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:

$S_{o}^{2}(Q)-1={\frac {4\pi \rho }{Q}}\int \limits _{0}^{\inf }r(g_{o}^{C}(r)-1)sin(Qr)\,dr$

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

1. Calculate the difference between the measured structure factor $S^{E}(Q)$ and the one calculated from the configuration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_o^C(Q)} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} is the experimental error.

1. Select and move one atom at random and calculate the new distribution function, structure factor and:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_n^2=\sum_i(S_n^C(Q_i)-S^E(Q_i))^2/\sigma(Q_i)^2}

1. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_n^2<\chi_o^2} accept the move and let the new configuration become the old. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_n^2>\chi_o^2} then the move is accepted with probability $exp(-(\chi _{n}^{2}-\chi _{0}^{2})/2)$ otherwiase rejected.
2. repeat from step 5

When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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, Mol. Simulation, 1 359-367 (1988)