The A Matrix in Molecular Vibration–Rotation Theory
Crawford's A matrix in the theory of molecular vibrations is, in a sense, the inverse of Wilson's B matrix, but is not unique because B is rectangular. We consider the general form of A and then use the Eckart conditions to obtain the solution A = M−1BTG−1, which has been widely used. Alth...
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| Published in: | Journal of molecular spectroscopy Vol. 205; no. 2; pp. 227 - 231 |
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01-02-2001
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| Abstract | Crawford's A matrix in the theory of molecular vibrations is, in a sense, the inverse of Wilson's B matrix, but is not unique because B is rectangular. We consider the general form of A and then use the Eckart conditions to obtain the solution A = M−1BTG−1, which has been widely used. Although the internal-coordinate harmonic force constants f = ATFXA, where FX are the Cartesian force constants, are superficially isotope-dependent, we show that this dependence vanishes. More generally, solutions of the form A = WBT(BWBT)−1, where W is an arbitrary nonsingular square matrix, are shown to give an f matrix that is independent of W. |
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| AbstractList | Crawford's A matrix in the theory of molecular vibrations is, in a sense, the inverse of Wilson's B matrix, but is not unique because B is rectangular. We consider the general form of A and then use the Eckart conditions to obtain the solution A = M(-1)B(T)G(-1), which has been widely used. Although the internal-coordinate harmonic force constants f = A(T)F(X)A, where F(X) are the Cartesian force constants, are superficially isotope-dependent, we show that this dependence vanishes. More generally, solutions of the form A = WB(T)(BWB(T))(-1), where W is an arbitrary nonsingular square matrix, are shown to give an f matrix that is independent of W. Copyright 2001 Academic Press. Crawford's A matrix in the theory of molecular vibrations is, in a sense, the inverse of Wilson's B matrix, but is not unique because B is rectangular. We consider the general form of A and then use the Eckart conditions to obtain the solution A = M−1BTG−1, which has been widely used. Although the internal-coordinate harmonic force constants f = ATFXA, where FX are the Cartesian force constants, are superficially isotope-dependent, we show that this dependence vanishes. More generally, solutions of the form A = WBT(BWBT)−1, where W is an arbitrary nonsingular square matrix, are shown to give an f matrix that is independent of W. |
| Author | Winnewisser, Brenda P. Watson, James K.G. |
| Author_xml | – sequence: 1 givenname: Brenda P. surname: Winnewisser fullname: Winnewisser, Brenda P. organization: Physikalisch-Chemisches Institut der Justus-Liebig-Universität, Heinrich-Buff-Ring 58, D-35392, Giessen, Germany – sequence: 2 givenname: James K.G. surname: Watson fullname: Watson, James K.G. organization: Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ontario, Canada, K1A 0R6 |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/11162208$$D View this record in MEDLINE/PubMed |
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| Cites_doi | 10.1063/1.1740158 10.1063/1.1750829 10.1021/jp993430o 10.1016/0022-2852(87)90053-1 10.1063/1.1700660 10.1063/1.1742730 10.1016/0022-2860(93)08009-S 10.1103/PhysRev.47.552 10.1016/0022-2852(75)90294-5 10.1063/1.1747981 |
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| Keywords | A matrix internal coordinates force constants ab initio force fields Cartesian coordinates |
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| References | Gussoni, Dellepiane, Abbate (RF7) 1975; 57 Fogarasi, Pulay (RF8) 1985 Polo (RF6) 1956; 24 Arfken, Weber (RF11) 1995 Crawford, Fletcher (RF4) 1951; 19 Malhiot, Ferigle (RF13) 1954; 22 Eckart (RF10) 1935; 47 Wilson, Decius, Cross (RF2) 1980 Crawford (RF5) 1952; 20 Woodward (RF1) 1972 Wilson (RF12) 1941; 9 Duncan, Lawie, Nivellini, Tullini, Ferguson, Harper, Tonge (RF14) 1987; 121 Califano (RF3) 1976 Winnewisser, Gambi, Winnewisser (RF9) 1994; 320 Guirgis, Zhu, Yu, Durig (RF15) 2000; 104 Califano (10.1006/jmsp.2000.8268_RF3) 1976 Duncan (10.1006/jmsp.2000.8268_RF14) 1987; 121 Eckart (10.1006/jmsp.2000.8268_RF10) 1935; 47 Winnewisser (10.1006/jmsp.2000.8268_RF9) 1994; 320 Arfken (10.1006/jmsp.2000.8268_RF11) 1995 Gussoni (10.1006/jmsp.2000.8268_RF7) 1975; 57 Malhiot (10.1006/jmsp.2000.8268_RF13) 1954; 22 Crawford (10.1006/jmsp.2000.8268_RF4) 1951; 19 Woodward (10.1006/jmsp.2000.8268_RF1) 1972 Guirgis (10.1006/jmsp.2000.8268_RF15) 2000; 104 Polo (10.1006/jmsp.2000.8268_RF6) 1956; 24 Crawford (10.1006/jmsp.2000.8268_RF5) 1952; 20 Wilson (10.1006/jmsp.2000.8268_RF12) 1941; 9 Fogarasi (10.1006/jmsp.2000.8268_RF8) 1985 Wilson (10.1006/jmsp.2000.8268_RF2) 1980 |
| References_xml | – volume: 47 start-page: 552 year: 1935 ident: RF10 publication-title: Phys. Rev. contributor: fullname: Eckart – year: 1980 ident: RF2 publication-title: Molecular Vibrations contributor: fullname: Cross – volume: 57 start-page: 323 year: 1975 ident: RF7 publication-title: J. Mol. Spectrosc. contributor: fullname: Abbate – volume: 19 start-page: 141 year: 1951 ident: RF4 publication-title: J. Chem. Phys. contributor: fullname: Fletcher – volume: 24 start-page: 1133 year: 1956 ident: RF6 publication-title: J. Chem. Phys. contributor: fullname: Polo – volume: 9 start-page: 76 year: 1941 ident: RF12 publication-title: J. Chem. Phys. contributor: fullname: Wilson – year: 1995 ident: RF11 publication-title: Mathematical Methods for Physicists contributor: fullname: Weber – volume: 20 start-page: 977 year: 1952 ident: RF5 publication-title: J. Chem. Phys. contributor: fullname: Crawford – year: 1972 ident: RF1 publication-title: Introduction to the Theory of Molecular Vibrations and Vibrational Spectroscopy contributor: fullname: Woodward – volume: 121 start-page: 294 year: 1987 ident: RF14 publication-title: J. Mol. Spectrosc. contributor: fullname: Tonge – volume: 104 start-page: 4383 year: 2000 ident: RF15 publication-title: J. Phys. Chem. A contributor: fullname: Durig – start-page: 125 year: 1985 end-page: 219 ident: RF8 publication-title: Vibrational Spectra and Structure contributor: fullname: Pulay – volume: 22 start-page: 717 year: 1954 ident: RF13 publication-title: J. Chem. Phys. contributor: fullname: Ferigle – year: 1976 ident: RF3 publication-title: Vibrational States contributor: fullname: Califano – volume: 320 start-page: 107 year: 1994 ident: RF9 publication-title: J. Mol. Struct. contributor: fullname: Winnewisser – year: 1980 ident: 10.1006/jmsp.2000.8268_RF2 contributor: fullname: Wilson – volume: 22 start-page: 717 year: 1954 ident: 10.1006/jmsp.2000.8268_RF13 publication-title: J. Chem. Phys. doi: 10.1063/1.1740158 contributor: fullname: Malhiot – year: 1972 ident: 10.1006/jmsp.2000.8268_RF1 contributor: fullname: Woodward – volume: 9 start-page: 76 year: 1941 ident: 10.1006/jmsp.2000.8268_RF12 publication-title: J. Chem. Phys. doi: 10.1063/1.1750829 contributor: fullname: Wilson – year: 1976 ident: 10.1006/jmsp.2000.8268_RF3 contributor: fullname: Califano – volume: 104 start-page: 4383 year: 2000 ident: 10.1006/jmsp.2000.8268_RF15 publication-title: J. Phys. Chem. A doi: 10.1021/jp993430o contributor: fullname: Guirgis – year: 1995 ident: 10.1006/jmsp.2000.8268_RF11 contributor: fullname: Arfken – volume: 121 start-page: 294 year: 1987 ident: 10.1006/jmsp.2000.8268_RF14 publication-title: J. Mol. Spectrosc. doi: 10.1016/0022-2852(87)90053-1 contributor: fullname: Duncan – volume: 20 start-page: 977 year: 1952 ident: 10.1006/jmsp.2000.8268_RF5 publication-title: J. Chem. Phys. doi: 10.1063/1.1700660 contributor: fullname: Crawford – volume: 24 start-page: 1133 year: 1956 ident: 10.1006/jmsp.2000.8268_RF6 publication-title: J. Chem. Phys. doi: 10.1063/1.1742730 contributor: fullname: Polo – volume: 320 start-page: 107 year: 1994 ident: 10.1006/jmsp.2000.8268_RF9 publication-title: J. Mol. Struct. doi: 10.1016/0022-2860(93)08009-S contributor: fullname: Winnewisser – volume: 47 start-page: 552 year: 1935 ident: 10.1006/jmsp.2000.8268_RF10 publication-title: Phys. Rev. doi: 10.1103/PhysRev.47.552 contributor: fullname: Eckart – volume: 57 start-page: 323 year: 1975 ident: 10.1006/jmsp.2000.8268_RF7 publication-title: J. Mol. Spectrosc. doi: 10.1016/0022-2852(75)90294-5 contributor: fullname: Gussoni – volume: 19 start-page: 141 year: 1951 ident: 10.1006/jmsp.2000.8268_RF4 publication-title: J. Chem. Phys. doi: 10.1063/1.1747981 contributor: fullname: Crawford – start-page: 125 year: 1985 ident: 10.1006/jmsp.2000.8268_RF8 contributor: fullname: Fogarasi |
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| Title | The A Matrix in Molecular Vibration–Rotation Theory |
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