Machine-learning of atomic-scale properties based on physical principles

We briefly summarize the kernel regression approach, as used recently in materials modelling, to fitting functions, particularly potential energy surfaces, and highlight how the linear algebra framework can be used to both predict and train from linear functionals of the potential energy, such as the total energy and atomic forces. We then give a detailed account of the Smooth Overlap of Atomic Positions (SOAP) representation and kernel, showing how it arises from an abstract representation of smooth atomic densities, and how it is related to several popular density-based representations of atomic structure. We also discuss recent generalisations that allow fine control of correlations between different atomic species, prediction and fitting of tensorial properties, and also how to construct structural kernels---applicable to comparing entire molecules or periodic systems---that go beyond an additive combination of local environments

On the short-time limit of ring polymer molecular dynamics.

We examine the short-time accuracy of a class of approximate quantum dynamical techniques that includes the centroid molecular dynamics (CMD) and ring polymer molecular dynamics (RPMD) methods. Both of these methods are based on the path integral molecular dynamics (PIMD) technique for calculating the exact static equilibrium properties of quantum mechanical systems. For Kubo-transformed real-time correlation functions involving operators that are linear functions of positions or momenta, the RPMD and (adiabatic) CMD approximations differ only in the choice of the artificial mass matrix of the system of ring polymer beads that is employed in PIMD. The obvious ansatz for a general method of this type is therefore to regard the elements of the PIMD (or Parrinello-Rahman) mass matrix as an adjustable set of parameters that can be chosen to improve the accuracy of the resulting approximation. We show here that this ansatz leads uniquely to the RPMD approximation when the criterion that is used to select the mass matrix is the short-time accuracy of the Kubo-transformed correlation function. In particular, we show that the leading error in the RPMD position autocorrelation function is O(t(8)) and the error in the velocity autocorrelation function is O(t(6)), for a general anharmonic potential. The corresponding errors in the CMD approximation are O(t(6)) and O(t(4)), respectively

Sum rule constraints on Kubo-transformed correlation functions

We show how the Kubo transform can be inverted in the time domain and then use this result to investigate the sum rule constraints on a Kubo-transformed correlation function c̃AB(t)=1β∫0βdλ〈A(- iλℏ)B(t)〉 that arise from the values of the static equilibrium properties cAB(n)(0)=[dn〈A(0)B(t)〉/dtn]t=0. We find, perhaps not surprisingly, that these sum rules only depend on the behavior of c̃AB(t) for times on the order of βℏ. The implications of this finding are discussed in light of the recent use of these sum rules to assess the quality of approximate Kubo-transformed correlation functions for liquid hydrogen at 14 K and liquid water at 298 K. © 2005 Elsevier B.V. All rights reserved