Reverse Monte Carlo methods
Most modelling techniques in the solid-state sciences are what would be called forward modelling. The starting point is an exact expression for a Hamiltonian or free energy, together with a way of using this to generate numerical quantities that can be compared with experiment. The starting expression could be a representation of the Schrödinger equation, or a numerical approximation to the forces between atoms. The models can also be based on differential equations to incorporate time-dependence (e.g. the Lagrangian of classical mechanics), or the partition function to obtain thermodynamic averages. The solutions to the models can be exact in some cases (particularly when the models are used to describe a system at zero temperature), but may frequently have some degree of statistical uncertainty. For example, in molecular dynamics, the dynamic equations are solved using discrete time steps over a limited time span, and in Monte Carlo simulations only a portion of the total phase space available to a system is sampled. Whatever approach is taken, the end point of forward modelling may be to provide new insights, to predict behaviour, or to interpret observations. A key point towards this end is to be able to reproduce some experimental data in order to check that the simulations are representative of reality, even though the true value of a simulation is to provide information, insight or understanding that cannot be extracted from experimental data.
In this chapter we will consider a second type of modelling technique, which is called inverse modelling. This is best described by comparison with forward modelling.
Figures & Tables
The present book shows the arguments which have been considered in the EMU School (No. 4), dealing with Energy Modelling in Minerals; these arguments have been selected in order to provide examples of application of the most advanced theories to several cases. It should be pointed out that although the ultimate solution of our problems should involve “ab initio” quantum-mechanical calculations, at present such sophisticated procedures are far from being routine. Therefore, although “ab initio” approaches will play an ever-increasing role in the future and some important and most recent examples of such approaches are illustrated here, the greatest part of the contributions is dealing with empirical atom-atom calculations. Remarkably enough, such “semi-empirical” applications are often quite successful, providing excellent (or comparatively excellent) results in spite of their more or less approximate nature. It often happens that the methods here illustrated are some steps ahead of the current level of empirical treatment, thereby indicating a possible way of improvement by figuring out routines to be adopted in practice. If some methods seem to be too speculative to be actually usable, here they also are shown, in view of their possible discussion, or just to indicate a way to obtain promising developments. Among the descriptions of practical methods and results, some purely theoretical arguments have been inserted; these arguments — although abstract — according to our opinion are fundamental for earth scientists. Owing to the present status of the art, in a number of arguments there is no unique opinion with respect to their theoretical treatment as it is explained by different authors. Instead of having all of them discarded except the one which looks to be the most appropriate to the Editor (who might sometimes be personally involved in the question), most of such controversial points have been left just as they are, in the original draft of their advocates. Accordingly, the reader might find some discrepancies between some articles and others, which may lead to some obscurity; there are, however, several good reasons in favour of our behaviour. First of all, with a few exceptions we apologize about, our attention in inviting the contributors has been extended to all the principal authors in the world, with no limitation to a group of particular friends; moreover, the presence of different opinions in the context might give rise to interesting debates and critical objections; a further point is that the validity of the different treatments is shown per se by either the level of the theory and most of all by the agreement with the corresponding experimental data. Since we have to do with an advanced school, and in line with what should be a scientific procedure, it is important to provide the user with the possibility of choosing what seems to be the most appropriate method among a number of selected possibilities, rather than yielding to the assertion that something is indeed the ultimate and unquestionable “truth”.