the nuts and bolts of first-principles simulation

CASTEP Workshop, Durham University, 6 – 13 December 2001

The Nuts and Bolts of First-Principles Simulation

3: Density Functional Theory

CASTEP Developers’ Groupwith support from the ESF k Network


Density functional theoryMike Gillan, University College London

• Ground-state energetics of electrons in condensed matter

• Energy as functional of density: the two fundamental theorems

• Equivalence of the interacting electron system to a non-interacting system in an effective external potential

• Kohn-sham equation

• Local-density approximation for exchange-correlation energy


The problem

• Hamiltonian H for system of interacting electrons acted on by electrostatic field of nuclei:

with T kinetic energy, U mutual interaction energy of electrons, V interaction energy with field of nuclei.

• To develop theory, V will be interaction with an arbitrary external field:

1

( )N

ii

V v

rwith ri position of electron i.

• Ground-state energy is impossible to calculate exactly, because of electron correlation. DFT includes correlation, but is still tractable because it has the form of a non-interacting electron theory.

H T U V


Energy as functional of density: the first theorem

For given external potential v(r), let many-body wavefunction be . Then ground-state energy Eg is:

0gE H V

and the electron density n(r) by: ^( ) ( )n n r r

where the density operator is defined as:

^

1

( ) ( )N

ii

n

r r r

^( )n r

Theorem 1: It is impossible that two different potentials give rise to the same ground-state density distribution n(r).

Corollary: n(r) uniquely specifies the external potential v(r) and hence the many-body wavefunction .


Convexity of the energy (1)

Theorem 1 expresses convexity of the energy Eg as function of external potential.

Convexity means: For two external potentials and , go along linear path between them; if is ground-state energy for then:

( )gE (0; )v r (1; )v r

( ; ) (1 (0; ) (1; )v v v r r r0 1 ,

( ) (1 ) (0) (1) .g g gE E E

Proof of follows from 2nd-order perturbation theory:

( ) (1 ) (0) (1)g g gE E E

0

2

02

0 0

/ ( ) ( )

( ) ( )/ 2 0 ,

( ) ( )

g

ng

n n

dE d V

Vd E d

E E

with and wavefns of ground and excited states, and their energies, and .

( ) ( )n 0 ( )E ( )nE (1) (0)V v v


Convexity of the energy (2)

Theorem 1 is equivalent to saying that a change of external potential cannot give a vanishing change of density

( )v r( )n r

This follows from convexity. Convexity implies that at is less than at . But , so that:

/gdE d 0 /gdE d 1

0/gdE d V

0 0 0 0(1) (0) < (0) (0)V V

so that:

( ) (1, ) < d ( ) (0, ) .d v n v n r r r r r r

Hence:

( ) ( ) < 0 ,d v n r r rwhich demonstrates that , and this is Theorem 1.( ) 0n r


Since ground-state energy Eg is uniquely specified by n(r), write it as Eg[n(r)]. It’s useful to separate out the interaction with the external field, and write:

Where F[n(r)] is ground-state expectation value of H0 when density is n(r).

[ ( )] ( ) ( ) [ ( )] ,gE n d v n F n r r r r r

DFT variational principlethe second theorem

Theorem 2 (variational principle): Ground-state energy for a given v(r) is obtained by minimising Eg[n(r)] with respect to n(r) for fixed v(r), and the n(r) that yields the minimum is the density in the ground state.

Where n’(r) is density associated with . This proves the theorem. The usual assumptions of non-degenerate ground state is needed.

Proof: Let v(r) and v’(r) be two different external potentials, with ground-state energies Eg and Eg’ and ground-state wavefns and . By Rayleigh-Ritz variational principle:

'

0< ' ' ( ) '( ) [ '( )] ,gE H V d v n F n r r r r

'


The Euler equation

Write F[n(r)] as:

where T[n] is kinetic energy of a system of non-interacting electrons whose density distribution is n(r). Then:

[ ] [ ] [ ] ,F n T n G n

[ ] ( ) ( ) [ ] [ ] .E n d v n T n G n r r r

Variational principle:

subject to constraint:0 ( ) ( ) ,

( ) ( )

T GE d v n

n n

r r r

r r

( ) 0 .d n r r

Handle the constant-number constraint by Lagrange undetermined multiplier, and get:

( ) ,( ) ( )

T Gv

n n

rr r

with undetermined multiplier the chemical potential.


Kohn-Sham equation

Rewrite the Euler equation for interacting electrons:

by defining , so that:

( )( ) ( )

T Gv

n n

rr r

eff ( ) ( ) / ( )v v G n r r r

eff ( )( )

Tv

n

rr

But this is Euler equation for non-interacting electrons in potential veff(r), and must be exactly equivalent to Schroedinger equation:

with n(r) given by:

22

eff ( ) ,2

vm r

2( ) 2 ( ) .n

r r

Then put n(r) back into G[n(r)] to get total energy:

tot[ ( )] ( ) ( ) [ ] [ ] .E n d v n T n G n r r r r


Self consistency

How to do DFT in practice???

• We don’t know G[n(r)], and probably never will, but suppose we know an adequate approximation to it.

• Make an initial guess at n(r), calculate and hence

for this initial n(r).

• Solve the Kohn-Sham equation with this veff(r) to get the KS orbitals

and hence calculate the new n(r):

• The output n’(r) is not the same as the input n(r). So iterate to reduce residual:

The whole procedure is called ‘searching for self consistency’.

/ ( )G n r

eff ( ) ( ) / ( )v v G n r r r

2'( ) 2 ( ) | .in

r r

1/22 d '( ) ( ) .n n n r r r


Exchange-correlation energy

• We have already split the total energy into pieces:

• Now separate out the Hartree energy:

• Then exchange-correlation energy Exc[n] is defined by:

So far, everything is formal and exact. If we knew the exact Exc[n], then we could calculate the exact ground-state energy of any system!

tot[ ] ( ) ( ) [ ]

[ ] [ ] [ ]

E n d v n F n

F n T n G n

r r r

2H

1 ( ) ( ')[ ( )] ' .

2 | ' |

n nE n e d d

r r

r r rr r

tot H xc[ ] ( ) ( ) [ ] [ ] [ ] .E n d v n T n E n E n r r r


Local density approximation

• There is one extended system for which Exc is known rather precisely: the uniform electron gas (jellium). For this system, we know exchange-correlation energy per electron as a function of density n.

• Local density approximation (LDA): assume the xc energy of an electron at point r is equal to for jellium, using the density n(r) at point r. Then total Exc for the whole system is:

• Some kind of justification can be given for LDA (see xxxxxxx). But the main justification is that it works quite well in practice.

LDAxc xc ( ( ))E d n n r r r

xc ( )n

xc ( ( ))n r


Kohn-Sham potential in LDA

The effective Kohn-Sham effective potential in general is:

The Hartree potential is:

Exchange-correlation potential in LDA:

Where:

So in LDA, everything can be straightforwardly calculated!

H xceff ( ) ( ) ( )

( ) ( ) ( )

G E Ev v v

n n n

r r rr r r

2 2H1 2

1 ( ) ( ) ( ') ' .

( ) ( ) 2 '

E n n ne d d e d

n n

1 2

1 2

r r rr r r

r r r r r r

xc 1 1 xc 1 xc( ) ( ) ( ( )) ( ( )) ,( )

v d n n nn

r r r r rr

xc xc( ) ( )d

n n ndn


Useful references

Here is a selection of references that contain more detail about DFT:

• P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964)

• W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965)

• N. D. Mermin, Phys. Rev. 137, A1441 (1965)

• R. O. Jones and O. Gunnarsson, Rev. Mod. Phys., 61, 689 (1989)

• M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias and J. D. Joannopoulos, Re. Mod. Phys., 64, 1045 (1992)

the nuts and bolts of first-principles simulation

Documents