subscript is used as a reminder that this measurement is made with respect to
the v0 vibrational state). The relationship between Deand D 0 , for diatomic
molecules, is
DeD 0 ^12 h (14.38)For polyatomic molecules, a factor of^12 hfrom all vibrations must be taken
into account. Since there are 3N6 vibrations in a (nonlinear) polyatomic
molecule, the relationship between Deand D 0 has a sum of 3N6 terms:
DeD (^0)
3 N 6
i 1
^12 hi
where the sum is over the 3N6 (or 3N5 for linear molecules) vibrations
of the polyatomic molecule.
There is no theoretical basis for a Morse potential energy curve. Its form
is empirical (that is, based on observation), but it is useful. First, it shows
a dissociation limit, just as real diatomic molecules experience. The disso-
ciation energy,De, appears in two places in the form of the Morse poten-
tial, as a premultiplicative term and as part of the definition of the constant
a. It accurately predicts the observed trend of closer-spaced vibrational levels
as the vibrational quantum number increases. Although diatomic molecules
(and larger molecules also) do not behave as perfect Morse oscillators, the
Morse potential is usually a better fit to the real vibrational behavior of
molecules.
But the form of the Morse potential also allows us to quantify the amount
of nonharmonic behavior of the molecule, or its anharmonicity.Because of
its form, a system whose potential energy is expressed in terms of a Morse
potential has a Hamiltonian that is solvable analytically. (It is one of the few
solvable systems that we did not cover in Chapter 12.*) One finds that the
energy of an oscillator having a Morse potential is quantized (no surprise here)
and has values given by
Ehe(v ^12 ) hexe(v ^12 )^2 (14.39)
where eis the harmonicvibrational frequency (notequal to of the classical
harmonic oscillator!),vis the vibrational quantum number, and xeis a di-
mensionless constant called the anharmonicity constant.Usually it is a small
number. The smaller the number, the less anharmonic the oscillator is. The an-
harmonicity xeis usually so small that in tabulations of anharmonicity data,
not just xebut xe
e(sometimes written exe) is tabulated, usually in units
of cm^1. The deviation from the ideal harmonic oscillator energy is related to
the squareof the vibrational quantum number.
As part of the solution to the Schrödinger equation, the anharmonicity con-
stant appears as a combination of other constants about the vibration. It is de-
fined as
xe
4
D
e
e(14.40)
14.12 Vibrational Spectroscopy of Diatomic and Linear Molecules 493*In case you’re curious, the wavefunctions for an oscillator that has a Morse-type po-
tential energy function are
nNn et/2 t n(t)where Nnis a normalization constant,tKeax,K(8De/^2 a^2 )1/2, and n(t) is a poly-
nomial function oftand Kwhose degree depends on the quantum number n. Compare this
with the eigenfunctions for the harmonic oscillator from Chapter 11.