{"term":{"id":"08644","title":"normal coordinate","longtitle":"IUPAC Gold Book - normal coordinate","doi":"10.1351\/goldbook.08644","code":"08644","status":"current","definitions":[{"id":1,"text":"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.","notes":{"1":"SI unit: \\(\\pu{kg^{-1\/2} m}\\). Common unit: \\(\\pu{u^{1\/2} Å \\approx 4.07497E-24 kg^{1\/2} m}\\)."},"links":[{"term":"normal coordinate analysis","url":"https:\/\/goldbook.iupac.org\/\/terms\/view\/08645"},{"term":"vibrational eigenvector","url":"https:\/\/goldbook.iupac.org\/\/terms\/view\/08705"}],"sources":["PAC, 2021, 93, 647. 'Glossary of methods and terms used in analytical spectroscopy (IUPAC Recommendations 2019)' on page 756 (https:\/\/doi.org\/10.1515\/pac-2019-0203)"]}],"altoutputs":{"html":"https:\/\/goldbook.iupac.org\/terms\/view\/08644\/html","xml":"https:\/\/goldbook.iupac.org\/terms\/view\/08644\/xml","plain":"https:\/\/goldbook.iupac.org\/terms\/view\/08644\/plain"},"citation":"Citation: 'normal coordinate' 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.08644","license":"The IUPAC Gold Book is licensed under Creative Commons Attribution-ShareAlike CC BY-SA 4.0 International (https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/) for individual terms.","collection":"If you are interested in licensing the Gold Book for commercial use, please contact the IUPAC Executive Director at executivedirector@iupac.org .","disclaimer":"The International Union of Pure and Applied Chemistry (IUPAC) is continuously reviewing and, where needed, updating terms in the Compendium of Chemical Terminology (the IUPAC Gold Book). Users of these terms are encouraged to include the version of a term with its use and to check regularly for updates to term definitions that you are using.","accessed":"2026-06-28T19:27:10+00:00"}}