configuration 3, and the B α values in Eqs. (3.2.10) and (3.2.11) for these configurations (B x ð0
Þ = 0.1237 a.u., B x ð90
Þ = 0.3728 a.u. and B y ð0
Þ = 0.00708 a.u.)
are in a good agreement with the values in Table 3.4. It is obvious, that the
functions B α in Eqs. (3.2.10) and (3.2.11) are periodic ones with the period of 180°
due to the homonuclearity of the N 2 molecule.
The dependence of the coefficients B α on the angle u
0
A u
0
A ¼ u A À 90
À
Á
for
fixed χ A = 0°, θ A = 45°, θ B = 0° and φ B = 0° is shown at Fig. 3.10.
The angle u
0
A is introduced to show the symmetry properties of the complex (and
the coefficients B α , respectively) more cleary when the the molecule CH 4 is rotated
by the angle φ A . For the considered configurations of the CH 4 –N 2 complex the
coefficient B α has only x and y components and has the properties: B x ð180
À
u
0
A Þ ¼ B x ð180
þ u
0
A Þ, B y ð180
À u
0
A Þ ¼ ÀB y ð180
þ u
0
A Þ. The analytical expressions for these coefficients have the following form (in a.u.)
B x u
0
A
À Á ¼ 0:6508 À 0:6558P
0
1 ðcos u
0
A Þ À 0:0515P
0
2 ðcos u
0
A Þ þ 0:6386P
0
3 ðcos u
0
A Þ
À 0:1913P
0
4 ðcos u
0
A Þ À À0:0655P
0
5 ðcos u
0
A Þ À 0:0778P
0
6 ðcos u
0
A Þ
ð3:2:13Þ
B y u
0
A
À Á ¼ À0:01062P
1
1 ðcos u
0
A Þ À 0:03098P
1
2 ðcos u
0
A Þ þ 0:01854P
1
3 ðcos u
0
A Þ
À 0:02633P
1
4 ðcos u
0
A Þ À 0:00127P
1
5 ðcos u
0
A Þ À 0:00485P
1
6 ðcos u
0
A Þ
þ 0:00185P
1
7 ðcos u
0
A Þ
:
ð3:2:14Þ
The coefficients B α in Eqs. (3.2.13) and (3.2.14) are the periodic functions of φ A
with the period of 360°. The coefficients B a u
0
A
À Á
(and B a u A
ð Þ, correspondently) are
also the smooth functions. So, the considered particular cases of molecular rotations
0
60
120
180
240
300
360
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Β
x
(a.u.)
ϕ' Α (Deg)
0
60
120
180
240
300
360
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Β
y
(a.u.)
ϕ' Α (Deg)
(a)
(b)
Fig. 3.10 Dependence of the coefficients B a on the angle u
0
A for fixed v A = 0°, h A = 45°, h B = 0°
and u B = 0° for the methane—dinitrogen complex [13]. (Reprinted with permission from Ref.
[64]. Copyright 2010 American Institute of Physics)
36
3 Interaction-induced Dipole Moment
Þ = 0.1237 a.u., B x ð90
Þ = 0.3728 a.u. and B y ð0
Þ = 0.00708 a.u.)
are in a good agreement with the values in Table 3.4. It is obvious, that the
functions B α in Eqs. (3.2.10) and (3.2.11) are periodic ones with the period of 180°
due to the homonuclearity of the N 2 molecule.
The dependence of the coefficients B α on the angle u
0
A u
0
A ¼ u A À 90
À
Á
for
fixed χ A = 0°, θ A = 45°, θ B = 0° and φ B = 0° is shown at Fig. 3.10.
The angle u
0
A is introduced to show the symmetry properties of the complex (and
the coefficients B α , respectively) more cleary when the the molecule CH 4 is rotated
by the angle φ A . For the considered configurations of the CH 4 –N 2 complex the
coefficient B α has only x and y components and has the properties: B x ð180
À
u
0
A Þ ¼ B x ð180
þ u
0
A Þ, B y ð180
À u
0
A Þ ¼ ÀB y ð180
þ u
0
A Þ. The analytical expressions for these coefficients have the following form (in a.u.)
B x u
0
A
À Á ¼ 0:6508 À 0:6558P
0
1 ðcos u
0
A Þ À 0:0515P
0
2 ðcos u
0
A Þ þ 0:6386P
0
3 ðcos u
0
A Þ
À 0:1913P
0
4 ðcos u
0
A Þ À À0:0655P
0
5 ðcos u
0
A Þ À 0:0778P
0
6 ðcos u
0
A Þ
ð3:2:13Þ
B y u
0
A
À Á ¼ À0:01062P
1
1 ðcos u
0
A Þ À 0:03098P
1
2 ðcos u
0
A Þ þ 0:01854P
1
3 ðcos u
0
A Þ
À 0:02633P
1
4 ðcos u
0
A Þ À 0:00127P
1
5 ðcos u
0
A Þ À 0:00485P
1
6 ðcos u
0
A Þ
þ 0:00185P
1
7 ðcos u
0
A Þ
:
ð3:2:14Þ
The coefficients B α in Eqs. (3.2.13) and (3.2.14) are the periodic functions of φ A
with the period of 360°. The coefficients B a u
0
A
À Á
(and B a u A
ð Þ, correspondently) are
also the smooth functions. So, the considered particular cases of molecular rotations
0
60
120
180
240
300
360
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Β
x
(a.u.)
ϕ' Α (Deg)
0
60
120
180
240
300
360
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Β
y
(a.u.)
ϕ' Α (Deg)
(a)
(b)
Fig. 3.10 Dependence of the coefficients B a on the angle u
0
A for fixed v A = 0°, h A = 45°, h B = 0°
and u B = 0° for the methane—dinitrogen complex [13]. (Reprinted with permission from Ref.
[64]. Copyright 2010 American Institute of Physics)
36
3 Interaction-induced Dipole Moment