Electrons in Materials
Density Functional Theory
Richard M. Martin
d orbitals
Electron density in La2CuO4 - difference from
sum of atom densities - J. M. Zuo (UIUC)
Comp. Mat. Science School 2001
1
Outline
• Many Body Problem!
• Density Functional Theory
Kohn-Sham Equations allow in principle exact
solution for ground state of many-body system using
independent particle methods
Approximate LDA, GGA functionals
• Examples of Results from practical calculations
• Pseudopotentials - needed for plane wave calculations
• Next Time - Bloch Theorem, Bands in crystals, Plane
wave calculations, Iterative methods
Comp. Mat. Science School 2001
2
Ab Initio Simulations of Matter
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Why is this a hard problem?
Many-Body Problem - Electrons/ Nuclei
Must be Accurate --- Computation
Emphasize here: Density Functional Theory
– Numerical Algorithms
– Some recent results
Comp. Mat. Science School 2001
3
Eigenstates of electrons
• For optical absortion, etc., one needs the
spectrum of excited states
• For thermodynamics and chemistry the
lowest states are most important
• In many problems the temperature is low
compared to characteristic electronic
energies and we need only the ground state
– Phase transitions
– Phonons, etc.
Comp. Mat. Science School 2001
4
The Ground State
• General idea: Can use minmization
methods to get the lowest energy state
• Why is this difficult ?
• It is a Many-Body Problem
• Yi ( r1, r2, r3, r4, r5, . . . )
• How to minimize in such a large space
Comp. Mat. Science School 2001
5
The Ground State
• How to minimize in such a large space
– Methods of Quantum Chemistry- expand in
extremely large bases - Billions - grows
exponentially with size of system
• Limited to small molecules
– Quantum Monte Carlo - statistical sampling of
high-dimensional spaces
• Exact for Bosons (Helium 4)
• Fermion sign problem for Electrons
Comp. Mat. Science School 2001
6
Quantum Monte Carlo
• Variational - Guess form for Y ( r1, r2, …)
• Minimize total energy with respect to all
parameters in Y
E0 =  dr1 dr2 dr3 … Y H Y
• Carry out the integrals by Monte Carlo
• Diffusion Monte Carlo - Start with VMC
and apply operator e-Ht Y to project out an
improved ground state Y0
• Exact for Bosons (Helium 4)
• Fermion sign problem for Electrons
Comp. Mat. Science School 2001
7
Density Functional Theory
• 1998 Nobel Prize in Chemistry to Walter
Kohn and John Pople
• Key point - the ground state energy for the
hard many-body problem can in principle
be found by solving non-interacting electron
equations in an effective potential
determined only by the density
D
H Yi (x,y,z) = Ei Yi (x,y,z) ,
h2
H=2m
2+
V(x,y,z)
• Recently accurate approximations for the
functionals of the
density have been found
Comp. Mat. Science School 2001
8
Density Functional Theory
• Must solve N equations, I = 1, N with a
self-consistent potential V(x,y,z) that
depends upon the density of the electrons
D
H Yi (x,y,z) = Ei Yi (x,y,z) ,
h2
H=2m
2+
V(x,y,z)
• Text-Book - Find the eigenstates
• More efficient Modern Algorithms
– Minimize total energy for N states subject to
the condition that they must be orthonormal
• Conjugate Gradient with constraints
– Recent “Order N” Linear scaling methods
Comp. Mat. Science School 2001
9
Examples of Results
• Hydrogen molecules - using the LSDA
(from O. Gunnarsson)
Comp. Mat. Science School 2001
10
Examples of Results
• Phase transformations of Si, Ge
• from Yin and Cohen (1982)
Needs and Mujica (1995)
Comp. Mat. Science School 2001
11
Enthalpy vs pressure
• H = E + PV - equilibrium structure at a
fixed pressure P is the one with minimum H
• Transition pressures slightly below
experiment 80 kbar vs ~100kbar
Needs and Mujica
(1995)
Simple
Hexagonal
Cubic Diamond
Comp. Mat. Science School 2001
12
Graphite vs Diamond
• A very severe test
• Fahy, Louie, Cohen calculated energy along a path
connecting the phases
• Most important - energy of grahite and diamond
essentially the same!
~ 0. 3 eV/atom barrier
Comp. Mat. Science School 2001
13
A new phase of Nitrogen
• Published in Nature this week. Reported in
the NY Times Dense, metastable
semiconductor
• Predicted by theory
Molecular
form
~10 years ago!
Mailhiot, et al 1992
“Cubic Gauche”
Polymeric form with
3 coordination
Comp. Mat. Science School 2001
14
The Great Failures
• Excitations are NOT well-predicted by the
“standard” LDA, GGA forms of DFT
The “Band Gap Problem”
Orbital dependent DFT is
more complicated but
gives improvements treat exchange better, e.g,
“Exact Exchange”
Ge is a
metal
in LDA!
M. Staedele et al, PRL 79, 2089 (1997)
Comp. Mat. Science School 2001
15
Conclusions
• The ground state properties are predicted
with remarkable success by the simple LDA
and GGAs.
Structures, phonons (~5%), ….
• Excitations are NOT well-predicted by the
usual LDA, GGA forms of DFT
The “Band Gap Problem”
Orbital dependant functionals increase
the gaps - agree well with experiment now a research topic
Comp. Mat. Science School 2001
16