Computational Chemistry

which can be written more compactly as

frð 1 ÞjK^jð 1 Þfsð 1 Þ

¼

Xm

t¼ 1

Xm

u¼ 1

c$tjcujðrujtsÞð 5 : 76 Þ

where of course (cf. (5.73))

ðrujtsÞ¼

ZZ

f$rð 1 Þfuð 1 Þf$tð 2 Þfsð 2 Þ

r 12

dv 1 dv 2 ð 5 : 77 Þ

Substituting Eqs.5.72and5.76for frð 1 ÞjJ^ð 1 Þfsð 1 Þ
and frð 1 ÞjK^ð 1 Þfsð 1 Þ
into Eq.5.71forFrswe get

Frs¼ frð 1 ÞjH^

core

ð 1 Þjfsð 1 Þ

DE

þ

Xn

j¼ 1

2

Xm

t¼ 1

Xm

u¼ 1

c$tjcujðrsjtuÞ#

Xm

t¼ 1

Xm

u¼ 1

c$tjcujðrsjtuÞ

"

i.e.

Frs¼Hrscoreð 1 Þþ

Xm

t¼ 1

Xm

u¼ 1

Xn

j¼ 1

c$tjcuj½ 2 ðrsjtuÞ#ðrujtsފð 5 : 78 Þ

where the integral of the operatorH^coreover the basis functions has been written

Hcorers ð 1 Þ¼ frð 1 ÞjH^

core

ð 1 Þjfsð 1 Þ

DE

ð 5 : 79 Þ

withH^coredefined by Eq.5.64¼5.19.
Equation5.78, with its ancillary definitions Eqs.5.73,5.77and5.79, is what we
wanted: the Fock matrix elements in terms of the basis functionsfand their weighting
coefficientsc, for a closed-shell molecule;mis the number of basis functions. We can
use Eq.5.78to calculate MO’s and energy levels (Section 5.2.3.6.2). Given a basis set
and molecular geometry (the integrals depend on molecular geometry, as will be
illustrated) and starting with an initial guess at thec’s, one (or rather the computer
algorithm) calculates the matrix elementsFrs, assembles them into the Fock matrixF,
etc. (Section 5.2.3.6.2 and Fig.5.6) Let us now examine certain details connected with
Eq.5.78and this procedure.

5.2.3.6.4 Using the Roothaan–Hall Equations to do ab initio Calculations – Some
Details

Equation5.78is normally modified by subsuming thec’s intoPtu, the elements of
the density matrixP:

5.2 The Basic Principles of the ab initio Method 209