What Is the RMC Method?

Reverse Monte Carlo (RMC) is a method for finding atomic configurations consistent with data obtained from scattering experiments and other measurements. It iteratively proposes changes to atomic positions and accepts or rejects them.

Agreement with Experimental Data

RMC evaluates the difference between scattering functions calculated from an atomic configuration and experimental data. For example, if the standard deviation at measurement point \(i\) is \(\sigma_i\), a measure of agreement can be written as

\[\chi^2 = \sum_i \frac{\left[S_{\mathrm{calc}}(Q_i)-S_{\mathrm{exp}}(Q_i)\right]^2}{\sigma_i^2}\]

Here, \(S_{\mathrm{calc}}\) is the structure factor calculated from the atomic configuration, and \(S_{\mathrm{exp}}\) is the measured value. When multiple datasets or structural constraints are used, their contributions also enter the evaluation. The precise objective function and coefficient definitions depend on the RMC_POT settings and version.

Acceptance and Rejection of Atomic Moves

After a trial modification of the atomic configuration, calculate the change in agreement:

\[\Delta\chi^2 = \chi^2_{\mathrm{new}}-\chi^2_{\mathrm{old}}\]

The acceptance rule given in the RMC_POT user guide is

\[P_{\mathrm{accept}} = \min\left[1,\exp(-\Delta\chi^2)\right]\]

Under this rule, moves that improve agreement (\(\Delta\chi^2\leq0\)) are accepted, while moves that worsen agreement can still be accepted with a finite probability.

Role of Standard Deviations

The standard deviation \(\sigma_i\) at each measurement point determines the weight assigned to the residual between experimental and calculated values. For the same residual, a point with smaller \(\sigma_i\) makes a larger contribution to \(\chi^2\).

References

• Orsolya Gereben, RMC_POT user guide. Consult the official manual for the definitions of acceptance probability and \(\chi^2\), as well as the specification of standard deviations for each dataset.

• https://www.szfki.hu/~nphys/rmc++/RMC_POT_user_guide.pdf