Reverse Monte Carlo

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Reverse Monte Carlo (RMC) [1] 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 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) (E=Experimental) 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 \geq \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.

References[edit]

Related reading