9 Exploring Potential Energy Surfaces: Critical Points and Molecular Dynamics

9.7 Intrinsic Reaction Coordinate

(June 30, 2021)

The concept of a reaction path is chemically intuitive (a pathway from reactants to products) yet somewhat theoretically ambiguous because most mathematical definitions depend upon the chosen coordinate system. Stationary points on a potential energy surface are independent of this choice, but the path connecting them is not, and there exist various mathematical definitions of a “reaction path”. Q-Chem uses the intrinsic reaction coordinate (IRC) definition, as originally defined by Fukui, 333 Fukui K.
J. Phys. Chem.
(1970), 74, pp. 4161.
Link
which has come to be a fairly standard choice in quantum chemistry. The IRC is essentially sequence of small, steepest-descent paths going downhill from the transition state.

The reaction path is most unlikely to be a straight line and so by taking a finite step length along the direction of the gradient you will leave the “true” reaction path. A series of small steepest descent steps will zig-zag along the actual reaction path (a behavior known as “stitching”). Ishida et al. 488 Ishida K., Morokuma K., Komornicki A.
J. Chem. Phys.
(1977), 66, pp. 215.
Link
developed a predictor-corrector algorithm, involving a second gradient calculation after the initial steepest-descent step, followed by a line search along the gradient bisector to get back on the path, and this algorithm was subsequently improved by Schmidt et al.. 972 Schmidt M. W., Gordon M. S., Dupuis M.
J. Am. Chem. Soc.
(1985), 107, pp. 2585.
Link
This is the method that Q-Chem adopts. It cannot be used for the first downhill step from the transition state, since the gradient is zero, so instead a step is taken along the Hessian mode whose frequency is imaginary.

The reaction path can be defined and followed in Z-matrix coordinates, Cartesian coordinates or mass-weighted Cartesian coordinates. The latter represents the “true” IRC as defined by Fukui. 333 Fukui K.
J. Phys. Chem.
(1970), 74, pp. 4161.
Link
If the rationale for following the reaction path is simply to determine which local minima are connected by a given transition state, which, is arguably the major use of IRC algorithms, then the choice of coordinates is irrelevant. In order to use the IRC code, the transition state geometry and the exact Hessian must be available. These must be computed via two prior calculations, with JOBTYPE = TS (transition structure search) and JOBTYPE = FREQ (Hessian calculation), respectively. Job control variables and examples appear below.

An IRC calculation is invoked by setting JOBTYPE = RPATH in the $rem section, and additional $rem variables are described below. IRC calculations may benefit from the methods discussed in Section 9.2 for obtaining good initial guesses for transition-state structures.

RPATH_COORDS
       Determines which coordinate system to use in the IRC search.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Use mass-weighted coordinates. 1 Use Cartesian coordinates. 2 Use Z-matrix coordinates.
RECOMMENDATION:
       Use the default. Note that use of Z-matrix coordinates requires that geometries be input in Z-matrix format.

RPATH_DIRECTION
       Determines the first direction of the eigenmode to follow. This will not usually be known prior to the Hessian diagonalization.
TYPE:
       INTEGER
DEFAULT:
        1
OPTIONS:
        1 Descend in the positive direction of the eigenmode, then restart in the negative direction. -1 Descend in the negative direction of the eigenmode, then restart in the positive direction.
RECOMMENDATION:
       It is usually not possible to determine in which direction to go a priori, so both directions are automatically considered. A job that reads in the final geometry from the reaction path job will use the final step from the second direction.

RPATH_MAX_CYCLES
       Specifies the maximum number of points to find on the reaction path.
TYPE:
       INTEGER
DEFAULT:
       20
OPTIONS:
       n User-defined number of cycles.
RECOMMENDATION:
       Use more points if the minimum is desired, but not reached using the default.

RPATH_MAX_STEPSIZE
       Specifies the maximum step size to be taken (in 0.001 a.u.).
TYPE:
       INTEGER
DEFAULT:
       150 corresponding to a step size of 0.15 a.u..
OPTIONS:
       n Step size = n/1000 a.u.
RECOMMENDATION:
       None.

RPATH_TOL_DISPLACEMENT
       Specifies the convergence threshold for the step. If a step size is chosen by the algorithm that is smaller than this, the path is deemed to have reached the minimum.
TYPE:
       INTEGER
DEFAULT:
       5000 Corresponding to 0.005 a.u.
OPTIONS:
       n User-defined. Tolerance = n/1000000 a.u.
RECOMMENDATION:
       Use the default. Note that this option only controls the threshold for ending the RPATH job and does nothing to the intermediate steps of the calculation. A smaller value will provide reaction paths that end closer to the true minimum. Use of smaller values without adjusting RPATH_MAX_STEPSIZE, however, can lead to oscillations about the minimum.

RPATH_PRINT
       Specifies the print output level.
TYPE:
       INTEGER
DEFAULT:
       2
OPTIONS:
       n
RECOMMENDATION:
       Use the default, as little additional information is printed at higher levels. Most of the output arises from the multiple single point calculations that are performed along the reaction pathway.

Example 9.18  Reaction path search. Note that there are three required jobs: a TS search, followed by a frequency (Hessian) calculation, and finally the IRC calculation.

$molecule
   0 1
   C
   H 1 1.20191
   N 1 1.22178 2 72.76337
$end

$rem
   JOBTYPE            ts
   BASIS              sto-3g
   METHOD             hf
$end

@@@

$molecule
   read
$end

$rem
   JOBTYPE            freq
   METHOD             hf
   BASIS              sto-3g
   SCF_GUESS          read
$end

@@@

$molecule
   read
$end

$rem
   JOBTYPE            rpath
   BASIS              sto-3g
   METHOD             hf
   SCF_GUESS          read
   RPATH_MAX_CYCLES   50
$end

View output