Nonlinear sequence transformations: A computational tool for quantum mechanical and quantum chemical calculations

Abstract

Sequence transformations, which transform a slowly convergent or divergent sequence {sn}∞n&=0 into a new sequence {sln}∞n=0 with hopefully better numerical properties, are useful computational tools to overcome convergence and divergence problems. In this article, sequence transformations are discussed that use explicit remainder estimates [E.J. Weniger, Comput. Phys. Rep. 10, 189 (1989)]. Because of the explicit incorporation of the information contained in the remainder estimates, these transformations are potentially very powerful and well suited for the summation of strongly divergent series. The Rayleigh-Schrödinger perturbation series for the ground-state energy of the quartic, sextic, and octic anharmonic oscillator is a typical example of a perturbation series that diverges quite strongly for every nonzero coupling constant. It can be summed efficiently even in the very challenging strong coupling regime, if sequence transformations are combined with a suitable renormalization technique described by F. Vinette and J. C˘íz˘ek [J. Math. Phys. 32, 3392 (1991)]. Moreover, a renormalized strong coupling expansion for the ground-state energy of an anharmonic oscillator can be constructed, which apparently converges for all coupling constants β ϵ [0,∞) and which makes the computation the ground-state energy almost trivial. Other applications of sequence transformations in quantum mechanical and quantum chemical calculations are also reviewed. © 1996 John Wiley & Sons, Inc.