Semiclassical Methods in Inelastic Scattering

A hierarchy of related semiclassical approximations is derived for treating molecular inelastic scattering. Starting from the coupled Schrödinger equations of motion, the radial wavefunctions are expressed as $u_i{\mathrm{(r)\hspace{0.17em}=\hspace{0.17em}u}}_{\text{1i}}{\mathrm{(r)X}}_i{\mathrm{(r)\hspace{0.17em}+\hspace{0.17em}u}}_{\text{2i}}{\mathrm{(r)Y}}_i\mathrm{(r)}$ , where $u_{\text{2i}}$ and $u_{\text{2i}}$ are two linearly independent approximate solutions for elastic scattering. A set of coupled first‐order equations for $X_i$ and $Y_i$ is obtained. By using simple WKB wavefunctions for $u_1$ and $u_2$ and by neglecting several terms, the coupled time‐dependent equations and several previous semiclassical approximations can be readily obtained. Inclusion of the neglected terms yields improved accuracy for these approximations. Use of modified WKB wavefunctions valid near the classical turning point extends these approximations to cases involving large energy transfer where the inelastic cross sections are sensitive to the region near the turning point. In the sudden and “classical” limits of the internal molecular motions the coupled equations in X and Y can be decoupled by using simple transformations. For cases close to these limits a set of coupled equations is obtained giving the deviation from the sudden or “classical” limit.