Normal Modes of Vibration¶

Once the Hessian matrix has been transformed into mass-weighted coordinates, Gaussian determines the natural ways in which the molecule can vibrate. These characteristic patterns of motion are called normal modes of vibration.

A normal mode represents a collective movement of all the atoms in the molecule. During a particular normal mode, every atom oscillates with the same frequency, although each atom may move by a different amount and in a different direction.

Instead of considering the motion of each atom independently, Gaussian decomposes the molecular motion into a set of independent vibrations known as normal modes.

Why Do We Need Normal Modes?¶

Consider a simple water molecule.

If one hydrogen atom moves,

the oxygen atom also moves.

The second hydrogen atom is affected as well.

Therefore,

the atoms do not vibrate independently.

Instead,

their motions are coupled through the chemical bonds.

Normal mode analysis separates these complicated coupled motions into independent vibrations that can be studied one at a time.

Coupled Motion of Atoms¶

Imagine three atoms connected by springs.

O ===== C ===== O

When one atom moves,

the springs transmit the motion to the neighboring atoms.

Instead of observing three independent atomic motions,

the entire molecule vibrates together.

This collective motion is much easier to understand when expressed as normal modes.

What is a Normal Mode?¶

A normal mode is an independent vibrational pattern in which every atom oscillates with the same frequency.

Each mode is characterized by

a unique vibrational frequency,
a specific pattern of atomic displacements,
a definite vibrational energy.

During a single normal mode,

all atoms move simultaneously,

but the relative motion of each atom remains fixed throughout the vibration.

Visualizing a Normal Mode¶

Consider the symmetric stretching vibration of carbon dioxide.

O  ←──  C  ──→  O

Both oxygen atoms move away from the carbon atom simultaneously.

Now consider the asymmetric stretching vibration.

O  ←──  C  ←──  O

One oxygen atom approaches the carbon while the other moves away.

Both are examples of normal modes.

Although the atomic motions differ,

each vibration occurs independently with its own characteristic frequency.

Independence of Normal Modes¶

One of the most important properties of normal modes is that they are mathematically independent.

This means

one normal mode does not influence another,
each vibration can be treated separately,
the total molecular vibration is simply the combination of all normal modes.

This greatly simplifies the analysis of molecular vibrations.

Number of Normal Modes¶

The number of normal modes depends on the number of atoms in the molecule.

A molecule containing N atoms possesses

Nonlinear Molecules¶

\[ \boxed{3N-6} \]

normal vibrational modes.

Linear Molecules¶

\[ \boxed{3N-5} \]

normal vibrational modes.

The remaining degrees of freedom correspond to

molecular translations,
molecular rotations.

These motions are not considered vibrational modes.

Degrees of Freedom¶

Every atom contributes three independent Cartesian coordinates.

Therefore,

a molecule containing N atoms has

\[ 3N \]

total degrees of freedom.

These are divided into

3N Coordinates
       │
       ├── 3 Translations
       ├── 3 Rotations (2 for linear molecules)
       └── Vibrational Modes

Thus,

only the remaining coordinates correspond to true molecular vibrations.

Examples¶

Water (H₂O)¶

Number of atoms

\[ N=3 \]

Total coordinates

\[ 3N=9 \]

Subtract

3 translations
3 rotations

Remaining vibrations

\[ 9-6=3 \]

The three normal modes are

symmetric stretching,
asymmetric stretching,
bending.

Carbon Dioxide (CO₂)¶

Carbon dioxide is linear.

For

\[ N=3 \]

Total coordinates

\[ 3N=9 \]

Subtract

3 translations
2 rotations

Remaining vibrations

\[ 9-5=4 \]

How Gaussian Determines Normal Modes¶

After constructing the mass-weighted Hessian,

Gaussian performs a matrix diagonalization.

Mass-Weighted Hessian
          │
          ▼
Matrix Diagonalization
          │
          ▼
Eigenvectors
          │
          ▼
Normal Modes

Each eigenvector corresponds to one normal mode.

The components of the eigenvector describe

which atoms move,
the direction of motion,
the relative magnitude of each atomic displacement.

Animation of Vibrations¶

Programs such as GaussView use the eigenvectors calculated by Gaussian to animate the molecular vibrations.

The animation shows

the direction of atomic motion,
the relative amplitude,
the phase of each atom.

These animations help identify

stretching vibrations,
bending vibrations,
rocking motions,
wagging motions,
twisting motions,
torsional motions.

Relationship to Vibrational Frequencies¶

Each normal mode possesses exactly one vibrational frequency.

Normal Mode
      │
      ▼
Characteristic Frequency

Stiffer vibrations produce

larger restoring forces,
higher frequencies.

Softer vibrations produce

smaller restoring forces,
lower frequencies.

Thus,

every vibrational frequency reported by Gaussian corresponds to one normal mode of the molecule.

Summary¶

Normal modes are the natural, independent vibrational patterns of a molecule. Instead of treating each atom separately, Gaussian describes molecular vibrations as coordinated motions involving all atoms simultaneously. A nonlinear molecule containing N atoms has 3N − 6 normal modes, while a linear molecule has 3N − 5. These modes are obtained by diagonalizing the mass-weighted Hessian matrix, and each mode is associated with a unique vibrational frequency and characteristic pattern of atomic motion. Normal mode analysis provides the foundation for interpreting vibrational spectra, infrared intensities, and thermodynamic properties.

Next Section¶

The next chapter introduces the Eigenvalue Problem, explaining how diagonalization of the mass-weighted Hessian produces the eigenvalues and eigenvectors that correspond to the vibrational frequencies and normal modes of the molecule.