Reverse Monte Carlo

Reverse Monte Carlo (RMC) is a variation of the standard Metropolis-Hastings algorithm to solve an inverse problem probing the configuration space though a random walk in search for set of parameters that is consistent with experimental data. An inverse problem is the task that often occurs in many branches of science and mathematics where the values of some model parameter(s) must be obtained from the observed data.

Applications

This method is used in condensed matter physics to produce structural models which are consistent with experimental data and subject to a set of constraints. A set of N atoms are randomly placed in a periodic boundary cell, and then a measurable quantity is calculated based on the current configuration. An iterative procedure is run where one randomly chosen atom is moved a random amount. The measurable quantity is recalculated. The move is rejected if the atom overlap with another atom or if the calculated quantity less similar to the experimental value than the previous configuration. Otherwise the move is accepted, and the procedure is executed again. As the number of accepted atom moves increases, the calculated quantity will get closer to the experimental value, until it reaches an equilibrium value, about which it will then oscillate. The resulting configuration should be a structure that is consistent with the experimental data within its errors.

Recently, Reverse Monte Carlo method (under the name of alternating least squares) was used to construct the mass spectrum of water isotopologue, HOD. [http://dx.doi.org/10.1016/j.jasms.2006.02.008]

Problematic Solutions

An outstanding problem in RMC modeling is the lack of unique solutions when too few constraints are imposed upon the simulation. With regards to traditional RMC modeling fitting pair correlation functions, although close fits to experimental data are obtained, the structure is unphysical when modeling materials contain significant three body contributions. The resulting structures tend to be very high in energy and contain unrealistically large populations of small member rings. One solution proposed was to employ a hybrid method of Metropolis Monte Carlo and RMC, in order to constrain the bond angles via accurate empirical potentials [http://www.informaworld.com/smpp/content~content=a713739907~db=all]

References

* R L McGreevy 2001 J. Phys.: Condens. Matter 13 R877-R913 doi:10.1088/0953-8984/13/46/201
* Phys. Rev. B 69, 195207 (2004) doi:10.1103/PhysRevB.69.195207 (http://link.aps.org/abstract/PRB/v69/e195207)
* EMU Notes in Mineralogy, Vol. 4 (2002), Chapter 4, 59–82