Chemistry Reference
In-Depth Information
This increasingly fine subdivision of the exact polarizability value into
different pseudostate contributions is of fundamental importance for the
increasingly refined evaluation of the London dispersion coefficients for
two H atoms interacting at long range, as we shall see in Chapter 11. The
calculation of the two-term pseudostate approximation is fully described
elsewhere (Magnasco, 2007) and will not be pursued further here. It can
only be said that, in general, the N-term approximation will involve
diagonalization of the (N
N) matrix M given by Equation 10.38, with
eigenvalues giving the excitation energies
«
i
and as eigenvectors the
; ...;
N. For a given atom
(or molecule), knowledge of the so-called N-term pseudospectrum
fa
i
; «
i
g
, i
¼
1
c
i
}, i
¼
1
;
correspondingN-termpseudostates {
2
; ...;
N, allows for the direct calculation of the dispersion
coefficients of the interacting atoms (or molecules).
Before ending this section, we notice that Equation 10.27 gives the exact
first-order solution of the RS dipole perturbation equation in un-normal-
ized form. From this result we can obtain the single normalized pseudos-
tate
;
2
c
equivalent to the exact
c
1
as
r
2
z
c
1
¼
CF
c; c ¼
Nz
þ
c
0
ð
10
:
58
Þ
Easy calculation shows that
s
s
8
43
2
43
9
2
18
43
¼
0
N
¼
; m ¼
¼
0
:
9705
; « ¼
:
4186
; a ¼
4
:
5
ð
10
:
59
Þ
so that the single optimized pseudostate (10.55) is seen to overestimate the
true
«
by only 5
10
4
E
h
and underestimate the true
m
0021ea
0
,
giving a dipole polarizability which differs from the exact one by just
0
by 0
:
02 a
0
. It must be admitted that the performance of this fully optimized
single function is exceptionally good, but the linearly optimized two-term
approximation does even better, giving the exact value for
:
a
(see
Table 10.1).
10.6 QUANTUM THEORY OF MAGNETIC
SUSCEPTIBILITIES
In this section we shall glance briefly at the use of RS perturbation
techniques in treating magnetic susceptibilities of atoms and molecules.
