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. [Pg.36]

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 [Pg.161]

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 [Pg.91]

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 [Pg.230]

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). [Pg.39]

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. [Pg.5]

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. [Pg.289]

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 [Pg.24]

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. [Pg.73]

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. [Pg.1591]

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. [Pg.42]

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. [Pg.121]

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 [Pg.87]

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. [Pg.1056]

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 [Pg.52]

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 [Pg.750]

See also in sourсe #XX -- [ Pg.18 ]

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