Nested Monte Carlo simulation of ionic systems with the primitive model using Debye-Hückel (DH) potential as an importance function and optimizing the DH potential with Kullback-Leibler divergence minimization

Abstract

In this work, Nested Monte Carlo (NMC) simulation was done for symmetric and asymmetric ionic systems where the energy function is described by a combination of hard-sphere and Coulomb interaction (the primitive model). In the NMC method, Monte Carlo (MC) moves for the primary chain (where the energy function is the primitive model) is given by running a short MC trajectory using an auxiliary potential reducing the computational cost significantly. The Debye-Hückel (DH) potential was used as the auxiliary potential in our work. It is shown that with a careful choice of Debye length in the DH potential, and length of the short MC run in the auxiliary chain, the NMC method gives the same result as the more expensive MC simulation using full Ewald summation in the primitive model. Implementing the minimization of the Kullback–Leibler (KL) divergence between the pair-correlation function of the standard (without any auxiliary potential) MC simulation and NMC, a simple algorithm was also presented to develop good DH-like potentials to be used as auxiliary potentials in the nested MC run. Overall, this technique significantly reduces the Ewald summation technique's computational cost and can be applied to any atomic and molecular system with coulombic interaction in the potential energy function.

Graphical abstract

Nested Monte Carlo (NMC) simulations for symmetric and asymmetric ionic systems are performed, where Monte Carlo (MC) moves for the primary chain are given by running a short MC trajectory using an auxiliary potential. Excellent agreement has been shown between the pair correlation functions calculated from NMC, and standard Ewald simulations.