Antisymmetry ofMany-electron Wavefunctions - Methods of Molecular Quantum Mechanics - page 93

is the eigenvalue of the S z operator

S z Y ¼ M S Y

ð 6 : 34 Þ

As a simple example, let us consider nowthe two-electronwavefunction

Y describing the bond between atoms A and B arising from the double

occupancy of the (normalized) bonding MO fð r Þ :

1
2

x 2 Þ¼jjffjj ¼ fð r 1 Þfð r 2 Þ Yð x 1 ; p ½ að s 1 Þbð s 2 Þbð s 1 Það s 2 Þ ð 6 : 35 Þ

with

fð r Þ¼x A ð r Þ c A þx B ð r Þ c B ¼ x A ð r Þþ l x B ð r Þ

p ð 6 :

36 Þ

1 þ l

2

þ 2 lS

where x B are two basis AOs centred on A and B respectively,

normalized but not orthogonal to each other, and

x A and

l ¼ c B = c A is the polarity

parameter of the MO. The integral

ð dr

A ð r Þx B ð r Þ

S ¼hx A jx B i¼ x ð 6 : 37 Þ

describes the overlap between the AOs x A and x B and, therefore, is called

the overlap integral. 4
Using definition (6.15) omitting for brevity the suffix 1, useless at the
moment, the integration over dx 2 gives the one-electron distribution
function

r 1 ð x ; x Þ¼fð r Þf ð r Þ½að s Þa ð s Þþbð s Þb ð s Þ ð 6 : 38 Þ

so that

r 1 ð r r Þ¼r 1 ð r ; ; r Þ¼fð r Þf ð r Þ¼ R ð r ; r Þ ð 6 : 39 Þ

4

S depends on the internuclear distance R, being one at R ¼ 0 and zero at R ¼ ¥ .