Page 275 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

266 Molecular structure

approximated by the energy E kí of equation (10.8) and the eigenfunction
Ø(Q, r) is approximated by the product
Ø(Q, r) ÷ kí (Q)ø k (Q, r) (10:9)
4
Perturbation terms in the Hamiltonian operator up to ë still lead to the
uncoupling of the nuclear and electronic motions, but change the form of the
electronic potential energy function in the equation for the nuclear motion.
Rather than present the details of the Born±Oppenheimer perturbation expan-
2
sion, we follow instead the equivalent, but more elegant procedure of M. Born
and K. Huang (1954).

Born±Huang treatment
Under the assumption that the Schrodinger equation (10.6) has been solved for
È
the complete set of orthonormal eigenfunctions ø k (r, Q), we may expand the
eigenfunction Ø(r, Q) of equation (10.5) in terms of ø k (r, Q)
X
Ø(r, Q) ˆ ÷ ë (Q)ø ë (r, Q) (10:10)
ë
where ÷ ë (Q) are the expansion coef®cients. Substitution of equation (10.10)
into (10.5) using (10.1) gives
X
^
^
(T Q ‡ V Q ‡ H e ÿ E)÷ ë (Q)ø ë (r, Q) ˆ 0 (10:11)
ë
where the operators have been placed inside the summation. Since the operator
^
H e commutes with the function ÷ ë (Q), we may substitute equation (10.6) into
(10.11) to obtain
X
^
[T Q ‡ å ë (Q) ÿ E]÷ ë (Q)ø ë (r, Q) ˆ 0 (10:12)
ë

We next multiply equation (10.12) by ø (r, Q) and integrate over the set of
k
electronic coordinates r, giving

X
^
ø (r, Q)T Q [÷ ë (Q)ø ë (r, Q)] dr ‡ [å k (Q) ÿ E]÷ k (Q) ˆ 0 (10:13)
k
ë
where we have used the orthonormal property (equation (10.7)). The operator
^
T Q acts on both functions in the product ÷ ë (Q)ø ë (r, Q) and involves the
second derivative with respect to the nuclear coordinates Q. To expand the
^
expression T Q [÷ ë (Q)ø ë (r, Q)], we note that
2 M. Born and K. Huang (1954) Dynamical Theory of Crystal Lattices (Oxford University Press, Oxford),
pp. 406±7.