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Partition of symmetric powers of \(G\)-sets into orbits: application to the enumeration of force constants. (English) Zbl 1320.81108

Summary: Using the theory of actions of groups on sets this paper describes an efficient method to obtain the partition of the symmetric powers of a \(G\)-set into orbits, where \(G\) is a finite group. In this method, a generating function is obtained for each representative of the conjugacy classes of subgroups of \(G\). The coefficients of the generating function corresponding to a representative subgroup \(H \leq G\) give the number of orbits isomorphic to the coset \(G/H\) that are contained in the successive symmetric powers of the \(G\)-set. A direct application of this approach is the attainment of the number and isotropy group of the vibrational force constants associated with a set of internal coordinates for a given molecule. As illustration, the method has been applied to \(XYZ _{3} (C _{3v })\) molecules.

MSC:

81V55 Molecular physics
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[1] Wilson E.B., Decius J.C., Cross P.C.: Molecular Vibrations. The Theory of Infrared and Raman Vibrational Spectra, pp. 54–61. McGraw-Hill, New York (1955)
[2] Watson J.K.G.: J. Mol. Spectrosc. 41, 229 (1972) · doi:10.1016/0022-2852(72)90136-1
[3] Sloane N.J.A.: Am. Math. Mon. 84, 82 (1977) · Zbl 0357.94014 · doi:10.2307/2319929
[4] Stanley R.P.: Bull. Am. Math. Soc. 1, 475 (1979) · Zbl 0497.20002 · doi:10.1090/S0273-0979-1979-14597-X
[5] Schmelzer A., Murrell J.: Int. J. Quantum Chem. 28, 287 (1985) · doi:10.1002/qua.560280210
[6] Collins M.A., Pearson D.F.: J. Chem. Phys. 99, 6756 (1993) · doi:10.1063/1.465819
[7] Cassam-Chenaï P., Patras F.: J. Math. Chem. 44, 938 (2008) · Zbl 1243.81237 · doi:10.1007/s10910-008-9354-y
[8] Humphreys J.F.: A Course in Group Theory, pp. 89–97. Oxford University Press, Oxford (1996) · Zbl 0843.20001
[9] Rose J.S.: A Course on Group Theory, pp. 68–87. Cambridge University Press, Cambridge (1978) · Zbl 0371.20001
[10] Solomon L.: J. Combin. Theory. 2, 603 (1967) · Zbl 0183.03601 · doi:10.1016/S0021-9800(67)80064-4
[11] W. Burnside, Theory of Groups of Finite Order, 2nd edn. (Cambridge University Press, Cambridge, 1911, reprinted by Dover, New York, 1955), pp. 246–247 · JFM 42.0151.02
[12] Hässelbarth W.: Theor. Chim. Acta. 67, 339 (1985) · doi:10.1007/BF00530087
[13] Martínez E.: J. Math. Chem. 35, 105 (2004) · Zbl 1045.92051 · doi:10.1023/B:JOMC.0000014307.75757.13
[14] Berge C.: Principles of Combinatorics, pp. 54–56. Academic Press, New York (1971) · Zbl 0227.05002
[15] Webb P.: Contemp. Math. 146, 441 (1993) · doi:10.1090/conm/146/01239
[16] Martínez E.: Chem. Phys. Lett. 506, 298 (2011) · doi:10.1016/j.cplett.2011.03.022
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