Page 266 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.6 Ground state of the helium atom 257
^
The operator H applies to He for Z ˆ 2, Li for Z ˆ 3, Be 2‡ for Z ˆ 4, and
‡
so forth.

Perturbation theory treatment
2
We regard the term e9 =r 12 in the Hamiltonian operator as a perturbation, so
that
e9 2
^
^
H9 ˆ H (1) ˆ (9:80)
r 12
In reality, this term is not small in comparison with the other terms so we
should not expect the perturbation technique to give accurate results. With this
choice for the perturbation, the Schrodinger equation for the unperturbed
È
Hamiltonian operator may be solved exactly.
The unperturbed Hamiltonian operator is the sum of two hydrogen-like
Hamiltonian operators, one for each electron
^
^
^
H (0) ˆ H (0) ‡ H (0)
1
2
where
" 2 Ze9 2
^
(0)
2
H 1 ˆÿ = ÿ
1
2m e r 1
" 2 Ze9 2
^
2
(0)
H 2 ˆÿ = ÿ
2
2m e r 2
If the unperturbed wave function ø (0) is written as the product
(0)
(0)
(0)
ø (r 1 , r 2 ) ˆ ø (r 1 )ø (r 2 )
1 2
and the unperturbed energy E (0) is written as the sum
E (0) ˆ E (0) ‡ E (0)
1 2
È
then the Schrodinger equation for the two-electron unperturbed system
(0)
(0)
(0)
^
H ø (r 1 , r 2 ) ˆ Eø (r 1 , r 2 )
separates into two independent equations,
(0)
(0)
(0)
H ø (0) ˆ E ø , i ˆ 1, 2
i i i i
which are identical except that one refers to electron 1 and the other to electron
2. The solutions are those of the hydrogen-like atom, as discussed in Chapter 6.
The ground-state energy for the unperturbed two-electron system is, then, twice
the ground-state energy of a hydrogen-like atom
2 2 2 2
Z e9 Z e9
E (0) ˆÿ2 ˆÿ (9:81)
2a 0 a 0
The ground-state wave function for the unperturbed two-electron system is the
product of two 1s hydrogen-like atomic orbitals