Cross correlation functions Cnm(E) via Lanczos algorithms without diagonalization
Hans Karlsson, Sverker Holmgren · The Journal of Chemical Physics · 2002
It is shown how the quasiminimal residual algorithm (QMR), based on the Lanczos algorithm, can be modified to compute cross-correlation functions Cnm(E)=〈Ψn|(E−H)−1|Ψm〉 without any diagonalization by recursively updating a small number of scalars. Only three Lanczos vectors need to be stored. Several left-hand side vectors 〈Ψn| and multiple shifts E can be considered simultaneously. The new method is termed the quasiminimal recursive residue generation method (QM-RRGM) and is applied to the collinear H+H2 problem to illustrate its convergence behavior. The properties of two different formulations of the Lanczos algorithm, the usual three-term and a coupled two-term recursion, are also discussed. The QM-RRGM exhibits smooth convergence behavior, and it is shown that the stopping criteria used in the QMR algorithm can also be used for computing correlation functions.