The overlap integral of two harmonic-oscillator wave functions

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Abstract
The overlap integral R(m, n) of two harmonic-oscillator wave functions ψm' and ψn'' is considered. It is shown that where [m, n] is the smaller of the two integers m and n, q = 2 (ν'ν'')1/2/(ν'' + ν') and ν' and ν'' are the frequencies associated with the wave functions ψm' and ψn'' respectively. It is also shown that if the quantities R(m, n), where m and n run from 0 to p, are considered as the elements of a square matrix R of order p + 1, then The reduction of the general expression for R(m, n) to special cases of importance in molecular spectroscopy is indicated.

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