Applications of higher order composite factorization schemes in imaginary time path integral simulations

Abstract

Suzuki’s higher order composite factorization which involves both the potential and the force is applied to imaginary time path integral simulation. The expression is more general than the original version and involves a free parameter α in the range of [0, 1]. Formal expressions are derived for statistical averages, based on both thermodynamic and quantum operator identities. The derived expressions are then tested for one-dimensional model systems using the numerical matrix multiplication method, which involves no statistical error. When an optimum choice of α is made, the higher order factorization approach is shown to be more efficient than primitive factorization by about a factor of 4 and better than other existing higher order algorithms with similar character. Actual path integral simulation tests are then made for an excess electron in supercritical helium and for bulk water, and these generally demonstrate the efficiency of the higher order factorization approach.

High order path integrals made easy

“…The latter class of estimators is generally simpler to derive and evaluate, but in the present manuscript we will compare potential and kinetic energy operators computed using both routes. Expressions for both TD and OP-method estimators have been derived and reported several times 25,31 , but we also list them in Appendix C for the sake of completeness. Evaluating the modified potential (7) only requires knowledge of the first derivative of the physical potential, which in a PIMD simulation has to be computed to evolve the dynamics.…”

“…The SC scheme could however be used to accelerate the convergence of the mean field centroid force in the case of fully adiabatic centroid molecular dynamics. 25 …”

“…Unfortunately, the higher order expansion introduces some cumbersome terms in the forces that depend formally on the Hessian of the physical potential. Therefore, rather than integrating the high-order PIMD equations of motion directly, research has focused on re-weighting schemes [25][26][27] , which are however affected by statistical inefficiency that worsens as system size increases 28 . A truncated cumulant expansion has recently shown considerable promise, although ad hoc estimators need to be devised specifically for different system properties 29 .…”

High order Chin actions in path integral Monte Carlo

“…(8) and (11,12) are not all independent and can be written as a function of only two, a 1 and t 0 , which are restricted to fulfill the conditions…”

“…Each one modifies the action in different directions: t 0 controls the weight of the different parts in which the kinetic part is splitted (8) and a 1 the weight of each part in which the double commutator is divided (12). Restricting first our analysis to distinguishable particles, the quantum partition function Z (1) can be obtained through a multidimensional integral of the M terms (beads) in which it is decomposed,…”