FRANCK-CONDON FACTORS FOR MULTIDIMENSIONAL HARMONIC-OSCILLATORS

We present a simple formula for the overlap integrals of two sets of m ulti-dimensional harmonic oscillators. The oscillators have in general different equilibrium points, force constants, and natural vibration modes. The formula expresses the overlap matrix in the one-dimensional case, [m'\n ''], as a so-called LU decomposition, [m'\n ''] = [0'\0 ' ']Sigma LmtUtn, where the summation index has a range 0 less than or e qual to t less than or equal to min(m,n), i.e., it is the matrix produ ct of a lower-triangular matrix L with an upper-triangular Li. These m atrices are obtained from simple recursion formulae. This form is esse ntially retained in the multi-dimensional case. General matrix element s are obtained by exact and finite expressions, relating them to matri x elements over a single set of harmonic oscillator wave functions. We present test calculations with error estimates, also comparing with l iterature examples. (C) 1998 Elsevier Science B.V.