X
A
abc ¼ X
ð0ÞA
abc þ E
A
d;abc F
A
d þ Á Á Á ;
ð3:1:6Þ
2
4 -pole (hexadecapole) moment
U
A
abcd ¼ U
ð0ÞA
abcd þ D
A
e;abcd F
A
e þ Á Á Á ;
ð3:1:7Þ
and 2
5 -pole moment
N
A
abcde ¼ N
ð0ÞA
abcde þ H
A
u;abcde F
A
u þ Á Á Á :
ð3:1:8Þ
If the other highest multipole moments are needed the procedure to obtain their
expressions is evident. So, in order to obtain the induced dipole moment of the
molecule A, we expand l
ind;A
a
in (3.1.4) using equations for the field and field
gradients (Eq. 2.2.5). Then Eqs. (3.1.4)–(3.1.18) for the dipole, quadrupole and
other moments should be also employed. And assuming F
0
a ¼ 0 for the expressions
obtained, the induction dipole moment of the molecule A can be written in the
general case for neutral systems as (see also [5])
l
ind;A
a
¼ a
A
ab l
B
c T bc þ
1
3
a
A
ab H
B
cd T bcd À
1
3
A
A
a;bc l
B
d T bcd À
1
9
A
A
a;bc H
B
de T bcde
þ
1
15
a
A
ab X
B
cde T bcde þ
1
15
E
A
a;bcd l
B
e T bcde þ
1
105
a
A
ab U
B
cdeu T bcdeu À
1
105
D
A
a;bcde l
B
u T bcdeu
À
1
45
A
A
a;bc X
B
deu T bcdeu þ
1
45
E
A
a;bcd H
B
eu T bcdeu þ
1
945
a
A
ab N
B
cdeum T bcdeum À
1
315
A
A
a;bc U
B
deum T bcdeum
À
1
315
D
A
a;bcde H
B
um T bcdeum þ
1
225
E
A
a;bcd X
B
eum T bcdeum þ
1
945
H
A
a;bcdeu l
B
m T bcdeum þ
1
2
b
A
abd l
B
c l
B
e T bc T de
þ a
A
ab l
A
c a
B
de T bd T ce þ
2
3
a
A
ab l
A
c A
B
d;eu T bu T cde À
1
3
a
A
ab H
A
cd a
B
eu T be T ucd À
1
3
A
A
a;bc l
A
c a
B
eu T ce T ubd
þ
1
3
b
A
abd l
B
d H
B
eu T bd T ceu À
1
3
B
A
a;b;cd l
B
e l
B
u T be T cdu þ Á Á Á :
ð3:1:9Þ
For simplicity, hereinafter for final formulas the upper index (0) for all operators
is omitted. Accordingly, using Eq. (2.3.5) the accurate dispersion contribution
through the order R
−7 to the dipole moment is written as:
l
disp;AB
a
¼ À
@E
AB
disp
@F 0
a
F 0
a ¼0
¼ l
disp;A
a
þ l
disp;B
a
;
where
l
disp;A
a
¼
1
2p
T bd T ce
Z 1
0
dxb
A
abc ð0; ix; ÀixÞa
B
de ðixÞ
þ
1
3p
T be T cdu
Z 1
0
dxB
A
a;e;du ð0; ix; ÀixÞa
B
bc ðixÞ À
1
3p
T be T cdu
Z 1
0
dxb
A
abc ð0; ix; ÀixÞA
B
e;du ðixÞ:
ð3:1:10Þ
3.1 General Backgrounds
19
A
abc ¼ X
ð0ÞA
abc þ E
A
d;abc F
A
d þ Á Á Á ;
ð3:1:6Þ
2
4 -pole (hexadecapole) moment
U
A
abcd ¼ U
ð0ÞA
abcd þ D
A
e;abcd F
A
e þ Á Á Á ;
ð3:1:7Þ
and 2
5 -pole moment
N
A
abcde ¼ N
ð0ÞA
abcde þ H
A
u;abcde F
A
u þ Á Á Á :
ð3:1:8Þ
If the other highest multipole moments are needed the procedure to obtain their
expressions is evident. So, in order to obtain the induced dipole moment of the
molecule A, we expand l
ind;A
a
in (3.1.4) using equations for the field and field
gradients (Eq. 2.2.5). Then Eqs. (3.1.4)–(3.1.18) for the dipole, quadrupole and
other moments should be also employed. And assuming F
0
a ¼ 0 for the expressions
obtained, the induction dipole moment of the molecule A can be written in the
general case for neutral systems as (see also [5])
l
ind;A
a
¼ a
A
ab l
B
c T bc þ
1
3
a
A
ab H
B
cd T bcd À
1
3
A
A
a;bc l
B
d T bcd À
1
9
A
A
a;bc H
B
de T bcde
þ
1
15
a
A
ab X
B
cde T bcde þ
1
15
E
A
a;bcd l
B
e T bcde þ
1
105
a
A
ab U
B
cdeu T bcdeu À
1
105
D
A
a;bcde l
B
u T bcdeu
À
1
45
A
A
a;bc X
B
deu T bcdeu þ
1
45
E
A
a;bcd H
B
eu T bcdeu þ
1
945
a
A
ab N
B
cdeum T bcdeum À
1
315
A
A
a;bc U
B
deum T bcdeum
À
1
315
D
A
a;bcde H
B
um T bcdeum þ
1
225
E
A
a;bcd X
B
eum T bcdeum þ
1
945
H
A
a;bcdeu l
B
m T bcdeum þ
1
2
b
A
abd l
B
c l
B
e T bc T de
þ a
A
ab l
A
c a
B
de T bd T ce þ
2
3
a
A
ab l
A
c A
B
d;eu T bu T cde À
1
3
a
A
ab H
A
cd a
B
eu T be T ucd À
1
3
A
A
a;bc l
A
c a
B
eu T ce T ubd
þ
1
3
b
A
abd l
B
d H
B
eu T bd T ceu À
1
3
B
A
a;b;cd l
B
e l
B
u T be T cdu þ Á Á Á :
ð3:1:9Þ
For simplicity, hereinafter for final formulas the upper index (0) for all operators
is omitted. Accordingly, using Eq. (2.3.5) the accurate dispersion contribution
through the order R
−7 to the dipole moment is written as:
l
disp;AB
a
¼ À
@E
AB
disp
@F 0
a
F 0
a ¼0
¼ l
disp;A
a
þ l
disp;B
a
;
where
l
disp;A
a
¼
1
2p
T bd T ce
Z 1
0
dxb
A
abc ð0; ix; ÀixÞa
B
de ðixÞ
þ
1
3p
T be T cdu
Z 1
0
dxB
A
a;e;du ð0; ix; ÀixÞa
B
bc ðixÞ À
1
3p
T be T cdu
Z 1
0
dxb
A
abc ð0; ix; ÀixÞA
B
e;du ðixÞ:
ð3:1:10Þ
3.1 General Backgrounds
19