Page 267 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

258 Approximation methods

Table 9.1. Ground-state energy of the helium
atom

Method Energy (eV) % error
Exact ÿ79.0
Perturbation theory:
E (0) ÿ108.8 ÿ37.7
E (0) ‡ E (1) ÿ74.8 ‡5.3
Variation theorem (E ) ÿ77.5 ‡1.9

3 3
1 Z 1 Z
(0) ÿ Zr 1 =a 0 ÿ Zr 2 =a 0 ÿ(r 1 ‡r 2 )=2
ø (r 1 , r 2 ) ˆ e e ˆ e (9:82)
ð a 0 ð a 0
where we have de®ned
2Zr i
r i , i ˆ 1, 2 (9:83)
a 0
The ®rst-order perturbation correction E (1) to the ground-state energy is
obtained by evaluating equation (9.24) with (9.80) as the perturbation and
(9.82) as the unperturbed eigenfunction
* + * +
2 2 2
e9 2Z e9 Ze9
E (1) ˆ ø (0) ø (0) ˆ ø (0) ø (0) ˆ I (9:84)
5 2
r 12 a 0 r 12 2 ð a 0
where r 12 ˆjr 2 ÿ r 1 j and where
ÿ(r 1 ‡r 2 )
e
2 2
I ˆ r r sin è 1 sin è 2 dr 1 dè 1 dj 1 dr 2 dè 2 dj 2 (9:85)
1 2
r 12
2
This six-fold integral I is evaluated in Appendix J and is equal to 20ð . Thus,
we have
5Ze9 2 5
E (1) ˆ ˆÿ E (0) (9:86)
8a 0 8Z
The ground-state energy of the perturbed system to ®rst order is, then
2
5Z e9
2
E ˆ E (0) ‡ E (1) ˆÿ Z ÿ (9:87)
8 a 0
Numerical values of E (0) and E (0) ‡ E (1) for the helium atom (Z ˆ 2) are
given in Table 9.1 along with the exact value. The unperturbed energy value
E (0) has a 37.7% error when compared with the exact value. This large
^
inaccuracy is expected because the perturbation H9 in equation (9.80) is not
small. When the ®rst-order perturbation correction is included, the calculated
energy has a 5.3% error, which is still large.