The Reverse Monte Carlo method

Random structure methods have proved useful in solving structures from X-ray powder diffraction patterns. The unit cell can usually be found from these patterns, but the normal single-crystal techniques for solving the structure cannot be used. A variation on this technique, the reverse Monte Carlo method, includes in the cost function the difference between the observed powder diffraction pattern and the powder pattern calculated from the model (McGreevy 1997). It is, however, always necessary to include some chemical information if the correct structure is to be found. Various constraints can be added to the cost function, such as target coordination numbers or the deviation between the bond valence sum and atomic valence (Adams and Swenson 2000b Swenson and Adams 2001). 

Tucker MG, Dove MT, Keen DA (2000) Application of the Reverse Monte Carlo method to ciystalhne materials. Journal of Applied Crystallography (submitted)... 

The Reverse Monte Carlo method was initially proposed by McGreevy and Pustzai [1]. The idea is to generate an atomic configuration of a system that matches the structural properties of the real system obtained by experiment. Throughout the simulation the differences between the simulation and experimental structural properties are minimized. The most conunonly used... 

In the Reverse Monte Carlo (RMC) method [5], the pair correlation function or the structure factor is calculated after each random move (Ssim(<]) or gsimfr)) and compared to the respective target function obtained from experimental diffraction data (Sexp(q) or gexp(r)). It is possible to calculate Ssm(q) with full periodicity from the atomic positions. This method is best in principle [10], but the computational cost is much greater than for any of the other available methods. It is also possible to obtain Ssm(q) by first calculating gsm(r) from the atomic positions and then Fourier transform this function and calculate Ssim(q). The disadvantage of this approach is that there is an additional computational cost associated with the Fourier transform of gsm(r) after each move. 

This paper does not intend to be a review rather comments and examples are given for some of the recent progress. The related literature is not searched exhaustively and the selection is rather arbitrary. Preliminary results of two new studies by the XD method are presented in order to demonstrate the capabilities of the method at new conditions. The reverse Monte Carlo (RMC) technique is also discussed in more detail to show a new perspective in the structural modelling of solutions. 

In spite of the great success of the computer simulation methods in the determination of the microscopic properties of the solutions, the capacity of the traditional MD and MC simulations is always limited by the choice of the suitable potential functions to describe the interatomic interactions. The potentials are most often checked by comparison of the structural properties calculated from the simulation with those determined experimentally. The reverse Monte Carlo (RMC) method, developed by McGreevy and Pusztai [41] does not rely upon knowledge of any interaction potential, instead it generates a large set of atomic configurations on the condition that the difference between the experimental and calculated structure functions (or pair-distribution functions) should be minimum. The same structural... 

The main advance in recent years has been the development of methods to obtain models of structures that are consistent with the total diffraction pattern. One method is the Reverse Monte Carlo (RMC) method (McGreevy and Pusztai 1988, McGreevy 1995, Keen 1997, 1998). In this method, the Monte Carlo technique is used to modify a configuration of atoms in order to give the best agreement with the data. This can be carried out using either S Q) or T(r) data, or both simultaneously. We also impose a... 

Proffen T, Welbeny TR (1997) Analysis of diffuse scattering via the reverse Monte Carlo technique A systematic investigation. Acta Crystallogr Sect A 53 202-216 Proffen T, Welbeny TR (1998) Analysis of diffuse scattering of single ciystals using Monte Carlo methods. Phase Transit 67 373-397... 

In the past four decades, we have witnessed the significant development of various methods to describe microporous solids because of their important contribution to improving of adsorption capacity and separation. Various models of different complexity have been developed [5]. Some models have been simple with simple geometry, such as slit or cylinder, while some are more structured such as the disk model of Segarra and Glandt [6]. Recently, there has been great interest in using the reverse Monte Carlo (MC) simulation to reconstruct the carbon structure, which produces the desired properties, such as the surfece area and pore volume [7, 8]. Much effort has been spent on studies of characterization of porous media [9-15]. In this chapter we will briefly review the classical approaches that still bear some impact on pore characterization, and concentrate on the advanced tools of density functional theory (DFT) and MC, which currently have wide applications in many systems. 

Unfortunately, there are no suitable isotopes for each element good elements are, for instance, chlorine and nickel. In many cases, differences between coefficients are not big enough for the matrix inversion i.e., the set of equations is ill defined. Sometimes, application of the Reverse Monte Carlo modeling method can help in separating partial structure... 

One way to avoid assuming a potential is the Reverse Monte Carlo numerical method (Nield et al., 1991). This technique employs an algorithm... 

The distribution function obtained is in good agreement both with the results in Refs. [7-9, 16], in which the ab-initio calculation methods were used, and with the results of the simulation by the reverse Monte-Carlo simulation [6]. In addition, it should be noted that the obtained results weakly depend on simulated d-metal (Fe, Ni, Cu, Au). The above allows to conclude that the melt local cluster structure is determined by central short-range repulsive forces to a greater extent and is universal for d-met-als with close-packed melting premelting structure. 

Ti02 itself does not form glass, but hydrated titania or the xerogel which is produced from titanium alkoxides or salts through the sol-gel reaction is usually amorphous. Petkov et al. (1998) reported a stmctural analysis on such titania xerogels. Reverse Monte Carlo method was employed to simulate the X-ray RDF curve. In the structure model proposed... 

Opletal G., Structural simulations using the hybrid reverse Monte Carlo method, PhD Thesis, RMIT University (2005). 

Instead of calculations, practical work can be done with scale models (33). In any case, calculations should be checked wherever possible by experimental methods. Using a Monte Carlo method, for example, on a shape that was not measured experimentaUy, the sample size in the computation was aUowed to degrade in such a way that the results of the computation were inaccurate (see Fig. 8) (30,31). Reversing the computation or augmenting the sample size as the calculation proceeds can reveal or eliminate this source of error. 

In all these examples, the importance of good simulation and modeling cannot be stressed enough. A variety of methods have been used in this field to simulate the data in the cases studies described above. Blander et al. [4], for example, used a semi-empirical molecular orbital method, MNDO, to calculate the geometries of the free haloaluminate ions and used these as a basis for the modeling of the data by the RPSU model [12]. Badyal et al. [6] used reverse Monte Carlo simulations, whereas Bowron et al. [11] simulated the neutron data from [MMIM]C1 with the Empirical Potential Structure Refinement (EPSR) model [13]. 

Swenson, J. and Adams, St. (2001). The application of the bond valence method to reverse Monte Carlo produced structural models of superionic glasses. Phys. Rev. B64, 024204. 

In general, Monte Carlo simulations are such calculations in which the values of some parameters are determined by the average of some randomly generated individuals.45-54 In chemistry applications, the most prevalent methods are the so called Metropolis Monte Carlo (MMC)55 and Reverse Monte Carlo (RMC) ones. The most important quantities in these methods are some kinds of U energy-type potentials (e.g. internal energy, enthalpy,... 

---