Reaction Path Following¶

Once the Minimum Energy Path (MEP) has been defined, Gaussian must determine how to move from one point on the path to the next. This process is known as reaction path following.

Rather than searching for a new stationary point, the IRC algorithm advances along the reaction coordinate using a series of small steps. After each step, the molecular geometry is corrected so that it remains on the Minimum Energy Path.

The complete reaction pathway is therefore generated as a sequence of closely spaced molecular structures connecting the transition state with the reactants and products.

The Basic Principle¶

The IRC calculation begins at the transition state.

Starting from this geometry, Gaussian

moves a small distance along the reaction coordinate,
corrects the geometry onto the Minimum Energy Path,
stores the optimized structure,
repeats the procedure.

This process continues until a stable minimum is reached.

Transition State
        │
        ▼
Move Along Reaction Coordinate
        │
        ▼
Correct Geometry
        │
        ▼
New IRC Point
        │
        ▼
Repeat

Following the Reaction Coordinate¶

The reaction coordinate is represented by a parameter

\[ s \]

which measures the distance travelled along the reaction pathway.

Each IRC point corresponds to

\[ \mathbf{R}(s), \]

where

\(s=0\) represents the transition state,
positive values move toward one minimum,
negative values move toward the opposite minimum.

Thus,

\[ \mathbf{R}(0) \]

is the transition state,

while

\[ \mathbf{R}(+s) \]

and

\[ \mathbf{R}(-s) \]

describe the two branches of the reaction path.

Small Incremental Steps¶

Instead of making one large displacement,

Gaussian advances using many small increments.

TS

●──●──●──●──●──●──●

Reactant

Each point represents a fully optimized IRC structure.

Using small step sizes improves numerical stability and ensures that the reaction path closely follows the true Minimum Energy Path.

Forward and Reverse Directions¶

An important feature of the IRC algorithm is that it follows the reaction coordinate in both directions.

Reactants
      ▲
      │
      │
Transition State
      │
      │
      ▼
Products

One calculation traces the path toward the reactants,

while a second traces the path toward the products.

Together,

these two calculations produce the complete reaction pathway.

Geometry Correction¶

After every displacement,

the molecular geometry no longer lies exactly on the Minimum Energy Path.

Gaussian therefore performs a correction step.

Predicted Geometry
        │
        ▼
Geometry Correction
        │
        ▼
IRC Geometry

The correction minimizes the energy in all directions perpendicular to the reaction coordinate while preserving movement along the reaction path.

As a result,

every stored IRC point remains on the Minimum Energy Path.

Energy Along the Path¶

As the reaction coordinate changes,

the molecular energy changes continuously.

Energy

        TS
         ●
        / \
       /   \
      /     \
     ●       ●

Reaction Coordinate →

The transition state is located at the maximum,

while both endpoints correspond to local minima.

The smooth variation in energy confirms that the IRC follows the physically meaningful reaction pathway.

Why Small Steps?¶

If the displacement between successive IRC points were too large,

the calculation could

leave the Minimum Energy Path,
miss important structural changes,
converge to an incorrect geometry.

Using small steps allows Gaussian to accurately follow the curvature of the Potential Energy Surface throughout the reaction.

Relation to Previous Mathematics¶

Reaction path following combines several concepts introduced earlier.

The energy and gradients evaluated at each IRC point are discussed in

Single Point Energy¶

📘 Single Point Energy Mathematics

The geometry corrections performed after each displacement are closely related to

Geometry Optimization¶

📘 Geometry Optimization Mathematics

The starting point of the reaction path is obtained from

Transition State Search¶

📘 Transition State Search Mathematics

Summary¶

Reaction path following is the fundamental procedure used during an Intrinsic Reaction Coordinate calculation. Beginning from the transition state, Gaussian advances in small steps along the reaction coordinate and corrects each new geometry so that it remains on the Minimum Energy Path. By repeating this process in both the forward and reverse directions, the complete reaction pathway connecting the reactants and products is obtained.

Next Section¶

The next chapter introduces Mass-Weighted Coordinates, explaining why Gaussian transforms ordinary Cartesian coordinates before following the reaction path and how this produces a physically meaningful reaction coordinate. ```