Molecular dynamics trajectory 268 Subject

Rather than solve a Schrodinger equation with the Nuclear Hamiltonian (above), a common approximation is to assume that atoms are heavy enough so that classical mechanics is a good enough approximation. Motion of the particles on the potential surface, according to the laws of classical mechanics, is then the subject of classical trajectory analysis or molecular dynamics. These come about by replacing Equation (7) on page 164 with its classical equivalent ... 

Fig. 3. Vibrational population distributions of N2 formed in associative desorption of N-atoms from ruthenium, (a) Predictions of a classical trajectory based theory adhering to the Born-Oppenheimer approximation, (b) Predictions of a molecular dynamics with electron friction theory taking into account interactions of the reacting molecule with the electron bath, (c) Born—Oppenheimer potential energy surface, (d) Experimentally-observed distribution. The qualitative failure of the electronically adiabatic approach provides some of the best available evidence that chemical reactions at metal surfaces are subject to strong electronically nonadiabatic influences. (See Refs. 44 and 45.)...

The most striking feature of AIMD and the novelty over traditional molecular dynamics simulations is the fact that the electronic structure is available all the time. Therefore, the analysis of the wavefunction in the course of the simulation or along the trajectory is one of the major subjects that can contribute to the understanding on the molecular and on the electronic level. The importance of analyzing the wavefunction in bioinorganic systems has been shown in many applications, see also Section... 

This point being made, we have not yet provided a description of how to follow a phase-space trajectory. This is the subject of molecular dynamics, upon which we now focus. 

Very striking results on the interactions of molecules with a catalyst have been recently reported in zeolite catalysis because of the well ordered structure of these materials it is worth mentioning the subjects of zeolite design [10] and of acidic properties of metallosilicates [11]. In other areas where polycrystallinic or even amorphous materials arc applied, highly interesting results are now numerously emerging (such as hydrocarbon oxidation on vanadium-based catalysts [12] location of transition metal cations on Si(100) [13] CO molecules on MgO surfaces [14] CH4 and O2 interaction with sodium- and zinc-doped CaO surfaces [15] CO and NO on heavy metal surfaces [16]). An illustration of the computerized visualization of molecular dynamics of Pd clusters on MgO(lOO) and on a three-dimensional trajectory of Ar in Na mordenitc, is the recent publication of Miura et al. [17]. 

The general principle of BD is based on Brownian motion, which is the random movement of solute molecules in dilute solution that result from repeated collisions of the solute with solvent molecules. In BD, solute molecules diffuse under the influence of systematic intermolecular and intramolecular forces, which are subject to frictional damping by the solvent, and the stochastic effects of the solvent, which is modeled as a continuum. The BD technique allows the generation of trajectories on much longer temporal and spatial scales than is feasible with molecular dynamics simulations, which are currently limited to a time of about 10 ns for medium-sized proteins. 

The basic idea behind an atomistic-level simulation is quite simple. Given an accurate description of the energetic interactions between a collection of atoms and a set of initial atomic coordinates (and in some cases, velocities), the positions (velocities) of these atoms are advanced subject to a set of thermodynamic constraints. If the positions are advanced stochastically, we call the simulation method Monte Carlo or MG [10]. No velocities are required for this technique. If the positions and velocities are advanced deterministically, we call the method molecular dynamics or MD [10]. Other methods exist which are part stochastic and part deterministic, but we need not concern ourselves with these details here. The important point is that statistical mechanics teUs us that the collection of atomic positions that are obtained from such a simulation, subject to certain conditions, is enough to enable aU of the thermophysical properties of the system to be determined. If the velocities are also available (as in an MD simulation), then time-dependent properties may also be computed. If done properly, the numerical method that generates the trajectories... 

Nowadays, computer simulations are treated as the third fundamental discipline of interface research in addition to the two classieal ones, namely theory and experiment. Based direetly on a microscopie model of the system, eomputer simulations can, in principle at least, provide an exact solution of any physicochemical problem. By far the most common methods of studying adsorption systems by simulations are the Monte Carlo (MC) technique and the molecular dynamics (MD) method. In this ehapter, a description of simidation methods will be omitted because several textbooks and review artieles on the subject are available [274-277]. The present discussion will be restricted to elementary aspects of simulation methods. In the deterministic MD method, the moleeular trajectories are eomputed by solving Newton s equations, and a time-correlated sequenee of configurations is generated. The main advantage of this technique is that it permits the study of time-dependent processes. In MC simulation, a stochastic element is an essential part of the method the trajectories are generated by random walk in configuration space. Struetural and thermodynamic properties are accessible by both methods. 

The general understanding of molecular dynamics rests mainly upon classical mechanics this holds true for full bimolecular collisions (see Trajectory Simulations of Molecular Collisions Classical Treatment) as well as half-collisions, i.e., the dissociation of a parent molecule into different products. The classical picture of photodissociation closely resembles the time-dependent picture the electronic transition from the ground to the excited electronic state is assumed to take place instantaneously so that the internal coordinates (Qi) and corresponding momenta (/, ) of the parent molecule remain unchanged during the excitation step (vertical transition). After the molecule is promoted to the PES of the upper state it starts to move subject to the classical equations of motion (Hamilton s equations)... 

Another fairly new method, using the electrostatic molecular potential, will not be discussed here since it is the subject of another contribution to this volume 50>. I will now consider methods that have had the widest application in the theoretical study of chemical reactivity, in order of increasing complexity a) molecular mechanics b) extended Htickel method c), d) empirical self-consistent field methods such as CNDO and MINDO e) the simplest ab initio approach f) the different S.C.F. methods, possibly including configuration interaction g) valence bond methods, and h) the dynamical approach, including the calculation of trajectories 61>. 

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