The convergence of energy levels
One approach to the calculation of the energy levels in the presence of anharmonicity is to use a function that resembles the true potential energy more closely. The Morse potential energy is
Where De is the depth of the potential minimum (Fig. 13.29). Near the well minimum the variation of V with displacement resembles a parabola (as can be checked by expanding the exponential as far as the first term) but, unlike a parabola, eqn 13.54 allows for dissociation at large displacements. The Schrödinger equation can be solved for the Morse potential and the permitted energy levels are 
The parameter xe is called the anharmonicity constant. The number of vibrational levels of a Morse oscillator is finite, and v = 0, 1, 2,..., vmax, as shown in Fig. 13.30 (see also Problem 13.26). The second term in the expression for G subtracts from the first with increasing effect as v increases, and hence gives rise to the convergence of the levels at high quantum numbers.
Although the Morse oscillator is quite useful theoretically, in practice the more general expression
G(v) = (v+
)v − (v +
)2xev + (v + )3yev + · · · (13.56) where xe, ye,...are empirical dimensionless constants characteristic of the molecule, is used to fit the experimental data and to find the dissociation energy of the molecule. When anharmonicities are present, the wavenumbers of transitions with ∆v =+1 are
∆G v+= v − 2(v + 1) xe v + · · ·
Equation 13.57 shows that, when xe > 0, the transitions move to lower wavenumbers as v increases.
Anharmonicity also accounts for the appearance of additional weak absorption lines corresponding to the transitions 2 ← 0, 3 ← 0,..., even though these first, second,...overtones are forbidden by the selection rule ∆v =±1. The first overtone, for example, gives rise to an absorption at
G(v +2)−G(v)=2v−2(2v+3)xev+ · · ·
The reason for the appearance of overtones is that the selection rule is derived from the properties of harmonic oscillator wavefunctions, which are only approximately valid when anharmonicity is present. Therefore, the selection rule is also only an approximation. For an anharmonic oscillator, all values of ∆v are allowed, but transitions with ∆v > 1 are allowed only weakly if the anharmonicity is slight.
Fig. 13.29 The dissociation energy of a molecule, D0, differs from the depth of the potential well, De, on account of the zero point energy of the vibrations of the bond.
Fig. 13.30 The Morse potential energy curve reproduces the general shape of a molecular potential energy curve. The corresponding Schrödinger equation can be solved, and the values of the energies obtained. The number of bound levels is finite.
