1.2 Intermolecular Interaction
7
a)
b1)
b2)
c)
Fig. 1.3 Examples of dipolar (a), quadrupolar (b1 and b2) and octopolar (c) distributions of charge
on a sphere. While b1 is axially symmetric similarly to the dipolar case (a), b2 has the twofold
symmetry
V 1 (R) =
1
4πε 0
R · p
|R| 3 .
(1.7)
A dipole moment is a vectorial quantity characterizing the polarization of the molecular charge distribution (for primarily neutral molecules). The electric field E associated with this potential is obtained in accordance with standard static electromagnetism as a gradient for R,
E 1 (R) = −∇ R V 1 (R)
= −
1
4πε 0
R · p∇ R
1
|R| 3 +
1
|R| 3 ∇ R R · p
=
1
4πε 0
1
|R| 5 [3(R · p)R − |R|
2 p].
(1.8)
To characterize the charge distribution of symmetrical molecules such as nitrogen
having the charge distribution essentially similar to b1 in Fig. 1.3, terms with n = 0
and n = 1 is insufficient because q = 0 and | p| = 0. The next term (n = 2) having
the form
V 2 (R) =
1
4πε 0
1
|R| 3
|r|
2 P 2 (cos θ Rr )ρ(r)dv
=
1
4πε 0
1
|R| 3
|r|
2
3 cos
2
θ Rr − 1
2
ρ(r)dv
=
1
4πε 0
1
|R| 5
t RQR
(1.9)