Density Functional Theory: a first look

Patrick Briddon

Theory of Condensed MatterDepartment of Physics, University of Newcastle,

UK.

ContentsDensity Functional Theory

– Hohenberg Kohn Theorems– Thomas Fermi Theory– Kohn-Sham Equations– Self Consistency

– Approximations for Exc.

Density Functional Theory

Work with n(r) instead of

Standard approach of QM :

rr nEESVext ,).(. DFT : work in terms of density :

rnEE N.B. : few IFs and BUTs here

3 Important Questions

Three important questions:

• Can we really write E[n]?

• If so, how can we find n(r)?

• What is the functional E[n] ?

1st Hohenberg Kohn Theorem

The external potential V(r) is determined to within a constant by the ground state charge density of a system.

This is an astonishing statement!

Why?

i.e. one-to-one relationship

rr extVn

1st Hohenberg Kohn Theorem

Proof:

Suppose we have two systems

Hamiltonians H1, H2

External potentials V1 , V2

GS wavefunctions: 12

But the same GS density n(r)

We clearly have 2121111 HHE

But:

212

212

11

ˆ

ˆ

VVH

VVVT

VTH

So:

rr dVVnEE

VVHE

2121

22122221

Swap 1 and 2: rr dVVnEE 2121

Contradiction!

Our starting point was wrong!

We cannot have two different systems with the same GS density.

Importance of this: we can write E[n].

Now move on the second question – how can we find the density?

Hohenberg – Kohn’s second theorem.

2nd Hohenberg Kohn Theorem

“The true ground state charge density is that which minimises the total energy.”

An equivalent to the usual variational principle of quantum mechanics.

Proof

We have (variational principle)

Define

nEnHnnE minminminGS

HE minGS

HnnE min

Then

Some problemsV-representability(only minimise over densities which can arise from GS wavefunctions of real systems

Levy showed that the densities need only satisfy a weaker condition that they can be obtained from an antisymmetric wavefunction (N-representable).

n must be +ve, continuous, normalised

Some extensions

Spin dependent potentials e.g. magnetic fields

E[n] E[n, n - n] or E[n, n]

Main advantage: better description of open shell systems.

3 Important Questions

Three important questions:

• Can we really write E[n]?

• If so, how can we find n(r)?

• What is the functional E[n] ?

Now for the last question.

What is the formula!

What is the functional?

Difficult : still not answered exactly!

termsdifficult

1

ext

ext

rrr

rrr

dVn

VTEji ji

Problem is that other two terms are very large - any attempt at approximation must be good.

Thomas-Fermi Method

rrrrrr

rrdd

nn

ji ji 211

Classical expression for electron-electron term:

Thomas-Fermi Method

3522

2ˆ An

mk

TFk

k

Statistical idea for KE based on uniform electron gas result:

KE per electron =32An

Thomas-Fermi Method

rr dAnT 35

What about a non uniform system?

A. Assume that things vary slowly:

Total is thus

r

V VnnT rr 32

Thomas-Fermi Method

Final energy is thus:

How useful is this?

rrrrrr

rrrrr

ddnn

dVndnAnE ext

21

35

Thomas-Fermi Method

What is the conclusion?

• Energies quite good (error < 1%).

• Difference of energies not good enough to describe bonding.

• How can we improve this?

Thomas-Fermi Method

Add exchange/correlation (missing do far).

• Try to take account of non-uniform system.

• Write T[n] as T[n, grad |n|]

• All failures!

Kohn-Sham methodPhys Rev 140, 1133A (1966)

Realised that approximation must be made to terms that are small: KE is big!

Improving T[n] did not work.

Need a completely different approach.

Second half of HK paper therefore discarded.

Kohn-Sham methodPhys Rev 140, 1133A (1966)

Introduce a system which:1. Is non-interacting2. Has same n(r) as the real system.

[non-interacting N-representability- an assumption! ]

Kohn-Sham contd.

TnTT s ˆ

where Ts[n] is the KE of the non-interacting system and the final term, T, is small.

nddnn

dVnnTE s

xc

ext

E 2

1

rr

rr

rr

rrr

Kohn-Sham contd.

Exc[n] includes both T and contributions to el-el energy beyond the Hartree term.

The key hope is that this is• small• less sensitive to external potential

These mean differences are accurate.

We now have two questions:(a) how to find Ts[n] ?(b) what is Exc[n] ?

For a non-interacting system it is exactly true that the many electron wavefunction is a single Slater determinant.

iN Nrrr det

!

1,,1

2rrn

and:

221nTs

For this:

The (r) must be found froma self consistent solution of:

2

xcext

221

rr

rr

rr

rrr

r

n

n

nEd

nVV

V

s

s

These are called the Kohn-Shamequations. Solve iteratively:

2out

rrn

Guess:

Construct

r

rrrr

rrnnE

dn

VVs xcext

Solve rsV2

21

Find new density:

rr innn

Look at rr inout nn

Form a better input and continue.

Self Consistent Cycle

• This process is called the self-consistent cycle.

• Starting guess is a superposition of atomic charge densities (or a restart dump).

• AIMPRO produces output showing how the energy converged and how the input and output densities come together.

AIMPRO SCF

etot,echerr 1 -1.1289007706 0.0547166884 0.884981 2.95 106.1 120.7 etot,echerr 2 -1.1319461182 0.0303263020 0.488911 3.05 106.1 120.7 etot,echerr 3 -1.1361047998 0.0000338275 0.000689 3.00 106.1 120.7 etot,echerr 4 -1.1360826939 0.0001723143 0.002700 2.99 106.1 120.7 etot,echerr 5 -1.1361076063 0.0000002649 0.000004 3.00 106.1 120.7

The numbers are:• Total energy (reduces to converged value)• 2 measures of• Time taken per iteration• Current memory being used (MB)• Max memory used so far (MB)

Kohn-Sham Levels

We got the Kohn Sham eqn:

rsV2

21

Q: what exactly are the and ?

A: the eigenvalues and eigenfunctions of a ficticious non-interacting system which has the same density as the real system.

KS Levels Contd.

They are not the energies of quasiparticles.

Typical semiconductor results:Lattice constant to 1%Bulk modulus to 1%Phonon frequencies to 5%

LDA “gap” for Si is 0.6eV; 0.1 eV for Ge!

KS Levels Contd.

Bandstructures are qualitatively correct. (Scissors operator).

Physical nature of the KS eigenfunctions sensible.

P in Si - get state just below conduction band

Dangling bonds - localised states in mid gap.

AIMPRO and KS levels

spin, kpoint : 1 1

1 -10.0938 1.2658 3.7422 3.7422 5.9056 8.7912

2.0000 0.0000 0.0000 0.0000 0.0000 0.0000

The KS levels in eV.

Used in “bandstructure” plots.

Occupancies also given (this is a spin averagedcalculation)

3 Important QuestionsThree important questions:

• Can we really write E[n]?• If so, how can we find n(r)?• What is the functional E[n] ?

• All remaining questions are in Exc[n]. Now finally we look at what this is.

Exchange correlation energy

Our DFT total energy is:

What about the last term?

nE

nn

nVnTE

xc

s

rr

rr

rr

rrr

dd2

1

dext

The Local Density Approximation (LDA)

rrr dnnE xcxcWrite

where xc(n) is the exchange-correlation energy per electron for a uniform electron gas.

This seems a bit rough and very similar to Thomas Fermi, but this term is now very much smaller.

Exc for Homogeneous electron gas

• By simple analytical treatment for the exchange energy.

• Using many body perturbation theory (for various limits of correlation energy)

• By looking at quantum Monte-Carlo calculations and parametrising them

• Intelligent interpolation between these

Exc for Homogeneous electron gas contd.

Simple analysis for exchange part gives

343431

x 43

23

nnE

Correlation is harder, see::• Perdew Zunger (1981)• Vosko, Wilk, Nusair (1980)• Perdew, Wang (1992)

Simple Tests : Molecules

Property Calc. Expt.

R(O–H) (Å) 0.967 0.957

(H–O–H) (deg) 105.7 104.5

1 (as str : cm-1) 3874 3757

2 (sym cm-1) 3773 3652

3 (bend cm-1) 1586 1596

dip. mom. (a.u.) 0.735 0.730

Example : water H20

Simple tests : solids

• Standard “bulk” calculations :– lattice constant (Si : 1%)– bulk modulus (Si : 2%)– phonon spectra (2 %)– formation energies (LDA : 20 %)– excitation energies (50 %)

Phonon Spectrum

Mode Calc (5) Calc (6) Expt 278 276 267,285TA(X) 88 90 79TA(L) 66 66 62LA(X) 224 225 227LA(L) 217 216 209TO(X) 256 255 252TO(L) 267 265 261LO(X) 246 244 241LO(L) 238 239 238

Material : GaAs. All frequencies in cm-1

How to Improve?

Next step is to move beyond the LDA:

rr dn

,2

34LDAxcxc

nnCEnnE

The Gradient expansion Approximation (GEA).

Early history of these was bad. Calculations made worse, not better.

Generalised Gradient Approximations (GGA)

• Sum rules are obeyed correctly

• scaling behaviour of exchange correlation energy correct

• Various limiting forms

• Bounds (Lieb-Oxford)

Idea is to ensure that

Popular GGAs

• B88 (empirical, chemistry)

• BLYP (chemistry)

• PW91 (physics, poor form)

• PBE96 (physics, easier to use)

HF LSD GGA expt

H2 84 113 105 109

CH4 328 462 420 419

H20 155 267 234 232

Cl2 17 81 63 58

Atomisation energies (kcal/mol)(1 eV = 23 kcal/mol)

Generalised Gradient Approximations (GGA)

In general, GGA weakens bonds slightly.

It improves results for:• binding energies of molecules• description of surfaces• H-bonding

THE CONCLUSION

• An absolutely huge success

• 1988 two groups in UK doing DFT– Cambridge (TCM)– Exeter

• Today: every department?

• Chemistry/engineering too!

• Applications in huge variety of areas.

Work to do!

• Kittel Ch 6: “Free Electron Fermi Gas”

• Hohenberg and Kohn, PR (1964)

• Kohn-Sham, PR (1965)

• Perdew Zunger, PRB (1981)

• Perdew, Wang, PRB (1992)

• Perdew, Burke, Enzerhoff PRL (1996)