Numerical Integrators

At its most basic level, molecular dynamics is about mapping out complicated point sets using trajectories of a system of ordinary differential equations (or, in Chaps. 6-8, a stochastic-differential equation system). The sets are typically defined as the collection of probable states for a certain system. In the case of Hamiltonian dynamics, they are directly associated to a region of the energy landscape. The trajectories are the means by which we efficiently explore the energy surface. In this chapter we address the design of numerical methods to calculate trajectories. 

Leimkuhler, C. Matthews, Molecular Dynamics, Interdisciplinary Applied 

In order to correctly model the different possible states of the system, it will be necessary to cover a large part of the accessible phase space, so either trajectories must be very long or we must use many initial conditions. There are many ways to solve initial value problems such as (2.1) combined with an initial condition z(0) = 5. The methods introduced here all rely on the idea of a discretization with a finite stepsize h, and an iterative procedure that computes, starting from zo =, a sequence zi,Z2. where z z(nh). The simplest scheme is certainly Euler s method which advances the solution from timestep to timestep by the formula  

The method is based on the observation that z(H-A) sa z(t)+hz(t), i.e. the beginning of a Taylor series expansion in powers of h, and the further observation that the solution satisfies the differential equation, hence z(f) may be replaced by f(z(t)). 

In order to be able to easily compare the properties of different methods in a unified way, we focus in this chapter primarily on a particular class of schemes, generalized one-step methods. Suppose that the system under study has a well defined flow map , defined on the phase space (which is assumed to exclude any singular points of the potential energy function). The solution of the initial value problem, z = /(z), z(0) = may be written z(f, ) (with z(0,( ) = ), and the flow-map satisfies = z(f, 5). A one-step method, starting from a given point, approximates a point on the solution trajectory at a given time h units later. Such a method defines a map % of the phase space as illustrated in Fig. 2.1. 

---

To generalize what we have just done to reactive and inelastic scattering, one needs to calculate numerically integrated trajectories for motions in many degrees of freedom. This is most convenient to develop in space-fixed Cartesian coordinates. In this case, the classical equations of motion (Hamilton s equations) are given... 

Numerical integration methods are widely used to solve these integrals. The Gauss-Miihler method [28] is employed in all of the calculations used here. This method is a Gaussian quadrature [29] which gives exact answers for Coulomb scattering. 

Once a point on the coexistence line has been found, one can trace out more of it using the approach of Kofke [177. 178] to numerically integrate die Clapeyron equation... 

Gear C W 1966 The numerical integration of ordinary differential equations of various orders ANL 7126... 

Ryckaert J-P, Ciccotti G and Berendsen H J C 1977 Numerical integration of the Cartesian equations of motion of a system with constraints molecular dynamics of n-alkanes J. Comput. Phys. 23 327-41... 

Molecular dynamics tracks tire temporal evolution of a microscopic model system tlirough numerical integration of tire equations of motion for tire degrees of freedom considered. The main asset of molecular dynamics is tliat it provides directly a wealtli of detailed infonnation on dynamical processes. 

The fonnal (or numerical) integration of this equation can be written as... 

A. Ahmad and L. Cohen. A numerical integration scheme for the A -body gravitational problem. J. Comp. Phys., 12 389-402, 1973. 

Essential Dynamics In most applications details of individual MD trajectories are of only minor interest. An illustrative example due to Grubmuller [10] is documented in Figure 3. It describes the dynamics of a polymer chain of 100 CH2 groups. Possible stepsizes for numerical integration are confined... 

E. Hairer. Backward analysis of numerical integrators and symplectic methods. Annals of Numerical Mathematics 1 (1994)... 

E. Hairer and Ch. Lubich. The life-span of backward error analysis for numerical integrators. Numer. Math. 76 (1997) 441-462... 

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

Another difference is related to the mathematical formulation. Equation (1) is deterministic and does not include explicit stochasticity. In contrast, the equations of motion for a Brownian particle include noise. Nevertheless, similar algorithms are adopted to solve the two differential equations as outlined below. The most common approach is to numerically integrate the above differential equations using small time steps and preset initial values. 

On the basis of these considerations it is suggested that the design of numerical integration techniques might employ the following two hypotheses ... 

Verlet, L. Computer Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules. Physical Review 159 (1967) 98-103 Janezic, D., Merzel, F. Split Integration Symplectic Method for Molecular Dynamics Integration. J. Chem. Inf. Comput. Sci. 37 (1997) 1048-1054 McLachlan, R. I. On the Numerical Integration of Ordinary Differential Equations by Symplectic Composition Methods. SIAM J. Sci. Comput. 16 (1995) 151-168... 

Thus we find that the choice of quaternion variables introduces barriers to efficient symplectic-reversible discretization, typically forcing us to use some off-the-shelf explicit numerical integrator for general systems such as a Runge-Kutta or predictor-corrector method. 

S. Reich, Dynamical Systems, Numerical Integration, and Exponentially Small Estimates, Habilitationsthesis, Konrad Zuse Center, Free University, Berlin (1997). 

Numerical Integrators for Quantum-Classical Molecular Dynamics... 

In this paper, we focus on numerical techniques for integrating the QCMD equations of motion. The aim of the paper is to systematize the discussion concerning numerical integrators for QCMD by ... 

It is the aim of this paper to take into account a wide range of systems to which QCMD is applied. For a precise understanding of the situation, it is necessary to recognize the differences between these applications, because these differences demand for specific features of the numerical integrator. In the following, we will describe a suitable classification of the application problems. 

Since we have discovered the underlying Hamiltonian structure of the QCMD model we are able to apply methods commonly used to construct suitable numerical integrators for Hamiltonian systems. Therefore we transform the QCMD equations (1) into the Liouville formalism. To this end, we introduce a new state z in the phase space, z = and define the nonlinear... 

Using splitting schemes of the exponential function allows for a generation of numerical integrators. For example [24, 22] ... 

---