Potential Energy Surface (PES)¶

Every molecular geometry corresponds to a unique electronic energy. As the positions of the nuclei change, the electronic energy also changes. The relationship between the nuclear coordinates and the electronic energy is known as the Potential Energy Surface (PES).

A Geometry Optimization calculation can therefore be viewed as a search for the lowest point on this energy surface.

Rather than thinking of a molecule as a rigid object, it is more useful to imagine it moving across an energy landscape. Some regions of the landscape are high in energy, while others correspond to stable molecular structures. The objective of Gaussian is to locate one of these stable regions efficiently.

A Simple Analogy¶

The Potential Energy Surface can be compared to a mountainous landscape.

Energy
  ^
  |
  |                /\        Transition State
  |               /  \
  |              /    \
  |             /      \
  |            /        \
  |      _____/          \_____
  |     /                      \
  |____/                        \______
       Minimum              Minimum

              Molecular Geometry →

In this landscape,

valleys correspond to stable molecular structures,
hills correspond to unstable geometries,
mountain passes correspond to transition states.

Geometry optimization attempts to move the molecule downhill until it reaches the bottom of a valley.

Energy as a Function of Nuclear Coordinates¶

Mathematically, the electronic energy depends on the positions of all nuclei,

\[ E = E(\mathbf{R}) \]

where

\(E\) is the electronic energy,
\(\mathbf{R}\) represents the complete set of nuclear coordinates.

For a molecule containing \(N\) atoms,

\[ \mathbf{R} = (x_1,y_1,z_1, x_2,y_2,z_2, \ldots, x_N,y_N,z_N) \]

Each atom contributes three Cartesian coordinates.

Consequently,

a molecule containing \(N\) atoms has

\[ 3N \]

geometrical coordinates.

After removing overall translation and rotation,

the number of independent internal coordinates becomes

\(3N-6\) for non-linear molecules,
\(3N-5\) for linear molecules.

Thus, the Potential Energy Surface is actually a function of many variables rather than just one or two.

A One-Dimensional Example¶

For a diatomic molecule, only the bond length changes.

The potential energy surface therefore reduces to a single curve.

Energy
  ^
  |
  |            *
  |         *     *
  |       *         *
  |     *            *
  |   *               *
  | *                  *
  +---------------------------->
          Bond Length

          Equilibrium Distance

The minimum of the curve corresponds to the equilibrium bond length.

If the atoms move closer together,

the energy increases because of nuclear repulsion.

If the atoms move farther apart,

the bond weakens and the energy also increases.

Potential Energy Surface of Polyatomic Molecules¶

For molecules containing many atoms,

every bond length,

every bond angle,

and every dihedral angle

can change simultaneously.

The potential energy surface therefore becomes extremely high dimensional (3N-6, for non-linear N atoms ).

For example,

Molecule	Number of Atoms	Internal Coordinates
H₂O	3	3
NH₃	4	6
Benzene	12	30
Formic Acid Dimer	10	24

Because the dimensionality increases rapidly,

the complete surface cannot be visualized directly.

Instead,

Gaussian explores only a tiny region of the surface during optimization.

Important Features of the PES¶

The Potential Energy Surface contains several important types of stationary points.

Local Minimum¶

      *
    *   *
  *       *
 *         *
*           *

A local minimum represents a stable molecular geometry.

Small movements in any direction increase the energy.

This is usually the desired result of a Geometry Optimization.

Transition State¶

      *
    *   *
  *       *
 *         *
*           *
 \         /
  \_______/

A transition state is a saddle point on the Potential Energy Surface.

The energy decreases in one direction but increases in all other directions.

Transition states are important in reaction mechanism studies.

Maximum¶

      *
     ***
    *****

A maximum is unstable in every direction.

Such structures are rarely encountered in molecular optimization.

What Does Gaussian Search For?¶

During a Geometry Optimization,

Gaussian does not attempt to construct the entire Potential Energy Surface.

Instead,

it performs a local search beginning from the initial molecular geometry supplied by the user.

At each optimization step,

Gaussian

calculates the electronic energy,
determines how the energy changes with geometry,
predicts a better structure,
repeats the process until a stationary point is reached.

Thus,

only a very small region of the Potential Energy Surface is explored during the optimization.

Why Is the PES Important?¶

Understanding the Potential Energy Surface explains why

different starting geometries may converge to different structures,
geometry optimizations sometimes fail,
transition-state searches are more difficult than minimum-energy searches,
frequency calculations are required after optimization.

Every optimization algorithm ultimately attempts to navigate this multidimensional energy landscape as efficiently as possible.

Summary¶

The Potential Energy Surface is a mathematical representation of how the electronic energy of a molecule changes as the atomic positions vary. Every molecular geometry corresponds to a unique point on this surface. Stable molecular structures are located at local minima, while transition states correspond to saddle points. Geometry Optimization is therefore the process of moving across the Potential Energy Surface until a stationary point is reached. Although the surface becomes extremely high-dimensional for large molecules, Gaussian efficiently explores only the local region surrounding the initial geometry to locate an optimized structure.

Next Section¶

In the next chapter, we introduce the Energy Gradient, which describes the slope of the Potential Energy Surface and determines the direction in which the atoms should move to lower the molecular energy during optimization.