On the quasiclassical calculation of fundamental and overtone intensities

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Abstract
Various approximations to the transition dipole moment matrix element <n’‖M‖n≳ are compared with each other and to exact (numerical) values of this overlap integral for different n→n’ transitions in a Morse potential with a linear dipole moment function. By partitioning the numerical integral into different contributions that involve the classically allowed and forbidden regions of each wave function, we have learned what conditions must be satisfied for validity of the different approximations. In particular, we consider the Landau approximation to the quasiclassical matrix element in which the exact wave function for the upper state is replaced by the Wentzel–Kramers–Brillouin (WKB) wave function in the classically allowed region of that state. We find that the Landau approximation is more accurate than might have been expected because of the compensation of the neglected tunneling contribution by the singular behavior of the WKB wave function in the classically allowed neighborhood of the turning point. Based on this study, we suggest an improved semiclassical approximation for transition dipole matrix elements that involve an arbitrary dipole moment function. This method is applied to the n’−0 transition of a Morse oscillator using a linear dipole moment function; it can reproduce the exact values of the transition dipole moment matrix element to better than 5% for n’=1 to n’=15. Under the condition that the dipole moment function is slowly varying or decreases monotonically with increasing internuclear separation, a simple expression is presented for estimating relative strengths of neighboring high overtone transitions.

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