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:

Closest approach between atoms (hard sphere potential)
Coordination numbers.
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:

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.
Calculate the total radial distribution function $g_{o}^{C}(r)$ for this old configuration (C=Calculated, o=Old).
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.
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.
Select and move one atom at random and calculate the new (n=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}
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 \geq \chi_o^2} then the move is accepted with probability 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 \exp(-(\chi_n^2-\chi_0^2)/2)} otherwise it is rejected.
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.

References

Reverse Monte Carlo

References

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