Construction of high-order force-gradient algorithms for integration of motion in classical and quantum systems

Abstract

A consequent approach is proposed to construct symplectic force-gradient algorithms of arbitrarily high orders in the time step for precise integration of motion in classical and quantum mechanics simulations. Within this approach the basic algorithms are first derived up to the eighth order by direct decompositions of exponential propagators and further collected using an advanced composition scheme to obtain the algorithms of higher orders. Contrary to the scheme by Chin and Kidwell [Phys. Rev. E 62, 8746 (2000)], where high-order algorithms are introduced by standard iterations of a force-gradient integrator of order four, the present method allows to reduce the total number of expensive force and its gradient evaluations to a minimum. At the same time, the precision of the integration increases significantly, especially with increasing the order of the generated schemes. The algorithms are tested in molecular dynamics and celestial mechanics simulations. It is shown, in particular, that the efficiency of the new fourth-order-based algorithms is better approximately in factors 5 to 1000 for orders 4 to 12, respectively. The results corresponding to sixth-and eighthorder-based composition schemes are also presented up to the sixteenth order. For orders 14 and 16, such highly precise schemes, at considerably smaller computational costs, allow to reduce unphysical deviations in the total energy up in 100 000 times with respect to those of the standard fourth-order-based iteration approach.

Numerical integration of variational equations

“…Symplectic integrators have already been applied successfully for the accurate integration of motion in multi-dimensional systems which are related for example, to problems of astronomical interest (e. g. [37]), of molecular dynamics (e. g. [29,38]) and dynamics of nonlinear lattices (e. g. [27]). Using the TM method these symplectic integration schemes can be extended to integrate also the corresponding variational equations.…”

“…The SBAB (SABA) integrators have already proved to be very efficient for the numerical study of different dynamical systems [21,26,27]. We note that several authors have used commutators for improving the efficiency of symplectic integrators (e. g. [28,29]). Setting ǫ = 1 we can apply the SBAB (SABA) integration schemes for the integration of Hamiltonian (5), since this Hamiltonian can be written as H = A + B, with…”

Stability of molecular dynamics simulations of classical systems

“…One might, however, avoid this overhead of computer time of the forces by using a higherorder force-gradient algorithm. 27 The limit of stability of the MD simulation can be extended with a factor of 2 behind the limit of stability of the energy by using a thermostat which removes the (energy) errors caused by the few ill-integrated modes. In the case of the KABLJ system the time-increment can be increased in total by a factor of eight from the traditional (conservative) choice h = 0.0025 18 to h = 0.020 without any significant impact on the equilibrium behavior and the mean square displacements (Figure 6), but the system collapses sooner or later for a larger time-increment.…”

Stiff convergence of force-gradient operator splitting methods

“…This way of computing the ground state is widely used by chemists, who call it imaginary time propagation. Force-gradient methods of order higher than four have been derived, e.g., by Omelyan et al [21]. These methods do however have some negative coefficients.…”