Analytical Derivation of Row-Orthonormal Hyperspherical Harmonics for Triatomic Systems

Abstract

Hyperspherical harmonics for triatomic systems as functions of row-orthonormal hyperspherical coordinates, (also called democratic hyperspherical harmonics) are obtained explicitly in terms of Jacobi polynomials and trigonometeric functions. These harmonics are regular at the poles of the triatomic kinetic energy operator, are complete, and are not highly oscillatory. They constitute an excellent basis set for calculating the local hyperspherical surface functions in the strong interaction region of nuclear configuration space. This basis set is, in addition, numerically very efficient and should permit benchmark-quality calculations of state-to-state differential and integral cross sections for those systems. The approach used for their derivation is new and should be applicable to systems of more than three atoms.

Reactive scattering with row-orthonormal hyperspherical coordinates. 4. Four-dimensional-space Wigner rotation function for pentaatomic systems

“…For the last two angles it is the G hyperspherical harmonics set. The latter functions are not presently known, but they may be determined by an extension of the method used previously to obtain them for triatomic 34 and tetraatomic 46 systems. Once they are obtained, all of the analytical tools needed for calculating accurate state-to-state differential and integral cross sections for pentaatomic reactions (such as H 2 + H + 3 and its isotopomers, which are important for interstellar processes 53 ) will be available.…”

“…The definition and properties of the ROHC for N-atom systems for N Z 3 have been given previously [29][30][31][32][33][34] and will only be summarized below succinctly. Let the corresponding set of N À 1 l-arrangement mass-scaled Jacobi vectors be r (1) l , r (2) l , .…”

“…In addition to Cartesian coordinates, hyperspherical coordinates of different varieties have been used to describe reactive scattering processes, [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] as well as bound states. 26,27 The row-orthonormal hyperspherical coordinate (ROHC) variety [28][29][30][31][32][33][34] has the useful property that the resulting many individual terms of a polyatomic system's Hamiltonian are invariant under a change of arrangement channel coordinates [30][31][32][33][34] (i.e., kinematic rotations), which makes them particularly appropriate for the study of chemical reactions. The formalism involved, for an N-atom system, introduces the concepts of rotations and angular momenta in spaces of dimension n = N À 1.…”