On the computation of matrix elements between numerical wave functions: The canonical functions method

The problem of the computation of the matrix elements\documentclass{article}\pagestyle{empty}\begin{document}$$ I(v,v';k) = \int_0^x {\Psi _v (r)(r - r_e )} ^k \Psi _{v'} (r)dr, $$\end{document} is considered when Ψ v ( r ) and Ψ v ( r ) are eigenfunctions related to a diatomic potential of the RKR type (defined by the coordinates of its turning points P i with polynomial interpolations). The eigenfunction Ψ( r ) is computed by the canonical functions method making use of the abscissas r i of P i uniquely. This limited number of points allows the storage of ψ v ( r i ) for all the required levels v , and reduces greatly the computational effort when v , ν′, and k are varying. The present method maintains all the advantages of a highly accurate numerical method (even for levels near the dissociation), and reduces greatly the computing time. Furthermore, it is shown that it may be extended to analytical potentials like Morse and Lennard‐Jones functions, to vibrational‐rotational eigenfunctions and to matrix elements between eigenfunctions related to two different potentials. Numerical applications are presented and discussed.