This knowledge can, subsequently, be used to perform the propagation of a set of classical coordinates and momenta forward in time. For any initial (i.e., t=0)momenta ps and p, one can use the above form for H to propagate the coordinates(8XK, 8S) and momenta, ps) forward in time. In this manner, one can use the rPh theory to follow the time evolution of a chemical reaction that begins(t=0) with coordinates and moment characteristic of reactants under specified laboratory conditions and moves through a Ts and onward to products. Once time has evolved long enough for product geometries to be realized, one can interrogate the values of 1/2 pk+ox(S)] to determine how much energy has been deposited into various product-molecule vibrations and of 1/2 ps to see what the final kinetic energy of the product fragments is. Of course, one also monitors what fraction of the trajectories, whose initial conditions are chosen to represent some experimental situation, progress to product geometries vs. returning to reactant geometries. In this way, one can determine the overall reaction probability D Classical Dynamics Simulation of Rates One can perform classical dynamics simulations of reactive events without using the reaction path Hamiltonian. Following a procedure like that outlined in Chapter 7 where condensed-media md simulations were discussed. one can time-evolve the Newton equations of motion of the molecular reaction species using, for example, the Cartesian coordinates of each atom in the system and with either a Born-Oppenheimer surface or a parameterized functional form. Of course. it is essential for whatever 16
16 This knowledge can, subsequently, be used to perform the propagation of a set of classical coordinates and momenta forward in time. For any initial (i.e., t = 0) momenta pS and pK , one can use the above form for H to propagate the coordinates {dXK, dS} and momenta {pK, pS} forward in time. In this manner, one can use the RPH theory to follow the time evolution of a chemical reaction that begins (t = 0) with coordinates and moment characteristic of reactants under specified laboratory conditions and moves through a TS and onward to products. Once time has evolved long enough for product geometries to be realized, one can interrogate the values of 1/2[pK 2 + wK 2 (S)] to determine how much energy has been deposited into various product-molecule vibrations and of 1/2 pS 2 to see what the final kinetic energy of the product fragments is. Of course, one also monitors what fraction of the trajectories, whose initial conditions are chosen to represent some experimental situation, progress to product geometries vs. returning to reactant geometries. In this way, one can determine the overall reaction probability. D. Classical Dynamics Simulation of Rates One can perform classical dynamics simulations of reactive events without using the reaction path Hamiltonian. Following a procedure like that outlined in Chapter 7 where condensed-media MD simulations were discussed, one can time-evolve the Newton equations of motion of the molecular reaction species using, for example, the Cartesian coordinates of each atom in the system and with either a Born-Oppenheimer surface or a parameterized functional form. Of course, it is essential for whatever
function one uses to accurately describe the reactive surface, especially near the transition With each such coordinate having an initial velocity( da/dt)o and an initial value go, one then uses Newtons equations written for a time step of duration St to propagate q and da/dt forward in time according, for example, to the following first-order propagation formul q(t+8 t)=g0+(da/dto 8t da/dt(t+ot)=(dq/dt)o-St[aE/Oq)/m Here ma is the mass factor connecting the velocity da/dt and the momentum pq conjugate to the coordinate q Pa=ma dq/dt, and-(aE/ag)o is the force along the coordianate g at the"initial"geometry go. Again, as in Chapter 7, I should note that the above formulas for propagating q and p forward in time represent only the most elementary approach to this problem. There are other, more sophisticated, numerical methods for effecting more accurate and longer-time propagations, but I will not go into them here. Rather, I wanted to focus on the basics of how these simulations are carried out
17 function one uses to accurately describe the reactive surface, especially near the transition state. With each such coordinate having an initial velocity (dq/dt)0 and an initial value q0 , one then uses Newton’s equations written for a time step of duration dt to propagate q and dq/dt forward in time according, for example , to the following first-order propagation formula: q(t+d t) = q0 + (dq/dt)0 dt dq/dt (t+dt) = (dq/dt)0 - dt [(¶E/¶q)0 /mq ]. Here mq is the mass factor connecting the velocity dq/dt and the momentum pq conjugate to the coordinate q: pq = mq dq/dt, and -(¶E/¶q)0 is the force along the coordianate q at the “initial” geometry q0 . Again, as in Chapter 7, I should note that the above formulas for propagating q and p forward in time represent only the most elementary approach to this problem. There are other, more sophisticated, numerical methods for effecting more accurate and longer-time propagations, but I will not go into them here. Rather, I wanted to focus on the basics of how these simulations are carried out
By applying the time-propagation proecess, one generates a set of"new coordinates q(t+ot)and new velocities da/dt(t+8t) appropriate to the system at time t+8t Using these new coordinates and momenta as go and (ddt)o and evaluating the forces HaE/aqo at these new coordinates, one can again use the Newton equations to generate another finite-time-step set of new coordinates and velocities. Through the sequential application of this process, one generates a sequence of coordinates and velocities that simulate the systems dynamical behavior n using this kind of classical trajectory approach to study chemical reactions, it is important to choose the initial coordinates and momenta in a way that is representative of the experimental conditions that one is attempting to simulate. The tools of statistical mechanics discussed in Chapter 7 guide us in making these choices. When one attempts, for example, to simulate the reactive collisions of an a atom with a bC molecule to produce AB+C, it is not appropriate to consider a single classical (or quantal) collision between A and BC. Why? Because in any laboratory setting 1. The a atoms are probably moving toward the bc molecules with a distribution of relative speeds. That is, within the sample of molecules( which likely contains 10 or more molecules), some A+ bC pairs have low relative kinetic energies when they collide, and others have higher relative kinetic energies. There is a probability distribution P(eke) for this relative kinetic energy that must be properly sampled in hoosing the initial conditions 2. The BC molecules may not all be in the same rotational ()or vibrational(v)state There is a probability distribution function P(J, v)describing the fraction of Bc molecules that are in a particular J state and a particular v state. Initial values of the bC molecule's 18
18 By applying the time-propagation proecess, one generates a set of “new” coordinates q(t+dt) and new velocities dq/dt(t+dt) appropriate to the system at time t+dt. Using these new coordinates and momenta as q0 and (dq/dt)0 and evaluating the forces –(¶E/¶q)0 at these new coordinates, one can again use the Newton equations to generate another finite-time-step set of new coordinates and velocities. Through the sequential application of this process, one generates a sequence of coordinates and velocities that simulate the system’s dynamical behavior. In using this kind of classical trajectory approach to study chemical reactions, it is important to choose the initial coordinates and momenta in a way that is representative of the experimental conditions that one is attempting to simulate. The tools of statistical mechanics discussed in Chapter 7 guide us in making these choices. When one attempts, for example, to simulate the reactive collisions of an A atom with a BC molecule to produce AB + C, it is not appropriate to consider a single classical (or quantal) collision between A and BC. Why? Because in any laboratory setting, 1. The A atoms are probably moving toward the BC molecules with a distribution of relative speeds. That is, within the sample of molecules (which likely contains 1010 or more molecules), some A + BC pairs have low relative kinetic energies when they collide, and others have higher relative kinetic energies. There is a probability distribution P(EKE ) for this relative kinetic energy that must be properly sampled in choosing the initial conditions. 2. The BC molecules may not all be in the same rotational (J) or vibrational (v) state. There is a probability distribution function P(J,v) describing the fraction of BC molecules that are in a particular J state and a particular v state. Initial values of the BC molecule's