Stability of the complex symmetric Lanczos algorithm for computing photodissociation cross sections using smooth exterior scaling or absorbing potentials

Abstract

The stability of the Lanczos algorithm for computing photodissociation cross sections is studied. The system is discretized on a grid and the discrete variable representation (DVR) is used to represent system operators. The Hamiltonian is augmented with an absorbing potential (AP) or smooth exterior scaling (SES), to enforce outgoing boundary conditions, making it complex symmetric. The main difference between the AP and the SES is that the former adds to the potential energy whereas the latter modifies the kinetic energy operator. Grozdanov et al (2004 J. Phys. B: At. Mol. Opt. Phys. 31 173) observed the fact that the Lanczos recursions could slow down and even stagnate for certain choices of parameters in the AP or SES. Here we show that for the SES, it is important that the maximum kinetic energy of the DVR is adapted to the physical problem or else the Lanczos recursions might be unstable. A similar result was found for the AP; that is, the Lanczos algorithm in order to converge the strength of the absorbing potential should be of the order of the scattering energy of interest. It is shown that with a discretization adopted to the physical problem at hand and a proper choice of parameters, the Lanczos recursions converge and provide accurate results for both the absorbing potential and the smooth exterior scaling.