Classical equations of motion

Time evolution in classical mechanics is described by the Newton equations 

The significance of this form of the Newton equations is its invariance to coordinate transformation. 

Another useful way to express the Newton equations of motion is in the Hamiltonian representation. One starts with the generalized momenta 

The mathematical operation done in (1.95) transforms the function L of variables x, x to a new function H of the variables x, The resulting function, 7T( x, / ), is the Hamiltonian, which is readily shown to satisfy 

The specification of all positions and momenta of all particles in the system defines the dynamical state of the system. Any dynamical variable, that is, a function of these positions and momenta, can be computed given this state. Dynamical variables are precursors of macroscopic observables that are defined as suitable averages over such variables and calculated using the machinery of statistical mechanics. 

Yi and p are the position and momentum vectors of particle i of mass mi, F/ is the force acting on the particle, U is the potential, and V/ is the gradient with respect to the position of this particle. These equations of motion can be obtained from the Lagrangian 

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Figure Al.6.1. Gaussian wavepacket in a hannonic oscillator. Note tliat the average position and momentum change according to the classical equations of motion (adapted from [6]).

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... 

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. 

While the Lorentz model only allows for a restoring force that is linear in the displacement of an electron from its equilibrium position, the anliannonic oscillator model includes the more general case of a force that varies in a nonlinear fashion with displacement. This is relevant when tire displacement of the electron becomes significant under strong drivmg fields, the regime of nonlinear optics. Treating this problem in one dimension, we may write an appropriate classical equation of motion for the displacement, v, of the electron from equilibrium as... 

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. 

A different approach is to represent the wavepacket by one or more Gaussian functions. When using a local harmonic approximation to the trae PES, that is, expanding the PES to second-order around the center of the function, the parameters for the Gaussians are found to evolve using classical equations of motion [22-26], Detailed reviews of Gaussian wavepacket methods are found in [27-29]. 

To deal with the problem of using a superposition of functions, Heller also tried using Gaussian wave packets with a fixed width as a time-dependent basis set for the representation of the evolving nuclear wave function [23]. Each frozen Gaussian function evolves under classical equations of motion, and the phase is provided by the classical action along the path... 

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 localized natme of the nucleai functions means that these reduce to classical equations of motion... 

In order to solve the classical equations of motion numerically, and, thus, to t)btain the motion of all atoms the forces acting on every atom have to be computed at each integration step. The forces are derived from an energy function which defines the molecular model [1, 2, 3]. Besides other important contributions (which we shall not discuss here) this function contains the Coulomb sum... 

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]. 

In this section we consider the classical equations of motion of particles in cases where the highest-frequency oscillations are nearly harmonic The positions y t) = j/i (t) evolve according to the second-order system of differential equations... 

The relationship between H and vibrational frequencies can be made clear by recalling the classical equations of motion in the Lagrangian formulation ... 

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). 

Equation (7) is a second-order differential equation. A more general formulation of Newton s equation of motion is given in terms of the system s Hamiltonian, FI [Eq. (1)]. Put in these terms, the classical equation of motion is written as a pair of coupled first-order differential equations ... 

Again, as in the previous section, we look for the stationary points of the path integral, i.e., the trajectories that extremize the Eucledian action (3.11) and thus satisfy the classical equation of motion in the upside-down barrier. 

Fixing the end-points at x t = 0) Xi and x t — t) — Xf, we recall that the classical equations of motion are obtained by setting the variation of the action with... 

In MD simulations we simply solve numerically the classical equations of motion, expressing the changes in coordinates and velocities at a time increment At by... 

The classical equations of motion can be integrated exactly and the solution for the position of the electron as a function of time,... 

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