A leap-frog algorithm-based efficient unified thermostat scheme for molecular dynamics

Abstract

An important challenge for modern molecular simulations is to develop robust algorithms for molecular dynamics (MD), which employ a time interval (i.e., step size) of evolution as large as possible to achieve high efficiency while maintaining accuracy for sampling the distribution of configuration and momentum, e.g., for the canonical ensemble where the number of particles ( N ), the volume ( V ), and the temperature ( T ) are constant. Starting from the velocity-Verlet algorithm, in the recent paper we have proposed a unified scheme—the middle scheme for constructing efficient thermostatting integrators for configurational sampling of the canonical ensemble. In the present paper we extend the middle scheme to derive new thermostatting integrators based on the leap-frog algorithm. The leap-frog middle scheme is able to recover the exact marginal distribution of configuration and that of momentum in the harmonic limit, regardless of the time interval as long as the evolution of the MD trajectory is stable. It is also proved that the leap-frog middle scheme leads to the same marginal distribution of configuration as the velocity- Verlet middle scheme does even for general anharmonic systems. As illustrated in numerical examples (such as a 1-dimensional quartic potential, liquid water at room temperature with or without bond-length constraints, etc.), while both middle schemes perform better than conventional approaches (e.g., the side scheme and the default Langevin dynamics method in AMBER) and can calculate configuration-dependent properties with the same accuracy by using a larger time interval for from at least 4 times to even an order of magnitude, the leap-frog middle scheme produces a more accurate marginal distribution of momentum than that the velocity-Verlet middle scheme does. The sampling efficiency analysis by employing the characteristic correlation time step number (of the potential or of the Hamiltonian) suggests that both middle schemes share the same efficiency in sampling the configuration space and lead to similar efficiency in sampling the whole phase space (i.e., both configuration and momentum). When both only-configuration- dependent and only-momentum-dependent physical observables are considered, we suggest that the leap-frog middle scheme should in principle offer a better approach. When we study properties involving cross-terms of configuration and momentum, e.g., the fluctuation of the Hamiltonian (related to the heat capacity) and the energy current and its fluctuation (related to the thermal conductivity), analytic results in the harmonic limit and numerical data for anharmonic systems indicate that which of the two middle schemes performs better depends on the system and on the specific property. Note that the leap-frog middle scheme is of first-order accuracy for sampling the whole phase space while the velocity-Verlet middle scheme is of second-order accuracy. It is reasonable to choose the velocity-Verlet middle scheme to perform simulations for such properties for general molecular systems. Because either the leap-frog or the velocity-Verlet algorithm is the default integrator in molecular dynamics simulation software packages, the two unified middle schemes can offer novel, simple, accurate, and efficient thermostatting integrators for performing MD (or path integral MD (PIMD)) for the classical (or quantum) canonical ensemble.