It should be emphasized again, that the multipole polarizabilities and moments in
Eqs. (5.1.10) and (5.1.15) are written there in the laboratory system of coordinates
and depend on the mutual orientation of the interacting molecules. As a result, the
first dipole hyperpolarizability of two interacting molecules is a function (surface)
of several variables: Euler angles (rotation of the first and the second molecule), the
intermolecular separation R and the internal coordinates when the molecules are
considered as nonrigid ones.
5.2 First Hyperpolarizabilitiy of the CH 4 –N 2 van der
Waals Complex
The references given in the preamble of this chapter give exhaustive enough representation of hyperplarizabilities for small van der Waals complexes including the
atom-atomic and atom-diatomic ones. Here, we dwell in more detail on the CH 4 –N 2
van der Waals complex which help us to illustrate the calculation technique to be
applied to the first hyperpolarizability of a molecule or any molecular complex. The
CH 4 –N 2 complex is very important and interesting complex that exists in
methane-nitrogen atmosphere of Titan.
The induction term b
ind
abc in Eq. (5.1.11) for the interacting molecules CH 4 and
N 2 , if the terms up to *R
−6 are taken into account, is written as [49]
b
ind
abc ¼ Sða;b; cÞ
1
2
b
A
abq T qr a
B
rc þ
1
6
a
A
aq T qre B
B
b;c;re À
1
6
a
B
aq T qre B
A
b;c;re þ
1
18
c
A
abcq T qre H
B
re
þ
1
30
b
A
abq T qreg E
B
c;reg À
1
18
A
A
a;qr T qreg B
B
b;c;eg þ
1
90
c
B
abcq T qreg X
A
reg
þ
1
30
M
A
a;b;qre T qreg a
B
gc À
1
54
N
A
a;b;c;qr T qreg H
B
eg þ
1
210
a
A
aq T qregu G
B
b;c;regu
À
1
210
a
B
aq T qregu G
A
b;c;regu þ
1
90
E
A
a;qre T qregu B
B
b;c;gu À
1
90
E
B
a;qre T qregu B
A
b;c;gu
þ
1
630
c
A
abcq T qregu U
B
regu À
1
630
c
B
abcq T qregu U
A
regu þ
1
270
P
A
a;b;c;qre T qregu H
B
gu
þ
1
2
a
B
aq T qr b
A
rbe T eu a
B
uc þ
1
2
a
A
aq T qr a
B
rg T gu b
A
ubc
!
ð5:2:1Þ
where again in accordance with Sect. 2.2.1
T ab. . .m
|fflffl ffl{zfflffl ffl}
n
¼ T
A!B
ab...m ¼ ðÀ1Þ
n T
B!A
ab...m :
ð5:2:2Þ
As mentioned earlier, the superscripts A and B denote the molecules CH 4 and
N 2 , respectively. The dispersion contribution to the first hyperpolarizability for this
complex is calculated using Eq. (5.1.15).
5.1 Theoretical Treatment
87
Eqs. (5.1.10) and (5.1.15) are written there in the laboratory system of coordinates
and depend on the mutual orientation of the interacting molecules. As a result, the
first dipole hyperpolarizability of two interacting molecules is a function (surface)
of several variables: Euler angles (rotation of the first and the second molecule), the
intermolecular separation R and the internal coordinates when the molecules are
considered as nonrigid ones.
5.2 First Hyperpolarizabilitiy of the CH 4 –N 2 van der
Waals Complex
The references given in the preamble of this chapter give exhaustive enough representation of hyperplarizabilities for small van der Waals complexes including the
atom-atomic and atom-diatomic ones. Here, we dwell in more detail on the CH 4 –N 2
van der Waals complex which help us to illustrate the calculation technique to be
applied to the first hyperpolarizability of a molecule or any molecular complex. The
CH 4 –N 2 complex is very important and interesting complex that exists in
methane-nitrogen atmosphere of Titan.
The induction term b
ind
abc in Eq. (5.1.11) for the interacting molecules CH 4 and
N 2 , if the terms up to *R
−6 are taken into account, is written as [49]
b
ind
abc ¼ Sða;b; cÞ
1
2
b
A
abq T qr a
B
rc þ
1
6
a
A
aq T qre B
B
b;c;re À
1
6
a
B
aq T qre B
A
b;c;re þ
1
18
c
A
abcq T qre H
B
re
þ
1
30
b
A
abq T qreg E
B
c;reg À
1
18
A
A
a;qr T qreg B
B
b;c;eg þ
1
90
c
B
abcq T qreg X
A
reg
þ
1
30
M
A
a;b;qre T qreg a
B
gc À
1
54
N
A
a;b;c;qr T qreg H
B
eg þ
1
210
a
A
aq T qregu G
B
b;c;regu
À
1
210
a
B
aq T qregu G
A
b;c;regu þ
1
90
E
A
a;qre T qregu B
B
b;c;gu À
1
90
E
B
a;qre T qregu B
A
b;c;gu
þ
1
630
c
A
abcq T qregu U
B
regu À
1
630
c
B
abcq T qregu U
A
regu þ
1
270
P
A
a;b;c;qre T qregu H
B
gu
þ
1
2
a
B
aq T qr b
A
rbe T eu a
B
uc þ
1
2
a
A
aq T qr a
B
rg T gu b
A
ubc
!
ð5:2:1Þ
where again in accordance with Sect. 2.2.1
T ab. . .m
|fflffl ffl{zfflffl ffl}
n
¼ T
A!B
ab...m ¼ ðÀ1Þ
n T
B!A
ab...m :
ð5:2:2Þ
As mentioned earlier, the superscripts A and B denote the molecules CH 4 and
N 2 , respectively. The dispersion contribution to the first hyperpolarizability for this
complex is calculated using Eq. (5.1.15).
5.1 Theoretical Treatment
87