Classical mechanics equations of motion

It is possible to parametarize the time-dependent Schrddinger equation in such a fashion that the equations of motion for the parameters appear as classical equations of motion, however, with a potential that is in principle more general than that used in ordinary Newtonian mechanics. However, it is important that the method is still exact and general even if the trajectories aie propagated by using the ordinary classical mechanical equations of motion. 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

The classical mechanical equations of motion for the 3N qk coordinates can be written in tenns of the above potential energy and the following kinetic energy function ... 

The I structure in liquid water cannot be inferred from the experimental methods listed in Table 2.1 because those methods provide data that are time averages over many I structure configurations. However, the technique of molecular dynamics (MD) computer simulation has led to reliable information about the I structure. In this technique, a computer is used to solve the classical mechanical equations of motion with a chosen intermolecular potential function for a few hundred water molecules constrained in space to maintaining the equilibrium liquid density, with data on the instantaneous position and velocity of the molecules provided both as numerical output and in the form of stereoscopic pictures. The principal features of the I structure determined in this fashion are ... 

The laws of physics needed to answer the question are well known. Since the potential energy surface is given, one knows the masses of the colliders and so one only needs to solve the SchrUdinger equation. The problem of course is that the number of coupled equations that need to be solved is enormous and not yet within reach of present day computers. Necessarily then the theorist is restricted to studying model systems and construction of approximations. One type of approximation is to solve the exact classical mechanical equations of motion. One selects initial conditions which correspond to the experimental initial state, integrates the equations of motion forward in time till the process is over and then obtains cross sections, product distributions etc. In essence, Hamilton s equations of motion serve as a black box , whose structure is determined by the masses and the potential energy surface. This black box provides the necessary transformation from initial conditions to final conditions. 

The above set of equations are those arising from the TDSCF treatment of the problem. We notice that no reference path R(t) appears in the equations, that the method can be extended to any number of degrees of freedom, and that the multidimensional problem would be reduced to a set of onedimensional ones. In the classical path theories discussed here we deviate from this scheme in a rather important fashion, namely by introducing one or more reference paths R(t), paths which formally follow classical mechanical equations of motion. 

The temporal behavior of molecules, which are quantum mechanical entities, is best described by the quantum mechanical equation of motion, i.e., the time-dependent Schrdd-inger equation. However, because this equation is extremely difficult to solve for large systems, a simpler classical mechanical description is often used to approximate the motion executed by the molecule s heavy atoms. Thus, in most computational studies of biomolecules, it is the classical mechanics Newtonian equation of motion that is being solved rather than the quantum mechanical equation. 

The fragmentation of a molecule in its ground electronic state is commonly known as unimolecular dissociation [26-28]. [For a recent review see Ref. 29 and the Faraday Discussion of the Chemical Society, vol. 102 (1995).] Because of its importance in several areas of physical chemistry, such as combustion or atmospheric kinetics, there is a high demand of accurate unimolecular dissociation rates. On the other hand, however, the calculation of reliable dissociation rates by dynamical methods (i.e., the solution of the classical or quantum mechanical equations of motion) is, for obvious technical problems, prohibited for all but a few simple molecules. For... 

In reality, of course, atoms obey quantum mechanics rather than classical mechanics. As you will discover in other chapters in this book, great advances have been made recently in the quantum mechanical treatment of molecular systems. However, one should realize just how much care has to go into the selection of correct coordinates and the necessity to choose appropriate systems for quantum mechanical study. For arbitrarily large systems, or for systems containing several heavy atoms, quantum methods are not yet readily applicable. It is in such cases that classical mechanical approaches can be utilized with profit. Furthermore, even in systems for which quantum mechanical treatments are now feasible, comparisons with classical data often help researchers to isolate those phenomena which arise solely in the quantum mechanics, yielding fundamental insight into the two different dynamics. In the classical approach, the motion of each atom is calculated by numerically solving the classical differential equations of motion (1), either second order with respect to time in the positions, x (Newton s law), or,... 

It is interesting to draw some analogy between a complex chemical reaction and a dynamic system, which consists of material points. The theoretical mechanics describes the state of such a system by means of the classical (canonical) equations of motion [63,64] based on a vast theoretical and experimental foundation... 

We first consider the option to set up a quantum mechanical equation of motion which obeys the correspondence principle. If we apply the correspondence principle to the classical nonrelativistic kinetic energy expression E = (2m) we arrive at the time-dependent Schrodinger equation, in which... 

At our most fundamental level of description we consider molecular systems to be composed of atomic nuclei and electrons, all obeying quantum mechanical laws. The question of the kinetics of a physicochemical event is therefore related to the time evolution of such composite systems. In the first sub-section we recall the basic quantum mechanical equation-of-motions relevant in this context. We then consider approximations that can be operated to simplify the nuclear-electronic dynamics, leading to the derivation of the mixed quantum-classical rate constant expression. 

The phase space of interest refers to the particle s position in space and its momentum. As momentum can be described by classical statistical mechanics, equations of motion can be expressed in terms of the particle s position in space and its momentum, hence the term Phase Space Dynamics. Considering that an ion beam is composed of a large number of charged particles, it then follows that the optical properties of the beam can be described as a collection of such parameters. 

Suppose we could, at least in principle, solve the (classic or quantum mechanical) equations of motion for such a huge number of microscopic entities, we would still need to specify initial conditions of the system, which, again, is a task of overwhelming complexity in practice. 

The solution of the dynamical problem for the gas and surface atoms requires in principle solution of the quantum mechanical equations of motion for the system. Since this problem has been solved only for 3-4 atomic systems we need to incorporate some approximations. One obvious suggestion is to treat the dynamics of the heavy solid atoms by classical rather than quantum dynamical equations. As far as the lattice is concerned we may furthermore take advantage of the periodicity of the atom positions. At the surface this periodicity is, however, broken in one direction and special techniques for handling this situation are needed. Lattice dynamics deals with the solution of the equations of motion for the atoms in the crystal. As a simple example we consider first a one-dimensional crystal of atoms with identical masses. If we include only the nearest neighbor interaction, the hamiltonian is given by ... 

The radiation reaction force. We return to a consideration of the classical atomic model which was introduced in sections 4.1 and 4.2. We found that there was a loss of energy in the form of radiation which occurred slow ly over many cycles of the electron s motion. However, this loss of energy was not taken into account in the mechanical equation of motion of the electron. This situation can be remedied by introducing a radiation reaction force, F, such that the work done by the reaction force in one cycle of the oscillation is equal to the energy emitted into the radiation field ... 

In classical mechanics, the state of the system may be completely specified by the set of Cartesian particle coordinates r. and velocities dr./dt at any given time. These evolve according to Newton s equations of motion. In principle, one can write down equations involving the state variables and forces acting on the particles which can be solved to give the location and velocity of each particle at any later (or earlier) time t, provided one knows the precise state of the classical system at time t. In quantum mechanics, the state of the system at time t is instead described by a well behaved mathematical fiinction of the particle coordinates q- rather than a simple list of positions and velocities. 

Vibrational motion is thus an important primary step in a general reaction mechanism and detailed investigation of this motion is of utmost relevance for our understanding of the dynamics of chemical reactions. In classical mechanics, vibrational motion is described by the time evolution and l t) of general internal position and momentum coordinates. These time dependent fiinctions are solutions of the classical equations of motion, e.g. Newton s equations for given initial conditions and I Iq) = Pq. 

The center of the wavepacket thus evolves along the trajectory defined by classical mechanics. This is in fact a general result for wavepackets in a hannonic potential, and follows from the Ehrenfest theorem [147] [see Eqs. (154,155) in Appendix C]. The equations of motion are straightforward to integrate, with the exception of the width matrix, Eq. (44). This equation is numerically unstable, and has been found to cause problems in practical applications using Morse potentials [148]. As a result, Heller inboduced the P-Z method as an alternative propagation method [24]. In this, the matrix A, is rewritten as a product of matrices... 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

The most commonly used method for applying constraints, particularly in molecula dynamics, is the SHAKE procedure of Ryckaert, Ciccotti and Berendsen [Ryckaert et a 1977]. In constraint dynamics the equations of motion are solved while simultaneous satisfying the imposed constraints. Constrained systems have been much studied in classics mechanics we shall illustrate the general principles using a simple system comprising a bo sliding down a frictionless slope in two dimensions (Figure 7.8). The box is constrained t remain on the slope and so the box s x and y coordinates must always satisfy the equatio of the slope (which we shall write as y = + c). If the slope were not present then the bo... 

The strength of molecular mechanics is that by treating molecules as classical objects, fliUy described by Newton s equations of motion, quite large systems can be modeled. Computations involving enzymes with thousands of atoms are done routinely. As computational capabilities have advanced, so... 

Depending on the desired level of accuracy, the equation of motion to be numerically solved may be the classical equation of motion (Newton s), a stochastic equation of motion (Langevin s), a Brownian equation of motion, or even a combination of quantum and classical mechanics (QM/MM, see Chapter 11). 

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