Page 276 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

10.1 Nuclear structure and motion 267
= á ÷ø ˆ ø= á ÷ ‡ ÷= á ø

:
2
2
2
= ÷ø ˆ ø= ÷ ‡ ÷= ø ‡ 2= á ÷ = á ø
á
á
á
Therefore, we obtain
Ù
X 1
^
2
T Q [÷ ë (Q)ø ë (r, Q)] ˆÿ" 2 = [÷ ë (Q)ø ë (r, Q)]
á
2M á
áˆ1
^
^
ˆ ø ë (r, Q)T Q ÷ ë (Q) ‡ ÷ ë (Q)T Q ø ë (r, Q)
Ù
X 1
:
ÿ " 2 = á ÷ ë (Q) = á ø ë (r, Q) (10:14)
M á
áˆ1
Substitution of equation (10.14) into (10.13) yields
X
^ ^
[T Q ‡ å k (Q) ÿ E]÷ k (Q) ‡ (c kë ‡ Ë kë )÷ ë (Q) ˆ 0 (10:15)
ë
^
where the coef®cients c kë (Q) and the operators Ë kë are de®ned by

^

c kë (Q) ø (r, Q)T Q ø ë (r, Q)dr (10:16)
k
Ù
X 1
^
:

Ë kë ÿ" 2 ø (r, Q)= á ø ë (r, Q)dr = á (10:17)
M á k
áˆ1
and equation (10.7) has been used. Since we have assumed that the electronic
eigenfunctions ø k (r, Q) are known for all values of the parameters Q, the
^
coef®cients c kë (Q) and the operators Ë kë may be determined. The set of
coupled equations (10.15) for the functions ÷ k (Q) is exact.
^
The integral I contained in the operator Ë kk is

I ø (r, Q)= á ø k (r, Q)dr
k
For stationary states, the eigenfunctions ø k (r, Q) may be chosen to be real
functions, so that this integral can also be written as

1
2
I ˆ = á [ø k (r, Q)] dr
2
According to equation (10.7), the integral I vanishes and, therefore, we have
^
Ë kk ˆ 0.
We now write equation (10.15) as
X
^ ^
[T Q ‡ U k (Q) ÿ E]÷ k (Q) ‡ (c kë ‡ Ë kë )÷ ë (Q) ˆ 0 (10:18)
ë(6ˆk)
where U k (Q) is de®ned by