Chemistry Reference
In-Depth Information
conjunction with c
0
¼
1
:
6875 for the 1s AO, yields the variational energy
bounds (E
0
¼
2
:
348 01E
h
, J
¼
0
:
240 29E
h
):
«
þ
¼
2
:
093 74E
h
«
¼
2
:
121 68E
h
ð
4
:
58
Þ
Either the excitation energies from the He ground state
D«
þ
¼
0
:
7539 E
h
D«
¼
0
:
7260 E
h
ð
4
:
59
Þ
or the splitting
2K
¼
0
:
ð
4
:
60
Þ
0279E
h
are seen to be in sufficiently good agreement with the experimental values
(Moore, 1971) of 0.7560, 0.7282, and 0.0292 respectively. It is worth
noting that hydrogenic AOs (with c
s
¼
1) would give a wrong order for
the energy levels (Magnasco, 2007).
4.4.2 The First 1s2p State of the He-like Atom
The same considerations can be made for the first 1s2p excited state.
Because of space degeneracy, in the absence of the electron interaction,
there are now six states having the same energy, and precisely
x
1
¼
1s2p
z
; x
2
¼
2p
z
1s
; x
3
¼
1s2p
x
; x
4
¼
2p
x
1s
; x
5
¼
1s2p
y
; x
6
¼
2p
y
1s
ð
4
:
61
Þ
where we have omitted for brevity the electron labels (electrons always
arranged in dictionary order). We omit the details of the calculation,
which follows strictly those already seen for the 1s2s state. We only say
that functions belonging to different (x,y,z) symmetries are orthogonal
andnotinteractingwitheachotherandwithrespecttoallSstates.The
(6
6) secular equation factorizes into three equivalent (2
2) blocks,
whose matrix elements are obtained from the previous ones simply by
replacing2sby2p,E
0
by E
0
0
,andJ, K by J
0
;
K
0
. The two roots are still
triply degenerate, so that the sixfold degeneracy occurring in the one-
electron approximation E
0
0
is not completely removed by consideration
of the electron repulsion. Complete removal of the residual degeneracy
