# Equations used in the DRC and IRC

## DRC

Force acting on any atom:

Acceleration due to force acting on each atom:

New velocity:

or:

V(i) = V(i) + V'(i)t + V"(i)t2 + V"'(i)t3

That is, the change in velocity is equal to the integral over the time interval of the acceleration.

New position of atoms:

X(i) = X(o) + V(o)t + (1/2) V't2 + (1/3) V"t3 + (1/4) V"'t4

That is, the change in position is equal to the integral over the time interval of the velocity.

The velocity vector is accurate to the extent that it takes into account the previous velocity, the current acceleration, the predicted acceleration, and the change in predicted acceleration over the time interval. Very little error is introduced due to higher order contributions to the velocity; those that do occur are absorbed in a re-normalization of the magnitude of the velocity vector after each time interval.

The size of Δt, the time interval, starts off at 0.1fs (1*10-16s), and changes depending on the factor needed to re-normalize the velocity vector. If it is significantly different from unity, Δt will be reduced; if it is very close to unity, Δt will be increased. The time interval is not printed, however typical values are between 0.1 and 1fs.

Even with all this, errors creep in and a system, started at the transition state, is unlikely to return precisely to the transition state unless an excess kinetic energy is supplied, for example 0.2 kcal/mol.

The calculation is carried out in Cartesian coordinates, and converted into internal coordinates for display. All Cartesian coordinates must be allowed to vary, in order to conserve angular and translational momentum.

## IRC

The calculation of the IRC is that of a fully damped DRC. Within the calculation, a fictional time interval is used to determine the degree of motion, but at the end of each step, all velocity components are removed.  The time interval, Δt, starts off at 0.1fs (1*10-16s), and changes depending on the factor needed to re-normalize the velocity vector. If it is significantly different from unity, Δt will be reduced; if it is very close to unity, Δt will be increased.  Typical values of Δt are between 0.1 and 1fs.