Reverse Monte Carlo: Difference between revisions

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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.).
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:
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)
#Closest approach between atoms (hard sphere potential)
#Coordination numbers.
#Coordination numbers.
#Angels in triplets of atoms.
#Angles in triplets of atoms.


The algorithm for RMC can be written:
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.
#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 <math>g_o^C(r)</math> for this old configuration.
#Calculate the total radial distribution function <math>g_o^C(r)</math> for this old configuration.
3. Transform to the total structure factor:
#Transform to the total structure factor:
<math>S_o^2 (Q)-1=\frac{4\pi\rho}{Q}\int\limits_{0}^{\inf} r(g_o^C(r)-1)sin(Qr)\, dr</math>
<math>S_o^2 (Q)-1=\frac{4\pi\rho}{Q}\int\limits_{0}^{\infty} r(g_o^C(r)-1)\sin(Qr)\, dr</math>


where ''Q'' is the momentum transfer <math>\rho</math> and the number density.
where ''Q'' is the momentum transfer <math>\rho</math> and the number density.
4. Calculate the difference between the measured structure factor <math>S^E(Q)</math> and the one calculated from the configuration <math>S_o^C(Q)</math>:
#Calculate the difference between the measured structure factor <math>S^E(Q)</math> and the one calculated from the configuration <math>S_o^C(Q)</math>:
<math>\chi_o^2=\sum_i(S_o^C(Q_i)-S^E(Q_i))^2/\sigma(Q_i)^2</math>
<math>\chi_o^2=\sum_i(S_o^C(Q_i)-S^E(Q_i))^2/\sigma(Q_i)^2</math>
this sum is taken over all experimental points <math>\sigma</math> is the experimental error.
this sum is taken over all experimental points <math>\sigma</math> is the experimental error.
5. Select and move one atom at random and calculate the new distribution function, structure factor and:
#Select and move one atom at random and calculate the new distribution function, structure factor and:
<math>\chi_n^2=\sum_i(S_n^C(Q_i)-S^E(Q_i))^2/\sigma(Q_i)^2</math>
<math>\chi_n^2=\sum_i(S_n^C(Q_i)-S^E(Q_i))^2/\sigma(Q_i)^2</math>
6. If <math>\chi_n^2<\chi_o^2</math> accept the move and let the new configuration become the old. If <math>\chi_n^2>\chi_o^2</math> then the move is accepted with probability <math>exp(-(\chi_n^2-\chi_0^2)/2)</math> otherwiase rejected.
#If <math>\chi_n^2<\chi_o^2</math> accept the move and let the new configuration become the old. If <math>\chi_n^2>\chi_o^2</math> then the move is accepted with probability <math>\exp(-(\chi_n^2-\chi_0^2)/2)</math> otherwiase rejected.
7. repeat from step 5.
#repeat from step 5.
When <math>\chi^2</math> have reached an equilibrium the configuration is saved and can be analysed.
When <math>\chi^2</math> have reached an equilibrium the configuration is saved and can be analysed.


Revision as of 18:55, 19 February 2007

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 chemical bonds etc.) can be applied. Some examples are:

  1. Closest approach between atoms (hard sphere potential)
  2. Coordination numbers.
  3. Angles 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 for this old configuration.
  3. Transform to the total structure factor:

where Q is the momentum transfer and the number density.

  1. Calculate the difference between the measured structure factor and the one calculated from the configuration :

this sum is taken over all experimental points is the experimental error.

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

  1. If accept the move and let the new configuration become the old. If then the move is accepted with probability otherwiase rejected.
  2. repeat from step 5.

When have reached an equilibrium the configuration is saved and can be analysed.

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)
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