Computational Chemistry

Equation4.74can be generalized tonbasis functions (cf. the matrix of Eq.4.62):

H 11 "ES 11 H 12 "ES 12 +++ H 1 n"ES 1 n

H 21 "ES 21 H 22 "ES 22 +++ H 2 n"ES 2 n

... ... +++ ...

Hn 1 "ESn 1 Hn 2 "ESn 2 +++ Hnn"ESnn

(^)
(^)

¼ 0 (4.73)

If we invoke the SHM simplification of orthogonality of theS integrals
(pp. 37–39), thenSii¼1 andSij¼0 and Eq.4.73becomes

H 11 "EH 12 +++ H 1 n

H 21 H 22 "E +++ H 2 n

... ... +++ ...

Hn 1 Hn 2 +++ Hnn"E

¼ 0 (4.74)

Substitutinga,band 0 for the appropriateH’s (Eqs.4.61a,b,c) we get

a"E b ... 0

ba"E ... 0

... ... ... ...

00 ... a"E

(^)
(^)
¼ 0 (4.75)
The diagonal terms will always bea"E, but the placement ofband 0 will
depend on whichi,jterms are adjacent and which are further-removed, which
depends on the numbering system chosen (see below). Since multiplying or divid-
ing a determinant by a number is equivalent to multiplying or dividing the elements
of one row or column by that number (Section 4.3.3), multiplying both sides of
Eq.4.75by 1/bntimes, i.e. by (1/b)ngives
ða"EÞ=b 1 ... 0
1 ða"EÞ=b ... 0
... ... ... ...
00 ... ða"EÞ=b
(^)
(^)
¼ 0 (4.76)
Finally, if we define (a"E)/b¼x, we get
x 1 ... 0
1 x ... 0
... ... ... ...
00 ... x

¼ 0 (4.77)

148 4 Introduction to Quantum Mechanics in Computational Chemistry