Hyperspherical harmonics for tetraatomic systems

Abstract

A recursion procedure for the analytical generation of hyperspherical harmonics for tetraatomic systems, in terms of row-orthonormal hyperspherical coordinates, is presented. Using this approach and an algebraic Mathematica program, these harmonics were obtained for values of the hyperangular momentum quantum number up to 30 ͑about 43.8 million of them͒. Their properties are presented and discussed. Since they are regular at the poles of the tetraatomic kinetic energy operator, are complete, and are not highly oscillatory, they constitute an excellent basis set for performing a partial wave expansion of the wave function of the corresponding Schrödinger equation 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.

Orthogonal coordinates for the dynamics of four bodies and for the representation of potentials of tetra‐atomic molecules

“…The interparticle distances from the Jacobi parameters can be calculated by r 2 ik = r 2 im + r 2 km + 2r im r km cos δ jmk (44) r 2 jk = r 2 jm + r 2 km − 2r jm r km cos δ jmk (45) r 2 kl = r 2 kn + r 2 ln − 2r kn r ln cos δ lnm (46) r 2 il = r 2 ln + r 2 im + r 2 mn + 2r mn (r im cos δ jmk − r ln cos δ lnm ) − 2r im r ln + mn (47) r 2 jl = r 2 ln + r 2 jm + r 2 mn − 2r mn (r jm cos δ jmk + r ln cos δ lnm ) + 2r jm r ln + mn (48) where + mn = cos ω mn sin δ jmk sin δ lnm + cos δ jmk cos δ lnm (49) From the interparticle distances to the Jacobi tree parameters, the r km and r ln distances can be calculated by…”

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

“…The F hyperspherical harmonics 32,46 for pentaatomic systems depends on 11 angles: the 3 Euler angles that rotate the space-fixed axes to the principal moment of inertia bodyfixed axes, the 6 internal Euler angles of a 4-dimensional space, and two principal moment of inertia angles. The basis set in the first 3 is the usual 3-dimensional Wigner rotation function, and in the next 6 it is the 4-dimensional space Wigner rotation function discussed in this paper.…”

“…32 As an example, it appears in the expressions for the hyperspherical harmonics for tetraatomic systems, which are eigenfunctions of the system's grand-canonical angular momentum operatorL 2 and of a set of other angular momentum operators that commute with it. 46 For that particular case, it appears twice: once as a function of the 3 Euler angles a l that rotate the space-fixed frame to the body-fixed principal momentum of inertia frame, and a second time as a function of the 3 internal Euler angles d l associated with the rotation matrix R (3) (d l ) of (2.2). In the rest of this section we discuss the definition of the Wigner rotation function and its determination for (n Z 3)-dimensional space.…”

“…Those equations make the D ð3Þl m 0 m useful basis functions for expanding the tetraatomic system's wave functions. 32,46 The fact that they stem from the matrix representation of the operator (5.3) is not central for this application.…”

Numerical generation of hyperspherical harmonics for tetra-atomic systems

“…10,18 We only summarize here their definitions and properties and refer the reader to these papers for further details. Let us call ͑i 1 , i 2 , i 3 ͒ the unit vectors of a reference frame for the three-dimensional physical space.…”