Basic Molecular Dynamics

Molecular dynamics [127] (MD) is a numerical technique to approximate the static and dynamic properties of classical many-body systems. MD techniques are commonly employed alongside Monte Carlo techniques because dense packing reduces the acceptance probability of translation and chain rotation moves to very low values, particularly when using all-atom models. 

The key idea is simple if U( rj ) is the total potential acting on particle i with mass nti and position r (t) at time t, then Newton s equations of motion (EOM)  

The Verlet algorithm uses the Taylor expansions of the position vectors in different time directions. With At the time-step of the simulation, adding the expansions for r (t- -At) and r (t—At) leads to  

A commonly applied modification is the velocity Verlet algorithm. It explicitly incorporates the particle s velocity, Vj, such that  

In predictor-corrector algorithms time derivatives of the position vectors at time t are used to predict the positions and their derivatives at time H- At. The predicted variables then are corrected according to the difference from those at time t, where a set of Gear constants are used. The latter are chosen to balance accuracy and stability, that is, short- and long-time conservation of energy. Optimized values depend on the order of the Taylor expansion ( order of the algorithm ). 

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A quantitative understanding of molecular electronic structure is vital to advances in chemical imaging. This understanding can be achieved through molecular dynamics (MD) simulations. In order to improve MD simulations, a number of specific areas need to be addressed in basic molecular dynamics theory. There is a need to develop a next generation of readily accessible, easy-to-use MD simulation packages. 

In order to improve MD simulations, a number of specific areas should be addressed in the area of basic molecular dynamics theory. These include (1) development of full quantum mechanical calculations on complex molecules and more robust ways to incorporate quantum mechanical calculations within larger-scale classical mechanics or statistical mechanics approaches (2) development and refinement of transferable force fields between arbitrary atoms and molecules, which are necessary building blocks for MD simulations of general systems and (3) development of multiscale theories and techniques for understanding systems. Moreover, the community must develop toolkits that allow general users to perform such simulations. 

A comprehensive introduction to the field, covering statistical mechanics, basic Monte Carlo, and molecular dynamics methods, plus some advanced techniques, including computer code. 

To demonstrate the basic ideas of molecular dynamics calculations, we shall first examine its application to adiabatic systems. The theory of vibronic coupling and non-adiabatic effects will then be discussed to define the sorts of processes in which we are interested. The complications added to dynamics calculations by these effects will then be considered. Some details of the mathematical formalism are included in appendices. Finally, examples will be given of direct dynamics studies that show how well the systems of interest can at present be treated. 

In this section, the basic theory of molecular dynamics is presented. Starting from the BO approximation to the nuclear Schrddinger equation, the picture of nuclear dynamics is that of an evolving wavepacket. As this picture may be unusual to readers used to thinking about nuclei as classical particles, a few prototypical examples are shown. 

One of the basic problems in molecular dynamics is how to sample infrequent events. Typically a reaction must pass over a barrier, and effort would be wasted if many trajectories are run that do not reach the reactant channel. 

P. Deuflhard, M. Dellnitz, O. Junge, and Ch. Schiitte. Computation of essential molecular dynamics by subdivision techniques I Basic concept. Preprint SC 96-45, Konrad Zuse Zentrum, Berlin (1996)... 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

The following sections cover the design goals, decisions, and outcomes of the first two major versions of NAMD and present directions for future development. It is assumed that the reader has been exposed to the basics of molecular dynamics [2, 3, 4] and parallel computing [5]. Additional information on NAMD is available electronically [6]. 

To have an overview of the algorithms and basic concepts used to perform molecular dynamics simulations... 

HyperChem run s the molecular dynain ics trajectory, averaging and analyzing a trajectory and creating the Cartesian coordinates and velocities, fhe period for reporting these coordinates and velocities is th e data collection period. At-2. It is a m iiltiplc of the basic time step. At = ii At], and is also referred to as a data step. The value 1I2 is set in the Molecular Dynamics options dialog box. 

HyperChem includes a number of time periods associated with a trajectory. These include the basic time step in the integration of Newton s equations plus various multiples of this associated with collecting data, the forming of statistical averages, etc. The fundamental time period is Atj s At, the integration time step stt in the Molecular Dynamics dialog box. 

There are basically two different computer simulation techniques known as molecular dynamics (MD) and Monte Carlo (MC) simulation. In MD molecular trajectories are computed by solving an equation of motion for equilibrium or nonequilibrium situations. Since the MD time scale is a physical one, this method permits investigations of time-dependent phenomena like, for example, transport processes [25,61-63]. In MC, on the other hand, trajectories are generated by a (biased) random walk in configuration space and, therefore, do not per se permit investigations of processes on a physical time scale (with the dynamics of spin lattices as an exception [64]). However, MC has the advantage that it can easily be applied to virtually all statistical-physical ensembles, which is of particular interest in the context of this chapter. On account of limitations of space and because excellent texts exist for the MD method [25,61-63,65], the present discussion will be restricted to the MC technique with particular emphasis on mixed stress-strain ensembles. 

In the next section we describe the basic models that have been used in simulations so far and summarize the Monte Carlo and molecular dynamics techniques that are used. Some principal results from the scaling analysis of EP are given in Sec. 3, and in Sec. 4 we focus on simulational results concerning various aspects of static properties the MWD of EP, the conformational properties of the chain molecules, and their behavior in constrained geometries. The fifth section concentrates on the specific properties of relaxation towards equilibrium in GM and LP as well as on the first numerical simulations of transport properties in such systems. The final section then concludes with summary and outlook on open problems. 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

The basic principles are described in many textbooks [24, 26]. They are thus only sketchily presented here. In a conventional classical molecular dynamics calculation, a system of particles is placed within a cell of fixed volume, most frequently cubic in size. A set of velocities is also assigned, usually drawn from a Maxwell-Boltzmann distribution appropriate to the temperature of interest and selected in a way so as to make the net linear momentum zero. The subsequent trajectories of the particles are then calculated using the Newton equations of motion. Employing the finite difference method, this set of differential equations is transformed into a set of algebraic equations, which are solved by computer. The particles are assumed to interact through some prescribed force law. The dispersion, dipole-dipole, and polarization forces are typically included whenever possible, they are taken from the literature. 

In Section II, the basic equations of OCT are developed using the methods of variational calculus. Methods for solving the resulting equations are discussed in Section III. Section IV is devoted to a discussion of the Electric Nuclear Bom-Oppenhermer (ENBO) approximation [41, 42]. This approximation provides a practical way of including polarization effects in coherent control calculations of molecular dynamics. In general, such effects are important as high electric fields often occur in the laser pulses used experimentally or predicted theoretically for such processes. The limits of validity of the ENBO approximation are also discussed in this section. 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

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