Gaussian Approximation Potentials

As in the case of other interatomic potentials, we base Gaussian Approximation Potentials on the assumption that the total energy of the system can be written as a 

The strict localization of c enables the independent computation of atomic energies. 

The Gaussian Approximation Potential scheme is similar to the Neural Network potentials introduced by Behler and Parinello [33], as both uses non-linear, non-parametric regression instead of fixed analytic forms. However, the representation of the atomic environments in GAP is complete and the Gaussian Process uses energies and forces for regression. Moreover, the training of the neural 

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A. Bart6k-Paitay, The Gaussian Approximation Potential, Springer Theses,... 

This approach for generating interatomic potentials, which we collectively refer to as Gaussian Approximation Potentials, has the favourable scaling and speed of analytic potentials, while the accuracy is comparable with the underlying quantum mechanical method. With Gaussian Approximation Potentials atomistic simulations can be taken to an entirely new level. 

Our first application of the Gaussian Approximation Potentials was a set of potentials for simple semiconductors. We calculated the total energies and forces of a number of configurations, which were generated by randomly displacing atoms in the perfect diamond structure. We included 8-atom and 64-atom supercells at different lattice constants and we perturbed the lattice vectors in some cases. The atoms were displaced at most by 0.3 A. 

DFT force components and the distribution of these differences is also displayed. The force and energy evaluation with the Gaussian Approximation Potential for diamond, in the current implementation, is about 4,000 times faster than Density Functional Theory in the case of a 216-atom supercell. 

Figure 6.7 shows the force errors of three Gaussian Approximation Potential models for diamond with cutoffs of 2.0, 2.75 and 3.7 A. The difference between the latter two models is negligible. However, the elastic moduli calculated from... 

The Gaussian Approximation Potential scheme is not limited to simple semiconductors. We demonstrate this by applying the scheme to a metallic system, namely the body-centred cubic (bcc) phase of iron. We included configurations in the training set where the lattice vectors of the one-atom primitive cell were randomised and where the positions of the atoms in 8 and 16-atom supercells were also... 

So far our tests of the Gaussian Approximation Potentials were limited to singlespecies systems, but the framework can be extended to multispecies systems. Here we report our first attempt to model such a system, the cubic phase of gallium nitride. Gallium nitride (GaN) is a two-component semiconductor with a wurtzite or zinc-blende structure. There is a charge transfer between the two species. 

To show that our scheme is not hmited to describing monoatomic semiconductors, I generated a potential for bcc iron, a metal, and for gallium nitride, an ionic semiconductor. Our preliminary tests, which were the comparison of the phonon dispersion and the elastic moduli with Density Functional Theory values, demonstrate that GAP models can easily be built for different kinds of materials. I also suggest that the Gaussian Approximation Potentials can be generated on the fly and used as auxiliary tools for example, in structure search applications. 

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