On (Andersen-)Parrinello-Rahman molecular dynamics, the related metadynamics, and the use of the Cauchy-Born rule (Q5962115)

The paper discusses scale-bridging criteria, establishing the link between molecular dynamics (MD) and continuum mechanics (CM), implicit in the version of MD known as the Parrinello-Rahman method, a version first proposed by Andersen in 1980. The following three aims of the papers are: (i) to discuss circumstances under which the kinetic energy (that is, the Lagrangian) takes exactly the form postulated in the paper of \textit{M. Parrinello} and \textit{A. Rahman} [``Polymorphic transitions in single crystals: A new molecular dynamics method'', J. Appl. Phys. 52, 7182--7190 (1981)]; (ii) to show how one of the assumptions (namely, irrotationality of cell motion) should be incorporated in the metadynamics governing the energy landscape of a crystalline substance or chemical reactions along paths connecting local minima; (iii) to point out a connection between zero-temperature MD and the Cauchy-Born rule for an elastic stored-energy mapping consistent with a given intermolecular potential. Using a cell geometry, the kinetic energy of the molecular population is examined and compared with the expression for the kinetic energy postulated in the above-cited paper by Parrinello and Rahman. Then, necessary and sufficient conditions are discussed for the identity of these energies for all admissible processes of a given molecular population occupying a given MD cell. On the base of a positional potential, the author derives the Lagrangian equations of motion and gives definitions of external-stress measures and internal stress. The consistent metadynamics (that is, a powerful molecular dynamics method for the study of pressure-induced structural transformations) are discussed. It is shown that, in order to prevent box rotations, one should assume that the tensor describing the cell deformation, in addition to its symmetry, is positive-definite. Finally, the author considers at zero temperature the Cauchy-Born rule being the standard recipe for computing a continuum -- mechanical stored-energy mapping corresponding to a given atomistic potential. The author derives a consistency condition between the ``microscopic'' intermolecular potential and the ``macroscopic'' stored-energy mapping. It is shown that the elasticity tensor is expressible (and computable at any instant of a MD run) in terms of the intermolecular potential.