Vibrational Frequency Analysis

Overview

The frequency analysis module computes vibrational normal modes by constructing and diagonalizing the mass-weighted Hessian matrix (the matrix of second derivatives of the energy with respect to nuclear coordinates, weighted by atomic masses). Each eigenvalue corresponds to the square of a vibrational frequency, and each eigenvector defines a normal mode of vibration.

From the computed normal mode frequencies, MAPLE derives thermochemical quantities using the rigid rotor-harmonic oscillator (RRHO) approximation, including zero-point vibrational energy (ZPVE), thermal corrections, enthalpy, entropy, and Gibbs free energy at the specified temperature and pressure.

Warning

The input structure should be properly optimized before running a frequency analysis. Frequencies computed at non-stationary points are not physically meaningful and will not yield correct thermochemical data.

Usage

The frequency analysis task is invoked by including the #freq header command in the input file:

#freq

When called without parameters, MAPLE uses the default settings: mass-weighted Hessian diagonalization at 298.15 K and 101.325 kPa (1 atm).

Input Example

Below is a complete input file for a frequency calculation on a previously optimized structure:

#model=ani1xnr
#freq
#device=gpu0

XYZ /path/to/optimized_structure.xyz

The XYZ directive reads atomic coordinates from an external file. This is convenient when using geometries produced by a prior optimization run.

Parameters

The following parameters can be adjusted for the frequency analysis task. Default values are shown.

Parameter	Type	Default	Description
`method`	str	`mw`	Hessian diagonalization method: `mw` (mass-weighted), `nonmw` (unweighted), or `both`.
`temperature`	float	`298.15`	Temperature in Kelvin for thermochemical evaluation.
`ilowfreq`	int	`2`	Low-frequency treatment: 0 = RRHO (no special treatment), 1 = Truhlar quasi-harmonic, 2 = Grimme quasi-RRHO, 3 = Minenkov interpolation.
`verbose`	int	`1`	Output verbosity level.
`treat_imag_as_real`	bool	`false`	Treat imaginary frequencies as real for thermochemistry.
`pressure_kpa`	float	`101.325`	Pressure in kPa (101.325 = 1 atm).
`device`	str	`cpu`	Compute device for Hessian evaluation.

Output

The frequency analysis output file reports the following quantities:

Normal mode frequencies -- all 3N-6 (or 3N-5 for linear molecules) vibrational frequencies in cm^-1. Imaginary frequencies are printed as negative values.
Zero-point vibrational energy (ZPVE) -- the sum of half-quanta for all real vibrational modes, reported in Hartree and kcal/mol.
Thermal corrections -- contributions from translational, rotational, and vibrational degrees of freedom at the specified temperature.
Enthalpy (H) -- total enthalpy including electronic energy, ZPVE, and thermal corrections.
Entropy (S) -- total entropy from translational, rotational, vibrational, and electronic partition functions.
Gibbs free energy (G) -- computed as G = H - TS at the specified temperature.

Important

Imaginary frequencies (printed as negative values) indicate that the structure is not a true minimum. One imaginary frequency corresponds to a transition state; multiple imaginary frequencies suggest the structure is a higher-order saddle point or has not been properly optimized. Re-optimize the structure before repeating the frequency analysis.

Thermochemistry

MAPLE evaluates thermochemical properties using the rigid rotor-harmonic oscillator (RRHO) approximation, which partitions the molecular partition function into independent contributions:

Translational partition function -- derived from the ideal gas model, depends on molecular mass, temperature, and pressure.
Rotational partition function -- computed from the principal moments of inertia assuming a rigid rotor, with separate handling for linear and nonlinear molecules.
Vibrational partition function -- each normal mode is treated as an independent quantum harmonic oscillator. Modes below the ilowfreq threshold are handled with a free-rotor interpolation to avoid numerical instability.
Electronic partition function -- assumes the electronic ground state dominates (multiplicity = 1 for closed-shell systems).

The total internal energy, enthalpy, entropy, and Gibbs free energy are assembled from these partition function components. All thermochemical values are reported at the user-specified temperature and pressure.