Symplectic integrators

R. D. Skeel, G. H. Zhang, and T. Schlick. A family of symplectic integrators Stability, accuracy, and molecular dynamics applications. SIAM J. Scient. COMP., 18 203-222, 1997. 

Robert D. Skeel, Jeffrey J. Biesiadecki, and Daniel Okunbor. Symplectic integration for macromolecular dynamics. In Proceedings of the International Conference Computation of Differential Equations and Dynamical Systems. World Scientific Publishing Co., 1992. in press. 

Robert D. Skeel and Jeffrey J. Biesiadecki. Symplectic integration with variable stepsize. Ann. Num. Math., 1 191-198, 1994. 

G. Benettin and A. Giorgilli. On the Hamiltonian interpolation of near to the identity symplectic mappings with applications to symplectic integration algorithms. J. Stat. Phys. 74 (1994)... 

This discussion suggests that even the reference trajectories used by symplectic integrators such as Verlet may not be sufficiently accurate in this more rigorous sense. They are quite reasonable, however, if one requires, for example, that trajectories capture the spectral densities associated with the fastest motions in accord to the governing model [13, 15]. Furthermore, other approaches, including nonsymplectic integrators and trajectories based on stochastic differential equations, can also be suitable in this case when carefully formulated. 

M. Zhang and R. D. Skeel. Cheap implicit symplectic integrators. Applied Numerical Mathematics, 25 297-302, 1997. 

D. Janezid and F. Merzel. An efficient symplectic integration algorithm for molecular dynamics simulations. J. Chem. Info. Comp. Set, 35 321-326, 1995. 

B. Leimkuhler and R. Skeel. Symplectic integrators in constrained Hamiltonian systems. J. Comp. Phys., 112 117-125, 1994. 

Thus these integrators are measure preserving and give trajectories that satisfy the Liouville theorem. [12] This is an important property of symplectic integrators, and, as mentioned before, it is this property that makes these integrators more stable than non-symplectic integrators. [30, 33]... 

This article is organized as follows Sect. 2 explains why it seems important to use symplectic integrators. Sect. 3 describes the Verlet-I/r-RESPA impulse MTS method, Sect. 4 presents the 5 femtosecond time step barrier. Sect. 5 introduce a possible solution termed the mollified impulse method (MOLLY), and Sect. 6 gives the results of preliminary numerical tests with MOLLY. 

Symplectic integration methods replace the t-flow (pt,H by the symplectic transformation which retains Hamiltonian features of They poses a backward error interpretation property which means that the computed solutions are solving exactly or, at worst, approximately a nearby Hamiltonian problem which means that the points computed by means of symplectic integration, lay either exactly or at worst, approximately on the true trajectories [5]. 

The explicit symplectic integrator can be derived in terms of free Lie algebra in which Hamilton equations (5) are written in the form... 

The Lie formalism used is the key in the development of symplectic integration. Symplectic integration consists in replacing by a product... 

Sanz-Serna, J. M. Symplectic Integrators for Hamiltonian Problems An Overview. Acta Numerica (1991) 243-286... 

Janezic, D., Merzel, F. An Efficient Symplectic Integration Algorithm for Molecular Dynamics Simulations. J. Chem. Inf. Comput. Sci. 35 (1995) 321-326... 

Forest, E., Ruth, R. D. Fourth-Order Symplectic Integration. Phys. D 43 (1990) 105-117... 

S. Reich, Symplectic integrators for systems of rigid bodies. Fields Institute Communications, 10, 181-191 (1996). 

A convenient and constructive approach to attain symplectic maps is given by the composition of symplectic maps, which yields again a symplectic map. For appropriate Hk, the splittings (6) and (7) are exactly of this form If the Hk are Hamiltonians with respect to the whole system, then the exp rLnk) define the phase flow generated by these Hk- Thus, the exp TL-Hk) are symplectic maps on the whole phase space and the compositions in (6) and (7) are symplectic maps, too. Moreover, in order to allow for a direct numerical realization, we have to find some Hk for which either exp(rL-Kfc) has an analytic solution or a given symplectic integrator. 

Fig. 1. Total energy (in kj/mol) versus time (in fs) for different integrators for a collinear collision of a classical particle with a harmonic quantum oscillator (for details see [2]). Dashed line Nonsymplectic scheme. Dotted Symplectic integrator of first order. Solid PICKABACK (symplectic, second order).

P. Nettesheim, F. A. Bornemann, B. Schmidt, and Ch. Schiitte An explicit and symplectic integrator for quantum-classical molecular dynamics. Chem. Phys. Lett. 256 (1996) 581-588... 

H. Yoshida Construction of higher order symplectic integrators. Physics Letters A 150 (1990) 262-268... 

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. 

In this paper, we consider the symplectic integration of the so-called Quantum-Classical Molecular Dynamics (QCMD) model. In the QCMD model (see [11, 9, 2, 3, 6] and references therein), most atoms are described by classical mechanics, but an important small portion of the system by quantum mechanics. This leads to a coupled system of Newtonian and Schrddinger equations. 

The last step is to find a symplectic, second order approximation st to exp StL ). In principle, we can use any symplectic integrator suitable for time-dependent Schrddinger equations (see, for example, [9]). Here we focus on the following three different possibilities corresponding to special properties of the spatially truncated operators H q) and V q). 

S.K. Gray and D..E. Manolopoulos. Symplectic integrators tailored to the time-dependent Schrddinger equation. J. Chem. Phys., 104 7099-7112, 1996. 

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