equivalent to the vibration of a mas l relative to infinitely heavy “wall” (Fig. 2.3a)
provided
l ¼
m 1 m 2
m 1 þ m 2
:
ð2:7Þ
l is called a reduced mass of masses m 1 and m 2 . Figure 2.3b is just a model for
the vibrating diatomic molecule—two heavy masses (atoms) connected via
weightless spring vibrate in a potential shown in Fig. 2.1a, which, for small displacements from the equilibrium position, can be approximated by a parabola
(n = 2). In addition, comparing Eqs. (2.6) and (2.1) it can be concluded that the FC
is just a second energy derivative with respect to displacement calculated at equilibrium, i.e., f ¼ U
2
ð Þ r e
ð Þ.
Solution of a Schrödinger equation with Hamiltonian
^
H ¼ À
h
2
2l
d
2
dr 2 þ
1
2
f r À r e
ð
Þ
2
ð2:8Þ
gives a well-known formula for the vibrational energy levels of harmonic oscillator,
namely
E t ¼ hm 0 t þ
1
2
ð2:9Þ
where t = 0, 1, 2, … is the vibrational quantum number, and m 0 is the classical
vibrational frequency, i.e.,
m 0 ¼
1
2p
ffiffiffi
f
l
s
:
ð2:10Þ
It follows that vibrational energy levels are equidistant with the energy gap equal
to DE ¼ hm 0 . Assuming that molecule behaves like harmonic oscillator (which is a
Fig. 2.3 One-dimensional oscillator. The descriptions of vibrations of: a mass l against infinitely
heavy wall, and b two masses m 1 and m 2 such that their reduced mass equals l are identical
2 Scaling Procedures in Vibrational Spectroscopy
55