The Hydrogen Bond - Models for Bonding in Chemistry - page 198

Since atom A (Rg) has no permanent electric moments, the expanded
electrostatic interaction energy and the polarization of HF by Rg are both
zero:

E ind
2

E 1 ð es Þ¼ ð Rg polarizes HF Þ¼ 0 ð 5 : 13 Þ

In this case, the possibility of forming a hydrogen bond comes from
second-order induction (polarization of the Rg atom by the permanent
electric moments of HF) and dispersion (the mutual interaction between
fluctuating induced moments on both partners).
As far as induction is concerned, the leading R 6 term describing
polarizationof aRg atomby the dipolarHFmolecule as given byEquation
(4.63) of Chapter 4:

2

a Rg ðm 1 HF Þ

3cos 2 uþ

1

E 2 ind ð 6 Þ¼ ð 5 : 14 Þ

R 6 2

180 Þ

ðu ¼

cannot discriminate between H-bonded and anti-H-bonded

0 Þ

ðu ¼

structures, giving in both cases:

2

2 a Rg

ðm 1 HF

Þ

E 2 ind

ð 6 Þ¼ ð 5 : 15 Þ

R 6

The next term in R 7 , implying further polarization of the Rg atom by
the mixed dipole-quadrupole moments of HF, contains a cos 3 u term:

3 a Rg m 1 HF m 2 HF
R 7

E 2 ind cos 3 u ð 7 Þ¼ ð 5 : 16 Þ

180 Þ which stabilizes the H-bonded structure Rg , so that we

can properly speak of formation of a H-bond between Rg and HF.
The same is true for dispersion, whose first two terms inCasimir-Polder
form (Magnasco, 2007, 2009a) are:

HF ðu ¼

R 6 ¥
0

1

E disp
2

6 a Rg

Þa HF

3cos 2 u

Þa Rg ÞDa HF ð Þ¼ ½ ð ð Þþð ð ð Þ

6 du iu iu 1 iu iu

2

p

ð 5 : 17 Þ

" #

p
3 ð

3

R 7 ¥
0

1

E disp
2

du a Rg cos 3 ua HF

210

2cos 2 uÞ cos ua HF

211

ð 7 Þ¼ ð iu Þ ð iu Þþ 3 ð iu Þ

2

p

ð 5 : 18 Þ