Classical Trajectories |
The movement of a point on a PES is associated with the changes in the potential energy and the kinetic energy of the system. The PES gives the changes in the potential energy alone and the changes in kinetic energies can be assessed from the velocities of the particles at each point. The position ( x ) and velocity ( v ) at any time (t + t) can be calculated from the position and velocity at a nearby time ( t ) by solving the equations of motion. |
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x ( t +
 t ) = x ( t ) + (dx / dt)
 t + 1/ 2(d 2 x / d t 2 )t 2 + ... |
(35.1) |
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| v ( t + t ) = v ( t ) + (dv / dt) t + 1/ 2 (d2v / dt2) t 2 + .... |
(35.2) |
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| In (35.1) and (35.2), dv/dt = d2x / dt2 = acceleration in the x direction which is nothing but force per unit mass, Fx /m. The force is given in terms of the PE ( V ) at the point (corresponding to time t ) by |
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Fx = -
 V / x ; V = PE |
(35.3) |
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Since the form of the PES is known,
 V /x can be calculated at each point. For small t, if we know x (t) and v( t ) , x ( t + t ) and v ( t +t) can be calculated from eq. (35.1) and (35.2) very easily if we neglect the t2 term in eq (35.2). This approach constitutes the simplest algorithm for the time evolution of the point on a PES. Other more sophisticated algorithms are available. A large number of trajectories can be generated starting from different initial conditions. From these, an energy or velocity dependent reactive cross sections can be calculated. One actually computes the probability of a reaction (i.e., a reactive cross section) for a given initial velocity and then averages the product [ ( R ) . v] to get a rate constant, as was done in the collision theory. This is a direct application of (35.1) and (35.2). These equations are not "symmetrical" in x and v. E.g., x ( t +t ) needs v(t), but v ( t + t ) does not need x(t). More symmetrical form of the equations of motion are the Hamiltons equations, which are |
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x / t =  = H / p x |
(35.4) |
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p x / t =  = - H / x |
(35.5) |
| Where H = hamiltonian = KE + PE. Usually these are solved for obtaining classical trajectories. These equations are completely equivalent to the Newtons Laws of motion. |
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