Atomic and Molecular Interactions - Methods of Molecular Quantum Mechanics - page 241

If all orientations were equally probable, then the average potential
energy h V i , and hence the average first-order electrostatic h C 3 i coefficient
(Magnasco et al., 1988; Magnasco, 2007), would be zero. In fact:

R 3 Ð W d

W F ðWÞ
Ð W

h V i W ¼ m A m B ¼ 0 ð 11 : 66 Þ

d W

since

ð ð ð p

A ð p
0

ð

dx A ð

2 1 1

p

d W ¼ d w d u

A sin

u d u

B sin

u ¼ 2 dx B ¼ 8

p p B

W 0 0 1 1

ð 11 : 67 Þ

where

x A ¼ cos

u ;

x B ¼ cos

u ð 11 :

68 Þ A B

2 3 0 1

2 2

ð ð ð

2 ð

ð

2 p 1 2 p 1

4

1 = 2

5 @ A

dx ð 1 x 2

d W F ðWÞ¼ d w cos w Þ d w dxx ¼ 0

W 0 1 0 1

ð 11 : 69 Þ

The vanishing of the average potential energy for free orientations is
true for all multipoles (dipoles, quadrupoles, octupoles, hexadecapoles,
etc.).
The Boltzmann probability for a dipole arrangement whose potential
energy is V is instead proportional to

W / exp ð V = kT Þ ð 11 : 70 Þ

We now average the quantity Vexp ð V = kT Þ over all possible orienta-
tions W assumed by the dipoles:

R 3 Ð W

h Vexp ð V = kT Þi ¼ m A m B

W F ðWÞ exp ½ aF ðWÞ
Ð W

d

ð 11 : 71 Þ

d W exp ½ aF ðWÞ