R-matrix

The term R-matrix has several meanings, depending on the field of study.

The term R-matrix is used in connection with the Yang–Baxter equation, first introduced in the field of statistical mechanics in the works of J. B. McGuire in 1964^[1] and C. N. Yang in 1967^[2] and in the group algebra $\mathbb {C} [S_{n}]$ of the symmetric group in the work of A. A. Jucys in 1966.^[3]

The classical R-matrix arises in the definition of the classical Yang–Baxter equation.^[4]

In quasitriangular Hopf algebra, the R-matrix is a solution of the Yang–Baxter equation.

The numerical modeling of diffraction gratings in optical science can be performed using the R-matrix propagation algorithm.^[5]

R-matrix method in quantum mechanics

There is a method in computational quantum mechanics for studying scattering known as the R-matrix. This method was originally formulated for studying resonances in nuclear scattering by Wigner and Eisenbud.^[6] Using that work as a basis, an R-matrix method was developed for electron, positron and photon scattering by atoms.^[7] This approach was later adapted for electron, positron and photon scattering by molecules.^[8]^[9]^[10]

R-matrix method is used in UKRmol^[11] and UKRmol+^[12] code suits. The user-friendly software Quantemol Electron Collisions (Quantemol-EC) and its predecessor Quantemol-N are based on UKRmol/UKRmol+ and employ MOLPRO package for electron configuration calculations.

References

^ McGuire, J. B. (1964-05-01). "Study of Exactly Soluble One-Dimensional N-Body Problems". Journal of Mathematical Physics. 5 (5). The American Institute of Physics (AIP): 622–636. Bibcode:1964JMP.....5..622M. doi:10.1063/1.1704156. ISSN 0022-2488.
^ Yang, C. N. (1967-12-04). "Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction". Physical Review Letters. 19 (23). American Physical Society (APS): 1312–1315. Bibcode:1967PhRvL..19.1312Y. doi:10.1103/PhysRevLett.19.1312. ISSN 0031-9007.
^ Jucys, A. A. (1966). "On the Young operators of the symmetric group" (PDF). Lietuvos Fizikos Rinkinys. 11 (23). Gos. Izd-vo Polit. i Nauch. literatury.: 163–180.
^ Kupershmidt, Boris A. (1999). "What a Classical r-Matrix Really Is". Journal of Nonlinear Mathematical Physics. 6 (4). Informa UK Limited: 448–488. arXiv:math/9910188. Bibcode:1999JNMP....6..448K. doi:10.2991/jnmp.1999.6.4.5. ISSN 1402-9251.
^ Li, Lifeng (1994-11-01). "Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings". Journal of the Optical Society of America A. 11 (11). The Optical Society: 2829–2836. Bibcode:1994JOSAA..11.2829L. doi:10.1364/josaa.11.002829. ISSN 1084-7529.
^ Wigner, E. P.; Eisenbud, L. (1947-07-01). "Higher Angular Momenta and Long Range Interaction in Resonance Reactions". Physical Review. 72 (1). American Physical Society (APS): 29–41. Bibcode:1947PhRv...72...29W. doi:10.1103/physrev.72.29. ISSN 0031-899X.
^ Burke, P G; Hibbert, A; Robb, W D (1971). "Electron scattering by complex atoms". Journal of Physics B: Atomic and Molecular Physics. 4 (2). IOP Publishing: 153–161. Bibcode:1971JPhB....4..153B. doi:10.1088/0022-3700/4/2/002. ISSN 0022-3700.
^ Schneider, Barry (1975). "R-matrix theory for electron-atom and electron-molecule collisions using analytic basis set expansions". Chemical Physics Letters. 31 (2). Elsevier BV: 237–241. Bibcode:1975CPL....31..237S. doi:10.1016/0009-2614(75)85010-x. ISSN 0009-2614.
^ Schneider, Barry I. (1975-06-01). "R-matrix theory for electron-molecule collisions using analytic basis set expansions. II. Electron-H₂ scattering in the static-exchange model". Physical Review A. 11 (6). American Physical Society (APS): 1957–1962. Bibcode:1975PhRvA..11.1957S. doi:10.1103/physreva.11.1957. ISSN 0556-2791.
^ C J Gillan, J Tennyson, and P G Burke, in Computational Methods for Electron-Molecule Collisions, eds. W M Huo and F A Gianturco, (Plenum, New York, 1995), p. 239
^ Carr, J.M.; Galiatsatos, P.G.; Gorfinkiel, J.D.; Harvey, A.G.; Lysaght, M.A.; Madden, D.; Mašín, Z.; Plummer, M.; Tennyson, J. (2012). "The UKRmol program suite". Eur. Phys. J. D (66): 58. doi:10.1140/epjd/e2011-20653-6.
^ Mašín, Zdeněk; Benda, Jakub; Gorfinkiel, Jimena D.; Harvey, Alex G.; Tennyson, Jonathan (2019-12-07). "UKRmol+: A suite for modelling electronic processes in molecules interacting with electrons, positrons and photons using the R-matrix method". Computer Physics Communications. 249: 107092. arXiv:1908.03018. doi:10.1016/j.cpc.2019.107092.

[1] McGuire, J. B. (1964-05-01). "Study of Exactly Soluble One-Dimensional N-Body Problems". Journal of Mathematical Physics. 5 (5). The American Institute of Physics (AIP): 622–636. Bibcode:1964JMP.....5..622M. doi:10.1063/1.1704156. ISSN 0022-2488.

[2] Yang, C. N. (1967-12-04). "Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction". Physical Review Letters. 19 (23). American Physical Society (APS): 1312–1315. Bibcode:1967PhRvL..19.1312Y. doi:10.1103/PhysRevLett.19.1312. ISSN 0031-9007.

[3] Jucys, A. A. (1966). "On the Young operators of the symmetric group" (PDF). Lietuvos Fizikos Rinkinys. 11 (23). Gos. Izd-vo Polit. i Nauch. literatury.: 163–180.

[4] Kupershmidt, Boris A. (1999). "What a Classical r-Matrix Really Is". Journal of Nonlinear Mathematical Physics. 6 (4). Informa UK Limited: 448–488. arXiv:math/9910188. Bibcode:1999JNMP....6..448K. doi:10.2991/jnmp.1999.6.4.5. ISSN 1402-9251.

[5] Li, Lifeng (1994-11-01). "Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings". Journal of the Optical Society of America A. 11 (11). The Optical Society: 2829–2836. Bibcode:1994JOSAA..11.2829L. doi:10.1364/josaa.11.002829. ISSN 1084-7529.

[6] Wigner, E. P.; Eisenbud, L. (1947-07-01). "Higher Angular Momenta and Long Range Interaction in Resonance Reactions". Physical Review. 72 (1). American Physical Society (APS): 29–41. Bibcode:1947PhRv...72...29W. doi:10.1103/physrev.72.29. ISSN 0031-899X.

[7] Burke, P G; Hibbert, A; Robb, W D (1971). "Electron scattering by complex atoms". Journal of Physics B: Atomic and Molecular Physics. 4 (2). IOP Publishing: 153–161. Bibcode:1971JPhB....4..153B. doi:10.1088/0022-3700/4/2/002. ISSN 0022-3700.

[8] Schneider, Barry (1975). "R-matrix theory for electron-atom and electron-molecule collisions using analytic basis set expansions". Chemical Physics Letters. 31 (2). Elsevier BV: 237–241. Bibcode:1975CPL....31..237S. doi:10.1016/0009-2614(75)85010-x. ISSN 0009-2614.

[9] Schneider, Barry I. (1975-06-01). "R-matrix theory for electron-molecule collisions using analytic basis set expansions. II. Electron-H₂ scattering in the static-exchange model". Physical Review A. 11 (6). American Physical Society (APS): 1957–1962. Bibcode:1975PhRvA..11.1957S. doi:10.1103/physreva.11.1957. ISSN 0556-2791.

[10] C J Gillan, J Tennyson, and P G Burke, in Computational Methods for Electron-Molecule Collisions, eds. W M Huo and F A Gianturco, (Plenum, New York, 1995), p. 239

[11] Carr, J.M.; Galiatsatos, P.G.; Gorfinkiel, J.D.; Harvey, A.G.; Lysaght, M.A.; Madden, D.; Mašín, Z.; Plummer, M.; Tennyson, J. (2012). "The UKRmol program suite". Eur. Phys. J. D (66): 58. doi:10.1140/epjd/e2011-20653-6.

[12] Mašín, Zdeněk; Benda, Jakub; Gorfinkiel, Jimena D.; Harvey, Alex G.; Tennyson, Jonathan (2019-12-07). "UKRmol+: A suite for modelling electronic processes in molecules interacting with electrons, positrons and photons using the R-matrix method". Computer Physics Communications. 249: 107092. arXiv:1908.03018. doi:10.1016/j.cpc.2019.107092.

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R-matrix method in quantum mechanics

See also

References