Page 265 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

256 Approximation methods
^
ÿc á1 E (1) ‡ c á2 H (1) ˆ 0
á
12
^
c á1 H (1) ÿ c á2 E (1) ˆ 0
á
12
(9:77)
ÿc á3 E (1) ˆ 0
á
ÿc á4 E (1) ˆ 0
á
(0)
To ®nd the `correct' set of unperturbed eigenfunctions ö , we substitute ®rst
á
E (1) ˆÿ3eE a 0 , then successively E (1) ˆ 3eE a 0 , 0, 0 into the set of equations
2 2
(9.77). The results are as follows
(1) ^ (1)
for E 2 ˆ H 12 ˆÿ3eE a 0 : c 1 ˆ c 2 ; c 3 ˆ c 4 ˆ 0
^
for E (1) ˆÿH (1) ˆ 3eE a 0 : c 1 ˆÿc 2 ; c 3 ˆ c 4 ˆ 0
2 12
(1)
for E ˆ 0: c 1 ˆ c 2 ˆ 0; c 3 and c 4 undetermined
2
Thus, the `correct' unperturbed eigenfunctions are
ö (0) ˆ 2 ÿ1=2 (j2si‡j2p 0 i)
1
ö (0) ˆ 2 ÿ1=2 (j2siÿj2p 0 i)
2
(9:78)
ö (0) ˆj2p 1 i
3
ö (0) ˆj2p ÿ1 i
4
The factor 2 ÿ1=2 is needed to normalize the `correct' eigenfunctions.

9.6 Ground state of the helium atom
In this section we examine the ground-state energy of the helium atom by
means of both perturbation theory and the variation method. We may then
compare the accuracy of the two procedures.
The potential energy V for a system consisting of two electrons, each with
mass m e and charge ÿe, and a nucleus with atomic number Z and charge ‡Ze
is
Ze9 2 Ze9 2 e9 2
V ˆÿ ÿ ‡
r 1 r 2 r 12
where r 1 and r 2 are the distances of electrons 1 and 2 from the nucleus, r 12 is
the distance between the two electrons, and e9 ˆ e for CGS units or
e9 ˆ e=(4ðå 0 ) 1=2 for SI units. If we assume that the nucleus is ®xed in space,
then the Hamiltonian operator for the two electrons is
" 2 Ze9 2 Ze9 2 e9 2
^ 2 2
H ˆÿ (= ‡ = ) ÿ ÿ ‡ (9:79)
1 2
2m e r 1 r 2 r 12