Equations of Motion and Initial Conditions

L is the CC length, / the CH length and y the HCH angle in the terminal groups, M the mass of a carbon atom, and the masses of,the substituents on Ci (C2) and C3 respectively. In the present study nti = /W2 = 1 and p = 1 / is the moment of inertia of the rotor which is formed of the two hydrogen atoms in a terminal methylene group. Eq. (51) leads to the partial derivatives  

The numerical integration of the three coupled second order differential Eq. (54), (55) requires six initial conditions. These are (i, ii, iii) the three values 2 a°, 

Five different values of have been studied, namely 61, 62,63, 64 and 65 kcal/mol. For tot = 61 kcal/mol, the only available chatmel is the synchronous conrotatory motion (transition state at 59.8 kcal/mol). For 62 kcal/mol, the rotation of a single group (transition state at 61.6 kcal/mol) and the concerted disrotatory motion (transition state at 61.9 kcal/mol) both become feasible motions, at least in principle. For each value of jEtot, j rot has been varied stepwise from 2 to 50 kcal/mol, with a step of 2 kcal/mol. In addition, for given values of E o and has been varied from 45° (conrotatory motion, ie. antisymmetric vibra- 

We first treat separately the trajectories conesponding to the particular values 6° = 45° and —45°. Indeed, the total symmetry of the problem is such that, whenever the motion of both rotors at the starting point is either purely conrotatory or purely disrotatory, it keeps this particularity throughout the trajectory. Then the trajectory can be drawn on a two-dimensional potential energy surface such as that pictured in Fig. 7 and 8. 

2 to 50 kcal/mol, with a step of 2 kcal/mol. In addition, for given values of E o and E°ot, 5° has been varied from 45° (conrotatory motion, ie. antisymmetric vibra- 

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Treatment of the equations of motion and initial conditions are discussed in Appendix B. 

In general the flow of a pure fluid is described by the equation of continuity, the three equations of motion, and the equation of energy balance. In addition, one has to specify boundary and initial conditions and also the dependence of p on p and T (the thermal equation of state) and the dependence of Cv or U on p and T (the caloric equation of state). 

Once a potential energy function is chosen or determined for a molecule, there are three major components to a trajectory study the selection of initial conditions for the excited molecule, the numerical integration of the classical equations of motion, and the analysis of the trajectories and their final conditions. The last item may include the time at which the trajectory decomposed to products, the nature of the trajectory s intramolecular motion, i.e., regular or irregular, and the vibrational, rotational and translational energies of the reaction products. 

The continuity equation, the equation of motion, and component mass conservation equation can be easily written with the above assumptions. Proper initial and boundary conditions can be imposed. These equations are general and apply to all resist systems. Differences exist in the constitutive equations describing material and transport properties. 

The system of MD particles is defined by a set of boundary and initial conditions and by interactions between particles represented by a force F(r,y) and the torques N(r,y). The particle system evolves according to the Newtonian equations of motion... 

The purpose and uses of the differential equations of motion and continuity, as mentioned previously, are to apply these equations to any viscous-flow problem. For a given specific problem, the terms that are zero or near zero are simply discarded and the remaining equations used in the solution to solve for the velocity, density, and pressure distributions. Of course, it is necessary to know the initial conditions and the boundary conditions to solve the equations. Several examples will be given to illustrate the general methods used. 

Second observation. Consider the infinite periodic array at a given time, say t = 0. Remove the walls. Now allow all the particles to move under the equations of motion, and the Infinite system will remain periodic forever. In other words, periodic boundary conditions are equivalent to an initial condition for the infinite system. 

Both Monte Carlo and molecular dynamics methods sample directly the phase space of a small but representative component of the crystal, the former by performing stochastic moves through configuration space, the latter by following a specified trajectory according to an equation of motion and chosen initial condition. A typical Hamiltonian for molecular dynamics simulation is [14] ... 

The solution of the equation of motion of the harmonic oscillator illustrates a general property of classical equations of motion. If the equation of motion and the initial conditions are known, the motion of the system is determined as a function of time. We say that classical equations of motion are deterministic. 

The second simulation technique is molecular dynamics. In this technique, which was pioneered by Alder, initial positions of theparticles of a system of several hundred particles are assigned in some way. Displacements of the particles are determined by numerically simulating the classical equations of motion. Periodic boundary conditions are applied as in the Monte Carlo method. The first molecular dynamics calculations were done on systems of hard spheres, but the method has been applied to monatomic systems having intermolecular forces represented by the square-well and Lennard-Jones potential energy functions, as well as on model systems representing molecular substances. Commercial software is now available to carry out molecular dynamics simulations on desktop computers. ... 

The methods for calculating classical trajectories of molecular collisions are well established, and they are straightforward to apply. Usually, the first step in a simulation is to represent the potential energy surface in an analytical form so that the forces can be rapidly calculated. Given the potential energy surface, the calculation can be viewed as having three parts selection of initial conditions, solution of the equations of motion, and analysis of the trajectories. 

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. 

A classical molecular dynamics trajectory is simply a set of atoms with initial conditions consisting of the 3N Cartesian coordinates of N atoms A(X, Y, Z ) and the 3N Cartesian velocities (v a VyA v a) evolving according to Newton s equation of motion ... 

The most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). 

By applying the equation of motion repeatedly, say for n times and making use of the initial conditions at t = t0, one obtains considering both j and j after some algebra, the result... 

Once the initial and boundary conditions are specified, the classical equations of motion are integrated as in any other simulation. From the start of the trajectory, the atoms are free to move under the influence of the potential. One simply identifies reaction mechanisms and products during the dynamics. For the case of sputtering, the atomic motion is integrated until it is no longer possible for atoms and molecules to eject. The final state of ejected material above the surface is then evaluated. Properties of interest include the total yield per ion, energy and angular distributions, and the structure and... 

In this section, we write down the explicit forms for the one-electron TDAN equations of motion for the situation where the Coulomb repulsion integral is set equal to zero. Also, we consider only the case where the band energy levels contain 2n electrons and the valence orbital of the scattered atom is initially empty. It will be obvious how to modify the equations to deal with other possible cases mostly this will involve just a change in the initial conditions. 

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