Chemistry Reference
In-Depth Information
which are respectively the eigenvalues and eigenvectors of matrix
H
and the best variational approximation to the eigenfunctions. The Ritz
method not only gives the best variational approximation to the ground-
state energy (the first eigenvalue
«
1
), but also approximations to the energy
of the excited states. A theorem due to MacDonald (1933) states further
that each of the ordered roots (4.47) gives an upper bound to the energy of
the respective excited state.
4.4 APPLICATIONS OF THE RITZ METHOD
An application to the first two excited states of theHe-like atomconcludes
this chapter.
4.4.1 The First 1s2s Excited State of the He-like Atom
We take as the orthonormal two-electron basis set the simple products of
one-electron functions
x
1
¼
1s
1
2s
2
;
x
2
¼
2s
1
1s
2
ð
4
:
50
Þ
which are assumed individually normalized and orthogonal.
6
The (2
2)
secular equation is
¼
0
H
11
«
H
12
ð
4
:
51
Þ
H
12
H
22
«
with the matrix elements
1
r
12
1s
1
2s
2
h
1
þ
h
2
þ
H
11
¼
1s
1
2s
2
ð
4
:
52
Þ
¼
h
1s1s
þ
h
2s2s
þð
1s
2
j
2s
2
Þ¼
E
0
þ
J
¼
H
22
1
r
12
1s
1
2s
2
h
1
þ
h
2
þ
H
12
¼
2s
1
1s
2
¼ð
1s 2s
j
1s 2s
Þ¼
K
ð
4
:
53
Þ
6
The appropriate 2s AO has the same form as
w
of Equation 4.14.
