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)'\[V = ½\sum_{k=1}^{3N-6}\lambda_{k}Q_{k}^{2}\ {\rm{and}}\ T = ½\sum_{k=1}^{3N-6}\dot Q_{k}^{2}\]where \(\lambda_{k}\) is the \(k^{\rm{th}}\) 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 \(k^{\rm{th}}\) column of the eigenvector matrix \(\boldsymbol{L}\). Normal coordinates are not independent when the potential energy is anharmonic, or the displacements are not infinitesimal.