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Mass-Weighted Coordinates

The Hessian matrix describes the curvature of the Potential Energy Surface and contains the force constants between all pairs of atomic coordinates. However, the Hessian alone is not sufficient to determine the vibrational frequencies of a molecule.

The reason is simple: atoms have different masses.

A hydrogen atom responds much more readily to an applied force than an oxygen atom because hydrogen is much lighter. Therefore, two atoms connected by identical chemical bonds will not vibrate in the same way if their masses differ.

To account for this effect, Gaussian transforms the Cartesian coordinates into mass-weighted coordinates before performing the vibrational analysis.

Why Are Masses Important?

Consider two simple diatomic molecules.

H ── H

and

I ── I

Suppose both molecules have identical bond stiffness.

Even though the force constants are the same,

the hydrogen molecule vibrates much faster than the iodine molecule.

This occurs because

  • hydrogen atoms are very light,
  • iodine atoms are much heavier.

Thus,

vibrational frequencies depend on

  • bond strength,
  • atomic masses.

The Hessian contains information about bond strength, but it does not explicitly account for the masses of the atoms.

The Mass Effect

Imagine trying to move two objects using the same force.

Light Object

Push  →
   ○

Moves easily

Heavy Object

Push →
 █████

Moves slowly

Although the applied force is identical,

the heavier object accelerates less.

The same principle applies to atoms in a molecule.

Heavy atoms vibrate more slowly than light atoms.

Cartesian Coordinates Are Not Enough

The Cartesian Hessian is constructed using the coordinates

x
y
z

for every atom.

These coordinates describe the geometry of the molecule but ignore one important property:

the mass of each atom.

Consequently,

the Cartesian Hessian alone cannot be used directly to calculate the vibrational frequencies.

A transformation is therefore required.

Mass-Weighted Coordinates

Gaussian introduces a new coordinate system in which every Cartesian coordinate is scaled according to the mass of the corresponding atom.

For each coordinate,

\[ \boxed{ Q_i = \sqrt{m_i}\,q_i } \]

where

  • \(q_i\) is the Cartesian displacement,
  • \(m_i\) is the atomic mass,
  • \(Q_i\) is the corresponding mass-weighted coordinate.

The heavier the atom,

the larger the scaling factor.

Physical Interpretation

Mass-weighting combines

  • the restoring force of the chemical bond,
  • the inertia of the atom,

into a single mathematical description.

Instead of treating force and mass separately,

the transformed coordinates automatically include both effects.

This greatly simplifies the vibrational equations.

Transforming the Hessian

Once the coordinates have been mass-weighted,

the Hessian is also transformed.

Mathematically,

the elements of the mass-weighted Hessian become

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

where

  • \(H_{ij}\) is the Cartesian Hessian,
  • \(m_i\) and \(m_j\) are the atomic masses.

The resulting matrix contains both

  • molecular force constants,
  • atomic mass information.

This is the matrix that Gaussian ultimately diagonalizes.

Why This Transformation Helps

Without mass-weighting,

the vibrational equations become unnecessarily complicated because force and mass appear separately.

After the transformation,

the equations become much simpler and can be solved using standard matrix diagonalization techniques.

This allows Gaussian to determine

  • normal modes,
  • vibrational frequencies,
  • vibrational energies,

directly from the transformed Hessian matrix.

Example

Consider a carbon–hydrogen bond.

H —— C

Hydrogen is much lighter than carbon.

During vibration,

the hydrogen atom moves considerably more than the carbon atom.

Mass-weighting naturally incorporates this difference,

ensuring that the calculated vibrational frequencies reflect the true physical motion of the molecule.

Relationship to the Frequency Calculation

The overall sequence is now

Optimized Geometry
        │
        ▼
Taylor Expansion
        │
        ▼
Hessian Matrix
        │
        ▼
Mass-Weighted Hessian
        │
        ▼
Matrix Diagonalization
        │
        ▼
Normal Modes

The transformation to mass-weighted coordinates is therefore an essential intermediate step between constructing the Hessian and determining the molecular vibrations.

Why Gaussian Uses Mass-Weighted Coordinates

The use of mass-weighted coordinates offers several advantages.

  • Accounts for differences in atomic masses.
  • Produces physically meaningful vibrational motions.
  • Simplifies the mathematical formulation.
  • Allows efficient matrix diagonalization.
  • Provides accurate vibrational frequencies.

Without this transformation, interpreting molecular vibrations would be significantly more difficult.

Summary

The Cartesian Hessian describes only the curvature of the Potential Energy Surface and does not account for the different masses of the atoms. Since vibrational motion depends on both the restoring forces and the atomic masses, Gaussian transforms the Cartesian coordinates into mass-weighted coordinates before performing the vibrational analysis. This transformation incorporates the effects of atomic inertia directly into the Hessian matrix, producing a mass-weighted Hessian that can be diagonalized to obtain the normal modes and vibrational frequencies of the molecule.

Next Section

The next chapter introduces Normal Modes of Vibration, explaining how the collective motion of atoms is obtained by diagonalizing the mass-weighted Hessian matrix and why every molecule possesses a characteristic set of independent vibrational modes.