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Usually the xii are small and negative, thus causing the vibrational levels to get closer together as the vibrational ladder is climbed. However, it is worth noting that unlike the diatomic case, the diagonal anharmonicity constants can also sometimes be positive. The so-called offdiagonal anharmonicity constants, xi j where i = j, arise from the mixing of normal modes. This mixing, caused by a breakdown of the harmonic oscillator approximation, is normally small for the lowest vibrational levels and can often be ignored.
4) is only valid when one of the atoms is essentially of infinite mass, in which case it is only the lighter atom, of mass m, which moves in a vibration. 5) and mA and mB are the masses of the two atoms. 4) can be solved, although it is a rather involved process; we focus here solely on the results. It is found, not surprisingly, that the energy is now quantized. 6) where ωe is the harmonic vibrational frequency (in Hz),2 and v is the vibrational quantum number, which can have the values 0, 1, 2, 3, .
6 Our concern in this book is with vibrational changes accompanying electronic transitions, but the use of symmetry in establishing vibrational selection rules is also extremely important in infrared and Raman spectroscopies. 20) that brings this about. 10), where you should substitute Qi for the diatomic normal coordinate η. When the vibrational quantum number vi is even (including zero), then only even powers of Qi appear in the Hermite polynomial. However, when vi is odd, only odd powers of Qi appear in the Hermite polynomial.