Harmonic oscillator/vibration

We expect, therefore, that a reasonable approximation to the vibrational energy levels vib of a diatomic molecule would be the harmonic-oscillator vibrational energy levels Eqs. (4.47) and (4.25) give... 

The (harmonic oscillator) vibrational frequencies are related by expression... 

We want to start and end the Numerov procedure in the classically forbidden regions. If we used the harmonic-osciUator approximation for the vibrational levels, the energies of the first six vibrational levels would be (r + )hvg, r = 0,1,..., 5. The reduced energy of the sixth harmonic-oscillator vibrational level would be 5.5hvJA = 5.5hvJ (253 axT )hc = 5.5(4403 cm 7/(253 cm 7 95.7. Because of anharmonicity (Section 4.3), the sixth vibrational level will actually occur below 95.7, so we are safe in using 95.7 to find the limits of the classically allowed region. We have 7>g, .(l — = 95.7, and with = 151.29, we find... 

The cooidinates Qi are determined in the process of a semiclassical treatment of molecular vibrations [3-6]. The principal aim of these calculations is to define the specific coordinates Qi, in die basis of which the Schrddinger wave functions for the vibrational motion of a molecule are reduced to 3N-6 simple linear hannonic oscillator wave functions. One of the mathematical expressions of this result is Eq. (1.25). 3N-6i the munber of vibrations in an N-atomic molecule (3N-5 in the case of linear molecules). Described in terms of Qi the complex vibrational motion of a molecule is expressed as a superposition of 3N-6 linear harmonic oscillator vibrations, each having specific form as described by Qj and own frequency of oscillation. More comments about normal coordinates will be given in the following section. For a complete description of the theory of normal vibrations the reader is referred to a number of monogr hs [3-6]. 

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