Reverse Monte Carlo

Reverse Monte Carlo (RMC) is a variation of the standard Metropolis Monte Carlo method. It is used to produce a 3 dimensional atomic model that fits a set of measurements (neutron diffraction, X-ray-diffraction, EXAFS etc.). In addition to measured data a number of constraints based on prior knowledge of the system (for example, chemical bonding 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 reverse Monte Carlo is not unique; it is a model consistent with the data and constraints provided.

The algorithm for reverse Monte Carlo can be written as follows:

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 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 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 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 \rho} the number density.

4. Calculate the difference between the measured structure factor 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^E(Q)} (E=Experimental) 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.

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 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.
7. 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

Related reading

• R. L. McGreevy, "Reverse Monte Carlo modelling", Journal of Physics: Condensed Matter 13 pp. R877-R913 (2001)
• 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 pp. 41-47 (2003)
• G. Evrard, L. Pusztai, "Reverse Monte Carlo modelling of the structure of disordered materials with RMC++: a new implementation of the algorithm in C++", Journal of Physics: Condensed Matter 17 pp. S1-S13 (2005)
• V. Sánchez-Gil, E. G. Noya, L. Temleitner, L. Pusztai "Reverse Monte Carlo modeling: The two distinct routes of calculating the experimental structure factor", Journal of Molecular Liquids 207 pp. 211-215 (2015)