Reverse Monte Carlo

Reverse Monte Carlo (RMC) [1] 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 chemocal bonds etc.) can be applied. Some examples are:

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

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. 3. Transform to the total structure factor: $S_o^2(Q)-1=\frac{4\pi \rho }{Q}\underset{0}{\overset{inf}{\int }}r(g_o^C(r)-1)sin(Qr)dr$

where Q is the momentum transfer $\rho$ and the number density. 4. Calculate the difference between the measured structure factor $S^E(Q)$ and the one calculated from the configuration $S_o^C(Q)$: $\underset{o}{\overset{2}{\chi }}={\sum }_i(S_o^C(Q_i)-S^E(Q_i){)}^2/\sigma (Q_i{)}^2$

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 distribution function, structure factor and: $\underset{n}{\overset{2}{\chi }}={\sum }_i(S_n^C(Q_i)-S^E(Q_i){)}^2/\sigma (Q_i{)}^2$ 6. If $\underset{n}{\overset{2}{\chi }}<\underset{o}{\overset{2}{\chi }}$ accept the move and let the new configuration become the old. If $\underset{n}{\overset{2}{\chi }}>\underset{o}{\overset{2}{\chi }}$ then the move is accepted with probability $exp(-(\underset{n}{\overset{2}{\chi }}-\underset{0}{\overset{2}{\chi }})/2)$ otherwiase 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)