(103i) Leveraging the Wolf Method for Electrostatics to Extend Time and Length Scales Accessible By Monte Carlo Simulations

Calculating the Coulombic potential through direct, pairwise r^-1 summation is known as the Madelung problem. Exact solutions, such as Ewald summation, Particle Mesh Ewald (PME), and Particle-Particle Particle Mesh (P3M), require large time and memory usage.[1-3] In Monte Carlo simulations, it is common to use the Ewald summation method, although an approximation known as Wolf summation is faster and more memory-efficient.[4] The original Wolf method was derived for point-charge simulations, in recent years, three different formulations to handle intra-molecular interactions in molecular systems have been developed. These approaches include 1) Gross et al, (2) Vlugt et al, and (3) Vlugt with intra-cutoff (as implemented in the Cassandra Monte Carlo Engine).[5-7] Two potentials, Dampened Shifted Potential and Dampened Shifted Force were implemented for each method.[8] This work is the first comparative analysis of the three approaches. The accuracy and performance of the Wolf method, relative to the Ewald summation, is evaluated through the prediction of the vapor-liquid equilibria for water, ethanol free energy of hydration, and the hydration of complex biological structures.

For the models tested, there was no distinguishable difference in the single point electrostatic energies calculated with the Vlugt and Vlugt with intra-cutoff methods. The relative error of the Vlugt method converged to 0.0 as alpha approached 1.0, while the Gross implementation diverged. In grand canonical Monte Carlo simulations, simulations of liquid-liked densities produced density histograms in exact agreement with reference calculations performed with the Ewald summation. However, significant deviations between vapor-phase density distributions calculated with Wolf and Ewald methods were observed. Using GCMC to hydrate large biological systems, the density produced using the Wolf method was indistinguishable from Ewald simulations, while displaying over twice the computational efficiency. Simulations of a water cube with a box length of 1000 Angstrom required nearly 1 Terabyte of RAM when using the Ewald summation, while Wolf summation required less than 32 GB. Using the CPU version of GOMC[9,10] the Wolf summation was approximately 8x faster than the Ewald summation method.

References:

[1] Ewald, Paul P. "Die Berechnung optischer und elektrostatischer Gitterpotentiale." Annalen der physik 369.3 (1921): 253-287.

[2] Darden, T.; York, D.; Pedersen, L. Particle mesh Ewald: an N·log(N) method for Ewald sums in large systems. Journal of Chemical Physics 1993, 98, 10089–10092.

[3] Hockney, R. W.; Eastwood, J. W. Computer Simulation Using Particles; McGraw-Hill: New York, 1981.

[4] Wolf, D., et al. "Exact method for the simulation of Coulombic systems by spherically truncated, pairwise r− 1 summation." The Journal of chemical physics 110.17 (1999): 8254-8282.

[5] Waibel, Christian, and Joachim Gross. "Modification of the Wolf method and evaluation for molecular simulation of vapor–liquid equilibria." Journal of Chemical Theory and Computation 14.4 (2018): 2198-2206.

[6] Hens, Remco, and Thijs JH Vlugt. "Molecular simulation of Vapor–Liquid Equilibria using the Wolf method for electrostatic interactions." Journal of Chemical & Engineering Data 63.4 (2017): 1096-1102.

[7] Cassandra: An Open Source Monte Carlo Package for Molecular Simulation", Jindal K. Shah, Eliseo Marin-Rimoldi, Ryan Gotchy Mullen,Brian P. Keene, Sandip Khan, Andrew S. Paluch, Neeraj Rai, Lucienne L. Romanielo, Thomas W. Rosch, Brian Yoo, and Edward J. Maginn, Journal of Computational Chemistry 2017, 38, 1727–1739

[8] Fennell, Christopher J., and J. Daniel Gezelter. "Is the Ewald summation still necessary? Pairwise alternatives to the accepted standard for long-range electrostatics." The Journal of chemical physics 124.23 (2006): 234104.

[9] Y. Nejahi, M. Barhaghi, J. Mick, B. Jackman, K. Rushaidat, Y. Li, L. Schwiebert and J. Potoff, "GOMC: GPU Optimized Monte Carlo for the simulation of phase equilibria and physical properties of complex fluids”, SoftwareX, vol. 9, p. 20–27, 2019.

[10] Y. Nejahi, M. Barhaghi, G. Schwing, L. Schwiebert and J. Potoff, "Update 2.70 to GOMC: GPU Optimized Monte Carlo for the simulation of phase equilibria and physical properties of complex fluids”, SoftwareX, vol. 13, p. 100627, 2021.