Hessian Matrix and Taylor Series Expansion¶

The Harmonic Approximation assumes that atoms undergo only small displacements about their equilibrium positions. To describe how the molecular energy changes during these small displacements, Gaussian approximates the Potential Energy Surface (PES) using a Taylor Series Expansion.

The second derivatives appearing in this expansion form the Hessian matrix, which contains all the information needed to determine molecular vibrational frequencies.

Thus, the Hessian matrix is the mathematical foundation of every Frequency Calculation.

Describing the Energy Near Equilibrium¶

Suppose the equilibrium geometry of a molecule is known.

If one or more atoms are displaced slightly, the molecular energy changes.

Rather than calculating the exact energy surface, Gaussian approximates the energy using a Taylor series around the equilibrium geometry.

For a displacement vector q, the energy can be written as

\[ E(\mathbf q) = E_0 + \left( \frac{\partial E}{\partial q} \right)_0 q + \frac12 \left( \frac{\partial^2E}{\partial q^2} \right)_0 q^2 +\cdots \]

where

\(E_0\) is the energy at the equilibrium geometry,
the first derivative represents the energy gradient,
the second derivative represents the local curvature of the Potential Energy Surface.

The Equilibrium Condition¶

At the optimized geometry,

the molecular forces vanish.

Therefore,

\[ \boxed{ \left( \frac{\partial E}{\partial q} \right)_0 = 0 } \]

This means the first-order term disappears.

The Taylor expansion therefore simplifies to

\[ \boxed{ E(\mathbf q) = E_0 + \frac12 \left( \frac{\partial^2E}{\partial q^2} \right)_0 q^2 } \]

This quadratic equation is precisely the mathematical expression of the Harmonic Approximation.

Multiple Atomic Coordinates¶

Real molecules contain many atoms.

Instead of one coordinate,

the energy depends on all nuclear coordinates simultaneously.

For a molecule containing N atoms,

there are

\[ 3N \]

Cartesian coordinates.

The Taylor expansion therefore becomes

\[ E = E_0 + \frac12 \sum_i \sum_j \frac{\partial^2E} {\partial q_i\partial q_j} q_iq_j \]

The second derivatives describe how the energy changes when any two coordinates are displaced.

The Hessian Matrix¶

The collection of all second derivatives forms the Hessian matrix.

It is written as

\[ \boxed{ H_{ij} = \frac{\partial^2E} {\partial q_i\partial q_j} } \]

or, in matrix notation,

\[ \mathbf H = \begin{bmatrix} \frac{\partial^2E}{\partial q_1^2} & \frac{\partial^2E}{\partial q_1\partial q_2} & \cdots \\ \frac{\partial^2E}{\partial q_2\partial q_1} & \frac{\partial^2E}{\partial q_2^2} & \cdots \\ \vdots & \vdots & \ddots \end{bmatrix} \]

For a molecule with N atoms, the Hessian has dimensions

\[ 3N \times 3N \]

Physical Meaning of the Hessian¶

Each element of the Hessian measures the curvature of the Potential Energy Surface.

Large Hessian Elements¶

steep potential
stiff chemical bonds
high vibrational frequencies

Small Hessian Elements¶

shallow potential
flexible molecular motions
low vibrational frequencies

Thus,

the Hessian determines how strongly each atom resists displacement.

Diagonal Elements¶

The diagonal elements

\[ H_{ii} = \frac{\partial^2E} {\partial q_i^2} \]

measure the curvature along a single coordinate.

They indicate how rapidly the energy changes when one coordinate is displaced independently.

Off-Diagonal Elements¶

The off-diagonal elements

\[ H_{ij} = \frac{\partial^2E} {\partial q_i\partial q_j} \]

describe the coupling between different coordinates.

These terms explain why atoms rarely vibrate independently.

Instead,

their motions are coupled,

producing collective molecular vibrations.

Symmetry of the Hessian¶

For ordinary molecular systems,

the Hessian is symmetric.

\[ \boxed{ H_{ij} = H_{ji} } \]

This property is important because symmetric matrices possess

real eigenvalues,
orthogonal eigenvectors,

making the vibrational problem mathematically well behaved.

What Does Gaussian Actually Compute?¶

During a Frequency Calculation,

Gaussian performs many electronic structure calculations after introducing small atomic displacements.

Optimized Geometry
        │
        ▼
Move One Coordinate Slightly
        │
        ▼
SCF Calculation
        │
        ▼
Calculate Energy Change
        │
        ▼
Repeat for All Coordinates
        │
        ▼
Construct Hessian Matrix

Once the Hessian is assembled,

it contains the complete local curvature of the Potential Energy Surface.

Why the Hessian is Important¶

The Hessian serves as the starting point for all remaining steps of the Frequency Calculation.

From the Hessian, Gaussian determines

normal modes of vibration,
vibrational frequencies,
infrared intensities,
Zero-Point Energy,
thermodynamic properties.

Without the Hessian,

none of these quantities could be calculated.

Summary¶

The Taylor Series Expansion provides a mathematical description of how the molecular energy changes when atoms are displaced from their equilibrium positions. Because the first derivatives vanish at the optimized geometry, the energy is dominated by the quadratic term, whose coefficients are the second derivatives of the energy. These second derivatives form the Hessian matrix, which describes the local curvature of the Potential Energy Surface. The Hessian is the central mathematical object in Frequency Analysis and serves as the foundation for calculating normal modes, vibrational frequencies, and thermodynamic properties.

Next Section¶

The next chapter introduces Mass-Weighted Coordinates, which transform the Hessian matrix into a form that separates the effects of atomic masses from the molecular force constants, making the vibrational problem much easier to solve.