Anharmonic vibrational energy levels

At higher levels of excitation anharmonicity has to be included to obtain accurate energy levels. Perturbation theory has been used to derive the following expression, often called a Dunham expansion (Hirst, 1985), for polyatomic anharmonic vibrational energy levels, which is similar to the Morse energy level expression Eq. (2.59), for a diatomic molecule ... 

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. 

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... 

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... 

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 ... 

Spectroscopists generally write (Og in place of v for diatomic molecules. The anharmonicity cogXg is usually tabulated as a single parameter. Higher vibrational energy levels are spaced closer together, just as in real molecules. The anharmonicity for a Morse oscillator is determined by... 

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