the efficiency of light scattering (index of the refraction n * α) including rotational
Raman scattering. Note, that the calculated values a
AB
ð1Þ and c
AB
ð1Þ are in a good
agreement with the values a
CH 4 þ a
N 2 = 28.16 a.u. and c
N 2 = 4.61 a.u. obtained by
G. Maroulis [99, 100].
It should be noted that at the formation of the complex the monomers CH 4 and
N 2 are deformed. As a result, the complex polarizability is changed due to the
nonrigidity of the monomers. However, these changes are not large. Indeed, the
calculations for the most stable configurations of the CH 4 –N 2 complex at full
optimization of their geometries have shown that accounting for the monomers
nonrigidity practically does not affect the mean polarizability a
AB
ðR e Þ and changes
slightly (several percent) the polarizability anisotropy c
AB
ðR e Þ.
Polarizability tensor invariants a
AB
ðRÞ and c
AB
ðRÞ for the family of the most
stable configurations of the CH 4 –N 2 complex can be also presented in the form of
Taylor series in the vicinity of R e = 6.84 a 0 :
a
AB
ðRÞ ¼ 27:940 þ 0:172ðR À 6:84Þ À 0:075ðR À 6:84Þ
2 þ 0:015ðR À 6:84Þ
3 ;
ð4:2:22Þ
c
AB
ðRÞ ¼ 3:421 À 0:018ðR À 6:84Þ þ 0:091ðR À 6:84Þ
2 À 0:020ðR À 6:84Þ
3 :
ð4:2:23Þ
These expressions allow estimating the derivatives of the polarizability tensor
invariants of the complex for the analysis of scattering processes in
methane-nitrogen gas media. Particularly, it is seen that the first derivatives
@a
AB
ðRÞ=@R ¼ 0:172 a.u. and @c
AB
ðRÞ=@R ¼ À0:018 a.u. are of different sign at R e
and considerably differ in their absolute value.
C. Polarizability of free orientated interacting molecules
The polarizability of the free oriented interacting molecules CH 4 and N 2 can be
easily calculated if the analytical expressions for polarizabilities a
AB
ab are averaged
over Euler angles v A , h A , / A , h B and / B . In this case, the polarizability tensor has
only two independent components:
a
AB
xx ¼ a
A þ a
B þ
4
R 3 a
A a
B þ
4
R 6 a
A a
B a
A þ a
B
À
Á þ
22a
A
45R 6 a
B
zz À a
B
xx
À
Á 2 þ
7C
AB
6
18R 6
c
A
a A þ
c
B
a B
ð4:2:24Þ
a
AB
yy ¼ a
AB
zz ¼ a
A
þ a
B
À
2
R 3 a
A
a
B
þ
1
R 6 a
A
a
B
a
A
þ a
B
À
Á
þ
19a
A
45R 6 a
B
zz À a
B
xx
À
Á 2 þ
2C
AB
6
9R 6
c
A
a A þ
c
B
a B
;
ð4:2:25Þ
72
4 Interaction-induced Polarizability
Raman scattering. Note, that the calculated values a
AB
ð1Þ and c
AB
ð1Þ are in a good
agreement with the values a
CH 4 þ a
N 2 = 28.16 a.u. and c
N 2 = 4.61 a.u. obtained by
G. Maroulis [99, 100].
It should be noted that at the formation of the complex the monomers CH 4 and
N 2 are deformed. As a result, the complex polarizability is changed due to the
nonrigidity of the monomers. However, these changes are not large. Indeed, the
calculations for the most stable configurations of the CH 4 –N 2 complex at full
optimization of their geometries have shown that accounting for the monomers
nonrigidity practically does not affect the mean polarizability a
AB
ðR e Þ and changes
slightly (several percent) the polarizability anisotropy c
AB
ðR e Þ.
Polarizability tensor invariants a
AB
ðRÞ and c
AB
ðRÞ for the family of the most
stable configurations of the CH 4 –N 2 complex can be also presented in the form of
Taylor series in the vicinity of R e = 6.84 a 0 :
a
AB
ðRÞ ¼ 27:940 þ 0:172ðR À 6:84Þ À 0:075ðR À 6:84Þ
2 þ 0:015ðR À 6:84Þ
3 ;
ð4:2:22Þ
c
AB
ðRÞ ¼ 3:421 À 0:018ðR À 6:84Þ þ 0:091ðR À 6:84Þ
2 À 0:020ðR À 6:84Þ
3 :
ð4:2:23Þ
These expressions allow estimating the derivatives of the polarizability tensor
invariants of the complex for the analysis of scattering processes in
methane-nitrogen gas media. Particularly, it is seen that the first derivatives
@a
AB
ðRÞ=@R ¼ 0:172 a.u. and @c
AB
ðRÞ=@R ¼ À0:018 a.u. are of different sign at R e
and considerably differ in their absolute value.
C. Polarizability of free orientated interacting molecules
The polarizability of the free oriented interacting molecules CH 4 and N 2 can be
easily calculated if the analytical expressions for polarizabilities a
AB
ab are averaged
over Euler angles v A , h A , / A , h B and / B . In this case, the polarizability tensor has
only two independent components:
a
AB
xx ¼ a
A þ a
B þ
4
R 3 a
A a
B þ
4
R 6 a
A a
B a
A þ a
B
À
Á þ
22a
A
45R 6 a
B
zz À a
B
xx
À
Á 2 þ
7C
AB
6
18R 6
c
A
a A þ
c
B
a B
ð4:2:24Þ
a
AB
yy ¼ a
AB
zz ¼ a
A
þ a
B
À
2
R 3 a
A
a
B
þ
1
R 6 a
A
a
B
a
A
þ a
B
À
Á
þ
19a
A
45R 6 a
B
zz À a
B
xx
À
Á 2 þ
2C
AB
6
9R 6
c
A
a A þ
c
B
a B
;
ð4:2:25Þ
72
4 Interaction-induced Polarizability