On the direct complex scaling of matrix elements expressed in a discrete variable representation: Application to molecular resonances

Abstract

We present an extension of a method initially proposed by Moiseyev and Corcoran [Phys. Rev. A 20, 814 (1978)] to a direct continuation of the matrix elements of a real Hamiltonian operator expressed in a contracted, discrete variable representation type basis set. It is based on the identity which relates the matrix elements of a complex scaled potential between real basis set functions to those of the unscaled potential between backward scaled basis functions. The method is first applied to the study of the resonances of a one dimensional model by means of complex scaling. It is shown that the resulting matrix elements of the scaled potential are no longer diagonal in the DVR. This paradox is discussed and shown to be of no practical consequence in the formulation. The scheme is then extended to the direct complex scaling of a two dimensional Hamiltonian operator expressed in a contracted basis set built through the successive adiabatic reduction method of Bac̆ić and Light. Results show that, due to the use of a numerical continuation, slightly larger grids have to be used as compared to the case of an analytic continuation where the potential is available.