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