a
AB
ab ¼ a
A
ab þ a
B
ab þ Da
AB
ab ;
ð4:1:3Þ
where a
A
ab and a
B
ab are the dipole polarizabilities of the atomic (or molecular)
systems A and B, and Da
AB
ab is the interaction polarizability. Here, the interaction
polarizability Da
AB
ab can be written as follows
Da
AB
ab ¼ a
ind
ab þ a
disp
ab þ a
exch
ab ;
ð4:1:4Þ
where a
ind
ab ¼ a
ind;A
ab þ a
ind;B
ab , a
disp
ab ¼ a
disp;A
ab
þ a
disp;B
ab
and a
exch
ab
are the induction,
dispersion and exchange contributions to the polarizability of interacting systems
A and B. Evidently, when the interacting systems are well separated, the interaction
polarizability Da
AB
ab is determined entirely by a well-known induction and dispersion
contributions a
ind
ab and a
disp
ab [43–45, 48, 56–58]. At shorter range, when the charge
distributions of interacting systems overlap, the exchange effects appear and the
additional contribution a
exch
ab begins to be important [59].
The induction contribution to the polarizability a
ind;A
ab
of the A molecule is
determined in accordance with (2.4.2) at F
0
a ¼ 0. Then, using analogous scheme
like for calculations of the induction dipole moment, the following expression can
be obtained for the induction polarizability of the molecule A:
a
ind;A
ab
¼ a
A
ac T cd a
B
db þ b
A
abc T cd l
B
d þ
1
3
a
A
ac T cde A
B
b;de À
1
3
A
A
a;cd T cde a
B
eb þ
1
3
b
A
abc T cde H
B
de
À
1
3
B
A
ab;cd T cde l
B
e þ
1
15
a
A
ac T cdeu E
B
b;deu þ
1
15
E
A
a;cde T cdeu a
B
ub À
1
9
A
A
a;cd T cdeu A
B
b;eu
þ
1
15
b
A
abc T cdeu X
B
deu À
1
9
B
A
ab;cd T cdeu H
B
eu þ
1
15
M
A
ab;cde T
A
cdeu l
B
u þ
1
105
a
A
ac T cdeum D
B
b;deum
À
1
105
D
A
a;cdeu T cdeum a
B
mb À
1
45
A
A
a;cd T cdeum E
A
b;eum þ
1
45
E
A
a;cde T cdeum A
B
b;um þ
1
45
M
A
ab;cde T
A
cdeum H
B
um
À
1
105
G
A
ab;cdeu T cdeum l
B
m þ
1
105
b
A
abc T cdeum U
B
deum À
1
45
B
A
ab;cd T cdeum X
B
eum þ
1
315
A
A
a;bc U
B
deum T bcdeum
þ a
A
ac T cd a
B
de T eu a
A
ub þ a
A
ac T cd b
B
dbe T eu l
A
u þ b
A
acd T ce l
B
e T du a
A
ub þ b
A
acd T cd a
B
de T eu l
A
u þ Á Á Á :
ð4:1:5Þ
Here T T
AB .
The dispersion contribution to the polarizability of the interacting molecules can
be found by calculation of the second derivative of the interaction energy
a
disp
ab ¼ À
@E
AB
disp
@F 0
a F
0
b
F 0 ¼0
¼ a
disp;A
ab
þ a
disp;B
ab
ð4:1:6Þ
where, restricting ourselves to the leading term *R
−6 , the expression for the dispersion contribution takes the form (see (2.3.5), [58])
54
4 Interaction-induced Polarizability
AB
ab ¼ a
A
ab þ a
B
ab þ Da
AB
ab ;
ð4:1:3Þ
where a
A
ab and a
B
ab are the dipole polarizabilities of the atomic (or molecular)
systems A and B, and Da
AB
ab is the interaction polarizability. Here, the interaction
polarizability Da
AB
ab can be written as follows
Da
AB
ab ¼ a
ind
ab þ a
disp
ab þ a
exch
ab ;
ð4:1:4Þ
where a
ind
ab ¼ a
ind;A
ab þ a
ind;B
ab , a
disp
ab ¼ a
disp;A
ab
þ a
disp;B
ab
and a
exch
ab
are the induction,
dispersion and exchange contributions to the polarizability of interacting systems
A and B. Evidently, when the interacting systems are well separated, the interaction
polarizability Da
AB
ab is determined entirely by a well-known induction and dispersion
contributions a
ind
ab and a
disp
ab [43–45, 48, 56–58]. At shorter range, when the charge
distributions of interacting systems overlap, the exchange effects appear and the
additional contribution a
exch
ab begins to be important [59].
The induction contribution to the polarizability a
ind;A
ab
of the A molecule is
determined in accordance with (2.4.2) at F
0
a ¼ 0. Then, using analogous scheme
like for calculations of the induction dipole moment, the following expression can
be obtained for the induction polarizability of the molecule A:
a
ind;A
ab
¼ a
A
ac T cd a
B
db þ b
A
abc T cd l
B
d þ
1
3
a
A
ac T cde A
B
b;de À
1
3
A
A
a;cd T cde a
B
eb þ
1
3
b
A
abc T cde H
B
de
À
1
3
B
A
ab;cd T cde l
B
e þ
1
15
a
A
ac T cdeu E
B
b;deu þ
1
15
E
A
a;cde T cdeu a
B
ub À
1
9
A
A
a;cd T cdeu A
B
b;eu
þ
1
15
b
A
abc T cdeu X
B
deu À
1
9
B
A
ab;cd T cdeu H
B
eu þ
1
15
M
A
ab;cde T
A
cdeu l
B
u þ
1
105
a
A
ac T cdeum D
B
b;deum
À
1
105
D
A
a;cdeu T cdeum a
B
mb À
1
45
A
A
a;cd T cdeum E
A
b;eum þ
1
45
E
A
a;cde T cdeum A
B
b;um þ
1
45
M
A
ab;cde T
A
cdeum H
B
um
À
1
105
G
A
ab;cdeu T cdeum l
B
m þ
1
105
b
A
abc T cdeum U
B
deum À
1
45
B
A
ab;cd T cdeum X
B
eum þ
1
315
A
A
a;bc U
B
deum T bcdeum
þ a
A
ac T cd a
B
de T eu a
A
ub þ a
A
ac T cd b
B
dbe T eu l
A
u þ b
A
acd T ce l
B
e T du a
A
ub þ b
A
acd T cd a
B
de T eu l
A
u þ Á Á Á :
ð4:1:5Þ
Here T T
AB .
The dispersion contribution to the polarizability of the interacting molecules can
be found by calculation of the second derivative of the interaction energy
a
disp
ab ¼ À
@E
AB
disp
@F 0
a F
0
b
F 0 ¼0
¼ a
disp;A
ab
þ a
disp;B
ab
ð4:1:6Þ
where, restricting ourselves to the leading term *R
−6 , the expression for the dispersion contribution takes the form (see (2.3.5), [58])
54
4 Interaction-induced Polarizability