Chemistry Reference
In-Depth Information
Table 11.1 N-term results for the dipole dispersion constant C
11
and C
6
London
dispersion coefficients for the H-H interaction
6
0
6
0
N
C
11
=
E
h
a
C
6
=
E
h
a
Accurate/%
1
1
6
92.3
2
1.080 357
6.482 1
99.7
3
1.083 067
6.498 4
99.99
4
1.083 167
6.499 00
99.999
5
1.083 170
6.499 02
100
In this case, we must know the dependence of the frequency-dependent
polarizabilities on the real frequency u, and the coupling occurs now via
the integration over the frequencies. When the necessary data are avail-
able, however, the London formula (11.30) is preferable because use of
the Casimir-Polder formula (11.32) presents some problems in the
accurate evaluation of the integral through numerical quadrature tech-
niques (Figari and Magnasco, 2003).
Using the London formula and some of the pseudospectra derived in
Chapter 10, we obtain for the leading term of the H-H interaction the
results collected in Table 11.1.
Table 11.1 shows that convergence is very rapid for the H-H interac-
tion. We give here the explicit calculation for N
¼
2:
2
5
1
2
5
þ
1
!
2
2
5
þ
!
2
1
8
25
6
1
8
2
6
1
2
25
6
2
6
C
11
ð
2-term
Þ¼
1
þ
5
5
25
2
2
4
36
5
þ
4
2
4
36
þ
5
5
2
5
36
5
7
¼
125
4
36
þ
1
2
36
þ
50
7
36
¼
1089
1008
¼
121
112
¼
1
¼
:
080 357
so that the two-term approximation gives the dispersion constant as the
ratio between two not divisible integers! However, this explicit calcula-
tion is no longer possible forN
>
2, wherewemust resort to the numerical
methods touched upon inChapter 10. Using a nonvariational technique in
momentum space, Koga and Matsumoto (1985) gave the three-term C
6
for H-H as the ratio of not divisible integers as
12 529
1928
¼
6
C
6
¼
:
498 443 983
...
