Computational Chemistry

5.2.3.6.3 Using the Roothaan–Hall Equations to do ab initio Calculations – the
Equations in terms of thec’s andf’s of the LCAO Expansion

The key process in the HF ab initio calculation of energies and wavefunctions is
calculation of the Fock matrix,i.e.ofthematrixelementsFrs(Section 5.2.3.6.2).
Equation (5.63)expressestheseintermsofthebasisfunctionsfand the
operatorsH^core,J^andK^,buttheJ^andK^operators (Eqs.5.28and5.31)are
themselves functions of the MO’scand therefore of thec’s and the basis
functionsf.ObviouslytheFrscan be written explicitly in terms of thec’s and
f’s; such a formulation enables the Fock matrix to be efficiently calculated from
the coefficients and the basis functions withoutexplicitlyevaluating the opera-
torsJ^andK^after each iteration. This formulation of the Fock matrix will now be
explained.
To see more clearly what is required, write Eq.5.63as

Frs¼ frð 1 ÞjH^

core

ð 1 Þjfsð 1 Þ

DE

þ

Xn

j¼ 1

2 frð 1 ÞjJ^jð 1 Þfsð 1 Þ

# frð 1 ÞjK^jð 1 Þjfsð 1 Þ

+,

ð 5 : 71 Þ

using the compact Dirac notation. The operatorH^core(1) involves only the Laplacian
differentiation operator, atomic numbers and electron coordinates, so we do not have
to consider substituting the Roothaan-hallc’s andf’s intoH^core. The operatorsJ^and
K^invoke two integrals which we now consider. The first integral, from Eq. (5.65), is

J^jð 1 Þfsð 1 Þ¼fsð 1 Þ

Z c$

jð^2 Þcjð^2 Þ

r 12

dv 2

Substituting forcj(2) the basis function expansion∑ctjf*t(2) and forcj(2) the
expansion∑cujfu(2) (cf. Eq.5.52):

J^jð 1 Þfsð 1 Þ¼fsð 1 Þ

Xm

t¼ 1

Xm

u¼ 1

c$tjcuj

Z

f$tð 2 Þfuð 2 Þ

r 12

dv 2

where the double sum arises because we multiply thec*sum by thecsum. To get
the desired expression forhfr(1)|J^(1)fs(1)i(usually writtenhfr(1)|J^(1)|fs(1)i) we
multiply this byf$r(1) and integrate with respect to the coordinates of electron 1,
getting:

frð 1 ÞjJ^jð 1 Þfsð 1 Þ

¼

Xm

t¼ 1

Xm

u¼ 1

c$tjcuj

ZZ

f$rð 1 Þfsð 1 Þf$tð 2 Þfuð 2 Þ

r 12

dv 1 dv 2

5.2 The Basic Principles of the ab initio Method 207