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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 +=+=+