Hyperspherical and related types of coordinates for the dynamical treatment of three-body systems

Progress in Theoretical Chemistry and Physics

Volumn 16, Issue , 2007, Pages 123-146

  Ragni, Mirco ^a   Bitencourt, Ana Carla Peixoto ^a   Aquilanti, Vincenzo ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

Author keywords

Hyperspherical Harmonic; Interparticle Distance; Jacobi Vector; Polyatomic System; Potential Energy Surface

Indexed keywords

EID: 78649707710     PISSN: 15677354     EISSN: 22150129     Source Type: Book Series    
DOI: 10.1007/978-1-4020-5460-0_4     Document Type: Chapter

References (121)

1. V. Aquilanti and S. Cavalli, Coordinates for molecular dynamics: orthogonal local systems. J. Chem: Phys., 85:1355–1361, 1986.
2. V. Aquilanti, S. Cavalli, and G. Grossi, Hyperspherical coordinates for molecular dynamics by the method of trees and the mapping of potential energy surfaces for triatomic systems. J. Chem. Phys., 86:1362–1375, 1986.
3. V. Aquilanti, S. Cavalli, G. Grossi, and R.W. Anderson, Representation in hyperspherical and related coordinates of the potential-energy surface for triatomic reactions. J. Chem. Soc. Faraday Trans, 86(s):1681–1687, 1990.
4. V. Aquilanti, G. Capecchi, and S. Cavalli, Hyperspherical coordinates for chemical reaction dynamics. Adv. Quant. Chem., 36:341–361, 1999.
5. V. Aquilanti and S. Tonzani, Three-body problem in quantum mechanics: Hyperspherical elliptic coordinates and harmonic basis sets. J. Chem. Phys., 120(9):4073, 2004.
6. R.G. Littlejohn, K.A. Mitchell, V. Aquilanti, and S. Cavalli, Body frames and frame singularities for three-atom systems. Phys. Rev. A, 58:3715–3717, 1998.
7. V. Aquilanti, G. Grossi, and A. Laganá, Hyperspherical diabatic and adiabatic representations for chemical reactions. Chem. Phys. Lett, 93:174–178, 1982.
8. V. Aquilanti, S. Cavalli, and A. Laganá, Hyperspherical adiabatic description of interference effects and resonances in collinear chemical reactions. Chem. Phys. Lett, 93:179–183, 1982.
9. V. Aquilanti, S. Cavalli, G. Grossi, and A. Laganá, A semiclassical approach to the dynamics of chemical reactions within the hyperspherical formalism. J. Mol. Struct., 93:319–323, 1983.
10. V. Aquilanti, S. Cavalli, G. Grossi, and A. Laganá, Nonadiabatic effects in the hyperspherical description of elementary chemical reactions. J. Mol. Struct., 107:95–99, 1984.
11. V. Aquilanti, Resonances in reactions: a semiclassical view. The Theory of Chemical Reaction Dynamics, D. C. Clary, ed. Reidel, Dordrecht, 383–413, 1986.
12. J.O. Hirschfelder, My adventures in theoretical chemistry. Annu. Rev. Phys. Chem., 34:1–30, 1983.
13. T.H. Gronwall, Ann. Math., 33:279, 1932.
14. T.H. Gronwall, The helium wave equation. Phys. Rev., 51:655–660, 1937.
15. P.M. Morse and H. Feshbach, Methods of Theoretical Physics. McGraw-Hill, New York, 1953.
16. V. Fock, K. Norske Vidensk. selsksk Forh., 41:138–145, 1958.
17. J.H. Macek, Properties of autoionizing states of He. J. Phys. B, 1:831–843, 1968.
18. −. Phys. Rev. A, 25:1535–1545, 1982.
19. L.M. Delves, Tertiary and general-order collisions (II). Nucl. Phys., 20:275–308, 1960.
20. L. Bianchi V. Gallina, P. Nata and G. Viano, Nuovo Cimento, 24:835, 1962.
21. W. Zickendraht, Construction of a complete orthogonal system for the quantum-mechanical threebody problem. Ann. Phys., 35:18–41, 1965.
22. W. Zickendraht, Configuration-space approach to three-particle scattering. Phys. Rev., 159: 1448–1455, 1967.
23. F.T. Smith, A symmetric representation for three-body problems. I. Motion in a plane. J. Math. Phys., 3:735–748, 1962.
24. R.E. Clapp, The binding energy of the triton. Phys. Rev., 76:873–874, 1949.
25. R.E. Clapp, A complete orthogonal expansion for the nuclear three-body problem. Ann. Phys., 13:187–236, 1961.
26. H. Klar, A model for triatomic large amplitude vibrations. Energy levels for the water molecule. Phys.Rev.A, 15:1452–1458, 1977.
27. Yu A. Simonov, Soviet J. Nucl. Phys., 3:461, 1966.
28. 2 O and ArC N. Phys. Chem. Chem. Phys., 2:4121–4129, 2000.
29. J.J. Soares Neto and F.V. Prudente, A novel finite element method implementation for calculating bound states of triatomic systems: Application to the water molecule. Theor. Chim. Acta, 89: 415–427, 1994.
30. R. Wallace, Chem. Phys., 34:93, 1979.
31. R. Wallace, A model for triatomic large amplitude vibrations. Energy levels for the water molecule. Chem. Phys. Lett., 67:442–443, 1979.
32. V. Aquilanti, G. Grossi, and A. Laganá, On hyperspherical mapping and harmonic expansions for potential energy surfaces. J. Chem. Phys., 76:1587–1588, 1982.
33. J. Manz, Molecular dynamics along hyperspherical coordinates. Comm. Atom. Mol. Phys., 17:91–113, 1985.
34. ′ ) + H in hyperspherical coordinates. Chem. Phys. Lett., 144:346–352, 1988.
35. P.G. Hipes and A. Kuppermann, Lifetime analysis of high-energy resonances in three-dimensional reactive scattering. Chem. Phys. Lett., 133:1–7, 1987.
36. 2 and Cl+HCl. Chem. Phys. Lett., 150:92–98, 1988.
37. G. Grossi, Angular parametrizations in the hyperspherical description of elementary chemical reactions. J. Chem. Phys., 81:3355–3356, 1984.
38. H. Nakamura, A. Ohsaki, and M. Baer, New implementation to approximate quantum mechanical treatment of atom-diatom chemical reactions. J. Phys. Chem., 90:6176–6184, 1986.
39. V. Aquilanti, G. Grossi, A. Laganá, E. Pelikan, and H. Klar, A decoupling scheme for a three-body problem treated by expansions into hyperspherical harmonics: the hydrogen molecular ion. Lett. Nuovo Cim., 41:541–544, 1984.
40. V. Aquilanti and G. Grossi, Discrete representations by artificial quantization in the quantum mechanics of anisotropic interactions. Lett. Nuovo Cim., 42:157–162, 1985.
41. V. Aquilanti, S. Cavalli, G. Grossi, and R.W. Anderson, Stereodirected states in molecular dynamics: A discrete basis representation for the quantum mechanical scattering matrix. J. Phys. Chem., 95:8184–8193, 1991.
42. V. Aquilanti, S. Cavalli, and G. Grossi, Discrete analogs of spherical harmonics and their use in quantum mechanics: The hyperquantization algorithm. Theor. Chim. Acta, 79:283–296, 1991.
43. V. Aquilanti, S. Cavalli, and D. De Fazio, Angular and hyperangular momentum coupling coefficients as hahn polynomials. J. Phys. Chem., 99:15694–15698, 1995.
44. V. Aquilanti, S. Cavalli, C. Coletti, D. De Fazio, and G. Grossi, Hyperangular momentum: Applications to atomic and molecular science. New Methods in Quantum Theory, eds. C.A. Tsipis, V.S. Popov, D.R. Herschbach, J.S. Avery, Kluwer, pages 233–250, 1996.
45. V. Aquilanti and G. Capecchi, Harmonic analysis and discrete polynomials from semiclassical angular momentum theory to the hyperquantization algorithm. Theor. Chem. Accounts, 104:183–188, 2000.
46. V. Aquilanti, S. Cavalli, and D. De Fazio, Hyperquantization algorithm: I. theory for triatomic systems. J. Chem. Phys., 109:3792–3804, 1998.
47. 2 reaction dynamics including open-shell and spin-orbit interaction. J. Chem. Phys., 109:3805–3818, 1998.
48. V. Aquilanti, S. Cavalli, A. Volpi, and D. De Fazio, The a + bc reaction by the hyperquantization algorithm: the symmetric hyperspherical parametrization for J>0. Adv. Quant. Chem., 39:103–121, 2001.
49. D. De Fazio, S. Cavalli, and V. Aquilanti, Orthogonal polynomials of a discrete variable as expansion basis sets in quantum mechanics. The hyperquantization algorithm. Int. J. Quant. Chem., 93:91–111, 2003.
50. K. Museth and A. Kuppermann, Asymptotic analysis of state-to-state tetraatomic reactions using row-orthonormal hyperspherical coordinates. J. Chem. Phys., 115:8285–8297, 2001.
51. A. Kuppermann, Reactive scattering with row-orthonormal hyperspherical coordinates. 2. Transformation properties and hamiltonian for tetraatomic systems. J. Phys. Chem., 101:6368–6383, 1997.
52. A. Kuppermann, Reactive scattering with row-orthonormal hyperspherical coordinates. 1. Transformation properties and hamiltonian for triatomic systems (vol 100, pg 2635, 1996). J. Phys. Chem., 100:11202–11202, 1996.
53. A. Kuppermann, Reactive scattering with row-orthonormal hyperspherical coordinates. 1. Transformation properties and hamiltonian for triatomic systems. J. Phys. Chem., 100:2621–2636, 1996(Erratum 100:11202–11202, 1996).
54. F.D. Colavecchia, F. Mrugala, G.A. Parker, and R.T. Pack, Accurate quantum calculations on threebody collisions in recombination and collision-induced dissociation. ii. the smooth variable discretization enhanced renormalized numerov propagator. J. Chem. Phys., 118:10387–10398, 2003.
55. 2 system. J. Chem. Phys., 117:6083–6102, 2002.
56. G.A. Parker, A. Laganá, S. Crocchianti, and R.T. Pack, A detailed 3-dimensional quantum study of the Li+FH reaction. J. Chem. Phys., 102:1238–1250, 1995.
57. C.Y. Yang, S.J. Klippenstein, J.D. Kress, and A. Laganá, Comparison of transition-state theory with quantum scattering-theory for the reaction Li+HF → LiF+H. J. Chem. Phys., 100:4917–4924, 1994.
58. A. Laganá, R.T. Pack, and G.A. Parker, Li+FH reactive cross-sections from J = 0 accurate quantum reactivity. J. Chem. Phys., 99:2269–2270, 1993.
59. G.A. Parker and R.T. Pack, Quantum reactive scattering in 3 dimensions using hyperspherical (APH) coordinates. 6. Analytic basis method for surface functions. J. Chem. Phys., 98:6883–6896, 1993.
60. G.A. Parker, R.T. Pack, and A. Laganá, Accurate 3d-quantum reactive probabilities of li + fh. Chem. Phys. Lett., 202:75–81, 1993.
61. 2 . J. Phys. Chem., 94:8055–8058, 1990.
62. 2 → HF+H with total angular-momentum J = 1. Chem. Phys. Lett., 170:306–310, 1990.
63. 2 . J. Chem. Phys., 92:2344–2361, 1990.
64. F.T. Smith, Generalized angular momentum in many-body collisions. Phys. Rev., 120:1058–1069, 1960.
65. D. Delande, PhD thesis, Université Pierre et Marie Curie, Paris, France, 1988.
66. V. Aquilanti, A. Caligiana, S. Cavalli, and C. Coletti, Hydrogenic orbitals in momentum space and hyperspherical harmonics elliptic sturmian basis sets. Int. J. Quant. Chem., 92:212–228, 2003.
67. O.I. Tolstikhin and H. Nakamura, Hyperspherical elliptic coordinates for the theory of light atom transfer reactions in atom-diatom collisions. J. Chem. Phys., 108:8899–8921, 1998.
68. O.I. Tolstikhin, V. N. Ostrovsky, and H. Nakamura, Cumulative reaction probability without absorbing potentials. Phys. Rev. Lett., 80:41–44, 1998.
69. O.I. Tolstikhin, I. Yu. Tolstikhina, and C. Namba, Interference effects in the decay of resonance states in three-body coulomb systems. Phys. Rev. A, 60:4673–4692, 1999.
70. O.I. Tolstikhin and C. Namba, Hyperspherical calculations of low-energy rearrangement processes in dtµ. Phys. Rev. A, 60:5111–5114, 1999.
71. K. Nobusada, O.I. Tolstikhin, and H. Nakamura, Quantum mechanical elucidation of reaction mechanisms of heavy-light-heavy systems: Role of potential ridge. J. Chem. Phys., 108:8922–8930, 1998.
72. K. Nobusada, O.I. Tolstikhin, and H. Nakamura, Quantum reaction dynamics of heavy-light-heavy systems: Reduction of the number of potential curves and transitions at avoided crossings. J. Phys. Chem. A, 102:9445–9453, 1998.
73. K. Nobusada, O.I. Tolstikhin, and H. Nakamura, Quantum reaction dynamics of Cl + HCl → HCl + Cl: vibrationally non-adiabatic reactions. J. Mol. Struct., Theochem 461-2:137–144, 1999.
74. O.I. Tolstikhin and M. Matsuzawa, Hyperspherical elliptic harmonics and their relation to the heun equation. Phys.Rev.A, 63:032510/1–032510/8, 2001.
75. O.I. Tolstikhin and M. Matsuzawa, Exploring the separability of the three-body coulomb problem in hyperspherical elliptic coordinates. Phys. Rev. A, 63:062705/1–062705/23, 2001.
76. V.K. Babamov and R.A. Marcus, Dynamics of hydrogen atom and proton transfer reactions. Symmetric case. J. Chem. Phys., 74:1790–1798, 1981.
77. A. Kuppermann, A useful mapping of triatomic potential energy surfaces. Chem. Phys. Lett., 32:374–375, 1975.
78. B.R. Johnson, On hyperspherical coordinates and mapping the internal configurations of a three body system. Chem. Phys., 73:5051–5058, 1980.
79. B.R. Johnson, The classical dynamics of three particles in hyperspherical coordinates. J. Chem. Phys., 79:1906–1915, 1983.
80. B.R. Johnson, The quantum dynamics of three particles in hyperspherical coordinates. J. Chem. Phys., 79:1916–1925, 1983.
81. L. Wolniewicz and J. Hinze, Atom-diatomic molecular reactive scattering: Investigation of the hyperangular integration. J. Chem. Phys., 85:2012–2018, 1986.
82. J.T. Muckerman, R.D. Gilbert, and G.D. Billing, A classical path approach to reactive scattering. i. use of hyperspherical coordinates. J. Chem. Phys., 88:4779–4787, 1988.
83. R.T. Pack, Coordinates for an optimum CS approximation in reactive scattering. Chem. Phys. Lett., 108:333–338, 1984.
84. M. Mishra and J. Linderberg, Hyperspherical representations of triatomic energy surfaces. Mol. Phys., 50:91, 1983.
85. J. Linderberg and B. Vessal, Reactive scattering in hyperspherical coordinates. Int. J. Quant. Chem., 31:65, 1987.
86. C.A. Mead, Superposition of reactive and nonreactive scattering amplitudes in the presence of a conical intersection. J. Chem. Phys., 72:3839–3840, 1980.
87. J. Robert and J. Baudon, A molecular description of molecular collisions. J. Phys. B, 19:171–184, 1986.
88. J. Robert and J. Baudon, J. Phys. (Paris), 47:631, 1986.
89. J. Robert and J. Baudon, Europhys. Lett., 2:363, 1986.
90. M. De Celles and B.T. Darling, J. Mol. Spectrosc., 29:66, 1969.
91. V. Aquilanti, A. Laganá, and R.D. Levine, On the all channels representation of the potential energy surface for reactive collisions. Chem. Phys. Lett., 158:87–94, 1989.
92. V. Aquilanti, S. Cavalli, and G. Grossi, Dynamics on reactive potential energy surfaces: the hyperspherical view. Advances in Molecular Vibrations and Collision Dynamics, ed. J.M. Bowman, JAI Press, Greenwhich (Conn), 2A:147–181, 1993.
93. V. Aquilanti, S. Cavalli, and G. Grossi, On the ridge effect in mode transitions: semiclassical analysis of the quantum pendulum. Chem. Phys. Lett., 110:43–48, 1984.
94. V. Aquilanti and S. Cavalli, Hyperspherical analysis of kinetic paths for elementary chemical reactions and their angular momentum dependence. Chem. Phys. Lett., 141:309–314, 1987.
95. 2 reaction. Chem. Phys. Lett., 162:179–184, 1989.
96. V. Aquilanti, G. Capecchi, S. Cavalli, C. Adamo, and V. Barone, Representation of potential energy surfaces by discrete polynomials: proton transfer in malonaldehyde. Phys. Chem. Chem. Phys., 2:4095–4103, 2000.
97. V. Aquilanti, S. Cavalli, C. Coletti, D. Di Domenico, and G. Grossi, Hyperspherical harmonics as sturmian orbitals in momentum space: a systematic approach to the few-body coulomb problem. Int. Rev. in Phys. Chem., 20:673–709, 2001.
98. +2 reaction: a dynamical test on potential energy surfaces for a system exhibiting a pronounced resonance pattern. Chem. Phys. Lett., 318:619–628, 2000.
99. → HeH+ + H reaction: a new potential energy surfaces and quantum mechanical cross sections. Mol. Phys., 98:1835– 1849, 2000,
100. 2, including long-range and spin-orbit effects. Physical Chem. Chem. Phys., 4:401–415, 2002.
101. V. Aquilanti, A. Beddoni, A. Lombardi, and R. Littlejohn, Hyperspherical harmonics for polyatomic systems: basis sets for kinematic rotations. Int. J. Quant. Chem., 89:277–291, 2002.
102. 2 reaction by the hyperquantization algorithm. J. Chem. Phys., 121:11675–11690, 2004.
103. 2 reaction dynamics and their splitting for nonzero angular momentum. J. Chem. Phys., 123(054314):1–15, 2005.
104. 2 → HF + H reaction. Chem. Phys. Lett., 371:504–509, 2003.
105. 2 reaction for various potential energy surfaces: features of the entrance channel and of the transition state, and low temperature reactivity. Chem. Phys., 308:237–253, 2005.
106. V. Aquilanti, L. Bonnet, and S. Cavalli, Kinematic rotations for four-center reactions: Mapping tetra-atomic potential energy surfaces on the kinetic sphere. Mol. Phys., 89:1–12, 1996.
107. V. Aquilanti and S. Cavalli, The quantum-mechanical hamiltonian for tetra-atomic systems in symmetric hyperspherical coordinates. J. Chem. Soc. Faraday Trans., 93:801–809, 1997.
108. R.G. Littlejohn, K.A. Mitchell, M. Reinsch, V. Aquilanti, and S. Cavalli, Internal spaces, kinematic rotations and body frames for four-atom systems. Phys. Rev., 58(A):3718–3738, 1998.
109. R.G. Littlejohn, K.A. Mitchell, and V. Aquilanti, Quantum dynamics of kinematic invariants in tetraand polyatomic systems. Phys Chem. Chem. Phys., 1:1259–1264, 1999.
110. V. Aquilanti, A. Beddoni, S. Cavalli, A. Lombardi, and R. Littlejohn, Collective hyperspherical coordinates for polyatomic molecules and clusters. Mol. Phys., 98:1763–1770, 2000.
111. V. Aquilanti, A. Lombardi, and R.G. Littlejohn, Hyperspherical harmonics for polyatomic systems: basis set for collective motions. Theor. Chem. Accounts, 111:400–406, 2004.
112. D.S. Wang and A. Kuppermann, Use of symbolic algebra in the calculation of hyperspherical harmonics. Int. J. Quant. Chem., 106:152–166, 2006.
113. A. Kuppermann, Hyperspherical harmonics for tetraatomic systems. 2. the weak interaction region. J. Phys. Chem. A, 108:8894–8904, 2004.
114. A. Kuppermann, Quantum reaction dynamics and hyperspherical harmonics. Isr. J. Chem., 43:229– 241, 2003.
115. D.S. Wang and A. Kuppermann, Hyperspherical harmonics for triatomic systems. J. Phys. Chem. A, 107:7290–7310, 2003.
116. D.S. Wang and A. Kuppermann, Hyperspherical harmonics for tetraatomic systems. J. Chem. Phys., 115:9184–9208, 2001.
117. V. Aquilanti, A. Lombardi, and E. Yurtsever, Global view of classical clusters: the hyperspherical approach to structure and dynamics. Phys. Chem. Chem. Phys., 4:5040–5051, 2002.
118. V. Aquilanti, A. Lombardi, M.B. Sevryuk, and E. Yurtsever, Phase-space invariants as indicators of the critical behavior of nanoaggregates. Phys. Rev. Lett., 93(4):113–402, 2004.
119. V. Aquilanti, A. Lombardi, and M.B. Sevryuk, Phase-space invariants for aggregates of particles: hyperangular momenta and partitions of the classical kinetic energy. J. Chem. Phys., 121:5579– 5589, 2004.
120. V. Aquilanti, E. Carmona-Novillo, E. Garcia, A. Lombardi, M.B. Sevryuk, and E. Yurtsever, Invariant energy partitions in chemical reactions and cluster dynamics simulations. Computational Materials Science, 35:187–191, 2005.
121. M.B. Sevryuk, A. Lombardi, and V. Aquilanti, Hyperangular momenta and energy partitions in multi-dimensional many-particle classical mechanics: the invariance approach to cluster dynamics. Phys.Rev.A, 72:033201, 2005.