where c
A and c
B are the mean hyperpolarizabilities of the methane and nitrogen
molecules that have the form
c
A
¼
3
5
c
A
zzzz þ 2c
A
zxzx
À
Á ;
ð4:2:26Þ
c
B
¼
1
15
3c
B
zzzz þ 12c
B
zxzx þ 8c
B
xxxx
À
Á :
ð4:2:27Þ
Note that when a
B
xx ¼ a
B
zz , Eqs. (4.2.24) and (4.2.25) become the well-known
expressions for the polarizabilities of two interacting spherically symmetric atoms
A and B [43, 44].
It is also seen from (4.2.24) and (4.2.25) that the dependence of a
AB
ii and the
anisotropy of the polarizability tensor a
AB
xx À a
AB
yy on R is defined mainly by the
terms *R
−3 , while the leading terms of the mean polarizability a
AB
¼
a
AB
xx þ 2 a
AB
yy
=3 are proportional to R
−6 :
a
AB
À a
A
À a
B
¼
2
R 6 a
A
a
B
a
A
þ a
B
À
Á þ
4a
A
9R 6 a
B
zz À a
B
xx
À
Á 2 þ
5C
AB
6
18R 6
c
A
a A þ
c
B
a B
:
ð4:2:28Þ
The results of calculations carried out in work [97] have also shown that the
induction and dispersion terms (*R
−6 ) of Eq. (4.2.28) give the leading contribution
(*99 %). The last part of contribution caused by the anisotropy of the polarizability tensor of the N 2 molecule is very small (*1 %).
D. Vibrational Raman spectrum
Some features of vibrational Raman spectrum for the most stable configuration 4
(see Fig. 3.6) of the CH 4 –N 2 complex are discussed in this section following to
[101]. This spectrum was obtained using the Gaussian 03 package at the
MP2/aug-cc-pVTZ level of theory. The automatic methods in Gaussian were used
to compute the Raman activities (R A ) for 90° scattering
R A ¼ ð45ða
0
Þ
2
k þ 7ðc
0
Þ
2
k Þ=45
ð4:2:29Þ
where ða
0
Þ
2
k is a square magnitude of derivative of the isotropic part of the polarizability tensor with respect to the k-th normal mode, and ðc
0
Þ
2
k is a square magnitude of derivative of the anisotropic part of the polarizability tensor with respect
to the k-th normal mode and the derivatives are evaluated at zero displacement. It
should be noted that R A is proportional to the Raman intensity I. The calculations of
R A [101] show that for all vibrational modes of the CH 4 –N 2 complex the values of
4.2 Polarizabilities of van der Waals Complexes
73
A and c
B are the mean hyperpolarizabilities of the methane and nitrogen
molecules that have the form
c
A
¼
3
5
c
A
zzzz þ 2c
A
zxzx
À
Á ;
ð4:2:26Þ
c
B
¼
1
15
3c
B
zzzz þ 12c
B
zxzx þ 8c
B
xxxx
À
Á :
ð4:2:27Þ
Note that when a
B
xx ¼ a
B
zz , Eqs. (4.2.24) and (4.2.25) become the well-known
expressions for the polarizabilities of two interacting spherically symmetric atoms
A and B [43, 44].
It is also seen from (4.2.24) and (4.2.25) that the dependence of a
AB
ii and the
anisotropy of the polarizability tensor a
AB
xx À a
AB
yy on R is defined mainly by the
terms *R
−3 , while the leading terms of the mean polarizability a
AB
¼
a
AB
xx þ 2 a
AB
yy
=3 are proportional to R
−6 :
a
AB
À a
A
À a
B
¼
2
R 6 a
A
a
B
a
A
þ a
B
À
Á þ
4a
A
9R 6 a
B
zz À a
B
xx
À
Á 2 þ
5C
AB
6
18R 6
c
A
a A þ
c
B
a B
:
ð4:2:28Þ
The results of calculations carried out in work [97] have also shown that the
induction and dispersion terms (*R
−6 ) of Eq. (4.2.28) give the leading contribution
(*99 %). The last part of contribution caused by the anisotropy of the polarizability tensor of the N 2 molecule is very small (*1 %).
D. Vibrational Raman spectrum
Some features of vibrational Raman spectrum for the most stable configuration 4
(see Fig. 3.6) of the CH 4 –N 2 complex are discussed in this section following to
[101]. This spectrum was obtained using the Gaussian 03 package at the
MP2/aug-cc-pVTZ level of theory. The automatic methods in Gaussian were used
to compute the Raman activities (R A ) for 90° scattering
R A ¼ ð45ða
0
Þ
2
k þ 7ðc
0
Þ
2
k Þ=45
ð4:2:29Þ
where ða
0
Þ
2
k is a square magnitude of derivative of the isotropic part of the polarizability tensor with respect to the k-th normal mode, and ðc
0
Þ
2
k is a square magnitude of derivative of the anisotropic part of the polarizability tensor with respect
to the k-th normal mode and the derivatives are evaluated at zero displacement. It
should be noted that R A is proportional to the Raman intensity I. The calculations of
R A [101] show that for all vibrational modes of the CH 4 –N 2 complex the values of
4.2 Polarizabilities of van der Waals Complexes
73