Reaction Coordinate¶

A Transition State is characterized by one unique direction along which the molecular energy decreases toward either the reactants or the products. This direction is known as the reaction coordinate.

The reaction coordinate represents the progress of a chemical reaction on the Potential Energy Surface (PES). It describes how the molecular geometry changes as chemical bonds break, new bonds form, or atoms rearrange during the reaction.

Unlike ordinary molecular vibrations, the reaction coordinate is the only direction in which the transition state is unstable.

What is a Reaction Coordinate?¶

A reaction coordinate is a mathematical parameter that measures the progress of a chemical reaction.

It can be represented as

\[ q \]

where

\(q < 0\) corresponds to structures closer to the reactants,
\(q = 0\) corresponds to the transition state,
\(q > 0\) corresponds to structures closer to the products.

Thus, every point along the reaction pathway has a unique value of the reaction coordinate.

Potential Energy Along the Reaction Coordinate¶

If the multidimensional Potential Energy Surface is projected onto a single reaction coordinate, the energy profile becomes

Energy
 ^
 |
 |                     ●
 |                   /   \
 |                 /       \
 |               /           \
 |             /               \
 |   ●─────────                 ─────────●
 |
 +-------------------------------------------------> Reaction Coordinate

 Reactants      Transition State         Products

The transition state lies at the highest point along the reaction coordinate and represents the activation barrier separating reactants and products.

A Special Direction on the Potential Energy Surface¶

Near a transition state, most nuclear motions increase the energy.

Only one direction decreases the energy.

                    Stable
                      ↑
                      │
              \       │       /
               \      │      /
                \     │     /
                 \    ●    /
------------------\---│---/------------------>
                     Reaction Coordinate
                 /    │    \
                /     │     \
               /      │      \
              /       │       \
                      │
                   Stable

The reaction coordinate is therefore the only unstable direction at a first-order saddle point.

Motion Along the Reaction Coordinate¶

Moving the nuclei slightly in opposite directions along the reaction coordinate produces different molecular structures.

Reactants
      ▲
      │
      │
Transition State
      │
      │
      ▼
Products

Small displacements from the transition state lead naturally toward either

the reactant structure, or
the product structure.

This is why the transition state connects both sides of the reaction.

Physical Meaning¶

The reaction coordinate usually corresponds to the dominant structural change occurring during the chemical reaction.

Examples include

breaking of a chemical bond,
formation of a new bond,
proton transfer,
hydrogen migration,
ring opening,
ring closure,
molecular rearrangement.

Although many atoms may move simultaneously, these motions combine to form a single collective coordinate describing the reaction.

Relation to Vibrational Analysis¶

After the transition state has been optimized, Gaussian performs a Frequency Calculation.

One vibrational mode possesses an imaginary frequency.

This vibrational mode is the reaction coordinate.

When animated in GaussView,

the atoms move along the expected chemical pathway.

Imaginary Frequency
          │
          ▼
Normal Mode
          │
          ▼
Reaction Coordinate

Therefore,

examining the imaginary vibrational mode is an essential step in verifying a transition state.

Why is the Reaction Coordinate Important?¶

The reaction coordinate allows chemists to

understand how reactants transform into products,
identify which bonds are breaking and forming,
determine the reaction mechanism,
calculate activation energies,
connect the transition state with the reactants and products.

Without identifying the reaction coordinate, it is impossible to determine whether the optimized transition state corresponds to the intended chemical reaction.

Relationship with Intrinsic Reaction Coordinate (IRC)¶

The reaction coordinate forms the basis of the Intrinsic Reaction Coordinate (IRC) method.

Starting from the transition state,

Gaussian can follow the reaction coordinate downhill on the Potential Energy Surface toward both

reactants,
products.

Reactants
      ▲
      │
      │
Transition State
      │
      │
      ▼
Products

An IRC calculation therefore confirms that the transition state connects the correct molecular structures.

(The IRC method will be discussed in a separate section of this documentation.)

Summary¶

The reaction coordinate is the unique direction that connects the reactants and products through the transition state. It represents the progress of the chemical reaction and corresponds to the single unstable direction on the Potential Energy Surface. In Gaussian, this direction is identified by the imaginary vibrational mode obtained from the Frequency Calculation. Understanding the reaction coordinate is essential for interpreting reaction mechanisms and verifying that the optimized transition state describes the intended chemical transformation.

Next Section¶

The next chapter examines the Hessian Matrix and Negative Curvature, explaining how Gaussian mathematically distinguishes a transition state from a local minimum through the eigenvalues of the Hessian matrix.