Chemistry Reference
In-Depth Information
solved problem (either physical or model) and (iii) H
1
is the small first-
order difference between H and H
0
, called the perturbation.
We now expand both eigenvalue E and eigenfunction
c
into powers
of
l
:
2
3
E
¼
E
0
þ
lE
1
þ
l
E
2
þ
l
E
3
þ
ð
10
:
3
Þ
c ¼ c
0
þ
l
c
1
þ
ð
10
:
4
Þ
where the coefficients of the different powers of
are, respectively, the
corrections of the various orders to energy and wavefunction (e.g. E
2
is
the second-order energy correction,
l
c
1
the first-order correction to the
wavefunction, and so on). It is often useful to define corrections up to a
given order, which we write, for example, as
E
ð
3
Þ
¼
E
0
þ
E
1
þ
E
2
þ
E
3
ð
10
:
5
Þ
meaning that we add corrections up to the third order.
By substituting the expansions into the Schroedinger Equation 10.1:
½ð
H
0
E
0
Þþ
l
ð
H
1
E
1
Þ
l
2
3
2
E
2
l
E
3
ðc
0
þ
l
c
1
þ
l
c
2
þÞ¼
0
ð
10
:
6
Þ
and separating orders, we obtain
<
:
ð
H
0
E
0
Þc
0
¼
0
0
l
ð
H
0
E
0
Þc
1
þð
H
1
E
1
Þc
0
¼
0
l
ð
10
:
7
Þ
ð
H
0
E
0
Þc
2
þð
H
1
E
1
Þc
1
E
2
c
0
¼
0
2
l
which are known as RS perturbation equations of the various orders
specified by the power of
.
Because of the Hermitian property of H
0
, bracketing Equations 10.7 on
the left by
hc
0
j
, all the first terms in the RS equations are zero, and we are
left with
l
<
hc
0
j
H
0
E
0
jc
0
i¼
0
0
l
hc
0
j
H
1
E
1
jc
0
i¼
0
l
ð
10
:
8
Þ
hc
0
j
H
1
E
1
jc
1
i
E
2
hc
0
jc
0
i¼
0
:
2
l
