DMRG control using an automated Richardson-type error protocol

Abstract

The density matrix renormalisation group (DMRG) algorithm solves the many-particle electronic Schrödinger equation as accurately as possible in a full-configuration-interaction (FCI) sense for a given one-electron basis. The FCI limit is approached by increasing the number of many-particle DMRG states which span the Hilbert space of interest. In this work we investigate an automated DMRG error protocol which extrapolates the electronic energy using Richardson's deferred approach to the limit. The key idea is to consider the result of a DMRG algorithm as an analytical function of an adjustable parameter like the number of DMRG states. We can then probe this analytical function by performing calculations for different set sizes of renormalised DMRG basis states. None of these calculations has to actually provide the desired accuracy but after we have collected enough information about the function's behaviour, we can represent it by an analytic rational function that may then be used to extrapolate to the converged energy. The advantage of this approach is that it delivers an error estimate for the electronic energy which can be used for accuracy control and as a convergence criterion. In addition, one may detect convergence to local energy minima by automatically increasing the analytic parameter (i.e. the number of renormalised DMRG states). The error estimates also allows us to aim at relative energies from extrapolated total energies of similar accuracy that is then independent of the number of renormalised DMRG states or of the truncation error of the reduced density matrix.