Chemistry Reference
In-Depth Information
gives the third-order energy correction in the form
E
3
¼hc
0
j
H
1
E
1
jc
2
i
E
2
hc
0
jc
1
i
3
l
ð
10
:
14
Þ
It will now be shown that it is possible to shift the order from the
operator to the wavefunction, and vice versa,
3
if we make repeated use of
the RS perturbation equations given in (10.7) and take into account the
fact that the operators are Hermitian. In fact, we can write
E
3
¼hð
H
1
E
1
Þc
0
jc
2
i
E
2
hc
0
jc
1
i
ð
10
:
15
Þ
and, using the complex conjugate of the first-order equation (the bra):
E
3
¼ hð
H
0
E
0
Þc
1
jc
2
i
E
2
hc
0
jc
1
i
¼hc
1
j
H
0
E
0
jc
2
i
E
2
hc
0
jc
1
i
ð
10
:
16
Þ
In this equation, the order has been shifted from the operator to the
wavefunction. If we now make use of the second-order RS differential
equation, the last term above can be written
E
3
¼hc
1
j
H
1
E
1
jc
1
i
E
2
½hc
0
jc
1
iþhc
1
jc
0
i
ð
10
:
17
Þ
Since the term in square brackets is identically zero, the last of
Equations 10.9 is recovered. The same can be done for E
2
.
Finally, it must be emphasized that the leading term of the RS perturba-
tion equations (10.7), the zeroth-order equation
ð
H
0
E
0
Þc
0
¼
0, must be
satisfied exactly, otherwise uncontrollable errors will affect the whole
chain of equations. Furthermore, it must be observed that only energy in
first order gives an upper bound to the true energy of the ground state, so
that the energy in second order, E
(2)
, may be below the true value.
4
10.2 FIRST-ORDER THEORY
First-order RS theory is useful, for instance, in explaining the Zeeman
effect and the splitting of the multiplet structure in atoms, or in giving the
3
This is known as the Dalgarno interchange theorem.
4
This is particularly true when the correct value of E
2
is determined.
