GULP

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Methods The starting point for the majority simulation techniques is the calculation of the energy, and so will it be for this article. Most methods are based around the initial determination of the internal energy, with subsequent treatment of the nuclear degrees of freedom in order to determine the appropriate free energy to the ensemble of interest. In principle, the internal energy of a solid is a manybody quantity that explicitly depends upon the positions and momenta of all electrons and nuclei. However, this is an intractable problem to solve at any level of theory, and thus approximations must be made to simplify the situation. To tackle this we assume that the effect of the electrons will largely be subsumed into an effective atom, and that the energy can be decomposed into an expansion in terms of interactions between different subsets of the total number of atoms, $N$: $$U=\sum_{i=1}^{N}U_{i}+\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N... ...rac{1}{6}\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{N}U_{ijk}+....$$ where the first term represents the self energies of the atoms, the second the pairwise interaction, etc. This decomposition is exact if performed to a high enough order. However, we know that the contribution from higher order terms becomes progressively smaller for most systems, and so we choose to neglect the terms beyond a certain point and introduce a degree of parameterisation of the remaining terms to compensate. Justification for this is forthcoming from quantum mechanics. It is well known that the Hartree-Fock method is a reasonable first approximation for the description of many systems, albeit with a systematic quantitative error for most observables. Here the highest term included is a four-centre integral, which indicates that including up to four-body terms should be reasonable approach, which is indeed found to be the case for most organic systems, for example. Furthermore, it is intuitively obvious that the further apart two atoms are, the weaker their interaction will be. Thus the introduction of distance cut-offs is a natural way to simplify the computational task. The form of the explicit interaction between atoms is usually chosen based on physical insights as to the nature of the forces between the particles. For instance, if considering a covalent diatomic molecule the natural representation of the potential energy surface would be a Morse potential since this is harmonic at the minimum and leads to dissociation at large bond lengths, in accord with spectroscopic observation. In the following sections we will review some of the common types of potential that are widely used, as well as some novel approaches which point towards the future of forcefield methods.