Hyperspherical harmonics for polyatomic systems: Basis set for kinematic rotations

In a symmetrical hyperspherical framework, the internal coordinates for the treatment of N-body systems are conveniently broken up into kinematic invariants and kinematic rotations. Kinematic rotations describe motions that leave unaltered the moments of the inertia of the N-body system and perform the permutation of particles. This article considers the corresponding expansions of the wave function in terms of hyperspherical harmonics giving explicit examples for the four-body case, for which the space of kinematic rotations (the “kinetic cube”) is the space SO(3)/V4 and then the related eigenfunctions will provide a basis on such manifold, as well as be symmetrical with respect to the exchange of identical particles (if any). V4 is also denoted as D2. The eigenfunctions are obtained studying the action of projection operators for V4 on Wigner D-functions. When n of the particles are identical, the exchange symmetry can be obtained using the projection operators for the Sn group. This eigenfunction expansion basis set for kinematic rotations can be also of interest for the mapping of the potential energy surfaces. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002

[1]  M. N. Hounkonnou,et al.  Quantum description of five-particle systems in principal-axis hyperspherical coordinates , 2000 .

[2]  M. Hamermesh.  Group theory and its application to physical problems , 1962 .

[3]  Richard N. Zare,et al.  Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics , 1988 .

[4]  L. Landau.  Quantum Mechanics-Nonrelativistic Theory , 1958 .

[5]  E. Wigner,et al.  Book Reviews: Group Theory. And Its Application to the Quantum Mechanics of Atomic Spectra , 1959 .

[6]  G. Arfken.  Mathematical Methods for Physicists , 1967 .

[7]  V. Aquilanti,et al.  The He + H2+ reaction: a dynamical test on potential energy surfaces for a system exhibiting a pronounced resonance pattern , 2000 .

[8]  V. Aquilanti,et al.  Representation in hyperspherical and related coordinates of the potential-energy surface for triatomic reactions , 1990 .

[9]  V. Aquilanti,et al.  Theory of electronically nonadiabatic reactions: Rotational, Coriolis, spin–orbit couplings and the hyperquantization algorithm , 2001 .

[10]  A. Zamorzaev,et al.  Rotations, quaternions and double groups by S. Altmann , 1988 .

[11]  K. Mitchell,et al.  Body frames and frame singularities for three-atom systems , 1998 .

[12]  Kinematic orbits and the structure of the internal space for systems of five or more bodies , 1999, physics/9910001.

[13]  Vincenzo Aquilanti,et al.  Hyperspherical harmonics as Sturmian orbitals in momentum space: A systematic approach to the few-body Coulomb problem , 2001 .

[14]  R. Littlejohn,et al.  Gauge fields in the separation of rotations and internal motions in the n-body problem , 1997 .

[15]  A. Kuppermann,et al.  Hyperspherical harmonics for tetraatomic systems , 2001 .

[16]  V. Aquilanti,et al.  Hyperquantization algorithm. I. Theory for triatomic systems , 1998 .

[17]  Sturmian expansions for quantum mechanical many-body problems, and hyperspherical harmonics , 2001 .

[18]  V. Aquilanti,et al.  The quantum-mechanical Hamiltonian for tetraatomic systems in symmetric hyperspherical coordinates , 1997 .

[19]  J. Lévy-Leblond,et al.  N‐Particle Kinematics and Group‐Theoretical Treatment of Phase Space I. Nonrelativistic , 1965 .

[20]  J. Lévy-Leblond.  Global and Democratic Methods for Classifying N‐Particle States , 1966 .

[21]  V. Aquilanti,et al.  Ab initio dynamics of the He + H+ 2 → HeH+ +H reaction: a new potential energy surface and quantum mechanical cross-sections , 2000 .

[22]  V. Aquilanti,et al.  Kinematic rotations for four-centre reactions: mapping tetra-atomic potential energy surfaces on the kinetic sphere , 1996 .

[23]  V. Aquilanti,et al.  Hyperspherical Coordinates for Chemical Reaction Dynamics , 2000 .

[24]  V. Aquilanti,et al.  The A+BC reaction by the hyperquantization algorithm: the symmetric hyperspherical parametrization for J > 0 , 2001 .

[25]  F. Smith.  Three-body collision rates in atomic recombination reactions , 1962 .

[26]  J. J. S. Neto,et al.  A WAVE FUNCTION EXPANSION FOR THE STUDY OF TETRA-ATOMIC MOLECULES IN HYPERSPHERICAL COORDINATE SYSTEMS , 1999 .

[27]  Vincenzo Aquilanti,et al.  Hyperspherical Symmetry of Hydrogenic Orbitals and Recoupling Coefficients among Alternative Bases , 1998 .

[28]  H. Mayer.  Construction of the hyperspherical functions for the quantum mechanics of three particles , 1975 .

[29]  Quantum description of rigidly or adiabatically constrained four-particle systems and supersymmetry , 1999 .

[30]  H. Galbraith.  Four‐Body Problem: A Complete System of Angular Functions , 1971 .

[31]  V. Aquilanti,et al.  Probabilities for the F+H2→HF+H reaction by the hyperquantization algorithm: alternative sequential diagonalization schemes , 1999 .

[32]  V. Aquilanti,et al.  Hyperquantization algorithm. II. Implementation for the F+H2 reaction dynamics including open-shell and spin-orbit interactions , 1998 .

[33]  Vincenzo Aquilanti,et al.  Quantum dynamics of kinematic invariants in tetra- and polyatomic systems , 1998 .

[34]  W. Zickendraht,et al.  Configuration‐Space Approach to the Four‐Particle Problem , 1969 .

[35]  K. Mitchell,et al.  INTERNAL SPACES, KINEMATIC ROTATIONS, AND BODY FRAMES FOR FOUR-ATOM SYSTEMS , 1998 .

[36]  C. Mead,et al.  On the determination of Born–Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei , 1979 .

[37]  Vincenzo Aquilanti,et al.  Coordinates for molecular dynamics: Orthogonal local systems , 1986 .

[38]  Littlejohn,et al.  Internal or shape coordinates in the n-body problem. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[39]  Vincenzo Aquilanti,et al.  Collective hyperspherical coordinates for polyatomic molecules and clusters , 2000 .

[40]  F. Smith.  Participation of Vibration in Exchange Reactions , 1959 .