Vibrational anharmonicity levels

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

In a diatomic molecule one of the main effects of mechanical anharmonicity, the only type that concerns us in detail, is to cause the vibrational energy levels to close up smoothly with increasing v, as shown in Figure 6.4. The separation of the levels becomes zero at the limit of dissociation. 

The much greater convergence in the R branch in the 1o3q band is due to a very much larger decrease of B in the upper state it is 75.69 x 10 cm less than the value of 1.478 221 834 cm in the lower state. This decrease is characteristic of vibrational overtone levels and is due, mostly, to anharmonicity which results in the molecule spending most of its time at much larger intemuclear distances than in the u = 0 level. 

When exposed to electromagnetic radiation of the appropriate energy, typically in the infrared, a molecule can interact with the radiation and absorb it, exciting the molecule into the next higher vibrational energy level. For the ideal harmonic oscillator, the selection rules are Av = +1 that is, the vibrational energy can only change by one quantum at a time. However, for anharmonic oscillators, weaker overtone transitions due to Av = +2, + 3, etc. may also be observed because of their nonideal behavior. For polyatomic molecules with more than one fundamental vibration, e.g., as seen in Fig. 3.1a for the water molecule, both overtones and... 

Fig. 6.4 Vibrational energy levels of a molecule with >, 3800 cm-1, v2wk 1600 cm-1, and p3 3900 cm-1. (These are roughly the values for H20.) The energy is calculated from (6.54), and does not include anharmonicity corrections.

Theory predicts that for a harmonic oscillator only a change from one vibrational energy level to the next higher is allowed, but for anharmonic oscillators weaker transitions to higher vibrational energy levels can occur. The resulting "overtones" are found at approximate multiples of the frequency of the fundamental. Combination frequencies representing sums... 

For diatomic molecules, corrections can be made for the assumption used in the derivation of the rotational partition function that the rotational energy levels are so closely spaced that they can be considered to be continuous. The equations to be used in making these corrections are given in Appendix 6. Also given are the equations to use in correcting for vibrational anharmonicity and nonrigid rotator effects. These corrections are usually small.22... 

Here E0 is the uth vibrational energy level with wave function rn10 is an harmonic frequency, and A" is the anharmonicity constant. Under certain circumstances a system of this land, initially in its ground state, and driven by a cw field... 

A similar interaction would be observed between all Fermi polyads containing sets of vibrational levels related by the selection rule A tv = 2, A tv = +1, and the hamiltonian matrix should be diagonalized for each Fermi polyad without the use of perturbation theory. If, on the other hand, the interaction (63) were smaller, or the separation between the unperturbed levels were larger, the interaction could be treated by perturbation theory it can be shown that, in second-order perturbation theory, equation (63) would contribute a term to the vibrational anharmonic constants... 

Although the vibrational motion of a diatomic molecule conforms quite closely to that of a harmonic oscillator, in practice the anharmonic deviations are quite significant and must be taken into account if vibrational energy levels are to be modelled accurately. A general form of the potential fimction V in equation (2.157) was proposed by Dunham... 

The anharmonic corrections to the vibrational energy levels can be derived from the solutions of the Schrodinger equation with a potential of the form... 

FIGURE 3. The solid line is a harmonic potential function, and the dashed line is a typical anharmonic function. The horizontal line represents the zero-point vibrational energy level, and the average value of r is shown. It is larger than the equilibrium value of r (vertical dashed line)... 

An improved treatment of molecular vibration must account for anharmonicity, deviation from a harmonic oscillator. Anharmonicity results in a finite number of vibrational energy levels and the possibility of dissociation of the molecule at sufficiently high energy. A very successful approximation for the energy of a diatomic molecule is the Morse potential ... 

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