Definitions
Example: Thomas-Fermi model of Atoms
Part I: Elements of Functional Analysis
Tomasz A. Wesolowski, Université de Genève
EPFL, Spring 2016
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Functional
By a functional, we mean a correspondence which assigns a definite
(real) number to each function (or a curve) belonging to some class.
I.M. Gelfand and S.V. Fomin, ”Calculus of Variations” Prentice-Hall, Inc. 1963, page 1
Examples:
A[f ] = max f (x, y , z)
B[f ] =
R
f (x, y , z)dxdydz
C [f ] =
R
f 2 (x, y , z)dxdydz
D[f ] =
1
2
RR
f (x,y ,z)f (x 0 ,y 0 ,z 0 )
dxdydzdx 0 dy 0 dz 0
[(x−x 0 )2 +(y −y 0 )2 +(z−z 0 )2 ]1/2
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Functional
In this lecture, mainly the functionals of the electron density denoted
with ρ will be considered i.e. f (x, y , z) = ρ(x, y , z).
The functional in Example B is the total number of electrons
Z
N = ρ(x, y , z)dxdydz
The functional in Example D can be identified with the Coulomb integral
(classical electron-electron repulsion).
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Functional derivative
F [ρ + δρ] − F [ρ] =
R
δρ δFδρ[ρ] dr + O(δρ)2
where δρ is an infinitesimally small function and O(δρ)2 denotes
all the terms proportional to the higher powers of δρ (quadratic,
cubic, etc.).
The linear term in F [ρ + δρ] − F [ρ] is called variation of the
functional F and is denoted as δF .
δF =
R
Tomasz A. Wesolowski, Université de Genève
δρ δFδρ[ρ] dr
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Functional derivative: example
For F [ρ] =
R
ρdr:
Z
F [ρ + δρ] − F [ρ] =
Z
(ρ + δρ) dr −
Z
ρdr =
1 · δρdr
its functional derivative is:
δF [ρ]
=1
δρ(r)
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Functional derivative: example
For F [ρ] =
R
v (r)ρ(r)dr:
Z
F [ρ + δρ] − F [ρ] =
Z
v (r) (ρ(r) + δρ(r)) dr −
v (r)ρ(r)dr
Z
=
v (r) · δρ(r)dr
its functional derivative is:
δF [ρ]
= v (r)
δρ(r)
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Functional derivative: example
For F [ρ] =
R
ρk (r)dr:
Z
F [ρ + δρ] − F [ρ] =
(ρ(r) + δρ(r)) r −
Z
=
k
Z
ρk (r)dr
kρk−1 (r) · δρ(r)dr + O(δ 2 ρ)
its functional derivative is:
δF [ρ]
= kρk−1 (r)
δρ(r)
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Functional derivative: exercise
Show that for the functional F [ρ] =
its functional derivative is:
R
|∇ρ|2
ρ dr
∇2 ρ |∇ρ|2
δF [ρ]
= −2
+
δρ(r)
ρ
ρ2
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Extremum of a functional
It is such a function ρ0 for which the functional F [ρ] is extremal
(minimum or maximum).
The necessary condition for the differentiable functional F [ρ] to
have an extremum for ρ(x, y , z) = ρ0 (x, y , z) is that its variation
vanish for ρ(x, y , z) = ρ0 (x, y , z).
δF = 0
This condition can be expressed using the introduced definition of
the functional derivative taking the form known as Euler equation:
δF [ρ]
δρ(r)
=0
(Note the analogy with extremum of a function in elementary
calculus.)
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Euler-Lagrange Equations
The function ρ might be subject of additional constraints of the
general form:
C [ρ] = 0
Example: The condition that the integral of the function ρ must
be equal to a given number (the total number of electrons) can be
written as:
Z
C [ρ] = ρ(r)dr − N = 0.0
The extremum of the functional F [ρ] in the presence of constraints
C [ρ] can be obtained from the Euler-Lagrange equation:
δF [ρ]
δρ(r)
[ρ]
− µ δC
δρ(r) = 0
where µ is a constant to be found (Lagrange multiplier).
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Historical Introduction
In the 1920’s, Thomas and Fermi realized that statistical
consideration can be used to approximate distribution of electrons
in an atom. The model called now the Thomas-Fermi model is the
first theory in which the description of an atom does not use the
wavefunction Ψ(x1 , x2 , ..., xN ) but it uses a much simpler quantity
- the electron density ρ(r). The Thomas-Fermi model is of little
practical use. It has serious known mathematical flaws. The
accuracy of the obtained energies is not sufficient. The
Thomas-Fermi model will be presented here because it can be seen
as the simplest Density Functional Theory, introducing the same
concepts and ideas as the more advanced contemporary theories.
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Kinetic energy in non-interacting uniform gas of electrons (1)
We recall at first the exact solution for the energy levels of the
particle in a three-dimensional infinite well:
(nx , ny , nz ) =
=
h2
2
2
2
n
+
n
+
n
x
y
z
8ml 2
h2 2
R
8ml 2
where l is the length of the box and nx , ny , nz are integers (1, 2,
3,...).
The number of the energy levels below a given level equals to the
number of such combinations of nx , ny , nz which satisfy:
nx2 + ny2 + nz2 ≤ Tomasz A. Wesolowski, Université de Genève
8ml 2
h2
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Kinetic energy in non-interacting uniform gas of electrons (2)
For sufficiently large l, the number of states satisfying the above
inequality (φ()) can be approximated by the volume of one octant
of the sphere with radius R.
φ() =
1
8
4πR 3
3
=
π
6
8ml 2 h2
3/2
As a consequence, the number of the energy states of the energies
between + ∆ and can be expressed as:
g ()∆ = φ( + ∆) − φ()
=
π
4
Tomasz A. Wesolowski, Université de Genève
8ml 2 h2
3/2
−1 ∆
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Kinetic energy in non-interacting uniform gas of electrons (3)
To obtain the total energy, Thomas and Fermi introduced the
statistical gas theory. For electrons, the probability of the state of
a given energy () to be occupied at a given temperature (T,
1
) is known as the Fermi-Dirac distribution (f ()):
β = kT
f () =
1
1 + exp[β( − µ)]
At zero temperature, the Fermi-Dirac distribution reduces to a
step function:
f () = 1 for < F
f () = 0 for > F
where F is called Fermi energy.
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Kinetic energy in non-interacting uniform gas of electrons (4)
The total energy of the electrons can be obtained as the integral
over energies from 0 to F and multiplying by 2 as each state can
be occupied by 2 electrons with opposite spins:
Z
Z
2m 3/2 3 F 3/2
l
d
E = 2 f ()g ()d = 4π
h2
0
8π 2m 3/2 3 5/2
=
l F
5
h2
We notice now that the total number of electrons in the
considered three-dimensional infinite well can be also expressed
using f () and g ():
Z
N=2
8π
g ()f ()d =
3
Tomasz A. Wesolowski, Université de Genève
2m
h2
3/2
l 3 F 3/2
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Kinetic energy in non-interacting uniform gas of electrons (5)
Combining the two equation for N and for E by eliminating F
leads to the relation between the energy (which equals to the
kinetic energy in the considered example) and the number of
electrons in the well:
2/3 5/3
N
3
3h2
3
l3 3
E =T =
NF =
5
10m 8π
l
Since l 3 is the volume of the well (V), the above equation relates
T
the density of the kinetic energy tk = V
with the density of the
N
electron density ρ = V .
3h2
tk =
10m
3
8π
Tomasz A. Wesolowski, Université de Genève
2/3
ρ5/3
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Local Density Approximation for kinetic energy functional (1)
The whole space can be formally divided into small cells such that
in which cell the electron density can be considered to be constant.
3h2 3 2/3 5/3
For each cell the relation tk = 10m
( 8π ) ρ
holds. Therefore,
the total kinetic energy of a slowly-varying electron density can be
obtained by integration over all cells:
T [ρ] ≈ TTF [ρ] = CF
where CF =
3
2 2/3
10 (3π )
R
ρ5/3 (r)dr
= 2.871 in atomic units.
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Local Density Approximation for kinetic energy functional (2)
The approximation in which the total energy is calculated as a sum
(integral) over volume elements such that in each of them the
electron density is considered to be uniform is known as Local
Density Approximation, LDA in modern density functional
theory. Local Density Approximation is used not only to the kinetic
energy functional but also to other functionals as well.
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Thomas-Fermi model of atoms (1)
The introduction of the kinetic energy functional by Thomas and
Fermi, was a fundamentally new concept. The electronic kinetic
energy is obtained in conventional quantum mechanics as an
expected P
value of the kinetic energy operator
1 2
T̂ = (Ψ| N
i [− 2 ∇i ]|Ψ) requiring the knowledge of the
wavefunction (Ψ) which as a complex function of 3N variables for
a system with N electrons. In the Thomas-Fermi model, the
electronic kinetic energy is expressed as a functional of the electron
density (ρ(r) which is a real function of 3 variables independent on
the number of electrons in the system. The underlying it Local
Density Approximation can not be expected to be good for
chemical molecules in which the electron density is characterized
by such features as cusps at nuclei and exponential behaviour far
from them.
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Thomas-Fermi model of atoms (2)
Applying the Thomas-Fermi functional to the energy expression of
an atom leads to the approximate functional of the total energy:
ETF [ρ(r)] = CF
R
ρ5/3 dr − Z
R
ρ(r)
r dr
+
1
2
RR
ρ(r)ρ(r0 )
0
|r−r0 | drdr
The Thomas-Fermi approximate functional leads to no-nonsense
energies of atoms, but it has a number of mathematical flaws such
as wrong asymptotic behaviour of the electrostatic potential. It is
not applicable for chemical problems because: this model predicts
that the chemical molecules are always unstable.
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Thomas-Fermi model of atoms (3)
For the Thomas-Fermi energy functional, the Euler-Lagrange
equation reads:
R
δ ETF [ρ] − µTF
ρ(r)dr − N
=0
δρ(r)
leading to
5
Z
µTF = CTF ρ2/3 (r) − φ(r) where φ(r) = −
3
r
Z
ρ(r0 )
dr
|r − r0 |
The function ρTF (r) solving the above equation has qualitatively
wrong properties:
ρTF (r) ∝
1
r 3/2
for r → 0
Tomasz A. Wesolowski, Université de Genève
and
ρTF (r) ∝
1
r6
for r → ∞
Part I: Elements of Functional Analysis
Definitions
Example: Thomas-Fermi model of Atoms
Kinetic energy as density functional: exercise
Follow the derivation of the Thomas-Fermi functional to obtain the
density functional for the kinetic energy in:
a) one-dimensional uniform electron gas;
b) two-dimensiomal uniform electron gas.
Tomasz A. Wesolowski, Université de Genève
Part I: Elements of Functional Analysis