Correlation energy estimators based on Møller-Plesset perturbation theory

Abstract

Some methods for the convergence acceleration of the Møller-Plesset perturbation series for the correlation energy are discussed. The order-by-order summation is less effective than the Feenberg series. The latter is obtained by renormalizing the unperturbed Hamilton operator by a constant factor that is optimized for the third order energy. In the fifth order case, the Feenberg series can be improved by orderdependent optimization of the parameter. Alternatively, one may use Padé approximants or a further method based on effective characteristic polynomials to accelerate the convergence of the perturbation series. Numerical evidence is presented that, besides the Feenberg-type approaches, suitable Padé approximants, and also the effective second order characteristic polynomial, are excellent tools for correlation energy estimation.

The size-extensivity of correlation energy estimators based on effective characteristic polynomials

“…Size-extensitivity is an important property that is helpful for obtaining reliable energy estimates for larger systems. It is hoped that correlation energy estimates on the basis of effective characteristic polynomials become more widespread since they offer a conceptually and computationally relatively simple but accurate computational tool as demonstrated in [10]. Only the MP series has to be computed, and its terms be combined to obtain an effective characteristic polynomial and the correlation energy as one of its roots as in Eq.…”

“…(7). Some error control is possible by comparison to some other methods for convergence acceleration of the perturbation series [10].…”

“…If only perturbation energies up to the fourth order are available then one can use the Π2 estimator [10]. The Π2 estimator is obtained as a root of a seconddegree effective characteristic polynomial.…”

Size dependence of Feenberg scaling

“…One of these conditions is the n th‐order generalization of Feenberg optimization 13. Homeier 10 pointed out that size extensivity for identical noninteracting subsystems holds in this case, since the dimensionless scaling parameter stems from a nonlinear algebraic equation, where all the coefficients are size extensive, as shown in Eq. (34) in 13.…”

“…These investigators scaled the initial partitioning by a single parameter, the value of which was determined by zeroing the third‐order correction in the scaled partitioning (or equivalently, by setting the energy up to the third order stationary with respect the scaling parameter) 1, 2. Feenberg's initial idea was widely exploited later in several studies, and some new criteria were set for establishing the value of the scaling parameter 3–16.…”