure 1
shows that the improvement of the LAA over the LAG is
small in the intermediate region, but pronounced outside the
surface.
The Airy exchange parametrizations are designed to accurately model the electron gas at a surface. Hence, they cannot
be assumed to successfully work for interior regions. The
subsystem functional approach8 uses an interpolation index
for the purpose of categorizing parts of the system as surface
or interior regions. We use a simple expression
X = 1 − ␣s2/共1 + ␣s2兲,
共9兲
where ␣ is determined below.
In the present work the LDA is used in the interior. In the
limit of low s, the LAG and LAA already approach the LDA
exchange. The end result for the interpolated exchange functional is therefore only slightly different from using the LAG
or LAA in the whole system. However, interpolation is
needed for the correlation and to enable future use of other
interior exchange functionals.
No “exact” correlation has been worked out for electrons
in a linear potential. To obtain a correlation functional, we
combine the LAA or LAG exchange with a correlation based
on the LDA, but with a multiplicative factor ␥. The numerical value of ␥ is given by a fit to jellium surface energies
␴xc. For a functional ⑀xc共r ; 关n兴兲, ␴xc = 兰n共z兲关⑀xc共r ; 关n兴兲
LDA
− ⑀xc
共n̄兲兴dz, where n共r兲 is from a self-consistent LDA calculation on a system with uniform background of positive
charge n̄ for z 艋 0 and 0 for z ⬎ 0 共Ref. 14兲. The value of n̄ is
commonly expressed in terms of rs = 关3 / 共4␲n̄兲兴1/3. The most
accurate XC jellium surface energies are given by the improved random-phase approximation scheme presented by
Yan et al.15 RPA+. We minimize a least-squares sum
approx
RPA+ 2
− ␴xc
兩 , using values for rs = 2.0, 2.07, 2.3, 2.66,
兺rs兩␴xc
3.0, 3.28, and 4.0. The surface placement ␣ and the LDA
correlation factor ␥ are fitted simultaneously16:
using a superscript LAA for the local Airy approximation,
the Lambert W function,13 and where c = 0.7168 is from a
085108-2
␣LAG = 2.843,
␥LAG = 0.8228,
共10兲
PHYSICAL REVIEW B 72, 085108 共2005兲
FUNCTIONAL DESIGNED TO INCLUDE SURFACE…
FIG. 2. Local surface XC energy for the rs = 2.66 jellium surface. The main figure shows the quantity that integrates to the surface energy ␴xc in ergs/ cm2. The upper inset shows the difference
between the functionals and LDA. The lower inset shows the interpolation indices X. Integration gives in ergs/ cm2 for LDA 1188, for
LAG 1121, and for LDA-LAG共LAA兲 the “exact” RPA+ value of
1214.
␣LAA = 2.804,
␥LAA = 0.8098.
共11兲
The resulting fit reproduces the jellium XC surface energies
with a mean absolute relative error 共MARE兲 less than half a
percent; cf. Fig. 2 and Table I.
The final form of the functional is
⑀ˆ x共r;关n兴兲 = ⑀ˆ LDA
„n共r兲…关X + 共1 − X兲Fx共s兲兴,
x
⑀c共r;关n兴兲 = ⑀LDA
„n共r兲…关X + 共1 − X兲␥兴,
c
共12兲
where Fx共s兲 is either from Eq. 共7兲 or from Eq. 共8兲, and ⑀LDA
c
is the PW LDA correlation.12
III. TESTS
Numerical tests were performed with the plane-wave code
SOCORRO.17,18 Pseudopotentials 共PP’s兲 were generated with
the FHI98PP code,19 modified to obtain the XC potential from
a numerical functional derivative. We use settings provided
by the included element library.18 The PP’s and code modi-
fications have been extensively tested. In addition to the
functionals presented by this paper, PP’s were generated for
the LDA, the GGA of Perdew and Wang 共PW91兲5, and the
GGA of Perdew, Becke, and Ernzerhof 共PBE兲6. For the latter,
bulk calculations with PP’s constructed with our numerical
functional derivatives agree with the results of PPs based on
analytical functional derivatives within 0.001%.6 We also obtain reasonable agreement with the all-electron bulk results
in Ref. 4. As the tools for PP analysis could not easily be
made to use numerical derivatives, an analysis was done for
PP’s of the above functionals with analytical derivatives using identical settings. These PP’s were found to have satisfactory logarithmic derivatives and pass the built-in ghoststate tests.18
The tests presented here have been chosen from a
condensed-matter point of view: three elements for which
the LDA and PBE give similar as well as different results.
The tests include materials where the GGA 共Al兲 and LDA
共Si兲 are considered to work well. Furthermore, we include a
transition metal, Pt, as a more complex material. Established
bulk properties are examined to make sure the new functionals do not significantly worsen established results. Then vacancy formation energies are studied, a property known to
include strong surface effects and which none of the presently established functionals describe correctly. No other
functional has been initially tested on this intricate property.
Bulk properties only include weak surface effects.
The equilibrium lattice constant a0 and bulk modulus
B0 = 兩 − V⳵2E / ⳵V2兩V0 are obtained from the energy minimum
given by a fit of seven points in a range about ±10% of the
cell volume at equilibrium V = V0 to the Murnaghan equation
of state.20 As seen in Table II our functionals improve on the
results of other functionals. A convincing sign of general
improvement is the tendency for values to stay between the
LDA and PBE, as they are known to overbind and underbind, respectively. As a measure of overall performance,
the table shows the mean absolute relative error x̄ and its
standard deviation ␴ = 关兺共xi − x̄兲2 / N兴1/2 for N absolute relative errors xi. The value of ␴ gives the spread of the errors
independently of their overall magnitude. If further testing
confirms the LDA-LAG共LAA兲’s robustness to be universal
for solid-state systems, they should be considered as a “first
TABLE I. Jellium XC surface energies in erg/ cm2. RPA+ values are from Ref. 15 and are taken as exact. The LDA-LAG and LDA-LAA
functionals are created using a two-parameter fit to values for rs up to 4.00.
rs
LDA
PW91
PBE
LAG
LDALAG
LDALAA
2.00
2.07
2.30
2.66
3.00
3.28
4.00
5.00
MARE
3354
2961
2019
1188
764
549
261
111
2%
3216
2837
1929
1131
725
521
247
104
7%
3264
2880
1960
1151
739
531
252
107
5%
3226
2842
1926
1121
714
509
236
96
9%
3414
3015
2058
1214
782
563
269
115
⬍1%
3414
3015
2058
1214
782
563
270
115
⬍1%
085108-3
RPA+
3413
3015
2060
1214
781
563
268
113
PHYSICAL REVIEW B 72, 085108 共2005兲
R. ARMIENTO AND A. E. MATTSSON
TABLE II. Results of electronic structure calculations for materials exhibiting widely different properties; Al, a free-electron metal; Pt,
a transition metal; and Si, a semiconductor. The LDA-LAG and LDA-LAA functionals are from this paper, Eq. 共12兲. Values given as percent
are relative errors as compared to experimental values. Values in boldface are mean absolute relative errors. The standard deviation of
absolute relative errors ␴ is defined in the text. LDA-LAG共LAA兲 are not fitted to any values shown in this table, but to jellium surface
energies.
PBE
LAG
LDALAG
LDALAA
Expt.
3.99
4.05
5.47
3.96
4.02
5.44
3.93
4.01
5.42
3.94
4.02
5.43
3.92a
4.03b
5.43c
−0.5%
+1.8%
−1.7%
+0.5%
−0.9%
+0.6%
1.0%
1.0%
␴
0.50
0.59
Bulk modulus of bulk crystal B0 关GPa兴
Pt
312
252
Al
81.7
72.6
Si
95.1
87.5
+1.8%
+0.5%
+0.7%
1.0%
0.57
+1.0%
−0.2%
+0.2%
0.5%
0.38
+0.3%
−0.5%
−0.2%
0.3%
0.12
+0.5%
−0.2%
0.0%
0.2%
0.21
254
74.9
86.8
272
76.8
88.7
294
82.1
91.5
291
81.7
90.5
Pt
Al
Si
−11.0%
−6.1%
−11.4%
9.5%
2.4
关eV兴
0.64
0.53
3.68
−10.2%
−3.1%
−12.1%
8.5%
3.9
−3.9%
−0.6%
−10.2%
4.9%
4.0
+3.9%
+6.2%
−7.4%
5.8%
1.5
+2.8%
+5.7%
−8.4%
5.6%
2.3
0.72
0.61
3.65
0.73
0.59
3.69
1.00
0.83
3.57
0.99
0.84
3.59
355.94
18.55
20.78
354.18
18.43
20.65
345.76
17.76
19.90
344.33
17.57
19.69
344.35
17.59
19.72
LDA
PW91
Lattice constant of bulk crystal a0 关Å兴
Pt
3.90
3.99
Al
3.96
4.05
Si
5.38
5.46
Pt
Al
Si
+10.2%
+5.7%
−3.7%
6.5%
␴
2.7
Monovacancy formation energy HFV
Pt
0.91
Al
0.67
Si
3.58
Atomic XC energies 关−hartree兴
Pt
343.92
Al
17.48
Si
19.60
283a
77.3b
98.8c
共1.35兲d
0.68e
共3.6兲f
aReference
24.
2.
25.
d1.35± 0.05 eV from Ref. 22.
e0.68± 0.03 eV from Ref. 2.
f3.6± 0.2 eV from Ref. 23.
bReference
cReference
choice” for such applications. Furthermore, an explicit trend
is seen in the sequence LDA, LAG, and LDA-LAG共LAA兲.
Throughout the table LAG shifts LDA values towards the
PW91/PBE values, while LDA-LAG共LAA兲 corrects them
back towards 共and occasionally even beyond兲 the LDA. This
behavior illustrates the importance of compatible correlation.
We now turn to tests of the strong surface effects manifest
in calculations of the monovacancy formation enthalpy HVF
= EV − 共N − 1兲E / N, where EV and E are total energies for the
system with and without a vacancy, and N is the number of
atoms in the fully populated supercell. Monovacancy energies are calculated using 64-atom cells. The vacancy cell is
geometrically relaxed, and both vacancy and bulk cells are
volume relaxed. The number of k points used is 43 for Pt, 63
for Al, and 33 for Si. The Si calculations are for the Td
structure.18 For Pt and Si the supercells are too small for the
results to be directly compared to experiment but are sufficient to allow for comparison between functionals.
Strong surface effects are seen for Al and Pt, but not in Si.
This is seen by the widely different results between functionals for the metals. Similar to the bulk properties, our surface
correlation corrects LAG results in the right direction, but it
is apparent that it is still too crude to give truly quantitative
results. The surprisingly good LDA result for Al might draw
some attention, but as has been pointed out before,2 it is not
reflected in any other property of Al and is thus coincidental.
085108-4
PHYSICAL REVIEW B 72, 085108 共2005兲
FUNCTIONAL DESIGNED TO INCLUDE SURFACE…
The unexpected discrepancy between PW91 and PBE monovacancy energies will be addressed in another publication.21
We examine only solid-state systems; we do not assess
performance for atoms and molecules. However, a hint is
provided by the atomic XC energies given from the allelectron calculations used for constructing PP’s 共cf. Table II兲.
The present functionals give results close to the LDA, with a
slight adjustment towards the PBE. For atoms, the PBE is
expected to be more accurate than the LDA.4
IV. CONCLUSIONS
In conclusion, we have presented two promising functionals for use in DFT calculations. The method of their construction is generic and could potentially be used with any
local approximation to ⑀ˆ xc in the interior region. The locality
criteria precludes using, e.g., the PBE for this region,8 and
*Electronic address: [email protected]
†Electronic
address: [email protected]
1 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 共1964兲; W.
Kohn and L. J. Sham, Phys. Rev. 140, A1133 共1965兲.
2
K. Carling, G. Wahnström, T. R. Mattsson, A. E. Mattsson, N.
Sandberg, and G. Grimvall, Phys. Rev. Lett. 85, 3862 共2000兲.
3
T. R. Mattsson and A. E. Mattsson, Phys. Rev. B 66, 214110
共2002兲.
4
S. Kurth, J. P. Perdew, and P. Blaha, Int. J. Quantum Chem. 75,
889 共1999兲.
5
J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R.
Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671
共1992兲.
6 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,
3865 共1996兲.
7 J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, Phys.
Rev. Lett. 91, 146401 共2003兲; V. N. Staroverov, G. E. Scuseria,
J. Tao, and J. P. Perdew, Phys. Rev. B 69, 075102 共2004兲.
8
R. Armiento and A. E. Mattsson, Phys. Rev. B 66, 165117
共2002兲.
9 L. Vitos, B. Johansson, J. Kollár, and H. L. Skriver, Phys. Rev. B
62, 10046 共2000兲.
10 W. Kohn and A. E. Mattsson, Phys. Rev. Lett. 81, 3487 共1998兲.
11
Equations 共4兲 and 共5兲 are given by an unconventional method of
integration and may be relevant also in other contexts: J. R.
Albright, J. Phys. A 10, 485 共1977兲; R. Armiento 共unpublished兲.
12 J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 共1992兲.
13 The Lambert W function is computed with just a few lines of
code; our implementation is available on request: R. M. Corless,
G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth,
the effect of a localized equivalent cannot be inferred from
GGA results. We are working on a gradient-corrected interior
functional and an improved surface correlation. The two varieties of edge treatment, LAG and LAA, behave similarly
but we recommend the LAA based on its better behavior far
outside the edge.
ACKNOWLEDGMENTS
We are grateful to Thomas R. Mattsson and Peter A.
Schultz for valuable help with the electronic structure calculations. R.A. was funded by the project ATOMICS at the
Swedish research council SSF. Sandia is a multiprogram
laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U. S. Department of Energy’s National
Nuclear Security Administration under Contract DE-AC0494AL85000.
Adv. Comput. Math. 5, 329 共1996兲.
D. Lang and W. Kohn, Phys. Rev. B 1, 4555 共1970兲.
15 Z. Yan, J. P. Perdew, and S. Kurth, Phys. Rev. B 61, 16430
共2000兲.
16
Note that this fit is accurate enough to be sensitive to the LDA
correlation used 共Ref. 12兲.
17
SOCORRO is developed at Sandia National Laboratories and available from http://dft.sandia.gov/Socorro/.
18 See EPAPS Document No. E-PRBMDO-72-020532 for details on
the electronic structure calculations. This document can be
reached via a direct link in the online article’s HTML reference
section or via the EPAPS homepage 共http://www.aip.org/
pubservs/epaps.html兲.
19
M. Fuchs and M. Scheffler, Comput. Phys. Commun. 119, 67
共1999兲; D. R. Hamann, Phys. Rev. B 40, 2980 共1989兲; N. Troullier and J. L. Martins, ibid. 43, 1993 共1991兲; X. Gonze, R.
Stumpf, and M. Scheffler, ibid. 44, 8503 共1991兲.
20
F. D. Murnaghan, Proc. Natl. Acad. Sci. U.S.A. 30, 244 共1944兲.
21 A. E. Mattsson, R. Armiento, P. A. Schultz, and T. R. Mattsson
共unpublished兲.
22
P. Ehrhart, P. Jung, H. Schultz, and H. Ullmaier, Atomic Defects
in Metal, Vol. 25 of Landolt-Börnstein, Group III: Condensed
Matter 共Springer-Verlag, Heidelberg, 1991兲.
23 G. D. Watkins and J. W. Corbett, Phys. Rev. 134, A1359 共1964兲;
E. L. Elkin and G. D. Watkins, Phys. Rev. 174, 881 共1968兲.
24 A. Khein, D. J. Singh, and C. J. Umrigar, Phys. Rev. B 51, 4105
共1995兲.
25
O. Madelung, Semiconductors, Vol. 17a of Landolt-Börnstein,
Group III: Condensed Matter 共Springer-Verlag, Berlin, 1982兲.
14 N.
085108-5
Paper 5
PBE and PW91 are not the same
A. E. Mattsson, R. Armiento, P. A. Schultz, and T. R. Mattsson,
to be submitted for publication.
5
SAND 2005-5379J
PBE and PW91 are not the same
Ann E. Mattsson,1, ∗ Rickard Armiento,2, † Peter A. Schultz,1, ‡ and Thomas R. Mattsson3, §
1
Multiscale Computational Materials Methods MS 1110,
Sandia National Laboratories, Albuquerque, New Mexico 87185-1110
2
Department of Physics, Royal Institute of Technology,
AlbaNova University Center, SE-106 91 Stockholm, Sweden
3
HEDP Theory/ICF Target Design MS 1186, Sandia National Laboratories, Albuquerque, New Mexico 87185-1186
(Dated: August 30, 2005)
Two of the most popular generalized gradient approximations in applied density functional theory,
PW91 and PBE, are generally regarded as essentially equivalent. They produce similar numerical
results for many simple properties, such as lattice constants, bulk moduli and atomization energies.
We examine more complex properties of systems with electronic surface regions, with the specific
application of the monovacancy formation energies of Pt and Al. A surprisingly large and consistent
discrepancy between PBE and PW91 results is obtained. This shows that despite similarities for
simpler properties, PBE and PW91 are not equivalent.
PACS numbers: 71.15Mb, 61.72Ji, 73.90+f
I.
INTRODUCTION
Kohn-Sham (KS) density-functional theory1 (DFT) is
a widely used and successful method for electronic structure calculations. The accuracy of DFT calculations
depends on the choice of approximation of the universal exchange-correlation (XC) functional. Besides the
simplest (but surprisingly effective) functional, the local
density approximation (LDA),1 many other functionals
have been suggested. Among the most popular functionals today are two generalized gradient approximations
(GGAs), PW912 and PBE,3 of J. P. Perdew and coworkers.
It is a commonly held view that PW91 and PBE are
mostly interchangable functionals and are expected to
produce virtually identical results. The original PBE
work3 presents a figure (Fig. 1) showing only minor differences between the exchange correlation refinement functions of the two functionals. Computer source code implementing PBE, distributed by K. Burke, states among
its comments: “PBE is a simplification of PW91, which
yields almost identical numerical results with simpler formulas from a simpler derivation.” Test calculations on
usual test systems, such as lattice constants, bulk moduli, and atomization energies, indeed give results that
are essentially identical. The view of PW91 and PBE
as interchangable is so deeply rooted that many DFT
codes implement only one of these functionals. Partially
because of this, it is rare for papers to present results
for both PW91 and PBE in otherwise equivalent calculations. It is thus hard to assess from the literature whether
the functionals indeed give equal results beyond simple
test systems.
Based on the view of PW91 and PBE as very similar, it is a common practice to mix results from these
functionals as if they were equivalent and to quote interchangably, “GGA.” While this might be justified for simpler properties, or if not too high an accuracy is needed,
this practice, in general, is not well founded. While test-
ing functionals for surface effects,4 two of us (RA and
AEM) recently encountered differences between PW91
and PBE results much larger than differences obtained by
using different codes and/or different types of pseudopotentials. In fact, PW91 and PBE results for the monovacancy formation energy can differ more than LDA and
PBE results. It is thus not generally appropriate to quote
just “GGA” for both PW91 and PBE results. In this article we will analyze the differences in results obtained by
PW91 and PBE with particular emphasis on properties,
like the monovacancy formation energy of metals, where
surface effects are known to exist.4–6
In Section II, we establish that there is a difference
in PW91 and PBE results for the monovacancy formation energy of Pt and Al. It has previously been shown5,6
that the differences in monovacancy formation energies of
metals obtained with different functionals is connected to
how well the different functionals describe surfaces, that
is, the size of their surface intrinsic error. In section III,
we examine the jellium surface model.7 We show that
a discrepancy between PW91 and PBE results is also
present in the jellium surface model and that PW91 and
PBE, thus, have different surface intrinsic errors. Section IV quantifies the difference in surface intrinsic error
between PW91 and PBE and revisits some previous results for monovacancy formation energies of metals where
the surface intrinsic errors have been corrected. We end
the article with a summary and conclusions.
II.
MONOVACANCY FORMATION ENERGIES
To ensure that the differences we obtain with PW91
and PBE are due to the functionals and not an artifact
due to other errors,8 we use several different codes with
different pseudopotentials and basis sets in our calculations. We compute the lattice constant, bulk modulus,
and monovacancy formation energies of Pt and Al. We
take great care in converging all our results, with respect
2
to basis sets and with respect to k -points. To the maximum possible extent we treat all functionals the same
within the same combination of code and type of pseudopotential.
We use three different DFT codes in our vacancy
calculations. Socorro is a plane-wave pseudopotential
DFT code, developed at Sandia National Laboratories.9
For these calculations, norm conserving separable pseudopotentials are used. With the fhi98pp software
package10–12 we created both Trouillier-Martin (TM)11
and Hamann12 type pseudopotentials with the default
settings and, for comparison, also TM type pseudopotentials intentionally made “harder” than default. VASP13
is a widely used plane-wave pseudopotential DFT code.
In the VASP calculations we use the provided projector augmented-wave (PAW) pseudopotentials14 and, for
comparison, we also used the provided ultra-soft (US)
pseudopotentials15 (which are not available for PBE).16
SeqQuest is a contracted-Gaussian basis set pseudopotential DFT code17 using norm-conserving non-separable
Hamann pseudopotentials. These pseudopotentials are
generated using Hamann’s GNCPP code (LDA) and the
fhi98pp code (PBE and PW91). In all calculations the
number of k -points used are 43 for Pt and 63 for Al, which
corresponds to 10 and 28 special k -points, respectively,
in the Monkhorst-Pack scheme.18 Additional details of
the calculations are given in the Appendix.
We first examine bulk properties of Pt and Al. Results
for the equilibrium lattice constant a0 and bulk modulus
B0 are shown in the upper two parts of Tables I and II.
As mentioned above, the results of PW91 and PBE are
virtually identical for these simple properties. Different
pseudopotentials and different codes give very similar results. Since most pseudopotentials and code implementations typically are tested against these properties, this
is expected.
We now turn to the monovacancy formation enthalpy
HVF = EV − (N − 1)E/N , where EV and E are total
energies for the system with and without a vacancy, and
N is the number of atoms in the fully populated (perfect crystal) supercell. The results for the Pt vacancy
are listed in the lower part of Table I. Although PW91
and PBE give similar results for the perfect Pt crystal,
the computed vacancy formation energies with the two
functionals are surprisingly different. Although the difference is not dramatic, it is significant, the PBE results
being almost 0.1 eV larger than the PW91 results, and
independent of code, pseudopotential, and basis set. The
exception is the VASP PAW results where the difference
is only 0.03 eV.22
All of the results, LDA, PW91, or PBE, significantly
underestimate the experimental vacancy formation energy of 1.35 eV.20 The 64-site cells used here are too small
for converged results for the Pt vacancy, but using larger
cells results in even smaller computed vacancy formation
energies.6 Despite the fact that the bulk properties are
converged, the electronic temperature, 0.015 Ry, used in
the Socorro and one of the SeqQuest calculations is too
TABLE I: Results from different DFT electronic structure
codes with different pseudopotentials for calculations of bulk
properties and the monovacancy formation energy of Pt. The
VASP calculations with ultrasoft pseudopotentials (US) are the
same as in Ref. 6.
LDA
PW91
PBE
Pt lattice constant of bulk crystal a0 [Å] (Exp: 3.92a )
AE FPd
3.90
−
3.97
Socorro TM
3.90
3.99
3.98
Socorro hard TM
3.90
3.99
3.98
Socorro Hamann
3.92
4.00
4.00
VASP PAW
3.91
3.99
3.98
VASP US
3.91
3.99
−
SeqQuest 0.015 Ry
3.88
3.97
3.96
SeqQuest 0.003 Ry
3.89
3.97
3.96
a
Pt bulk modulus of bulk crystal B0 [GPa], (Exp: 283 )
AE FPd
312
−
247
Socorro TM
313
252
255
Socorro hard TM
313
254
255
Socorro Hamann
317
252
254
VASP PAW
305
242
246
VASP US
291
230
−
SeqQuest 0.015 Ry
318
259
260
SeqQuest 0.003 Ry
316
257
259
Pt monovacancy formation energy HVF [eV] (Exp: 1.35b )
Socorro TM
0.91
0.64
0.72
Socorro hard TM
0.91
0.66
0.73
Socorro Hamann
0.92
0.64
0.69
VASP PAW
0.93
0.66
0.69
VASP US
0.99
0.72
−
SeqQuest 0.015 Ry
1.18
0.88
0.96
SeqQuest 0.003 Ry
1.10
0.82
0.89
a Ref.
19, b Ref. 20, d All electron, full potential results from Ref. 21
large for the vacancy calculations to be converged. Reducing the temperature to 0.003 Ry, a more reasonable
value, in SeqQuest calculations causes the vacancy formation energy to get (significantly) smaller, rather than
larger. The difference between PBE and PW91 results is
still evident.
The SeqQuest vacancy formation energies are substantially larger (and hence in better agreement with experiment) than results from the other codes. The local
atomic orbital basis set used in these calculations has
been augmented with a extensive set of floating orbitals
(see Appendix) to achieve basis convergence and, therefore, we expect only a small portion of the difference is
due to basis set insufficiency (less than 0.02 eV). The
SeqQuest vacancy calculations froze the volume of the
vacancy cell at the optimal crystal volume, while the
other calculations relaxed the volume of the vacancy cell.
The volume relaxation reduces EV by less than 0.05 eV.
Other differences between the calculations are responsible for the remaining discrepancy and will be the subject of another article. Despite these variations between
the different calculations, the difference between the results with the PW91 and PBE functionals is the same.
3
TABLE II: Results from different DFT electronic structure
codes for calculations of bulk properties and the monovacancy
formation energy of Al.
LDA
PW91
PBE
Al lattice constant of bulk crystal a0 [Å] (Exp: 4.03c )
AE FPd
3.98
−
4.04
Socorro
3.96
4.05
4.05
VASP PAW
3.99
4.05
4.04
Al bulk modulus of bulk crystal B0 [GPa], (Exp: 77.3c )
AE FPd
84
−
77
Socorro
82
73
75
VASP PAW
84
74
78
Al monovacancy formation energy HVF [eV] (Exp: 0.68c )
Socorro
0.67
0.53
0.61
VASP PAW
0.68
0.54
0.63
c Ref.
5, d All electron, full potential results from Ref. 21
The variability in the monovacancy formation energies
reported in Table I illustrates the point in Ref. 8 that is
important to document all salient details about a calculations for it to reproducible, and for the results to be
potentially useful for later analyses such as this one.
In Table II, the results for Al are presented. Just as
for Pt, the bulk properties using PBE and PW91 are essentially the same. But, once again, the PBE and PW91
values for the monovacancy formation energy are different, with the PBE value being almost 0.1 eV larger than
the PW91 value. Note that for Al, contrary to for Pt,
the substantial difference between PBE and PW91 is seen
also in the VASP PAW results. This might not appear to
be a dramatic difference between two different functionals, but it is larger than expected for functionals that are
commonly regarded as more or less identical. Indeed, the
difference between PBE and PW91 is a good fraction of
the difference between LDA and PBE, in particular for
Al. Is it thus clear that is as important to distinguish if
PW91 or PBE has been used in a calculation as it is to
distinguish either of them from LDA.
For Al, the cell size and electronic temperatures used
give converged monovacancy formation energies. As seen
in Table II LDA gives the monovacancy formation energy closest to the experimental value. However, the
bulk properties are clearly best calculated with PW91 or
PBE. This has been previously discussed and explained
in Ref. 5, but will be revisited in the following sections.
We will now return to the main focus of this article, the
differences in results obtained with the PW91 and the
PBE functionals.
Removing an atom to create a vacancy in a bulk metal
can be seen as creating an internal surface. Thus, it
is reasonable to expect some similarities in the physics
of vacancies and the physics of surfaces.4–6 That PBE
and PW91 give consistently different results for vacancies suggest that the cause lies in their treatment of surface regions. Next, we examine a model surface system,
and investigate the performance of PW91 and PBE for
TABLE III: Jellium XC surface energies, in erg/cm2 , calculated with LDA, PW91, and PBE, and mean absolute relative
errors (mare) compared to the RPA+ values that are taken
as exact.
rs (bohr)
2.00
2.07
2.30
2.66
3.00
3.28
4.00
5.00
mare
2.00
2.07
2.30
2.66
3.00
3.28
4.00
5.00
mare
2.00
2.07
2.30
2.66
3.00
3.28
4.00
5.00
mare
LDA
PW91
PBE
Total exchange-correlation
3354
3216
3264
2961
2837
2880
2019
1929
1960
1188
1131
1151
764
725
739
549
521
531
261
247
252
111
104
107
2%
7%
5%
Exchange
3037
2402
2437
2674
2094
2126
1809
1371
1394
1051
755
769
669
454
464
477
308
316
222
124
128
92
38
40
30%
15%
13%
Correlation
317
815
827
287
742
754
210
558
567
136
376
382
95
271
275
72
212
215
39
123
124
19
66
67
63%
7%
9%
RPA+
3413
3015
2060
1214
781
563
268
113
2624
2296
1521
854
526
364
157
57
789
719
539
360
255
199
111
56
surface energy calculations.
III.
SURFACE MODEL: THE JELLIUM
SURFACE
To better understand the difference between PW91
and PBE results for the monovacancy formation energy
of metals, we will now examine a more abstract model
system, the jellium surface. For a functional
xc (r ; [n]),
R
the jellium surface energy σxc =
n(z)[xc (r ; [n]) −
LDA
xc (n̄)]dz, where n(r ) is from a self-consistent LDAcalculation on a system with uniform background of positive charge n̄ for z ≤ 0 and 0 for z > 0 (Ref. 7).
The value of n̄ is commonly expressed in terms of rs =
[3/(4πn̄)]1/3 . The most accurate XC jellium surface energies are given by the “improved random-phase approximation” (RPA+).23 In Table III we show the results for
LDA, PW91, PBE and RPA+.
A first observation in Table III is that LDA performs
better than both PW91 and PBE for this system de-
4
spite the individual exchange and correlation components being far off. Thus, there is a very large cancellation of errors between exchange and correlation for
LDA. The causes are well known. The LDA exchangecorrelation combination is derived from a real model system (the uniform electron gas), making exchange and correlation approximations “compatible.” To be more specific, there is a system, the uniform electron gas, where
LDA’s exchange-correlation is exact. When LDA is applied to a non-uniform system, the errors in exchange
and correlation tend to cancel. This benefit from using model systems as a basis for functional development
is central in the subsystem functional approach for constructing functionals.4,24 The basic principle is to divide
a system into subsystems where one type of physics dominates the behavior and, in each subsystem, to use a
functional based on a model system that captures the
essential physics. In Ref. 4 a functional is presented that
can be used where parts of a solid state system can be
considered to exhibit typical surface behavior, vacancies
being a good example. In contrast, PW91 and PBE are
constructed from other principles. LDA fulfills a number of “exact constraints” that also hold for the exact
exchange-correlation functional. The approach to functional design of J. P. Perdew and coworkers is based on
using the extra degrees of freedom in the functional expressions beyond LDA to fulfill even more of the constraints that have been derived for the exact exchangecorrelation functional. For example, PW91 and PBE satisfy additional exact constraints beyond those of LDA.
Focusing on the performance of PW91 and PBE, we see
that some cancellation of errors is present also for these
functionals. Their performance at surfaces are different,
however, both for exchange and correlation. Judging
from the RPA+ values, PBE’s performance at surfaces
is better than PW91’s, but still not as good as LDA’s
performance. As has been mentioned above, the differences in jellium surface energies are closely connected to
the differences found in the monovacancy results. In the
following, this connection is further enlightened by using the jellium data in Table III to derive simple PW91
and PBE surface intrinsic error25 corrections to be used
for correcting monovacancy formation energies of metals
calculated with PW91 and PBE. We are using methods
similar to those used in Refs. 5 and 6.
IV.
SURFACE INTRINSIC ERROR
CORRECTIONS
A functional’s surface intrinsic error, evident in Table III, was first discussed in Ref. 25, where a scheme
for correcting this error was also outlined. In modified
form, this correction scheme has been used to correct
monovacancy formation energies5,6 and the work of adhesion of Pd on α-alumina.26 However, it was assumed
that PW91 and PBE had the same surface intrinsic error
and PBE corrections were applied to PW91 results. In
this section we will derive new, simpler corrections for
LDA, PW91, and PBE, and apply these to monovacancy
formation energy results presented here and in previous
publications. Note that all major conculsions of previous
work still hold.
The key concept of the correction scheme for the surface intrinsic error is to use the known error of a functional in one system as a correction for the results, using the same functional, in a similar system with an unknown error. Here, we will use the known errors in surface energies for the jellium surface model system presented in Table III, that is, the surface intrinsic errors or
RPA+
∆σxc = (σxc − σxc
), as corrections for surface energies in general. For this purpose we construct functions
that take r̃s = rs /a0 as input, where a0 is the bohr radius, and give ∆σxc as output. The expression to fit to
the numbers in Table III is based on Ref. 27’s assertion
−7/2
−5/2
that σxc ∼ r̃s
+ O(r̃s
) for low r̃s , and Ref. 6’s assertion that in this limit the relative difference vanishes,
RPA+
RPA+
(σxc − σxc
)/σxc
→ 0. Using the two lowest or−5/2
−3/2
der terms gives the form: ∆σxc (r̃s ) = A r̃s
+ B r̃s
.
Least squares fits give for LDA: A = 448.454 erg/cm2 and
B = −55.845 erg/cm2 , for PW91: A = 1577.2 erg/cm2
and B = −231.29 erg/cm2 , and for PBE: A = 1193.7
erg/cm2 and B = −174.37 erg/cm2 . Figure 1 shows
the relative jellium surface energy error vs. rs , and it
is indeed seen that PW91 and PBE have quite different surface intrinsic errors and that different surface energy corrections are needed for these two functionals. A
transformation of units from erg/cm2 to eV/Å2 results
in Fig. 2, where we have also renamed the jellium surface
model system’s surface energy error to a general surface
energy correction. The dimensionless parameter r̃s can
be transformed to a density which is the electron density
inside the jellium system very far from the surface. We
call this density the “bulk density” and in Fig. 2 we use
Å−3 as its unit.28
In order to be able to correct monovacancy formation energies, two additional quantities need to be determined. First, we need to decide what rs , or “bulk
density”, we should use to obtain a value for the surface intrinsic error correction (see Fig. 2). We have argued5,6,26 that the actual bulk density is a good value
to use in a metal vacancy system. Second, we need to
estimate a surface area for the vacancy, to transform the
surface energy correction to a vacancy formation energy
correction. Here, we use the same estimates for these
quantities that we have used previously.
The bulk density corresponding to Pt is 0.669 Å−3
(Ref. 6). The corresponding surface energy corrections
2
2
are 0.038 eV/Å for PW91, and 0.028 eV/Å for PBE.
Using the rather rough vacancy area estimates of Ref. 6
yields formation energy corrections of 0.64 eV for PW91,
and 0.47 eV for PBE. Hence, the theoretically predicted
difference between PW91 and PBE Pt monovacancy formation energies is 0.17 eV. This is larger than the actual
difference found in the DFT calculations (see Table I),
but this is not surprising since we are operating at the
5
relative error, DΣxc ΣRPA+
xc
0.08
TABLE IV: Corrected Al monovacancy formation energy (in
eV). The correction is applied to the values in Table II, for
details see the text. The experimental value is 0.68 ± 0.03
eV.5 We estimate that the DFT calculation based value is
0.75 ± 0.03 eV.
0.06
0.04
0
0
1
2
3
4
jellium bulk density parameter, rs HbohrL
FIG. 1: Relative jellium surface energy error of LDA (solid),
PBE (dashed), and PW91 (dash-dotted) functionals. The
error bars represent the roundoff errors of the integer RPA+
values. While one can be certain that the data is not more
accurate than this, actual errors are likely larger. We use the
interpolation/extrapolation formula of Ref. 27 for the values
RPA+
of σxc
.
correction HeVÅ2 L
0.05
0.04
0.03
0.02
0.01
0
LDA
0.73
0.74
Socorro
VASP PAW
0.02
0
0.2
0.4
0.6
bulk density HÅ-3 L
0.8
1
2
FIG. 2: Surface energy correction per area (eV/Å ) for LDA
(solid), PBE (dashed), and PW91 (dash-dotted).
limit of accuracy for this rather simple correction scheme.
The fact that the correction is in the right direction and
on the correct energy scale is a clear indication that the
differences in monovacancy formation energy and jellium
surface energy are strongly correlated.
The simple correction scheme should, however, work
very well for the free-electron-like Al charge density, and
in Table IV we show corrected values for all three functionals and two different codes. All corrected monovacancy formation energies are between 0.05 and 0.1 eV
larger than the experimental value (which has an errorbar
of ±0.03 eV). The small spread in the corrected monovacancy formation energies indicates that the surface intrinsic error of the present functionals is the main culprit
for errors in this quantity.
In Ref. 6 a correction derived for PBE was applied to
PW91 monovacancy formation energy results. In Table V
PW91
0.73
0.74
PBE
0.76
0.78
TABLE V: PW91 monovacancy formation energies (in eV)
from Ref. 6 when re-corrected using the PW91 correction derived in the present paper. For comparison, unmodified LDA
values are cited from the reference.
relax
ELDA
0.95
1.50
2.89
Pt
Pd
Mo
corrected
ELDA
1.15
1.71
3.00
relax
EPW91
0.68
1.20
2.67
corrected
EPW91
1.34
1.85
3.05
we instead use the PW91 correction derived in this paper to correct the PW91 monovacancy formation energy
results of that paper. Note, however, that these monovacancy formation energies are calculated using ultrasoft
pseudopotentials,15 which possibly have affected the vacancy formation energy results as much as the difference
between PW91 and PBE corrections (see Table I). We
do not apply any corrections to the Pt monovacancy formation energies presented in Table I since the Pt cell
size we use in this work is too small for the result, even
corrected, to be compared to the experimental value.
Finally, we want to point out that the PW91 results
in Refs. 5 and 26 are corrected with the PBE correction,
which results in too low corrected values for the PW91
monovacancy formation energy and the PW91 work of
adhesion, respectively. This does not, however, affect
any of the major conclusions in either paper.
V.
DISCUSSION AND CONCLUSIONS
In this article we have established that PW91 and PBE
are not the same. In particular we have presented surprisingly large discrepancies in results using PW91 and
PBE for calculation of properties where surface effects
are present. Specifically, we have studied the monovacancy formation energy of Pt and Al and jellium surface
energies. Furthermore, we have shown how the results for
these two types of systems are connected. In view of the
fact that PW91 and PBE do not give the same results in
all calculations, we conclude that: 1) for calculations to
be reproducible, the use of PW91 or PBE must be clearly
documented, i.e., to only state “GGA” is not sufficient;
2) the functionals are not similar enough to motivate the
use of pseudopotentials constructed for one of them in
6
calculations with the other; 3) when testing functionals,
one should include test systems where surface effects are
present.
R. A. was funded by the project ATOMICS at the
Swedish research council SSF. Sandia is a multiprogram
laboratory operated by Sandia Corporation, a Lockheed
Martin Company, for the United States Department of
Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.
APPENDIX: DETAILS OF THE CALCULATIONS
1.
Socorro
The Perdew-Wang correlation29 is used in the LDA
calculations. For the pseudopotentials (PPs) we used
a scalar-relativistic calculation on an ordinary non-ionic
reference configuration. No non-linear core correction
was used. For Al we use a Hamann type PP with l = 2
as the local component. The s, p, and d core cutoff radii
in bohr for Al are 1.2419, 1.5469, and 1.3692. For Pt we
use two Trouiler-Martin (TM) type PPs and one Hamann
type PP. The l = 0 is used as the local component for
all three types of Pt PPs. The s, p, and d core cutoff
radii in bohr are 2.4935, 2.6182, and 2.4935 for Pt TM,
2.4935, 2.6182, and 1.7719 for Pt hard TM, and 1.4226,
1.7719, and 0.7543 for Pt Hamann. The equilibrium lattice constant a0 and bulk modulus B0 = −V ∂ 2 E/∂V 2 |V0
are obtained from the energy minimum given by a fit
of 7 points, in a range of about ±10% of the cell volume at equilibrium V = V0 , to the Murnaghan equation of state.30 The vacancy cell is geometrically relaxed, and both vacancy and bulk cells are volume relaxed. The structural optimization was terminated when
the root-mean-square of the force components was below
5.0×10−4 Ryd/bohr. Wavefunction/density cutoffs were
60 Ry/240 Ry for Pt and 20 Ry/80 Ry for Al. The number of bands used in the Pt calculation needed to be very
high in order to converge the calculations. We used 430
bands for Pt and 144 for Al. We used a Fermi smearing temperature of 1.5 × 10−2 Ry for Pt and 3 × 10−3
Ry for Al. All bulk property and the Al vacancy calculations used a density based convergence criteria for the
electronic iterations. The self-consistent loop was terminated when the root-mean-square distance between the
new and old density fields was less than 1 × 10−6 bohr−3 .
For the Pt vacancy calculations we used an energy based
convergence criteria; the SC loop was terminated when
the cell energy of consecutive steps changed less than
1 × 10−5 Ryd.
2.
VASP
The Perdew-Zunger correlation31 is used in the LDA
calculations. The official VASP pseudopotentials are
used. The equilibrium lattice constant a0 and bulk modulus B0 are obtained from the energy minimum given by
a fit of at least 7 points, centered around the cell volume at equilibrium V = V0 , to the Murnaghan equation
of state.30 The vacancy cell is geometrically relaxed and
both vacancy and bulk cells are volume relaxed.
Common settings for all the Pt PAW calculations are:
plane wave cutoff 300 eV, augmentation 600 eV, electronic iteration cutoff 10−5 eV, and a Fermi smearing
of 0.10 eV. The calculations use a PAW potential with
recommended cutoff energy (ENMAX) 230.228 eV for
LDA, ENMAX 230.277 eV for PW91, and a PAW potential dated 05Jan2001 with ENMAX 230.283 eV for PBE.
We here use ENMAX to identify the potential used. The
LDA and PW91 calculations use an ionic relaxation cutoff of 0.005 eV/Å while for PBE 0.01 eV/Å was used.
Remaining forces on the ions were less than 0.006 eV/Å,
even for PBE, and thus this difference does not explain
the deviating result for VASP PAW PBE in Table I. The
Pt US calculations are taken from Ref. 6.
For the Al PAW calculations, common settings are:
plane wave cutoff 320 eV, augmentation 640 eV, electronic iteration cutoff 10−5 eV, a Fermi smearing of 0.10
eV, and an ionic relaxation cutoff of 0.005 eV/Å. The calculations use a PAW potential with ENMAX 240.957 eV
for LDA, ENMAX 240.437 eV for PW91, and the Al h
08Apr2002 potential with ENMAX 294.838 eV for PBE.
3.
SeqQuest
We used SeqQuest only for Pt calculations. The
Perdew-Zunger correlation31 is used in the LDA calculations. The atomic configuration for the PP generation
is d9s0.5 (i.e. net charge +0.5). We include up to l = 2,
with the l = 2 channel used as the local potential. The
l = 0 and l = 2 channels use Hamann’s default settings.
For the l = 1 channel a linearization energy ep = 0.01 Ry
is used with Rp = 1.56 bohr for LDA and 1.57 bohr for
PBE and PW91. The basis set used is a “valence double
zeta plus polarization” (DZP) one, that is, two radial degrees of freedom are used for s and d, while one is used for
p. The Pt basis, designated 4s2p5d/2s1p2d, consists of
4 s-gaussians contracted into 2 independent functions, 2
p-gaussians contracted into 1 independent function, and
5 d-gaussians contracted into 2 independent functions.
This equals 15 total basis functions/atom (2s+3p+10d).
The specific gaussians are different for LDA, PBE, and
PW91, but are approximately equal. For all functionals the outermost (smallest) gaussian is for s ∼ 0.08, for
p ∼ 0.12, and for d ∼ 0.16. A floating basis was added in
the vacant site in the vacancy calculations. The floating
basis consists of two sets of single gaussians. The first set
roughly consists of the outermost gaussians of the missing Pt atom (s: 0.08, p: 0.12 and d: 0.16), while the
second set of single gaussians have 2.5 times the exponents of the first set (s: 0.20, p: 0.30, and d: 0.40). Various improvements (Pt triple-zeta d, more-zeta s and/or
7
p, and other modifications of floating orbitals) on top of
this all change the results by no more than ∼ 0.01 eV.
The bulk lattice parameter a0 was optimized in 1-atom
cells with a k -mesh=163 and an r-mesh=183 , equivalent
to a k -mesh=43 and an r -mesh =723 for the 64-atom
cell. By performing 64-atom bulk crystal reference calculations at optimal a0 for a given PP/functional/basis
∗
†
‡
§
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Electronic address: [email protected]
Electronic address: [email protected]
Electronic address: [email protected]
Electronic address: [email protected]
P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964);
W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1964).
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M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev.
46, 6671 (1992); 48, 4978 (1993).
J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.
Lett. 77, 3865 (1996).
R. Armiento and A. E. Mattsson, Phys. Rev. B 72, 085108
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3862 (2000).
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A. E. Mattsson, P. A. Schultz, M. P. Desjarlais,
T. R. Mattsson, and K. Leung, Modelling Simul. Mater.
Sci. Eng. 13, R1 (2005).
Socorro is developed at Sandia National Laboratories and
available from http://dft.sandia.gov/Socorro/.
M. Fuchs and M. Scheffler, Comput. Phys. Commun. 119,
67 (1999); X. Gonze, R. Stumpf, and M. Scheffler, Phys.
Rev. B 44, 8503 (1991).
N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993
(1991).
D. R. Hamann, Phys. Rev. B 40, 2980 (1989).
G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993); 49,
14251 (1994); G. Kresse and J. Furthmüller, Phys. Rev. B
54, 11169 (1996).
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D. Vanderbilt, Phys. Rev. B 41, 7892 (1990); G. Kresse
and J. Hafner, J. Phys.: Condens. Matter. 6, 8245 (1994).
Note, however, that the use of the US pseudopotentials in
VASP is discouraged in favor of the PAW ones by the VASP
developers.
P. A. Schultz, SeqQuest code,
http://dft.sandia.gov/quest/.
H. J. Monkhorst, and J. D. Pack, Phys. Rev. B 13, 5188
we verified that E(64-atom cell)/ 64 ∼ E(1-atom cell)
with a difference less than 10 µRy/Pt. A Fermi smearing temperature of 0.003 Ry was used. Increasing the
temperature from 0.003 Ry to the 0.015 Ry used in the
Socorro plane-wave calculations increases the monovacancy formation energy, see Table I.
19
20
21
22
23
24
25
26
27
28
29
30
31
(1976).
A. Khein, D. J. Singh, and C. J. Umrigar, Phys. Rev. B
51, 4105 (1995).
P. Ehrhart, P. Jung, H. Schultz, and H. Ullmaier, Atomic
Defects in Metal, vol. 25 of Landolt-Börnstein - Group III
Condensed Matter (Springer-Verlag, Heidelberg, 1991).
S. Kurth, J. P. Perdew, and P. Blaha, Int. J. Quantum
Chem. 75, 889 (1999).
The PW91 implementation in VASP is somewhat different
from standard implementations, and VASP PW91 results
should in general not be compared to other PW91 results.
This is, in particular, true for spin-resolved calculations. It
seems unlikely, though, that this is the only reason for the
small difference in PW91 and PBE results for VASP PAW,
compared to results from other codes, for the monovacancy
formation energy of Pt. In fact, comparing to the results
from the other codes it instead seems like it is the VASP
PBE monovacancy formation energy for Pt that is somewhat low. Note also that all VASP PAW monovacancy formation energies for Al are in agreement with the Socorro
results.
Z. Yan, J. P. Perdew, and S. Kurth, Phys. Rev. B 61,
16430 (2000).
R. Armiento and A. E. Mattsson, Phys. Rev. B 66, 165117
(2002); W. Kohn and A. E. Mattsson, Phys. Rev. Lett. 81,
3487 (1998).
A. E. Mattsson and W. Kohn, J. Chem. Phys. 115, 3441
(2001).
A. E. Mattsson and D. R. Jennison, Surf. Sci. Lett. 520,
L611 (2002).
L. M. Almeida, J. P. Perdew, and C. Fiolhais, Phys. Rev.
B 66, 075115 (2002).
A web calculator where the input parameter “bulk density” can be given in several different units and the output “surface energy corrections for LDA, PW91, and
PBE” are given in several different units is avalable at
http://dft.sandia.gov.
J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).
F. D. Murnaghan, Proc. Natl. Acad. Sci. U.S.A 30, 244
(1944).
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Paper 6
Numerical integration of functions originating from quantum mechanics
R. Armiento,
Technical report (2003).
6
Numerical integration of functions originating from
quantum mechanics
R. Armiento
Department of Physics, Royal Institute of Technology, AlbaNova University Centre,
SE-106 91 Stockholm, Sweden
Applications in quantum physics commonly involve large batches of integrals of smooth but very
oscillatory functions. The purpose of this work is to benchmark and compare different numerical
algorithms for evaluating such integrals. The routines studied include: two from the QUADPACK
package based on Gauss-Kronrod quadrature; one routine based on Patterson’s improvements
of Gauss-Kronrod quadrature; and two routines that use a non-standard algorithm of applying
quadrature-like rules of unrestricted order. The last algorithm has been seen in previous works,
but is not in wide-spread use. The present work includes optimized implementations of this
algorithm for both serial and parallel computation.
1.
BACKGROUND
Applications performing quantum physics based calculations usually involve numerical treatment of ‘wave functions’. These functions originate from the wave-like
Schödinger’s differential equation and are smooth but very oscillatory. Although it
is possible for these functions to involve difficult or singular points, the locations
of such points relate closely to physical properties of the treated problem, and are
assumed to be known in advance. It is not uncommon for applications to calculate
large batches of integrals involving such functions, and hence it is of interest to
perform these integrations as efficient as possible. The focus of the present work
is to compare some implementations for performing such integrations over finite
intervals.
Almost all readily available integration routines are based on different kinds of
Gauss quadrature rules [1]. A traditional Gauss quadrature rule involves m evaluations of the integrand and integrates all polynomials of order 2m exactly. The
extensions by Kronrod [2] sacrifice the exact integration of some of the polynomials
of highest order for the ability to reuse integrand evaluations from a lower order
formula. Furthermore, Patterson has given an algorithm for deriving formulas of
increasing orders which reuse all prior integrand evaluations [3].
Application of quadrature formulas of high order on a generic non-polynomial
integrand works well only if the integrand is very smooth. In contrast, most generalpurpose integration routines put effort in detecting and treating badly behaved
functions. This usually means that they are based on quadrature formulas of lower
orders, and are less than optimal for the integrands of study in the present work,
which are known to be perfectly smooth.
This work will compare and benchmark some readily available integration routines. The following routines will be studied:
DQNG: A routine in the QUADPACK [4] package. This routine successively
applies a set of Gauss-Kronrod rules which reuse all prior integrand evaluations.
The routines used are of order 21, 43, and 87. If order 87 is not enough, the routine
2
gives up and reports an error.
QUAD: a routine created by Krogh and Snyder [5], based on a previous routine
by Patterson [3]. The routine uses a set of Gauss-Kronrod-Patterson rules of orders
1, 3, 5, 7, 15, 31, 63, 127 and 255. If order 255 is not enough, the routine gives up
and reports an error.
DQAG: a routine in the QUADPACK [4] package. This routine uses only one
Gauss-Kronrod rule (of order selectable between 15, 21, 31, 41, 51 and 61) and
adaptively subdivides the interval of integration until the required accuracy is fulfilled.
TINT and DEFINT: two routines that are based on a non-standard derivation
of quadrature-like rules, applying rules of higher and higher orders until sufficient
accuracy is found. DEFINT is available in the JCAM software collection [6]. The
TINT routine was developed as a part of a recent work involving the author [7] and
has thus not been readily available or thoroughly tested previously. The algorithm
and its implementation will be described in the next section.
In this suite, TINT is the only routine that is trivially expandable to apply
rules of arbitrary order without doing interval subdivision. However, it would be
theoretically possible to create a routine that indefinitely applies successive GaussKronrod-Patterson rules with no interval subdivision, but no such routine has been
available to the author. However, such a routine should be rather easy to construct
by combining Patterson two works [3] with a lookup table similar to the one used
in the TINT routine.
Here follows a list of references to other routines that have not been included in
the tests but aim for similar integrands as this work. The list should be of relevance to projects searching for a suitable routine for massive numerical integration
of smooth integrals. Most of these and other routines are referenced from GAMS
[8], and available from there or from Netlib [9]. 1) The Numerical Algorithms
Group (NAG) library [10] includes routines for quadrature, and is available in both
a serial and parallel version. The D01AHF routine uses the same Gauss-KronrodPatterson rules as QUAD, but also subdivides the interval if the accuracy is not
enough, much like DQAG does. Its description also lists a few other adjustments
aimed to improve performance and reliability. D01AUFP is a routine for parallel integration. 2) IMSL Math and Stat Libraries [11] has two routines, QDAG
and QDNG, which are aimed for similar applications as the DQAG and DQNG
routines of QUADPACK. 3) The GNU Scientific Library (GSL) [12] is an open
source alternative to commercial libraries. However, the quadrature routines are
just reimplementations in C of the QUADPACK algorithms. 4) The NMS Numerical library [13] and CMLIB [14] both include a routine DQ1DAX by D. Kahaner
that aims at doing efficient numerical integration. 5) the SLATEC library [15] includes the QUADPACK routines but also has two additional routines, DGAUS8
and QNC79 that are aimed at integration of smooth integrals. The first one is
based on an adaptive use of a 8-point Legendre-Gauss algorithm and the second
one on a 7-point Newton-Cotes quadrature rule. 6) The ACM Toms library [16]
includes the QUAD routine included in the tests (as algorithm number 699), but
also has some other relevant routines. DQPSRT, algorithm number 691 [17], use
Gauss-Kronrod rules for quadrature based on recursive monotone stable formulas.
3
INTHP, Algorithm 614 [18], is based on a derivation of optimal quadrature points
for a certain class of functions. 7) The archive of Harwell subroutine library [19]
includes a routine QA04 that “Integrate to specified accuracy using adaptive Gaussian Integration”. 8) IBM:s Engineering and Scientific Subroutine Library (ESSL)
[20] includes a set of different quadrature routines. 9) The ParInt [21] research
group provides a freely available parallel integration routine for download. 10) W.
Gander and W. Gautschi have worked on two routines ADAPTSIM and ADAPTLOB [22] to replace the quadrature routines in MATLAB [23] before version 6.
However, none of the mentioned routines seems to be as trivial to extend to use
rules of arbitrary order as TINT.
2.
THE ALGORITHM OF DEFINT AND TINT
Different variations of the algorithms of the TINT and DEFINT routines have
been explored through papers of various authors [24; 25]. Specifically, the routine
DEFINT was developed by M. Mori and is based on work of M. Mori and H.
Takahasi [25]. A related variation of the algorithm was rediscovered independently
during the creation of a routine aiming for efficient parallel numerical integration
of integrands originating from quantum mechanics. This was done in a recent work
involving the author [7] and resulted in the routine TINT. The algorithm will be
described in the following.
Integration over any finite range can be substituted into an integration over a
range from 0 to 1, so only that case will be discussed here. Consider a well behaved
function f (x) in which we perform an integral substitution, x = w(x0 ) which fulfill
w(0) = 0 and w(1) = 1,
Z
1
f (x)dx =
0
Z
1
f (w(x))w0 (x)dx.
(1)
0
where w0 (x) is the derivative of w(x). Now, consider w(x) to fulfill the additional
requirement that its right derivatives, to any order, equal zero as x → +0 and its
left derivatives, to any order, equal zero as x → 1. In this case the integration of the
combination f (w(x))w0 (x) can be seen as an integration of one period of a periodic
function, as the function values and all derivatives match at the borders. The main
idea here is that for such integrands ordinary trapezoid integration is known to
converge rapidly, because of a cancellation of errors. This argument assumes that
w0 (x) going to zero in the integration limits also makes f (w(x))w0 (x) go to zero.
A sufficient (but not necessary) requirement is that f (x) is finite in these limits.
Similar assumptions are made for the derivatives of f (x).
A possible choice for w(x) that fulfills the requirements is
w(x) =
Z
x
2
ce−1/(z−z ) dz,
c=
µZ
0
0
2
w0 (x) = c e−1/(x−x
)
1
2
e−1/(z−z ) dz
¶−1
,
(2)
(3)
Trapezoid integration of the substituted f (x) can now be recast on a form similar
4
to a Gaussian quadrature rule:
Z
1/h−1
1
f (x)dx ≈ h
0
X
vn f (xn ),
xn = w(hn),
vn = w0 (hn),
(4)
n=1
where h is a chosen step length, and since by construction the integrand goes to
zero on the limits of the integration, the two outermost terms have been dropped.
For each step length the values of vn and xn can be pre-calculated with some
other simple numerical integration algorithm during the program initialization. The
algorithm is now based on reducing h in iterative steps until the relative difference
between results from two consecutive steps is less than some error bound ². A major
benefit inherited from the trapezoid integration is that the number of function
evaluations needed for each step can be halved if h is reduced with a factor of 2 in
each step, since the previous computed approximation can be reused.
Despite the fact that Eq. (4) does not include the end points of the interval and
thus is formally open, the nature of the function w(x) brings x1 and xn−1 extremely
close to 0 and 1. Hence, when implemented with numbers of limited resolution, the
formula is effectively a closed one.
The TINT routine uses the w(x) of Eq. (2), whereas the works of H. Takahasi
and M. Mori [25] focus on another transformation called the DE-rule, which consequently is used in the routine DEFINT. The DEFINT routine also has a more
refined error estimate than only estimating the error as the relative difference of
two consecutive iterations.
3.
IMPLEMENTATION
The implementation of TINT in ANSI Fortran 77 [26] is present in the Appendix.
The algorithm relies on fixed values of the primitive function of Eq. (3) and the
routine uses an initialization subroutine, TINIT, which calculates these values by
numerical integration and stores them in a lookup table. These numerical integrations are done by calling the external QUADPACK DQK61 routine which applies
a 61 points Gauss-Kronrod rule, which has been observed to give enough accuracy.
In this way the weights and abscissae for decreasing step sizes are calculated and
put in the lookup table for later use during applications of Eq. (4). The weights and
abscissae are stored intermixed in one long array TINTDT to ensure optimal use
of the cache memory. To keep track of start and stop points for different step sizes
in this array, another small lookup table is used, TINTRG. This saves a few mathematical operations compared to computing the start and stop points each time we
use the routine, for the small cost of an array of only a few elements. A further
possible optimization which is not done here, is to use the symmetry of Eq. (3)
around x = 0.5 to halve the size of the lookup table. The lookup table currently
ends at 217 interval divisions, however, this can be trivially adjusted through the
parameters MAXORD and DTSIZE (the latter should just be set to 2MAXORD+1 ).
Once the lookup table has been initialized any number of calls to the integration
routine, TINT, can be performed. This routine is just a straightforward application
of the pre-calculated abscissae and weights to the function according to the formula
Eq. (4).
For the parallel version of the routines (TINITP, TINTP), some adjustments
5
Table I. Integrands used to benchmark the routines in this work. The primitive functions are
used to pre-calculate a normalization constant making the value of the integrals exactly 1. The
integrands are all constructed to be heavily oscillatory and descending. Integrals and primitive
functions have been produced by taking derivatives of suitable primitive functions. The numerical
constants have been chosen to level the difficulty of the integrands.
Name
I1
Integrand
e−0.01x (0.01 cos(0.3cx) + 0.3c sin(0.3cx))
2
I2
I3
I4
I5
I6
I7
I8
I9
I10
2
2c cos(0.001cx )
2 sin(0.001cx )
−
x
x3
2c cos(0.003cx2 )
(1+ln(x)) sin(0.003cx2 )
0.003
−
ln(x)
(x ln(x))2
0.2c(1+x) cos(0.2cx)−sin(0.2cx)
(1+x)2
(1+x2 )−2 ((1+x2 ) cos(x) sin(0.05cx)+
2
(0.05c(1+x ) cos(0.05cx)−2x
sin(0.05cx)) sin(x))
√
80c sin( 1+80cx)
√
2 1+80cx
√
sin(0.5cx)
e− 1+x (0.5c cos(0.5cx) − 2√1+x )
e−0.01x (0.002cx cos(0.001cx2 ) − 0.01 sin(0.001x2 ))
0.001
2 cos(x)+0.002cx2 cos(0.001cx2 )+x sin(x)−2 sin(0.001cx2 )
x3
0.05cx cos(cos(0.05cx)) sin(0.05cx)+sin(cos(0.05cx))
x2
Primitive function
− cos(0.3cx)e−0.01x
sin(0.001cx2 )
x2
sin(0.003cx2 )
x ln(x)
sin(0.2cx)
1+x
sin(0.05cx) sin(x)
1+x2
√
− cos( 1 + 80cx)
√
sin(0.5cx)e− 1+x
sin(0.001cx2 )e−0.01x
sin(0.001cx2 )−cos(x)
x2
sin(cos(0.05cx))
−
x
have been made. The starting step size is now chosen as to make the integrands
evaluations evenly divisible between the parallel nodes. To avoid load balancing
issues for integrands which are unevenly hard to evaluate for different abscissae,
the values of the lookup table are distributed among the nodes to make all nodes
compute values throughout the whole integration interval. The parallelization of the
integration routine is then done in the straightforward way of distributing the work
of the loops over quadrature coefficients. The implementation in the Appendix has
been made with as few deviations from the ANSI Fortran 77 standard as allowed
by the MPI standard [27].
In this paper the routine was run with the MINORDER parameter set to 5
to ensure at least 31 evaluations of the integrand. This helped eliminate some
unreliability, and it is advisable to use this choice unless the routine is applied to a
batch of significantly easier integrands.
4.
BENCHMARKING SERIAL ROUTINES
As explained, the focus of this work is integration over finite intervals of smooth
oscillatory functions. Such functions will be simulated using sine, cosine and exponential functions. They will be normalized with a known exact solution, so that
they integrate to exactly 1. The 10 unnormalized integrands and their primitive
functions are tabulated in Table I. All integrands have a free parameter c. We also
refer to the limits of the integration as a to b. Integrations are performed with a
set to 10 and the parameters b and c taking on wide range of values to average out
any localized behavior of the routines.
The first test is to evaluate all integrals for 250 x 250 evenly spaced parameter
values with b going from 20 to 100 and c going from 1 to 2. This gives an ’easy’
set of integrals that only includes functions of a few oscillations which can be
integrated within the limited refinements used by the routines QNG and QUAD. In
addition, integrals whose normalization constant becomes a number of magnitude
6
‘Easy’ tests, 62447 integrals
6
15
x 10
Unreliability, 1000000 ‘easy’ integrals
700
600
Unsuccessful integrals
Integrand evaluations
DQNG
QUAD
DEINT
DQAG
TINT
10
5
US
RF
UF
TOT
500
400
300
200
100
0
I1
I2
I3
I4
I5 I6 I7
Integrand
I8
I9
I10 Avg
0
DQNG
QUAD
DEINT
DQAG
TINT
Fig. 1. (A) 250 x 250 ’easy’ variations of the integrands of table I integrated by the different
integration routines (except for certain troublesome variations). The routines based on usual
quadrature rules outperform TINT and DEFINT for this kind of integrals. The required accuracy
of DEFINT is adjusted with a factor 0.05 to make its number of successful returns be on the
same order of magnitude as other routines. (B) Measurement of reliability for 1000 x 1000 ’easy’
integrals. Unsuccessful integrals are classified in three categories: 1) US (unreliable success): the
routine returns an error or warning, but the returned value still fulfills the accuracy requirements.
RF (reliable failure): the routine reports an error or warning and the returned value do not fulfill
requested accuracy. ’UF’ (unreliable failure): a value not fulfilling the accuracy goal is returned,
without any errors or warnings from the routine. This graph is somewhat unfair to DEFINT,
since 543 of its unsuccessful integrals come from I10 alone. If these are excluded its reliability is
about the same as TINT (i.e., after adjusting its accuracy requirement with a 0.05 factor).
below 10−5 are removed to avoid too unconditioned integrals. Furthermore all
difficult parameter values, for which any routine either reported trouble or did not
return a result of sufficient accuracy, are also removed. The motivation behind this
is that even just a few such points may lead to many extra integrand evaluations,
and since the troublesome points may be different for different routines and are
rare this may affect the test unfairly. However, while employing this scheme it was
noticed that DEFINT is much more aggressive in its error estimate than the other
routines, which lead to the removal of a huge number of points and skewed the test.
To get a more fair comparison, the error bound on DEFINT is therefore increased
with a factor of 0.05. This made its number of successful returns be on the same
order of magnitude as TINT.
To test the reliability of the routines, a full set of 1000 x 1000 integrals in the
easy parameter range is used. The outcome of the routine is classified in one of four
categories. The categories are either ’success’ or ’failure’ depending on whether the
routine delivers a result fulfilling the required accuracy; and ’reliable’ or ’unreliable’
depending on whether warnings issued are correct or unnecessary/missing.
The results are shown in Fig. 1. It is seen that for these easy integrals TINT
and DEFINT require about twice as many integrand evaluations as other routines.
They are also somewhat less reliable than the other routines.
The next test is done for ’hard’ integrals, which excludes DQNG, QUAD and
DEFINT, since these routines give up before having done enough integrand evaluations. A set of 50 x 50 parameter values is used, with b going from 100 to 200 and
c going from 2 to 25. The test is done for three options for the required accuracy,
10−8 , 10−10 , 10−12 . Again, parameter values that give integrals that are uncon-
7
‘Hard’ tests, 1e−6, 2500 integrals
6
DQAG
TINT
5
4
3
2
1
0
I1
I2
I3
I4
I5
I6
I7
I8
6
x 10
6
DQAG
TINT
5
4
3
2
1
0
I9 I10 Avg
‘Hard’ tests, 1e−8, 2499 integrals
I1
I2
I3
I4
I5
I6
I7
I8
I9 I10 Avg
6
Number of integrand evaluations
Number of integrand evaluations
x 10
Number of integrand evaluations
6
6
x 10
‘Hard’ tests, 1e−10, 2389 integrals
DQAG
TINT
5
4
3
2
1
0
I1
I2
I3
I4
I5
I6
I7
I8
I9 I10 Avg
Fig. 2. The integrands of table I integrated by the different integration routines on an ’hard’ grid
of 50 x 50 parameter values giving integrands with a huge number of oscillations. It is seen how
TINT outperform DQAG for this kind of integrals.
Unsuccessful integrals
120
100
80
60
40
20
0
Unreliability, 1000000 ‘hard’ integrals, 1e−8
700
US
RF
UF
TOT
600
Unsuccessful integrals
US
RF
UF
TOT
500
400
300
200
100
DQAG
TINT
0
DQAG
TINT
Unreliability, 1000000 ‘hard’ integrals, 1e−10
6
Unsuccessful integrals / 10 000
Unreliability, 1000000 ‘hard’ integrals, 1e−6
140
US
RF
UF
TOT
5
4
3
2
1
0
DQAG
TINT
Fig. 3. The reliability of the routines on the ’hard’ 1000 x 1000 grid of parameter values. Unsuccessful integrals are classified as in Fig. 1b.
ditioned or for which TINT or DQAG have difficulties are excluded. The results
are shown in Fig. 2. In these tests which are closer to an actual application, TINT
requires significantly less integrand evaluation than DQAG. The reliability for the
’hard’ set of parameters is also investigated by running a full set of 1000 x 1000
integrals. The results are shown in Fig. 3.
Figure 4 shows the time spent in each routine per integrand evaluation. This is
an attempt to do a fair measurement of the efficiency of the code of the routines.
The routines perform mostly equal, which is no surprise since the central part of
all routines is a straightforward loop over quadrature points.
5.
BENCHMARKING PARALLEL IMPLEMENTATION OF TINT
The parallelized version of TINT is tested in the ’hard’ parameter range for 1, 2,
4, 8, and 16 computer nodes. This is done with and without ’load’. The case
without load is 300 x 300 of the usual integrands. The case with load is 5 x 5
integrals where the integrand is simulated to be hard to compute by forcing the
integrand function to multiply 25000 double precision numbers before returning. A
logarithmic execution time graph is shown in Fig. 5.
It is seen that the unloaded version have scalability problems, and already for 4
nodes the communication overhead makes the routine go slower than for 2 nodes.
For the loaded case, the scalability becomes much better since a less fraction of
the cpu time is spent on handling communications. This means that one usually
8
−6
4
Time per evalutation
x 10
Time (s)
3
2
1
0
1e−6
1e−8
1e−10
DQNG
QUAD
DEFINT
DQAG
TINT
DQAG
TINT
Fig. 4. Comparison of execution time per integrand evaluation. The five bars to the left show the
results for execution of ’easy’ integrals, and the three to the right show the results for execution
of ’hard’ integrals.
3
Logarithmic execution time diagram
10
2
Time(s)
10
1
10
0
10
1
TINTP
Perfect scalability
TINTP LOAD
Perfect scalability
2
4
Nodes
8
16
Fig. 5. A logarithmic executions graph for parallized versions of the routines running on an
increasing number of nodes.
don’t want to use the parallel code on integrands that are too simple to compute,
but rather on eg. integrands that consist of other integrals or are otherwise time
consuming. However, the added load has here been completely evenly distributed
over the integrand, which will not be the usual case. The more unbalanced the
load is, the worse the scaling will be, since nodes with lighter loads will have to
idle-wait for others to complete. The way this is usually handled is by dynamically
redistributing the work (i.e., dynamic load balancing).
All data in this paper are from a HP Itanium2 Cluster of rx2600/zx6000 nodes
with two 900 Mhz Itanium 2 ”McKinley” processors per node and with myrinet as
their interconnection (Myricom M3F-PCIXD-2 cards). Some incomplete runs have
also been done on other architectures, but no major deviations have been observed
from what is seen in the published data.
9
6.
CONCLUSIONS
This work has provided extensive benchmarks for some common integration algorithms and their implementations, in the context of integration of functions originating from quantum mechanics. The results of these benchmarks are useful when
implementing applications that perform larger batches of such integrals. The work
has also put forward an implementation of an algorithm that is clearly better than
the “standard” DQAG for the applications this work focus on.
It seems as when DQAG starts to subdivide the interval of integration it is
outperformed by TINT which does not have to resort to any subdivisions. However,
since TINT is outperformed by the Gauss-Kronrod-Patterson routines for simple
integrands, it is possible that a routine applying these rules indefinitely also would
perform significantly better than DQAG does.
The reliability graphs are somewhat hard to interpret. TINT has good reliability
when the accuracy requirement is low. However, it also seems that TINT suffers
more than DQAG when the requirement is set higher. However, for 10−8 and
10−10 DQAG returns many superfluous warnings, probably because both these
cases are somewhat on the edge for where the limited numerical precision of the
floating point numbers starts to affect the results. It is however surprising that
DQAG is slightly more reliable with an accuracy requirement of 10−8 than 10−6 . In
actual applications the unreliability must be tackled with some careful supervision
of the routines, since one bad evaluation can destroy much of the work done for
other integrals. It is possible to increase the reability of TINT by adjusting the
minimum number of integrand evaluations through the MINORD parameter to fit
the integrals it is used on.
The simplicity of TINT puts it in a good position for parallel implementation.
For integrands that are not too simple or too unbalanced, the provided parallel
implementation should be useful.
7. ACKNOWLEDGMENTS
The author wish to acknowledge support from the project ATOMICS at the Swedish
research council SSF and from the Göran Gustafsson Foundation. The computer
calculations was done on the Lucidor cluster at PDC in Stockholm.
REFERENCES
1. Golub, G. H. and Welsch, J. H. Calculations of Gauss quadrature rules. Math. Comput. 22
(1969), 221-230; Arfken, G. ”Appendix 2: Gaussian Quadrature.” Mathematical Methods for
Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 968-974, 1985.
2. Nodes and Weights for Quadrature Formulae. Sixteen Place Tables. Nauka, Moscow, 1964;
English translation by the Consultants Bureau, New York, 1965.
3. ACM Trans. Math. Soft. 15 (1989) 123; Comm. ACM 16 689 (1973), Acm algorithm 468.
4. R. Piessens, E. De Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, QUADPACK, a
Subroutine Package for Automatic Integration (Springer, Berlin, 1983). QUADPACK routines
are available from Netlib [9].
5. ACM Trans. Math. Soft. 17 (1991) 457
6. JCAM is a collection of FORTRAN programs published in the Journal of Computational and
Applied Mathematics. Routine DEFINT is available for download through a link from GAMS
[8].
7. R. Armiento and A. E. Mattsson, Phys. Rev. B 66, 165117 (2002).
8. NIST Guide to Available Mathematical Software (GAMS), http://gams.nist.gov/
10
9. Netlib Repository at University of Tennessee and Oak Ridge National Laboratory,
http://www.netlib.org/
10. NAG Fortran Library Manual, Mark 20, (Numerical Algorithms Group, Ltd., Oxford, 2002).
11. IMSL MATH/LIBRARY User’s Manual, Version 3.0, (Visual Numerics, Inc., Houston, Texas
USA, 1994). http://www.absoft.com/imsl
12. GNU Scientific Library Reference Manual - Second Edition, M. Galassi, J. Davies,
J. Theiler, B. Gough, G. Jungman, M. Booth, F. Rossi, (Network Theory Ltd., 2003).
http://www.gnu.org/software/gsl/
13. D.K. Kahaner, C. Moler, and S. Nash. Numerical Methods and Software. Prentice Hall,
Englewood Cliffs, NJ, 1989. Routines available as links from GAMS [8].
14. Numerical library compiled by R. Boisvert, S. Howe, and D. Kahaner of NIST, mostly from
externally available program packages. Routines available as links from GAMS [8].
15. SLATEC Common Mathematical Library, Version 4.1, July 1993. “A comprehensive software library containing over 1400 general purpose mathematical and statistical routines written
in Fortran 77.”. Available from Netlib [9]
16. Algorithms Policy, ACM Transactions on Mathematical Software, vol. 12, no. 2 (1986) 171.
Routines available from Netlib [9]
17. P. Favati et al., ACM TOMS 17 (1991) 218
18. K. Sikorski, F. Stenger, and J. Schwing, ACM TOMS 10 (1984) 152
19. Harwell subroutine library: A catalogue of subroutines, Tech. Report AERE R 9185, Harwell
Laboratory. http://www.cse.clrc.ac.uk/nag/hsl/
20. ESSL for AIX V4.1, ESSL for Linux on pSeries V4.1, Guide and Reference, IBM Document
numbers SA22-7904-01 and SA22-7906-00.
http://www.ibm.com/servers/eserver/pseries/library/sp_books/essl.html
21. E. de Doncker, K. Kaugars, L. Cucos and R. Zanny. Proceedings of the Second Computational Particle Physics Symposium (CPP’01), pp. 110-119 (2001).
http://www.cs.wmich.edu/~parint/
22. W. Gander and W. Gautschi, BIT Vol. 40, No. 1 (2000) 84.
http://www.inf.ethz.ch/personal/gander/
23. MATLAB : the language of technical computing, (The Math Works, Inc., Natick, Massachusetts USA, 2002), http://www.mathworks.com
24. C. Schwartz, J. Comput. Phys. 4 (1969) 19; S. Haber, SIAM J. Numer. Anal. 14 (1977) 668.
25. H. Takahasi and M. Mori, Numer. Math. 21 (1973); H. Takahasi and M. Mori Publ. RIMS.
Kyoto Univ. 10 (1974) 721; M. Mori, J. Comp. Appl. Math. 12-13 (1985) 119; T. Ooura and
M. Mori J. Comput. Appl. Math. 38 (1991) 353.
26. ANSI. 1978, “American National Standard for Information Processing: Programming Language FORTRAN,” ANSI X3.9-1978 (ISO 1539) (New York: American National Standards
Institute, Inc.).
27. L. Clark, I. Glendinning, and R. Hempel. The MPI message passing interface standard.
Technical report, Edinburgh Parallel Computing Centre, The University of Edinburgh, 1994.
A. APPENDIX: IMPLEMENTATION TINT
subroutine tint(f,a,b,epsabs,epsrel,minord,result,abserr,
*
ier,order)
c***date written
2004-01-12 (yyyy-mm-dd)
c***revision date
2004-02-18 (yyyy-mm-dd)
c***keywords automatic integrator
c***author rickard armiento, kth, albanova university center,
c
kth physics, theory of materials, se-106 91 stockholm,
c
sweden
c***purpose the routine approximates i = integral of f over (a,b)
c
for a smooth integrand f, trying to satisfy absolute and
c
relative claims for accuracy
11
c***references physical review b 66, 165117 (2002)
c***input arguments
c
f
- double precision
c
integrand function f(x). the actual function must
c
be declared /external/ in the calling program.
c
a
- double precision
c
lower limit of integration
c
b
- double precision
c
upper limit of integration
c
epsabs - double precision
c
requested absolute accuracy
c
epsrel - double precision
c
relative accuracy requested
c
minord - integer
c
force integration to make at least
c
(2**minord)-1 subdivisions
c***output arguments
c
result - double precision
c
approximation to integral over f from a to b
c
abserr - double precision
c
estimate of the absolute error
c
ier
- integer
c
ier = 0 normal termination. /result/ should
c
approximate integral within requested
c
accuracy.
c
ier = 1 subroutine stopped as maximum number of
c
subdivisions was reached. by increasing
c
/maxord/ and /dtsize/ parameters inside
c
the code more subdivisions can be used.
c
however, it may also be advisable to
c
investigate the integrand for
c
difficulties.
c
order - integer
c
on return, the number of subintervals
c
produced in the subdiviosion process was
c
2*2**order
c***subroutine parameters
external f
double precision a,abserr,b,epsabs,epsrel,f,result
integer ier,minord,order
c***adjustable parameters
c if changed, make sure to syncronize throughout file
c maxord: give up after 2**maxord subdivisions
c dtsize: 2*(2**maxord), max size of lookup table
integer maxord, dtsize
parameter( maxord = 17 )
parameter( dtsize = 262144 )
c***common block
common /cmtint/ tintdt, stdx, tintrg, stord
double precision tintdt(dtsize), stdx
integer tintrg(maxord), stord
c***local variables
double precision dx, oldsum
double precision diff
integer i, startp, endp
c***first executable statement tint
12
ier = 0
diff = b-a
c do first batch with lowest interval division
oldsum = 0.0d0
endp = tintrg(stord)-2
do 10 i=1,endp,2
oldsum = oldsum + f(a + diff*(tintdt(i)))*
*
diff*tintdt(i+1)
10
continue
oldsum = oldsum*stdx
order = stord+1
dx = stdx
c begin loop for increasing interval division (orders)
20
startp = tintrg(order-1)
endp = tintrg(order)-2
c loop over all subdivisions
result = 0.0d0
do 30 i=startp,endp,2
result = result + f(a + diff*(tintdt(i)))*
*
diff*tintdt(i+1)
30
continue
result = 0.5d0*(oldsum + result*dx)
abserr = abs(result-oldsum)
c exit if accuracy is fulfilled
if( (order .ge. minord) .and. ((abserr .le. epsabs) .or.
*
(abserr .le. epsrel*abs(result) )) ) goto 40
order = order + 1
dx = dx*0.5d0
oldsum = result
c loop if not order .gt. maxorder
if(order .le. maxord) goto 20
c abnormal exit, accuracy goal not fulfilled
ier = 1
40
return
end
c***function used for substitutions in trapetzoid integration
double precision function subfp(x)
c***subroutine parameters
double precision x
c***first executable statement subfp
subfp = 142.2503757770958682d0*exp(-1.0d0/(x-x*x))
return
end
c***subprogram for initializing lookup table
subroutine tinit
c***adjustable parameters
c if adjusted, make sure to syncronize with subroutines
c loword: do fist run with 2**loword subdivisions
integer loword, maxord, dtsize
parameter( loword = 3 )
parameter( maxord = 17 )
parameter( dtsize = 262144 )
c***common block
common /cmtint/ tintdt, stdx, tintrg, stord
13
double precision tintdt(dtsize), stdx
integer tintrg(maxord), stord
c***local variables
integer n, i, j, order
double precision x, dx, nxtdx, abserr, resabs, resasc
c***subprograms
external subfp
double precision subfp
c***first executable statement tinit
stord = loword
stdx = 1.0d0/2**stord
n = 2**stord
dx = stdx
nxtdx = stdx*0.5d0
j = 1
c do first batch with stepsize dx
x = dx
do 110 i=1,n-1,1
call dqk61(subfp,0.0d0,x,tintdt(j),abserr,resabs,resasc)
tintdt(j+1) = subfp(x)
x = x + dx
j = j + 2
110 continue
c do following batches starting with stepsize
c dx and offset 0.5*dx, and use half
c stepsize each consecutive step
do 130 order=stord+1,maxord
tintrg(order-1) = j
x = nxtdx
do 120 i=1,n,1
call dqk61(subfp,0.0d0,x,tintdt(j),abserr,resabs,resasc)
tintdt(j+1) = subfp(x)
j = j + 2
x = x + dx
120
continue
n = n*2
dx = nxtdx
nxtdx = nxtdx * 0.5d0
130 continue
tintrg(maxord) = j
return
end
B. APPENDIX: IMPLEMENTATION TINTP
subroutine tintp(f,a,b,epsabs,epsrel,minord,result,abserr,
ier,order)
implicit none
c***date written
2004-01-12 (yyyy-mm-dd)
c***revision date
2004-02-18 (yyyy-mm-dd)
c***keywords automatic integrator
c***author rickard armiento, kth, albanova university center,
c
kth physics, theory of materials, se-106 91 stockholm,
c
sweden
c***purpose the routine approximates i = integral of f over (a,b)
c
for a smooth integrand f, trying to satisfy absolute and
*
14
c
relative claims for accuracy
c***references physical review b 66, 165117 (2002)
c***input arguments
c
f
- double precision
c
integrand function f(x). the actual function must
c
be declared /external/ in the calling program.
c
a
- double precision
c
lower limit of integration
c
b
- double precision
c
upper limit of integration
c
epsabs - double precision
c
requested absolute accuracy
c
epsrel - double precision
c
relative accuracy requested
c
minord - integer
c
force integration to make at least
c
([total number of computer nodes] *
c
2**minord)-1 subdivisions
c***output arguments
c
result - double precision
c
approximation to integral over f from a to b
c
abserr - double precision
c
estimate of the absolute error
c
ier
- integer
c
ier = 0 normal termination. /result/ should
c
approximate integral within requested
c
accuracy.
c
ier = 1 subroutine stopped as maximum number of
c
subdivisions was reached. by increasing
c
/maxord/ and /dtsize/ parameters inside
c
the code more subdivisions can be used.
c
however, it may also be advisable to
c
investigate the integrand for
c
difficulties.
c
order - integer
c
on return, the number of subintervals
c
produced in the subdiviosion process was
c
2*2**order
c***subroutine parameters
external f
double precision a,abserr,b,epsabs,epsrel,f,result
integer ier,minord,order
c***include files
include "mpif.h"
c***adjustable parameters
c if changed, make sure to syncronize throughout file
c maxord: give up after 2**maxord subdivisions
c dtsize: 2*(2**maxord), max size of lookup table
integer maxord, dtsize
parameter( maxord = 17 )
parameter( dtsize = 262144 )
c***common block
common /cmtinp/ tintdt, stdx, tintrg, stord, size, rank, commid
double precision tintdt(dtsize), stdx
integer tintrg(maxord), stord, size, rank, commid
c***local variables
15
double precision dx, oldsum, psum
double precision diff
integer i, startp, endp, mpierr
c***first executable statement tintp
ier = 0
diff = b-a
c do first batch with lowest interval division
psum = 0.0d0
endp = tintrg(stord)-2
do 10 i=1,endp,2
psum = psum + f(a + diff*(tintdt(i)))*
*
diff*tintdt(i+1)
10
continue
call mpi_allreduce(psum,oldsum,1,mpi_double_precision,
*
mpi_sum,commid,mpierr)
oldsum = oldsum*stdx
order = stord+1
dx = stdx
c begin loop for increasing interval division (orders)
20
startp = tintrg(order-1)
endp = tintrg(order)-2
c loop over all subdivisions
psum = 0.0d0
do 30 i=startp,endp,2
psum = psum + f(a + diff*(tintdt(i)))*
*
diff*tintdt(i+1)
30
continue
call mpi_allreduce(psum,result,1,MPI_DOUBLE_PRECISION,MPI_SUM,
*
commid,mpierr)
result = 0.5d0*(oldsum + result*dx)
abserr = abs(result-oldsum)
c exit if accuracy is fulfilled
if( (order .ge. minord) .and. ((abserr .le. epsabs) .or.
*
(abserr .le. epsrel*abs(result) )) ) goto 40
order = order + 1
dx = dx*0.5d0
oldsum = result
c loop if not order .gt. maxorder
if(order .le. maxord) goto 20
c abnormal exit, accuracy goal not fulfilled
ier = 1
40
return
end
c***function used for substitutions in trapetzoid integration
double precision function subfpp(x)
c***subroutine parameters
double precision x
c***first executable statement subfp
subfpp = 142.2503757770958682d0*exp(-1.0d0/(x-x*x))
return
end
c***subprogram for initializing lookup table
subroutine tinitp(cid)
implicit none
16
integer cid
c***adjustable parameters
c if adjusted, make sure to syncronize with subroutines
c loword: do first run with loword*(nbr of nodes) subdivisions
integer loword, maxord, dtsize
parameter( loword = 1 )
parameter( maxord = 17 )
parameter( dtsize = 262144 )
c***common block
common /cmtinp/ tintdt, stdx, tintrg, stord, size, rank, commid
double precision tintdt(dtsize), stdx
integer tintrg(maxord), stord, size, rank, commid
c***local variables
integer n, i, j, order, mpierr
double precision x, dx, offset, abserr, resabs, resasc
c***subprograms
external subfpp
double precision subfpp
c***first executable statement tinit
commid = cid
call mpi_comm_size(cid,size,mpierr)
call mpi_comm_rank(cid,rank,mpierr)
if((size .gt. 1) .or. (loword .gt. 1)) then
stord = loword
else
stord = 2
endif
stdx = 1.0d0/(size*stord)
n = stord
dx = stdx*size
j = 1
c do first batch with stepsize dx; the first step on the first node (rank0)
c is skipped unbalancing the load somewhat this first step, to make sure
c following steps are equally distributed.
x = stdx*rank
do 110 i=1,stord
if((rank+i) .gt. 1) then
call dqk61(subfpp,0.0d0,x,tintdt(j),abserr,resabs,resasc)
tintdt(j+1) = subfpp(x)
j = j + 2
endif
x = x + dx
110 continue
c do following batches starting with stepsize
c dx and offset 0.5*dx, and use half
c stepsize each consecutive step
offset = stdx*0.5d0
order = stord
120 tintrg(order) = j
order = order + 1
x = offset*(2*rank+1)
do 130 i=1,n,1
call dqk61(subfpp,0.0d0,x,tintdt(j),abserr,resabs,resasc)
tintdt(j+1) = subfpp(x)
j = j + 2
x = x + dx
17
130
continue
n = n*2
dx = dx * 0.5d0
offset = offset * 0.5d0
if(order .lt. maxord) goto 120
tintrg(order) = j
return
end