An efficient time-stepping scheme for ab initio molecular dynamics simulations

In ab initio molecular dynamics simulations of real-world problems, the simple Verlet method is still widely used for integrating the equations of motion, while more efficient algorithms are routinely used in classical molecular dynamics. We show that if the Verlet method is used in conjunction with pre- and postprocessing, the accuracy of the time integration is significantly improved with only a small computational overhead. The validity of the processed Verlet method is demonstrated in several examples including ab initio molecular dynamics simulations of liquid water. The structural properties obtained from the processed Verlet method are found to be sufficiently accurate even for large time steps close to the stability limit. This approach results in a 2x performance gain over the standard Verlet method for a given accuracy.Comment: 32 pages, 11 figure

Integrable discretization of the vector/matrix nonlinear Schr\"odinger equation and the associated Yang-Baxter map

The action of a B\"acklund-Darboux transformation on a spectral problem associated with a known integrable system can define a new discrete spectral problem. In this paper, we interpret a slightly generalized version of the binary B\"acklund-Darboux (or Zakharov-Shabat dressing) transformation for the nonlinear Schr\"odinger (NLS) hierarchy as a discrete spectral problem, wherein the two intermediate potentials appearing in the Darboux matrix are considered as a pair of new dependent variables. Then, we associate the discrete spectral problem with a suitable isospectral time-evolution equation, which forms the Lax-pair representation for a space-discrete NLS system. This formulation is valid for the most general case where the two dependent variables take values in (rectangular) matrices. In contrast to the matrix generalization of the Ablowitz-Ladik lattice, our discretization has a rational nonlinearity and admits a Hermitian conjugation reduction between the two dependent variables. Thus, a new proper space-discretization of the vector/matrix NLS equation is obtained; by changing the time part of the Lax pair, we also obtain an integrable space-discretization of the vector/matrix modified KdV (mKdV) equation. Because B\"acklund-Darboux transformations are permutable, we can increase the number of discrete independent variables in a multi-dimensionally consistent way. By solving the consistency condition on the two-dimensional lattice, we obtain a new Yang-Baxter map of the NLS type, which can be considered as a fully discrete analog of the principal chiral model for projection matrices.Comment: 33 pages; (v2) minor corrections (v3) added one paragraph on a space-discrete matrix KdV equation at the end of section