Formal mathematical description of a normal vibration. When the potential energy, $V$, is harmonic and the kinetic energy, $T$, is calculated in the limit of infinitesimally small displacements, the normal coordinates are defined as independent entities such that '(for non-linear molecules)'$$V=½\sum _{k=1}^{3N-6}{\lambda }_k{Q}_k^2\mathrm{a}\mathrm{n}\mathrm{d}T=½\sum _{k=1}^{3N-6}{\dot{Q}}_k^2$$where ${\lambda }_k$ is the $k^{\mathrm{t}\mathrm{h}}$ eigenvalue and $\dot{Q}$ is $\partial Q/\partial t$. Normal coordinate $Q_k$ is related to the internal displacement coordinates, $R_i$, through the elements, $L_{ik}=\partial R_i/\partial Q_k$ of the $k^{\mathrm{t}\mathrm{h}}$ column of the eigenvector matrix $\mathit{L}$. Normal coordinates are not independent when the potential energy is anharmonic, or the displacements are not infinitesimal.