configuration of the slow subsystem corresponds to a set of eigenvalues of the
excitation energies of the fast subsystem U k (Q).
The next step of the adiabatic approximation is that the motion of the slow
subsystem does not change the wave function of the fast one, i.e., the fast subsystem
follows the slow one without any inertia and always ‘tracks’ the change in Q.
In order to find out to which equation the eigenfunctions and eigenvalues of the
slow subsystem satisfy, we must substitute the wave function (3.2.2) into the
wave equation (3.2.1b):
b
T e ðrÞ þ b
T N ðQÞ þ b
V ðr; QÞ
h
i
U k ðr; QÞvðQÞ ¼ E Á U k ðr; QÞvðQÞ:
ð3:2:4Þ
Let us substitute U k ðQÞ Á U k ðr; QÞ Á vðQÞ into the left and right sides of (3.2.4)
and distribute the terms in the left and right sides with the same variables:
b
T e ðrÞ þ b
V ðr; QÞ
h
i
U k ðr; QÞvðQÞ þ b
T N ðQÞ þ U k ðQÞ
h
i
U k ðr; QÞvðQÞ
¼ U k ðQÞU k ðr; QÞvðQÞ þ E U k ðr; QÞvðQÞ
ð3:2:5Þ
The underlined parts of (3.2.5) is the wave equation for the fast subsystem
(3.2.3) (the term v(Q) is canceled here because the operators on the lhs do not act on
it, and it is a factor), the rest is the wave equation for the slow subsystem similar to
(3.2.3); (the term U(r,Q) here is also canceled for the same reason):
b
T N ðQÞ þ U k ðQÞ
h
i
vðQÞ ¼ E k vðQÞ
ð 3:2:6Þ
A comparison of (3.2.6) with the Schrödinger equation (3.2.3) shows that the
eigenvalues of the fast subsystem U k (Q) (the rhs of 3.2.3) are the potential energy
of the slow subsystem for the k-th state of the fast subsystem (lhs of 3.2.6).
To further understand the physical meaning of U k (Q) and E k , we represent
b
V ðr; QÞ as:
b
V ðr; QÞ ¼ b
V e ðr; QÞ þ b
V N ðQÞ
ð 3:2:7Þ
(Q is a parameter in c
V e ðr; QÞ!), and
E k ¼ E
el
i þ E
v
k ;
ð3:2:8Þ
then the wave (3.2.1b) can be reduced to the form:
b
T e ðrÞ þ b
T N ðQÞ þ b
V e ðr; QÞ þ b
V N ðQÞ
h
i
U k ðr; QÞvðQÞ ¼ E
el
i þ E
v
k
À
Á Á U k ðr; QÞvðQÞ
ð3:2:9Þ
3.2 Adiabatic Approximations. Potential Energy Curves and Surfaces
45