Reverse Monte Carlo

Reverse Monte Carlo (RMC) [1-4] is a variation of the standard Metropolis Monte Carlo (MMC) method. It is used to produce a 3 dimensional atomic model that fits a set of measurements (Neutron-, X-ray-diffraction, EXAFS etc.). In addition to measured data a number of constraints based on prior knowledge of the system (like chemical bonds etc.) can be applied. Some examples are:

1. Closest approach between atoms (hard sphere potential)
2. Coordination numbers.
3. Angles in triplets of atoms.

The 3 dimensional structure that is produced by RMC is not unique, it is a model consistent with the data and constraints provided.

The algorithm for RMC can be written:

1. Start with a configuration of atoms with periodic boundary conditions. This can be a random or a crystalline configuration from a different simulation or model.
2. Calculate the total radial distribution function $g_{o}^{C}(r)$ for this old configuration (C=Calculated, o=Old).
3. Transform to the total structure factor:

   $S_{o}^{C}(Q)-1={\frac {4\pi \rho }{Q}}\int \limits _{0}^{\infty }r(g_{o}^{C}(r)-1)\sin(Qr)\,dr$

   where Q is the momentum transfer and $\rho$ the number density.

4. Calculate the difference between the measured structure factor $S^{E}(Q)$ and the one calculated from the configuration $S_{o}^{C}(Q)$:

   $\chi _{o}^{2}=\sum _{i}(S_{o}^{C}(Q_{i})-S^{E}(Q_{i}))^{2}/\sigma (Q_{i})^{2}$

   this sum is taken over all experimental points $\sigma$ is the experimental error.

5. Select and move one atom at random and calculate the new (n=New) distribution function, structure factor and:

   $\chi _{n}^{2}=\sum _{i}(S_{n}^{C}(Q_{i})-S^{E}(Q_{i}))^{2}/\sigma (Q_{i})^{2}$

6. If $\chi _{n}^{2}<\chi _{o}^{2}$ accept the move and let the new configuration become the old. If $\chi _{n}^{2}>\chi _{o}^{2}$ then the move is accepted with probability $\exp(-(\chi _{n}^{2}-\chi _{0}^{2})/2)$ otherwise it is rejected.
7. repeat from step 5.

When $\chi ^{2}$ have reached an equilibrium the configuration is saved and can be analysed.

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References

1. R. L. McGreevy and L. Pusztai, "Reverse Monte Carlo Simulation: A New Technique for the Determination of Disordered Structures", Molecular Simulation, 1 pp. 359-367 (1988)
2. R. L. McGreevy, "Reverse Monte Carlo modelling", J.Phys.:Cond. Matter 13 pp. R877-R913 (2001)
3. R. L. McGreevy and P. Zetterström, "To RMC or not to RMC? The use of reverse Monte Carlo modelling", Current Opinion in Solid State and Materials Science. 7 no. 1 (2003) pp. 41-47 Elsevier Science
4. G. Evrard, L. Pusztai, "Reverse Monte Carlo modelling of the structure of disordered materials with RMC++: a new implementation of the algorithm in C++", J.Phys.:Cond. Matter 17 pp. S1-S13 (2005)