Computational Chemistry

electron densities of the individual atoms of the molecule, at the molecular
geometry. The KS Fock matrix elementshrs¼ frjh^

KS

jfs

DE

are calculated and
the KS Fock matrix is orthogonalized and diagonalized, etc., to give initial
guesses of thec’s in the basis set expansion of Eq.7.26(and also initial values
of thee’s). Thesec’s are used in Eq.7.26to calculate a set of KS MOs which
with Eq.7.22are used to calculate a betterr.Thisnewdensityfunctionisused
to calculate improved matrix elementshrswhich in turn give improvedc’s and
then an improved density function, the iterative process being continued until
the electron density etc. converge. The final density and KS orbitals are used to
calculate the energy from Eq.7.21.
The KS Fock matrix elements are integrals of the Fock operator over the basis
functions. Because useful functionals are so complicated, these integrals, specifi-
cally thehifrjvXCjfs integrals, unlike the corresponding ones in Hartree–Fock
theory, cannot be solved analytically. The usual procedure is to approximate the
integral by summing the integrand in steps determined by agrid. For example,
suppose we want to integratee"x
2
from" 1 to 1. This could be done approxi-
mately, using a grid of widthDx¼0.2 and summing from"2 to 2 (limits at which
the function is small):

Z 1

" 1

e"x

2

dx¼

Z 1

" 1

fðxÞdx’ 0 : 2 fð" 2 þ 0 : 2 Þþ 0 : 2 fð" 2 þ 0 : 4 Þ

þ+++þ 0 : 2 fð 2 Þ¼ 0 : 2 ð 9 : 80 Þ¼ 1 : 96

The integral is actuallyp1/2¼1.77. For a functionf(x,y) the grid would define
the steps inxandyand actually look like a grid or net, approximating the integral as
a sum of the volumes of parallelepipeds, and for the DFT functionf(x,y,z) the grid
specifies the steps ofx,yandz. Clearly the finer the grid the more accurately the
integrals are approximated, and reasonable accuracy in DFT calculations requires
(but is not guaranteed by) a sufficiently fine grid.
Here is a summary of the steps in obtaining the Kohn–Sham orbitals and energy
levels:

1. Specify a geometry (and charge and multiplicity; electron spin can be handled
   in DFT by using separatea- andb-spin density functions).


2. Specify a basis set {f}.


3. Make an initial guess ofr(e.g. by superposing atomicrs).


4. Use the guess ofrto calculate an initial guessvxc(r) fromvxc(r)¼functional
   derivativedExc/dr(Eq.7.24). This uses the approximate functionalExcwe have
   chosen for the calculation.


5. Use the initial guesses ofrandvxc(r) to calculate the K-S operatorh^KS

"

1

2

r^2 i"

X

nuclei A

ZA

r1A

þ

Z

rðr 2 Þ

r 12

dr 2 þvXCð 1 Þ

(see Eq.7.23).

458 7 Density Functional Calculations