Computational Chemistry

ðPSÞ¼

ðPSÞ 11 ðPSÞ 12 ðPSÞ 13 (((ðPSÞ 1 m

ðPSÞ 21 ðPSÞ 22 ðPSÞ 23 (((ðPSÞ 2 m

... ... ((( ...

ðPSÞm 1 ðPSÞm 2 ðPSÞm 3 (((ðPSÞmm

0

B

B

B

@

1

C

C

C

A

ð 5 : 221 Þ

has elements

ðPSÞrs¼PrsSrs¼ 2

Xn

i¼ 1

cricsiSrs ð 5 : 222 Þ

Note that (PS) isnotthe matrixPSobtained by matrix multiplication ofPandS;
each element ofthatmatrix would result from series multiplication: a row ofP
times a column ofS(Section 4.3.3).
The diagonal elements of (PS) are

ðPSÞrr¼PrrSrr¼ 2

Xn

i¼ 1

c^2 ri ð 5 : 223 Þ

Compare this with Eq. (5.213): for a ground-state closed-shell molecule there are
two electrons in each occupied MO and Eq. (5.213) can be written

nr¼ 2

Xn

i¼ 1

c^2 ri ð 5 : 224 Þ

i.e.

nr¼ðPSÞrr ð 5 : 225 Þ

The off-diagonal elements of (PS) are given by Eq. (5.222),r 6 ¼s. Compare this
with Eq. (5.214): for a ground-state closed-shell molecule there are two electrons in
each occupied MO and Eq. (5.214) can be written

nr=s¼ 2

Xn

i¼ 1

ð 2 cricsiSrsÞð 5 : 226 Þ

i.e.

nr=s¼ 2 ðPSÞrs ð 5 : 227 Þ

Thus the matrix (PS) can be written

ðPSÞ¼

n 1 1 = 2 n 1 = 2 1 = 2 n 1 = 3 (((^1 = 2 n 1 =m

(^1) = 2 n 2 = 1 n 2 1 = 2 n 2 = 3 ((( (^1) = 2 n 2 =m
... ... ((( ...
(^1) = 2 nm= 1 1 = 2 nm= 2 1 = 2 nm= 3 ((( nm

0

B

B

B

@

1

C

C

C

A

ð 5 : 228 Þ

The matrix (PS) (or sometimes 2(PS)) is called apopulation matrix.

5.5 Applications of the Ab initio Method 349