First principles electronic structure: density functional theory

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First principles electronic structure density
functional theory
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Electronic Schroedinger equation
Atomic units, classical nuclei, Born-Oppenheimer
approximation. Describes 99 of condensed
matter, materials physics, chemistry, biology,
psychology, sociology, .... 3 basic parameters
electronic mass, proton mass, electric charge
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• Prediction in principle, just need to know what
  atoms are, can predict all properties of quantum
  system to very high accuracy

No parameters, no garbage in, garbage out
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Solving the Schroedinger Equation

• For example, consider that we have n electrons
  populating a 3D space. Lets divide 3D space
  into NxNxN2x2x2 grid points. To reconstruct ?,
  how many points must we keep track of?

n electrons ? (N3n) ? (N3)
1 8 8
10 109 8
100 1090 8
1000 10900 8
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Two solution philosophies (approximate)
Density functional theory Variational wave functions (CI, CC, VMC, DMC)
Basic object Density Many-body wave function
Basic equation EFp H?E?
Efficiency strategy Density is only a 3D function Efficient parameterization of the wave function
Accuracy limitation Dont know the functional Run out of computer time to add parameters
Argument for accuracy Argue functional is accurate, compare to experiment Variational theorem lower energy is closer
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BEFORE density functional theory

• Thomas-Fermi (TF) theory (1927)
• Find expression for energy as a function of
  electron density
• Kinetic energy is hard
• Approximate as non-interacting gas
• Energy functional

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Thomas-Fermi what is it missing?
Charge density interaction
Kinetic energy
External field
Model fluid of charge with additional resistance
to high density

• Missing
• Exclusion principle
• Electron correlation
• Proper kinetic energy

Cannot describe chemical bonding!
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Density functional theory

• Hohenberg and Kohn (Phys. Rev. 1964) introduced
  two theorems
• Electron density lt-gt the external potential
  constant
• Total energy of any density is an upper bound to
  the ground state energy, if we know the
  functional
• But mapping is unknown

Very clear and detailed proofs of these theorems
can be found in Electronic Structure, Richard
M. Martin (Cambridge University Press 2004 ,ISBN
0 521 78285 6).
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First theorem

• For each external potential, there is a unique
  ground state electronic density
• proof outline
• Suppose there are two different potentials
• Suppose they have the same ground state density
• Show that this leads to an inconsistency

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Hohenberg-Kohn I
- The external potential corresponds to a unique
ground state electron density. - A given ground
state electron density corresponds to a unique
external potential - In particular, there is a
one to one correspondence between the external
potential and the ground state electron density
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Second theorem

• Variational principle with respect to the density
• instead of finding an unknown function of 1023
  variables, we need only find an unknown function
  of 3, the charge density.
• incredible simplification

• Still need to find the functional!

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Hohenberg-Kohn II
There exists a universal functional of the
density F?(r) such that the ground state energy
E is minimized at the true ground state
density. Note how very useful this is. We now
have a variational theorem to obtain the ground
state density (and, correspondingly, the
energy) A functional is a mapping from a function
(the electron density) to a number (the ground
state energy). The equation that we need to solve
comes from taking a functional derivative
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KS II
(compared to

• while these theorems are important, they dont
  solve the problem
• we simply dont know, a priori, the explicit form
  of the functional
• this aspect of the problem was addressed by Kohn
  and Sham in 1965
• Walter Kohn was awarded the 1998 Nobel Prize in
  Chemistry for his development of density
  functional theory

Walter Kohns Nobel Lecture can be found
at http//nobelprize.org/nobel_prizes/chemistry/la
ureates/1998/kohn-lecture.html
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What might F?(r) look like?
From simple inspection
UNIVERSAL!
Naively, we might expect the functional to
contain terms that resemble the kinetic energy of
the electrons and the coulomb interaction of the
electrons
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Density Functional Theory

• Completely rigorous approach to any interacting
  problem in which we can map, exactly, the
  interacting problem to a non-interacting one.

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Ingredients of Density Functional Theory

• Ingredients
• Note that what differs from one electronic system
  to another is the external potential of the ions
• Hohenberg-Kohn I one to one correpondence
  between the external potential and a ground state
  density
• Hohenberg-Kohn II Existence of a universal
  functional such that the ground state energy is
  minimized at the true ground state density
• The universality is important. This functional
  is exactly the same for any electron problem. If
  I evaluate F for a given trial orbital, it will
  always be the same for that orbital - regardless
  of the system of particles.
• Kohn-Sham a way to approximate the functional F

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Euler-Lagrange System
The Hohenberg-Kohn theorems give us a variational
statement about the ground state density
the exact density makes the functional
derivative of F exactly equal to the negative of
the external potential (to within a constant)
If we knew how to evaluate F, we could solve all
Coulombic problems exactly. However, we do not
know how to do this. We must, instead,
approximate this functional. This is where
Kohn-Sham comes in.
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Kohn-Sham Approach
Kohn and Sham said
Separate kinetic energy coulomb energy, and
other Kinetic energy of the system of
non-interacting electrons at the same
density. Coulomb is the electrostatic term
(Hartree) Exchange-correlation is everything else
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Kohn-Sham Approach
The next step to solving for the energy is to
introduce a set of one-electron orthonormal
orbitals.
Now the variational condition can be applied, and
one obtains the one-electron Kohn-Sham equations.
Where VXC is the exchange correlation functional,
related to the xc energy as
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Comparison with Hartree-Fock
Naturally, youre remembering the Hartree-Fock
equations and realizing that this equation is in
fact quite similar
So, just as with Hartree-Fock, the approach to
solving the Kohn-Sham equations is a
self-consistent approach. That is, an initial
guess of the density is fed into the equation,
from which a set of orbitals can be derived.
These orbitals lead to an improved value for the
density, which is then taken in the next
iteration to recompute better orbitals. And so on.
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The Exchange-Correlation Functional
The exchange-correlation functional is clearly
the key to success of DFT. One of the great
appealing aspects of DFT is that even relatively
simple approximations to VXC can give quite
accurate results. The local density
approximation (LDA) is by far the simplest and
used to be the most widely used functional.
Approximate as the xc energy of homogeneous
electron gas Ceperley and Alder performed
accurate quantum Monte Carlo calculations for the
electron gas Fit energies(e.g., Perdew-Zunger)
to get f(p)
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Solving the Kohn-Sham System
To solve the Kohn-Sham equations, a number of
different methods exist. Basis set expansion of
the orbitals -Localized orbitals molecules,
etc -Plane waves solids, metals,
liquids Meaning of the orbitals -Kohn-Sham
In principle meaningless, only representation of
the density -Hartree-Fock Electron
distributions in the non-interacting
approximation There is some ad-hoc justification
for using K-S orbitals as approximate
quasiparticle distributions, but its qualitative
at best (see Fermi liquid theory).
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self-consistent approach

• like Hartree, LDA-DFT equations must be solved
  self-consistently
• great effort to develop efficient and scalable
  algorithms
• remarkably successful
• widely available
• can download computer codes that perform these
  calculations

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Local Density Approximation
In the original Kohn-Sham paper, the authors
themselves cast doubt on its accuracy for many
properties. We do not expect an accurate
description of chemical bonding. And yet, not
until at least 10 years later (the 70s), time
and time again it was shown that LDA provided
remarkably accurate results. LDA was shown to
give excellent agreement with experiment for,
e.g., lattice constants, bulk moduli, vibrational
spectra, structure factors, and much more. One
of the reasons for its huge success is that, in
the end, only a very small part of the energy is
approximated.
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Local Density Approximation
LDA also works well because errors in the
approximation of exchange and correlation tend to
cancel. For example, in a typical LDA atom,
theres a 10 underestimate in the exchange
energy. This error in exchange is compensated by
a 100-200 overestimate of the correlation
energy.
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Local Density Approximation
In theory, the LDA method should work best for
systems with slowly varying densities (i.e., as
close to a homogeneous electron gas as
possible). However, it is interesting that even
for many systems where the density varies
considerably, the LDA approach performs well!
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Good Bad - Local Density Approximation
Total energies
Structures of highly correlated systems
(transition metals, FeO, NiO, predicts the
non-magnetic phase of iron to be ground
state) Doesnt describe weak interactions
well. Makes hydrogen bonds stronger than they
should be. Band gaps (shape and position is
pretty good, but will underestimate gaps by
roughly a factor of two will predict metallic
structure for some semiconductors)

• Ground state densities well-represented

Cohesive energies are pretty good LDA tends to
overbind a system (whereas HF tends to
underbind) Bond lengths are good, tend to be
underestimated by 1-2 Good for geometries,
vibrations, etc.
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Local Density Approximation
One place where LDA performs poorly is in the
calculation of excited states.
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Beyond LDA
Description
LDA Value of the density
GGA (PBE, BLYP, PW91) Dependence on gradients
meta-GGA (TPSS) Laplacian
DFTU Ad-hoc correction for localized orbitals
Hybrid DFT (B3LYP, HSE) Hartree-Fock overestimates gaps. Mix in 20 of HF, get gaps about right.
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• There are many different density functionals
• Each works best in different situations
• Difficult to know if it worked for the right
  reasons
• Practically, very good accuracy/computational cost