First principles electronic
structure: density functional
theory
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
2
• 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’
3
Solving the Schroedinger Equation
• For example, consider that we have n electrons populating a 3D space. Let’s divide
3D space into NxNxN=2x2x2 grid points. To reconstruct , how many points must
we keep track of?
Y = Y( r1,...,rn )
N
N
N
# of points = N
n=#
electrons
 (N3n)
 (N3)
1
8
8
10
109
8
100
1090
8
1000
10900
8
3n
4
Two solution philosophies (approximate)
Density functional
theory
Variational wave
functions (CI, CC, VMC,
DMC)
Basic object
Density
Many-body wave
function
Basic equation
E=F[p]
Hψ=Eψ
Efficiency strategy
Density is only a 3D
function
Efficient
parameterization of the
wave function
Accuracy limitation
Don’t 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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Density functional theory
• Hohenberg and Kohn (Phys. Rev. 1964)
introduced two theorems:
– Electron density <-> 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).
First theorem
• For each external potential, there is a unique
ground state electronic density
• proof outline:
1. Suppose there are two different potentials
2. Suppose they have the same ground state density
3. Show that this leads to an inconsistency
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
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:
Etot [n(r )] = T [n(r )] + Eint [n(r )] + ò dr Vext (r ) n(r ) + Eion-ion
– Still need to find the functional!
What might F[ r)] look like?
From simple inspection:
E=
ò Y (V
*
ext
+ T + Vint )Yd r
3n
UNIVERSAL!
E[ r(r)] = ò Vext (r)r(r)dr + F [ r(r)]
Naively, we might expect the functional to contain terms that resemble the kinetic
energy of the electrons and the coulomb interaction of the electrons
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
E = min { ò Vext (r) r(r)dr + F [ r(r)]}
r
UNIVERSAL!
• 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
Euler-Lagrange System
The Hohenberg-Kohn theorems give us a variational statement about
the ground state density:
d
dF
E - m ò r( r ) dr Þ
+ Vext = m
dr
dr
(
)
“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.
Kohn-Sham Approach
Kohn and Sham said:
F [ r(r)] = EKE [ r(r)] + E H [ r(r)] + EXC [ r(r)]
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
Kohn-Sham Approach
The next step to solving for the energy is to introduce a set of one-electron
orthonormal orbitals.
N
r(r) = å j i (r)
2
i=1
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:
æ dEXC [ r(r)] ö
VXC [r ] = ç
÷
dr
(r
)
è
ø
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)
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 it’s qualitative at best (see Fermi liquid theory).
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
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 70’s), 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.
For example, here are various energy
contributions for a Mn atom:
• Hartree (ECV, EVV)
• Kinetic (T0,V)
• Exchange (EX)
• Correlation energy is about EC ~ 0.1EX
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, there’s a ~10% underestimate in the
exchange energy.
This error in exchange is compensated by a ~100-200% overestimate of the
correlation energy.
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!
Slow varying
Faster varying
Good & Bad - Local Density Approximation
Total energies
• Ground state densities well-represented
Structures of highly correlated systems
(transition metals, FeO, NiO, predicts
the non-magnetic phase of iron to be
ground state)
Doesn’t describe weak interactions well.
Makes hydrogen bonds stronger than
they should be.
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.
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)
Local Density Approximation
One place where LDA performs poorly is in the calculation of excited states.
Beyond LDA
Description
LDA
Value of the density
GGA (PBE, BLYP, PW91)
Dependence on gradients
meta-GGA (TPSS)
Laplacian
DFT+U
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
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