Molecular dynamics integration scheme

Move the nuclei according to the molecular dynamics integration scheme. 

Abstract. The overall Hamiltonian structure of the Quantum-Classical Molecular Dynamics model makes - analogously to classical molecular dynamics - symplectic integration schemes the methods of choice for long-term simulations. This has already been demonstrated by the symplectic PICKABACK method [19]. However, this method requires a relatively small step-size due to the high-frequency quantum modes. Therefore, following related ideas from classical molecular dynamics, we investigate symplectic multiple-time-stepping methods and indicate various possibilities to overcome the step-size limitation of PICKABACK. 

Abstract. We present novel time integration schemes for Newtonian dynamics whose fastest oscillations are nearly harmonic, for constrained Newtonian dynamics including the Car-Parrinello equations of ab initio molecular dynamics, and for mixed quantum-classical molecular dynamics. The methods attain favorable properties by using matrix-function vector products which are computed via Lanczos method. This permits to take longer time steps than in standard integrators. 

In this paper we present a number of time integrators for various problems ranging from classical to quantum molecular dynamics. These integrators share some common features they are new, they are second-order accurate and time-reversible, they improve substantially over standard schemes in well-defined model situations — and none of them has been tested on real applications at the time of this writing. This last feature will hopefully change in the near future [20]. 

Similiar problems are known in classical MD simulations, where intramolecular and intermolecular dynamics evolve on different time scales. One possible solution to this problem is the method of multiple time scale propagators which is describede in section 5. Berne and co-workers [21] first used different time steps to integrate the intra- and intermolecular degrees of freedom in order to reduce the computational effort drastically. The method is based on a Trotter-factorization of the classical Liouville-operator for the time evolution of the classical system, resulting in a time reversible propagation scheme. The multiple time scale approach has also been used to speed up Car-Parrinello simulations [20] and ab initio molecular dynamics algorithms [21]. 

Molecular dynamics examines the temporal evolution of a collection of atoms on the basis of an explicit integration of the equations of motion. From the point of view of diffusion, this poses grave problems. The time step demanded in the consideration of atomic motions in solids is dictated by the periods associated with lattice vibrations. Recall our analysis from chap. 5 in which we found that a typical period for such vibrations is smaller than a picosecond. Hence, without recourse to clever acceleration schemes, explicit integration of the equations of motion demands time steps yet smaller than these vibrational periods. 

Despite the reservations set down above, to carry out a molecular dynamics study of the diffusion process itself one resorts to a computational cell of the type described earlier. The temperature is assigned and maintained via some scheme such as the Nose thermostat (Frenkel and Smit, 1996), and the atomic-level trajectories are obtained via a direct integration of the equations of motion. In fig. 3.22, we showed the type of resuiting trajectories in the case of surface... 

As an intermediate between deterministic integration and Monte-Carlo, one may consider Hybrid Monte-Carlo (HMC) methods [110, 278]. These schemes use, for the proposal distribution, a moderate length path obtained from deterministic molecular dynamics (usually computed by the Verlet method). Thus, at each step of an HMC scheme, we obtain s 1 timesteps... 

In the course of time, however, a rather sophisticated scheme has developed of quantitative treatments of solute-solvent interactions in the framework of LSERs. The individual parameters employed were imagined to correspond to a particular solute-solvent interaction mechanism. Unfortunately, as it turned out, the various empirical polarity scales feature just different blends of fundamental intermolecular forces. As a consequence, we note at the door to the twenty-first century, alas with melancholy, that the era of combining empirical solvent parameters in multiparameter equations, in a scientific context, is beginning to fade away. As a matter of fact, solution chemistry researeh is increasingly being occupied by theoretical physics in terms of molecular dynamics (MD) and Monte Carlo (MC) simulations, the integral equation approach, etc. 

One of the most common numerical methods used in molecular dynamics to solve Newton s equations of motions is the Velocity Verlet integrator. This is typically implemented as a second order method, and we find that it can become numerically tmstable dtuing the course of hyperthermal collision events, where the atom velocities are often far from equilibrium. As an alternative, we have implemented a fifth/ sixth order predictor-corrector scheme for our calculations. Specifically, the driver we chose utilizes the Adams-Bashforth predictor method together with the Adams-Moulton corrector method for approximating the solution to the equations of motion. 

Skeel RD (1999) Integration schemes for molecular dynamics and related applications. In Ainsworth M, Levesley J, Marietta M (eds) The graduate student s guide to numerical analysis. Springer, New York, pp 119-176... 

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