Vibrational Frequencies¶

The primary goal of a Gaussian Frequency Calculation is to determine the natural vibrational frequencies of a molecule. These frequencies describe how rapidly the atoms oscillate about their equilibrium positions and provide direct information about the strength of chemical bonds and the stability of the molecular structure.

Each vibrational frequency corresponds to one normal mode of vibration obtained from the diagonalization of the mass-weighted Hessian matrix.

Gaussian reports these frequencies in units of wavenumbers (cm⁻¹), which are commonly used in infrared (IR) and Raman spectroscopy.

From Eigenvalues to Frequencies¶

After solving the eigenvalue problem,

Gaussian obtains a set of eigenvalues,

\[ \lambda_1,\lambda_2,\lambda_3,\ldots \]

Each eigenvalue represents the curvature of the Potential Energy Surface along one normal mode.

The vibrational frequency is calculated as

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

where

\( \nu_i \) is the vibrational frequency,
\( c \) is the speed of light,
\( \lambda_i \) is the eigenvalue of the corresponding normal mode.

Gaussian automatically performs this conversion and reports the frequencies in cm⁻¹.

Why Use Wavenumbers?¶

Instead of reporting frequency in hertz (Hz),

molecular spectroscopy traditionally uses wavenumbers.

The relationship is

\[ \boxed{ \tilde{\nu} = \frac{\nu}{c} } \]

where

\( \tilde{\nu} \) is the wavenumber (cm⁻¹),
\( \nu \) is the frequency (Hz),
\( c \) is the speed of light.

Using wavenumbers produces convenient numerical values for molecular vibrations, which typically range from tens to several thousand cm⁻¹.

Typical Frequency Ranges¶

Different types of molecular vibrations occur in different frequency regions.

Vibrational Motion	Approximate Range (cm⁻¹)
Heavy atom torsions	20–200
Skeletal deformations	200–600
Bond bending	600–1500
C–C stretching	800–1300
C=C stretching	1500–1700
C=O stretching	1650–1800
O–H stretching	3200–3700
N–H stretching	3300–3500

These ranges are approximate and depend on the molecular environment.

Interpretation of Frequencies¶

The sign of the calculated frequency is extremely important.

Positive Frequencies¶

500 cm⁻¹

1250 cm⁻¹

1740 cm⁻¹

3400 cm⁻¹

Positive frequencies indicate

stable vibrational motion,
upward curvature of the Potential Energy Surface,
a physically meaningful normal mode.

A molecule with only positive vibrational frequencies corresponds to a stable minimum on the Potential Energy Surface.

Near-Zero Frequencies¶

Very small frequencies,

for example,

0 cm⁻¹

5 cm⁻¹

12 cm⁻¹

usually correspond to

molecular translations,
molecular rotations,
numerical noise.

These modes are not true molecular vibrations.

Imaginary Frequencies¶

Gaussian represents imaginary frequencies as negative values.

Example

-450 cm⁻¹

Although printed as negative numbers,

the actual mathematical frequency is imaginary because the corresponding eigenvalue is negative.

Imaginary frequencies indicate

an unstable direction,
downward curvature,
a transition state.

Stationary Point Identification¶

The number of imaginary frequencies determines the nature of the optimized geometry.

Imaginary Frequencies	Structure
0	Stable minimum
1	First-order transition state
2 or more	Higher-order saddle point

Consequently, one of the most important purposes of a Frequency Calculation is to verify the character of the optimized structure.

What Determines the Frequency?¶

Several physical factors influence the vibrational frequency.

Bond Strength¶

Strong chemical bonds

possess large force constants,
vibrate more rapidly,
produce higher frequencies.

Examples

O–H
N–H
C≡N

Atomic Mass¶

Heavy atoms vibrate more slowly than light atoms.

For example,

H–H

vibrates much faster than

I–I

because hydrogen atoms are significantly lighter than iodine atoms.

Molecular Environment¶

Hydrogen bonding,

electronic effects,

and neighboring functional groups

can shift the vibrational frequencies by altering the local force constants.

Frequency Scaling Factors¶

The harmonic approximation generally overestimates experimental vibrational frequencies.

This occurs because

anharmonic effects are neglected,
electronic structure methods are approximate,
basis sets introduce small errors.

Therefore,

calculated harmonic frequencies are often multiplied by an empirical frequency scaling factor.

Typical scaling factors are

Method	Typical Scaling Factor
HF/6-31G	~0.89
B3LYP/6-31G	~0.96
B3LYP/6-31G(d)	~0.96

The exact scaling factor depends on the chosen level of theory.

Vibrational Spectrum¶

Each calculated frequency corresponds to one peak in the vibrational spectrum.

Frequency Calculation
         │
         ▼
Normal Modes
         │
         ▼
Vibrational Frequencies
         │
         ▼
Infrared Spectrum

Whether a particular vibration actually appears in the infrared spectrum depends on its IR intensity, which is discussed in the next chapter.

Relationship to Gaussian Output¶

In the Gaussian output,

the frequencies appear as

Frequencies --  145.6  486.3  1758.2

Each value corresponds to

one normal mode,
one eigenvalue,
one characteristic molecular vibration.

These frequencies form the basis for interpreting molecular structure, spectroscopy, and thermodynamic properties.

Summary¶

Vibrational frequencies describe the natural oscillations of a molecule about its equilibrium geometry. They are obtained by converting the eigenvalues of the mass-weighted Hessian matrix into observable frequencies expressed in wavenumbers (cm⁻¹). Positive frequencies correspond to stable vibrational modes, while imaginary frequencies indicate unstable directions on the Potential Energy Surface and are characteristic of transition states. The magnitude of each frequency depends primarily on the bond strength, atomic masses, and molecular environment. Together, the calculated frequencies provide essential information for molecular spectroscopy, structural verification, and thermodynamic analysis.

Next Section¶

The next chapter introduces Infrared Intensities, explaining why some vibrational modes absorb infrared radiation while others remain inactive, and how Gaussian predicts the intensity of each absorption band.