Normalcoordinate

Normal coordinates are generalized coordinates used in the description of small-amplitude vibrations of molecules and solids. They are linear combinations of Cartesian displacements that diagonalize the vibrational motion, producing independent harmonic oscillators known as normal modes.

In a polyatomic molecule with N atoms, there are 3N Cartesian displacements. After removing the three translations and three rotations for nonlinear molecules (linear molecules have 3N−5 vibrational degrees of freedom), the remaining motions can be described by a set of normal coordinates Qk, each corresponding to a distinct vibrational pattern.

To obtain them, one writes the kinetic energy in mass-weighted coordinates and the potential energy as a quadratic form about equilibrium. The dynamical matrix F = M−1/2 K M−1/2 has eigenvectors ek and eigenvalues ωk^2. The normal coordinates are Qk = ∑i ei,k sqrt(mi) qi, where qi are Cartesian displacements and mi the atomic masses. Conversely, the actual displacements are qi = ∑k ei,k Qk / sqrt(mi). The frequencies are ωk = sqrt(λk), with λk the eigenvalues.

Classically, each normal coordinate behaves as an independent simple harmonic oscillator with energy proportional to Qk^2, while quantum mechanically the vibrational energy is E = ∑k (nk + 1/2) ħωk, where nk are mode quantum numbers.

Normal-coordinate analysis is foundational in vibrational spectroscopy, infrared and Raman activity studies, and computational chemistry, providing a compact, symmetry-aware description of complex molecular motions and easing inter-mode coupling. Degeneracies and symmetry can cause multiple coordinates to share the same frequency.