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Complete Frequency Calculation Algorithm

A Gaussian Frequency Calculation is the culmination of several mathematical and computational procedures. Starting from an optimized molecular geometry, Gaussian determines the vibrational properties of the molecule by analyzing the curvature of the Potential Energy Surface (PES). From this analysis, it predicts vibrational frequencies, infrared intensities, and thermodynamic properties.

Unlike a Geometry Optimization, which searches for the lowest-energy molecular structure, a Frequency Calculation assumes that the geometry already corresponds to a stationary point on the Potential Energy Surface. The program then investigates how the molecular energy changes when the atoms are displaced by very small amounts.

The complete algorithm combines concepts from electronic structure theory, molecular mechanics, linear algebra, and statistical thermodynamics.

Overall Workflow

The complete Frequency Calculation performed by Gaussian can be summarized as

Optimized Molecular Geometry
             │
             ▼
Electronic Structure Calculation (SCF)
             │
             ▼
Compute Energy and Nuclear Forces
             │
             ▼
Small Atomic Displacements
             │
             ▼
Evaluate Second Derivatives
             │
             ▼
Construct Cartesian Hessian Matrix
             │
             ▼
Transform to Mass-Weighted Coordinates
             │
             ▼
Diagonalize Mass-Weighted Hessian
             │
             ▼
Obtain Eigenvalues and Eigenvectors
             │
      ┌──────┴─────────┐
      ▼                ▼
Normal Modes     Vibrational Frequencies
      │                │
      └──────┬─────────┘
             ▼
Calculate IR Intensities
             │
             ▼
Compute Thermodynamic Properties
             │
             ▼
Generate Frequency Output

Each stage of this workflow builds upon the previous one, ultimately producing the vibrational and thermodynamic information reported in the Gaussian output file.

Step 1 — Optimized Molecular Geometry

The Frequency Calculation begins with an optimized molecular structure.

Optimized Geometry

This structure should correspond to a stationary point on the Potential Energy Surface.

At this geometry,

  • the molecular energy is minimized (or stationary),
  • the nuclear forces are approximately zero,
  • only small atomic vibrations remain.

Frequency calculations should generally be performed only after a successful Geometry Optimization.

Step 2 — Electronic Structure Calculation

Gaussian first performs a standard electronic structure calculation using the chosen level of theory.

Examples include

  • Hartree–Fock (HF)
  • Density Functional Theory (DFT)
  • MP2
  • Coupled Cluster (CC)

This calculation determines

  • molecular orbitals,
  • electron density,
  • electronic energy.

The resulting wavefunction serves as the basis for all subsequent calculations.

Step 3 — Evaluate the Energy Gradient

Using the converged wavefunction,

Gaussian evaluates the first derivatives of the molecular energy with respect to all nuclear coordinates.

At the optimized geometry,

these gradients should be approximately zero,

confirming that the structure corresponds to a stationary point.

Step 4 — Small Atomic Displacements

Each atomic coordinate is displaced by a very small amount in both the positive and negative directions.

Original Position

      ●

Positive Displacement

      ↑
      ●

Negative Displacement

      ●
      ↓

These displacements are extremely small so that the molecule remains within the harmonic region of the Potential Energy Surface.

Step 5 — Calculate the Hessian Matrix

For every displaced geometry,

Gaussian performs additional electronic structure calculations.

The resulting energy changes are used to evaluate the second derivatives of the molecular energy.

These second derivatives form the Cartesian Hessian Matrix.

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

The Hessian describes the curvature of the Potential Energy Surface around the equilibrium geometry.

Step 6 — Transform to Mass-Weighted Coordinates

Since atoms have different masses,

the Cartesian Hessian cannot be used directly.

Gaussian therefore transforms the Hessian into mass-weighted coordinates.

\[ F_{ij} = \frac{H_{ij}} {\sqrt{m_im_j}} \]

This transformation incorporates both

  • molecular force constants,
  • atomic masses,

into a single matrix.

Step 7 — Matrix Diagonalization

The mass-weighted Hessian is diagonalized by solving the eigenvalue equation

\[ F\mathbf{x} = \lambda\mathbf{x} \]

Diagonalization produces

  • eigenvalues,
  • eigenvectors.

These quantities completely describe the molecular vibrations.

Step 8 — Determine the Normal Modes

Each eigenvector corresponds to one normal mode of vibration.

These modes represent the independent vibrational motions of the molecule.

For a molecule containing N atoms

  • nonlinear molecules possess 3N − 6 vibrational modes,
  • linear molecules possess 3N − 5 vibrational modes.

Each mode describes

  • the direction of atomic motion,
  • the relative displacement of every atom,
  • the pattern of vibration.

Step 9 — Calculate Vibrational Frequencies

The eigenvalues are converted into vibrational frequencies using

\[ \nu_i = \frac{1}{2\pi c} \sqrt{\lambda_i} \]

Gaussian reports these frequencies in

cm⁻¹

Positive frequencies indicate stable molecular vibrations,

whereas negative (imaginary) frequencies indicate unstable directions on the Potential Energy Surface.

Step 10 — Calculate Infrared Intensities

For every normal mode,

Gaussian evaluates the change in the molecular dipole moment.

If

\[ \frac{\partial\mu}{\partial Q} \neq 0 \]

the vibration is infrared active.

The calculated intensities determine the heights of the peaks in the predicted infrared spectrum.

Step 11 — Compute Thermodynamic Properties

The calculated vibrational frequencies are then used within the framework of statistical thermodynamics.

Gaussian evaluates

  • Zero-Point Energy,
  • Thermal Energy,
  • Enthalpy,
  • Entropy,
  • Gibbs Free Energy,
  • Heat Capacity.

These quantities are reported for the specified temperature and pressure.

Step 12 — Generate the Output

Finally,

Gaussian prints the results,

including

  • Vibrational frequencies.
  • Reduced masses.
  • Force constants.
  • Normal modes.
  • IR intensities.
  • Zero-Point Energy.
  • Thermal corrections.
  • Enthalpy.
  • Entropy.
  • Gibbs Free Energy.

These quantities form the basis for interpreting molecular vibrations, infrared spectra, and thermodynamic behavior.

Complete Algorithm Summary

The complete sequence followed by Gaussian is

Optimized Geometry
        │
        ▼
SCF Calculation
        │
        ▼
Electronic Wavefunction
        │
        ▼
Energy Gradient
        │
        ▼
Small Atomic Displacements
        │
        ▼
Second Derivatives
        │
        ▼
Cartesian Hessian
        │
        ▼
Mass-Weighted Hessian
        │
        ▼
Matrix Diagonalization
        │
        ├──────────────► Eigenvalues
        │                    │
        │                    ▼
        │           Vibrational Frequencies
        │
        ▼
Eigenvectors
        │
        ▼
Normal Modes
        │
        ├──────────────► IR Intensities
        │
        ▼
Statistical Thermodynamics
        │
        ▼
ZPE, Enthalpy, Entropy,
Gibbs Free Energy
        │
        ▼
Gaussian Frequency Output

Summary

A Gaussian Frequency Calculation transforms an optimized molecular structure into a complete description of its vibrational and thermodynamic properties. Beginning with a converged electronic wavefunction, Gaussian evaluates the second derivatives of the molecular energy to construct the Hessian matrix. After transforming this matrix into mass-weighted coordinates and solving the associated eigenvalue problem, the program obtains the normal modes and vibrational frequencies of the molecule. These frequencies are then used to predict infrared spectra and calculate thermodynamic quantities such as Zero-Point Energy, enthalpy, entropy, and Gibbs free energy. Together, these steps provide a comprehensive characterization of the molecular structure and form an essential part of most computational chemistry workflows.