Page 269 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

260 Approximation methods
* +

2 3 1
(Z9 ÿ Z)e9 2 1 Z9 2
e
ö 1 ö 1 ˆ (Z9 ÿ Z)e9 r ÿ1 ÿ2 Z9r 1 =a 0 4ðr dr 1
1
1

r 1 ð a 0 0
Z9
ˆ (Z9 ÿ Z)e9 2
a 0
where equations (A.26) and (A.28) have been used. The fourth term equals the
second. The ®fth term is identical to E (1) of the perturbation treatment given by
equation (9.86) except that Z is replaced by Z9 and therefore this term equals
2
5Z9e9 =8a 0 . Thus, the quantity E in equation (9.91) is
2 2 2 2 2
Z9 e9 Z9(Z9 ÿ Z)e9 5Z9e9 e9
5
2
E ˆ 2 ÿ ‡ 2 ‡ ˆ [Z9 ÿ 2(Z ÿ )Z9]
16
2a 0 a 0 8a 0 a 0
(9:92)
We next minimize E with respect to the parameter Z9
dE e9 2
5
ˆ 2[Z9 ÿ (Z ÿ )] ˆ 0
dZ9 16 a 0
so that
5
Z9 ˆ Z ÿ
16
Substituting this result into equation (9.92) gives
e9 2
5 2
E ˆÿ(Z ÿ ) > E 0 (9:93)
16
a 0
as an upper bound for the ground-state energy E 0 .
When applied to the helium atom (Z ˆ 2), this upper bound is
2 2 2
27 e9 e9
E ˆÿ ˆÿ2:85 (9:94)
16 a 0 a 0
The numerical value of E is listed in Table 9.1. The simple variation function
(9.88) gives an upper bound to the energy with a 1.9% error in comparison
with the exact value. Thus, the variation theorem leads to a more accurate
result than the perturbation treatment. Moreover, a more complex trial function
with more parameters should be expected to give an even more accurate
estimate.

Problems
9.1 The Hamiltonian operator for a hydrogen atom in a uniform external electric
®eld E along the z-coordinate axis is
ÿ" 2 e9 2
^ 2
H ˆ = ÿ ÿ eEz
2ì r