Page 268 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.6 Ground state of the helium atom 259

Variation method treatment
As a normalized trial function ö for the determination of the ground-state
energy by the variation method, we select the unperturbed eigenfunction
(0)
ø (r 1 , r 2 ) of the perturbation treatment, except that we replace the atomic
number Z by a parameter Z9

ö ˆ ö 1 ö 2
3=2
1 Z9
ö 1 ˆ e ÿ Z9r 1 =a 0 (9:88)
ð 1=2 a 0
3=2
1 Z9
ö 2 ˆ e ÿ Z9r 2 =a 0
ð 1=2 a 0
The parameter Z9 is an effective atomic number whose value is determined by
the minimization of E in equation (9.2). Since the hydrogen-like wave func-
tions ö 1 and ö 2 are normalized, we have
hö 1 jö 1 iˆhö 2 jö 2 iˆ 1 (9:89)
The quantity E is obtained by combining equations (9.2), (9.79), (9.88), and
(9.89) to give
* + * +
2 2 2 2
" "
2 Ze9 2 Ze9

E ˆ ö 1 ÿ = ÿ ö 1 ‡ ö 2 ÿ = ÿ ö 2

1
2

2m e r 1 2m e r 2
* +

2
e9
‡ ö 1 ö 2 ö 1 ö 2 (9:90)

r 12
Note that while the trial function ö ˆ ö 1 ö 2 depends on the parameter Z9, the
Hamiltonian operator contains the true atomic number Z. Therefore, we rewrite
equation (9.90) in the form
* + * +

" 2 2 2
2 Z9e9 (Z9 ÿ Z)e9

E ˆ ö 1 ÿ = ÿ ö 1 ‡ ö 1 ö 1
1
2m e r 1 r 1
* + * +

" 2 2 2
2 Z9e9 (Z9 ÿ Z)e9

‡ ö 2 ÿ = ÿ ö 2 ‡ ö 2 ö 2
2
2m e r 2 r 2
* +

2
e9
‡ ö 1 ö 2 ö 1 ö 2 (9:91)

r 12
The ®rst term on the right-hand side is just the energy of a hydrogen-like atom
2
2
with nuclear charge Z9, namely, ÿZ9 e9 =2a 0 . The third term has the same
value as the ®rst. The second term is evaluated as follows