Computational Chemistry

simple H€uckel method (Section 4.3.3) and the extended H€uckel (Section 4.4.1)
method. Here, however, we have seen (in outline) how the equation may be
rigorously derived. Also, unlike the case in the H€uckel methods the Fock matrix
elements are rigorously defined theoretically: from Eqs.5.55

Frs¼

Z

frF^fsdv ð 5 : 61 ¼ 4 : 54 Þ

and Eq.5.36

F^¼H^coreð 1 Þþ

Xn

j¼ 1

ð 2 J^jð 1 Þ#K^jð 1 ÞÞð 5 : 62 ¼ 5 : 36 Þ

it follows that

Frs¼

Z

fr H^

core

ð 1 Þþ

Xn

j¼ 1

ð 2 J^jð 1 Þ#K^jð 1 ÞÞ

"

fsdv ð 5 : 63 Þ

where

H^coreð 1 Þ¼#^1

2

r^21 #

X

allm

Zm

rm 1

ð 5 : 64 ¼ 5 : 19 Þ

J^jð 1 Þ¼

Z

c$jð 2 Þ

1

r 12



cjð 2 Þdv 2 ð 5 : 65 ¼ 5 : 29 Þ

and

K^ið 1 Þcjð 1 Þ¼cið 1 Þ

Z

c$ið 2 Þ

1

r 12



cjð 2 Þdv 2 ð 5 : 66 ¼ 5 : 30 Þ

To use the Roothaan–Hall equations we want them in standard eigenvalue-like
form so that we can diagonalize the Fock matrixFof Eq.5.57to get the coefficients
cand the energy levelse, just as we did in connection with the extended H€uckel
method (Section 4.4.1). The procedure for diagonalizingFand extracting thec’s
ande’s and is exactly the same as that explained for the extended H€uckel method
(although here the cycle is iterative, i.e. repetitive, see below):

1. The overlap matrixSis calculated and used to calculate an orthogonalizing
   matrixS#1/2, as in Eq. 4.107:

S!D!S#^1 =^2 ð 5 : 67 Þ

2.S#1/2is used to convertFtoF^0 (cf. Eq. 4.104):

F^0 ¼S#^1 =^2 FS#^1 =^2 ð 5 : 68 Þ

204 5 Ab initio Calculations