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. Let’s divide

3D space into NxNxN=2x2x2 grid points. To reconstruct Ψ, how many points must we keep track of?

N

N

N

n = # electrons

Ψ (N3n) ρ (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 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 accuracyArgue functional is accurate, compare to experiment

Variational theorem: lower energy is closer

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 potentials2. Suppose they have the same ground state density3. 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:

– Still need to find the functional!

What might F[ρr)] look like?

UNIVERSAL!

From simple inspection:

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

• 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

UNIVERSAL!

Euler-Lagrange SystemThe 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.

Kohn-Sham ApproachKohn 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

Kohn-Sham ApproachThe 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:

The Exchange-Correlation FunctionalThe 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).

• 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

self-consistent approach

Local Density ApproximationIn 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 ApproximationLDA 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 ApproximationIn 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

• 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.

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.

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)

Total energies

Local Density ApproximationOne place where LDA performs poorly is in the calculation of excited states.

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Beyond LDADescription

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