Chemistry Reference
In-Depth Information
In Equation 10.51, the first expression on the right-hand side is the
transition moment in Dirac form and the second in charge density
notation. These two integrals are evaluated in spherical coordinates in
the Appendix.
E
2
can be further minimized with respect to the nonlinear parameter c
entering both
. It is seen that best c is obtained as a solution of the
cubic equation (Magnasco, 1978)
m
and
«
7c
3
9c
2
þ
9c
5
¼
0
ð
10
:
53
Þ
which has the real root c
¼
0
7970.
We then have the cases exemplified in Table 10.1.
:
H
0
10.5.1.1 Eigenstate of
In this case we put c
¼
1
2 in expression (10.48). Evaluating the integrals
by means of Equations 10.51 and 10.52 gives, in atomic units, the results
of the first row of Table 10.1:
=
3
8
¼
0
96a
0
m ¼
0
:
7449ea
0
;
« ¼
:
375E
h
; a ¼
2
:
ð
10
:
54
Þ
is only 66%of the correct value 4.5. Including higher np
z
eigenstates with c
¼
1
This value of
a
=
n only improves this result a little. Including terms
up to n
¼
7 gives
660 being
reached for n
¼
30. This is only 81.3% of the exact value, the remaining
18.7% coming from the contribution of the continuous part of the
spectrum. These results show that the expansion in eigenstates of H
0
is
disappointingly poor, the correct value being obtained only through
difficult calculations.
a ¼
3
:
606, the asymptotic value of
a ¼
3
:
10.5.1.2 Single optimized pseudostate
The best c
¼
0
7970 is obtained in this case from the real root of
Equation 10.53. We then obtain the results of the second row of
Table 10.1:
:
m ¼
0
:
9684
;
« ¼
0
:
4191
; a ¼
4
:
48
ð
10
:
55
Þ
