POSTECH

Special Lecture on Density Functional Theory: (05) Kohn-Sham Equations for DFT

by Prof. Hyun M. Jang ( )Dept. of Materials Science and Engineering, and Division of Advanced Materials Science, Pohang University of Science and Technology

(POSTECH), Republic of Korea. also at Dept. of Physics, Pohang University of Science and Technology

(POSTECH), Republic of Korea.

Kohn-Sham Equations for DFT

The Kohn-Sham formulation centers on mapping the full

interacting system with the real potential onto a fictitious non-

interacting system whereby the electrons move within an effective

Kohn-Sham single-particle potential, The Kohn-Sham

method is still exact since it yields the same ground-state density as

the real system, but greatly facilitates the calculation.

* Kohn-Sham Equations

The Kohn-Sham equation is based on the following assumption

(called Kohn-Sham ansatz): The exact ground-state density can be

represented by the ground-state density of an auxiliary system of non-

interacting particles, called non-interacting V-representability.

We write the variational problem for the Hohenberg-Kohn (H-K)

density functional, introducing a Lagrange multiplier to constrain

the number of electrons to be N.

).(rKSV

Kohn-Sham Equations

H-K density functional:

Thus,

The corresponding Euler equation:

Kohn-Sham separated F[n(r)] into three parts.

where Ts[n(r)] is the kinetic energy of a non-interacting electron gas

of density n(r) (not the same as that of the interacting system), and the

second term of Eq. (3) is the classical electrostatic (Hartree) energy.

Exc[n(r)] is the exchange-correlation energy and contains (i) the

difference between the exact and non-interacting kinetic energies and

(ii) the non-classical contribution to the electron-electron interactions,

of which the exchange is a significant part. Exc[n] is usually a small

fraction of the total energy. is unaffected by mapping.

+= )]([)()()]([ rrrrr nFdnVnE ext

( )[ ] )1(..........0)()()()]([ = + NdnnVdnF ext rrrrrr )2(.......)(

)(

)]([r

r

rextV

n

nF+=

)3(...............)]([)()(

2

1)]([)]([ r

rr

rrrrrr nE

nnddnTnF

XCS+

+=

)(rextV

Kohn-Sham Equations

Applying the Euler equation [i.e., Eq. (2)] to Eq. (3), one obtains.

in which the Kohn-Sham potential, Vks(r), is given by

where the exchange-correlation potential Vxc(r) is defined by

The crucial point to note here is that Eq. (4) is precisely the same

equation which would be obtained for a non-interacting system of

particles moving in an external potential, Vks(r).

(Original Paper)W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133-1138 (1965).

)4(...............................)(),(

)]([

=+ r

r

rks

S Vn

nT

)5(........................)()()(

)( rrrr

rrr

extxcksVV

ndV ++

=

)6(..............................),(

)]([)(

r

rr

n

nEV xcxc

=

Kohn-Sham Equations

Considering the relation given in Eq. (4), one can construct the

following hamiltonian for the auxiliary independent particle (non-

interacting) system:

The density of this auxiliary non-interacting particle system can be

constructed by the sum of squares of the orbitals for each spin.

where is the spin variable. The independent-particle kinetic energy is given by

)8(..............)()(2

)()()(),()(

1

2

1

2

11

2

2

=

=

===

==

==

occN N

ii

ii

N

iii

N

ii

nrn

rr

rrrr*

)7(.....)(2

1)(

2 2

][

2

2

rr ksksauxVV

mHH

AUks++=

h

. +==

NNNN

= =

+= ==

==

===

N

iii

N

iii

N

ii

N

iii

N

iiiS

dd

dT

1

2*2

1

2*

2

11

2*

1

2

)9(...............)()(2

1)()(

2

1)()(

2

1

2

1

rrrrrr

rrr

The density of this auxiliary

particle system is the same as the

ground-state density of the full

interacting system as Vext remains

unchanged under the mapping.

Kohn-Sham Equations

From the 2nd expression of Eq. (9):

From Eq. (8):

The eigenstates for can be found by the variational

principle subjected to the constraint of orthonormality, namely,

Thus, variation of the bra leads to:

where Using Eqs. (7) and (13), one can establish

)10(......)(2

1

)(

2

*r

r

i

i

ST

=

)11(.................................................)()(

)(*

rr

r

i

i

n=

( )i )(

ksauxHH =

i

( )[ ] )12(.............01 = iiiauxi i

H

)13(...... iiiaux

H =

. ksaux

HH =

Kohn-Sham Equation

)14(....................)()()(2

1

)()()(2

2

2

2

rrr

rrr

iiiks

iiks

V

Vm

AU

i

=

+

=

+

h

Kohn-Sham Equations

Here is called the Kohn-Sham (K-S) wave-function or orbital.

The ground-state density is obtained by solving these N non-

interacting Schrdinger-like equations (N-independent particle

equations). These equations would lead to the exact ground-state

density and energy for the real interacting system if the exact

functional Vxc(r) (or Exc[n]) were known.

The most important property (or experimental observable) is the

total energy. From this quantity, one can obtain various properties,

such as equilibrium atomic structures, band structures, density of

states, phonon dispersion curves ( vs. k), etc.

i

)16(..................)()(

)()(

2

1)]([)]([

)15(.........................)]([)()(][

rrr

rr

rrrrrr

rrrr

dnV

nnddnTnFand

nFdnVnE

xc

s

ext

+

+=

+=

Kohn-Sham Equations

Using the last expression of Eq. (9), Eq. (15) can be rewritten as

The sum of the single-particle K-S energy does not give the

total energy (E) because this overcounts the Hartree electron-electron

interaction energy.

where

and

where Enn(R) represents the interaction between ions, and .

)17(......................)]([)()(

2

1

)()()(2

)()]([1

2

*

rrr

rrrr

rrrrrrr

nEnn

dd

dnVdnE

xc

exti

Nocc

ii

+

+

+

==

( )==

occ

ii

1

)18(..............)()()(

2

1

1

Rrr

rrrr

nn

Nocc

ii

Enn

ddE +

=

=

)19(.....)(

)(,2 )(2

1

potentialHartreen

dVNocc

i

N

iHii

=

= rr

rrr

)20(............................................)(,

=

RRR

zzEnn

.NNocc=

Kohn-Sham Equations

The infinite sum in Eq. (20) converges very slowly since the Coulomb

interaction is very long ranged. There is, however, a useful technique

(a trick due to Ewald) that allows us to circumvent this problem and

to evaluate Eq. (20). I will describe this in a later chapter on the k-

space formalisms of the total energy.

Since Eq. (21) is the formula actually

implemented in most DFT codes. This expression of (or )

would be exact if the exact functional were known.

++

+==

=

occN

innxcxcHiksEEnVVdEE

1

)21.....()()()()(2

1 Rrrrr

.)()()]([ = rrrr dnVnE xcxcksE

i

][nExc

Kohn-Sham Equations

Schematic

representation

of the self-

consistent loop

for the solution

of Kohn-Sham

equation. In

general, one

must iterate

two such loops

simultaneously

for the two

spins, with the

potential for

each spin.

initial guess

)(),()( rrr oonnorn

o

Compute effective potential

Solve K-S Eqn.

Compute the electron density

[ ]++= nnVVVVxcextHKS

,)()()( rrr

Converged (self-consistent) ?

end

if yes

if no

)()()(2

1 2rrr

iiiKS

V =

+

2)()( =

N

ii

n rr

Kohn-Sham Equations

* Detailed Explanations of the Computational Procedure

(1) Supply an adequate model density to start the iterative procedure.

In a solid-state system or a molecule, one could construct no(r) from a

sum of atomic densities, namely,

where R represents the position of the nucleus and n is the atomic density of the nucleus . (2) The external potential is typically a sum of nuclear potentials

centered at the atomic positions.

V could simply be the Coulomb attraction between the bare nucleusand the electrons, namely, where is the nuclear

charge. In most cases, however, the use of the Coulomb potential

renders the calculation unfeasible, and one has to resort to pseudo-

potentials.

( ) )22(..........)(

Rrr = nno

,/)( rzrV = z

)23(.......)()(

Rrr = VVext

Kohn-Sham Equations

(3) The Hartree potential is given by the following integral form;

We have a couple of techniques to evaluate this integral, either by

direct integration or by solving the equivalent Poissons equation,

namely,

(4) Finally, has to be evaluated.

Numerous approximate xc functionals have appeared in the literature

over the past 30 years. Among these, the local density approximation

(LDA) is simplest of all and most commonly used.

where is the exchange-correlation energy per electron in a

homogeneous electron gas of the density, n(r).

)19(........................)()(

)( 3

=

=

rr

r

rr

rrr

nrd

ndV

H

)24(.................................)(4)(2 rr nVH

=

)(rxcV )6(.....

),(

)]([)(

r

rr

n

nEV xcxc

[ ])(rnxc

= )25(........)()]([)()]([

3)( rrrrr nnrdnndE

xcxcxcLDA

Kohn-Sham Equations

The exchange-correlation potential, Vxc(r), then takes the following

form:

(5) Now that we have the Kohn-Sham potential, we are able to solve

the Kohn-Sham equation and to obtain the p lowest eigenstates of the

Hamiltonian, In most cases (except for an atom with a

1-D differential equation), one has to diagonalize Conventional

diagonalization schemes scale N3 with the dimension of the matrix N

which is roughly proportional to the number of atoms in the

calculations. A significant improvement in the diagonalization of

matrix (more exactly a direct minimization of the total energy) had

been achieved by Payne et al. Their iterative method scales much

better with the dimension of the matrix. Nonetheless, diagonalizing

the hamiltonian ( ) is usually the most time-consuming part of an

ordinary Kohn-Sham calculation.

( ) ( ) )26(.................),(

][)(][

)(

)]([)(

rr

r

rr

n

nnn

n

nEV xc

xc

xc

xc

+=

).(auxks

HH =.

ksH

ksH

ksH

Kohn-Sham Equations

(References for diagonalization or minimization) (1) M. C. Payne, M. P. Teter, D.

C. Allan, T. A. Arias, and J. D. Joannopoulos, Iterative Minimization Techniques for

ab initio Total-Energy Calculations: Molecular Dynamics and Conjugate Gradients,

Review of Modern Physics, Vol. 64, pp. 1045-1097 (1992). (2) G. Kresse and D.

Joubert, From Ultrsoft Pseudopotentials to the Projector Augmented-Wave Method,

Phys. Rev. B, Vol. 59, pp. 1758-1775 (1999).

(6) Once the K-S equation is solved, one can compute the electronic

density by The self-consistency cycle is stopped

when some convergence criterion is reached. The two most common

criteria are based on the difference of total energies or densities from

iteration i and i-1. The cycle is stopped when

or where designate user defined tolerances.

If, on the contrary, the criteria have not been fulfilled, one has to

restart the self consistency cycle with a new density. The simplest

but useful approach is a linear mixing scheme given by

.)()( 2=Nocc

ii

n rr

E

ii EE

Kohn-Sham Equations

This is the best choice in the absence of other information.

(7) At the end of the calculations, one can evaluate several

observables, the most important of which is the total energy given by

Eq. (21). From this quantity, one can obtain many useful physical

observables that include equilibrium atomic configurations, band

structures, orbital-resolved density of states, 3-D electron-density

contours, phonon dispersion curves, dielectric responses, and

ionization potentials, to name a few.

)27(.......)()1(1

in

i

out

i

in

i

in

i

out

i

in

innnnnn +=+=+