noticable, that the numerical values of B α form the smooth functions depending on
the angle θ B , which can be expanded in series in terms of Legendre polynomials of
low order. The analytical expressions for the coefficients B α , obtained by fitting to
the numerical values deduced from the ab initio calculation, have the forms (in a.u.):
B x h B
ð Þ ¼ 0:2732 À 0:1601P
0
2 ðcos h B Þ þ 0:0333P
0
4 ðcos h B ÞÀ0:0227P
0
6 ðcos h B Þ;
ð3:2:10Þ
B y h B
ð Þ ¼ 0:00696 þ 0:01060P
0
2 ðcos h B Þ À 0:00718P
0
4 ðcos h B Þ
À 0:00330P
0
6 ðcos h B Þ;
ð3:2:11Þ
B z h B
ð Þ ¼ À0:1076P
1
2 ðcos h B Þ À 0:0307P
1
4 ðcos h B Þ À 0:0011P
1
6 ðcos h B ÞÁ ð3:2:12Þ
Here P
m
l ðcos h B Þ are the associated Legendre polynomials. It should be pointed
out, that at θ B = 0° the complex CH 4 –N 2 is in configuration 4 and at θ B = 90° in
0
30
60
90
120
150
180
0.1
0.2
0.3
0.4
θ Β (Deg)
Β
x (a.u.)
0
30
60
90
120
150
180
0.000
0.004
0.008
0.012
0.016
θ Β ,(Deg)
B
y (a.u.)
0
30
60
90
120
150
180
-0.24
-0.16
-0.08
0.00
0.08
0.16
0.24
θ Β (Deg)
Β
z (a.u.)
(a)
(b)
(c)
Fig. 3.9 Dependence of the coefficients B α on the angle θ B for the configurations of the CH 4 –N 2
complex with fixed χ A = 0°, θ A = 45°, φ A = (180/π) arcsin 1=
ffiffi ffi
3
p
À
Á
and φ B = 0° [13]. Points—
values of B α deduced from the ab initio calculation (see text); lines—analytical calculation by
Eqs. (3.2.10) and (3.2.11) (Reprinted with permission from Ref. [64]. Copyright 2010 American
Institute of Physics.)
3.2 Dipole Moment of van der Waals Complexes
35