Title: vibrational kinetic energy
Long Title: IUPAC Gold Book - vibrational kinetic energy
DOI: 10.1351/goldbook.08706
Status: current

Definition
Kinetic energy of the molecule as a function of the displacements of the atoms from equilibrium positions. For Cartesian displacement coordinates (\(x, y, z\)), \[T = \sum_{\alpha}½m_{\alpha}(\dot x_{\alpha}^{2} + \dot y_{\alpha}^{2} + \dot z_{\alpha}^{2})\] where the sum is over all atoms and \(\dot{x}_{\alpha}\), etc., are the displacement velocities. For internal coordinates \[T = ½\boldsymbol{P}^{t}\boldsymbol{GP} = ½\dot{\boldsymbol{R}}^{t}{\boldsymbol{G}}^{-1}\dot {\boldsymbol{R}} = ½\sum_{ij}G_{ij}^{-1}\dot R_{i}\dot R_{j}\] where \(\dot {\boldsymbol{R}}^{t}\) and \(\dot {\boldsymbol{R}}\) are the row and column vector, respectively, of the \(\partial R_{i}/\partial t\), \(\boldsymbol{P}^{t}\) and \(\boldsymbol{P}\) are the row and column vector, respectively, of the momenta conjugate to the \(R_{i}\), \(\boldsymbol{G}^{-1}\) is the inverse of the \(\boldsymbol{G}\) matrix and \(G_{ij}^{-1}\) is the \(ij^{\rm{th}}\) element of \(\boldsymbol{G}^{-1}\). The elements of the \(\boldsymbol{G}\) matrix are defined by \[G_{ij} = \sum_{\alpha} \frac{1}{m_{\alpha}} B_{i\alpha} B_{j\alpha}\] where \(B_{i\alpha}\) and \(B_{j\alpha}\) relate the \(i^{\rm{th}}\) and \(j^{\rm{th}}\) internal coordinate to the \(\alpha^{\rm{th}}\) Cartesian coordinate through the equation \(\boldsymbol{R} = \boldsymbol{BX}\) in which \(\boldsymbol{X}\) is the column vector of the Cartesian coordinates.


Source
- PAC, 2021, 93, 647. 'Glossary of methods and terms used in analytical spectroscopy (IUPAC Recommendations 2019)' on page 768 (https://doi.org/10.1515/pac-2019-0203)

Other Outputs
- html: https://goldbook.iupac.org/terms/view/08706/html
- json: https://goldbook.iupac.org/terms/view/08706/json
- xml: https://goldbook.iupac.org/terms/view/08706/xml

Citation: Citation: 'vibrational kinetic energy' in IUPAC Compendium of Chemical Terminology, 5th ed. International Union of Pure and Applied Chemistry; 2025. Online version 5.0.0, 2025. 10.1351/goldbook.08706
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Accessed: 2026-05-31T08:18:41+00:00