Reverse Monte Carlo
Encyclopedia
Introduction
The Reverse Monte Carlo (RMC) modelling method is a variation of the standard Metropolis-Hastings algorithmMetropolis-Hastings algorithm
In mathematics and physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult...
to solve an inverse problem
Inverse problem
An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in...
whereby a model is adjusted until its parameters have the greatest consistency with experimental data. Inverse problems
Inverse problem
An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in...
are found in many branches of science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...
and mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, but this approach is probably best known for its applications in condensed matter physics
Condensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
and solid state chemistry.
Basic method
This method is often used in condensed matter sciencesCondensed matter physics
Condensed matter physics deals with the physical properties of condensed phases of matter. These properties appear when a number of atoms at the supramolecular and macromolecular scale interact strongly and adhere to each other or are otherwise highly concentrated in a system. The most familiar...
to produce atom-based structural models that are consistent with experimental data
Experimental data
Experimental data in science is data produced by a measurement, test method, experimental design or quasi-experimental design. In clinical research any data produced as a result of clinical trial...
and subject to a set of constraints.
An initial configuration is constructed by placing atoms in a periodic boundary
Periodic boundary conditions
In mathematical models and computer simulations, periodic boundary conditions are a set of boundary conditions that are often used to simulate a large system by modelling a small part that is far from its edge...
cell, and one or more measurable quantities
Physical quantity
A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.-Definition of a physical quantity:Formally, the International Vocabulary of Metrology, 3rd edition defines quantity as:...
are calculated based on the current configuration. Commonly used data include the pair distribution function and its Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
, the latter of which is derived directly from neutron or x-ray total scattering data. Other data that are used included Bragg diffraction data for crystalline materials, and EXAFS
Extended X-Ray Absorption Fine Structure
X-ray Absorption Spectroscopy includes both Extended X-Ray Absorption Fine Structure and X-ray Absorption Near Edge Structure . XAS is the measurement of the x-ray absorption coefficient of a material as a function of energy...
data. The comparison with experiment is quantified using a function of the form
where and are the observed (measured) and calculated quantities respectively, and is a measure of the accuracy of the measurement. The sum is over all independent measurements, which will include the sum over all points in a function such as the pair distribution function.
An iterative procedure is run where one randomly chosen atom is moved a random amount, followed by a new calculation of the measurable quantities. Such a process will cause to either increase or decrease in value by an amount . The move is accepted with the probability according to the normal Metropolis-Hastings algorithm
Metropolis-Hastings algorithm
In mathematics and physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult...
, ensuring that moves that give better agreement with experimental data are accepted, and moves that worsen agreement with experimental data can be accepted to a greater or lesser extent corresponding to how much the agreement has worsened. Moreover, the move may also be rejected if it breaks certain constraints, even if the agreement with data is improved. An example would be if two atoms become closer than a pre-set limit.
Following the acceptance/rejection test, the procedure is repeated. As the number of accepted atom moves increases, the calculated quantities will become closer to the experimental values until they reach an equilibrium state. From then onwards the RMC algorithm will simply generate a small oscillation in the value of . The resulting atomic configuration should be a structure that is consistent with the experimental data within its errors.
Applications
The RMC method for condensed matter problems was initially developed by McGreevy and Pusztai in 1988, with application to liquidLiquid
Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...
argon
Argon
Argon is a chemical element represented by the symbol Ar. Argon has atomic number 18 and is the third element in group 18 of the periodic table . Argon is the third most common gas in the Earth's atmosphere, at 0.93%, making it more common than carbon dioxide...
(Note that there were earlier independent applications of this approach, for example those of Kaplow et al and Gerold and Kern; it is, however, the McGreevy and Pusztai implementation that is best known). For several years the primary application was for liquids and amorphous materials, particularly because this provides the only means to obtain structural models from data, whereas crystallography
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...
has analysis methods for both single crystal and powder diffraction
Powder diffraction
Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials.-Explanation:...
data. More recently, it has become clear that RMC can provide important information for disordered crystalline materials also.
Issues with the RMC method
The RMC method suffers from a number of potential problems. The most notable problem is that often more than one qualitatively different model will give similar agreement with experimental data. For example, in the case of amorphous silicon, the integral of the first peak in the pair distribution function may imply an average atomic coordination number of 4. This might reflect the fact that all atoms have coordination number of 4, but similarly having half the atoms with coordination number of 3 and half with 5 will also be consistent with this data. Unless a constraint on the cordination number is employed, the RMC method will have no means of generating a unique coordination number and most likely a spread of coordination numbers will result. Since the RMC method follows the normal rules of statistical mechanics, its final solution will be the one with the highest degree of disorder (entropyEntropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
) possible. A second problem comes from the fact that without constraints the RMC method will typically have more variables than observables. One result from this will be that the final atomic configuration may have artefacts that arise from the method attempting to fit noise in the data.
One should remark, however, that most applications of the RMC approach today take account of these problems by appropriate use of implicit or explicit constraints.
Implementations of the RMC method
There are two publicly-available implementations of the RMC method.RMCProfile
RMCProfile is a significantly developed version of the original RMC code, written in Fortran 95 with some Fortran 2003Fortran
Fortran is a general-purpose, procedural, imperative programming language that is especially suited to numeric computation and scientific computing...
features. It has maintained the ability to model liquids and amorphous materials using the pair distribution function, total scattering and EXAFS
Extended X-Ray Absorption Fine Structure
X-ray Absorption Spectroscopy includes both Extended X-Ray Absorption Fine Structure and X-ray Absorption Near Edge Structure . XAS is the measurement of the x-ray absorption coefficient of a material as a function of energy...
data, but also includes the capability of modelling crystalline materials by explicitly using the information contained within the Bragg diffraction data. RMCProfile gives users a range of constraints, including the inclusion of molecular potentials and distance windows, which exploit possibilities afforded by the lack of significant diffusion in crystalline materials. RMCProfile allows simulation of magnetic materials, using the magnetic component of total scattering data, and also allows simulation of materials where atoms are allowed to swap positions (as found in many solid solution
Solid solution
A solid solution is a solid-state solution of one or more solutes in a solvent. Such a mixture is considered a solution rather than a compound when the crystal structure of the solvent remains unchanged by addition of the solutes, and when the mixture remains in a single homogeneous phase...
s).
RMC++
RMC++ a rewritten version of the original RMC code in C++ . RMC++ is designed specifically for the study of liquids and amorphous materials, using pair distribution function, total scattering and EXAFSExtended X-Ray Absorption Fine Structure
X-ray Absorption Spectroscopy includes both Extended X-Ray Absorption Fine Structure and X-ray Absorption Near Edge Structure . XAS is the measurement of the x-ray absorption coefficient of a material as a function of energy...
data.