Incorporating the Geometric Phase Effect in Triatomic and Tetraatomic Hyperspherical Harmonics

Abstract

Hyperspherical harmonics in the democratic row-orthonormal hyperspherical coordinates are very appropriate basis sets for performing reactive scattering calculations for triatomic and tetraatomic systems. The mathematical conditions for incorporating the geometric phase effect in these harmonics are given. These conditions are implemented for triatomic systems, and their explicit analytical expressions in terms of Jacobi polynomials, in both the absence and presence of the geometric phase effect, are given.

Analytical Derivation of Row-Orthonormal Hyperspherical Harmonics for Triatomic Systems

“…Therefore, we are in complete agreement with those authors for all the J = 2 cases we tested. It should be noted that their hyperspherical coordinates, principal axes of inertia, and Wigner rotation functions are different from those in the present paper, and this agreement takes this difference into account …”

The H3+ rovibrational spectrum revisited with a global electronic potential energy surface

“…Equation ͑3͒ is solved by expanding the surface functions on a basis of principal-axes-of-inertia hyperspherical harmonics F Md ⌸nLJD ͑⍀͒. These harmonics 36,43,44 are simultaneous eigenfunctions of the nuclear angular momentum operator, its projection on a space fixed axis and the parity, and are labeled by the corresponding quantum numbers J, M, and ⌸. They are also eigenfunctions of the grand canonical angular momentum squared, ⌳ 2 , as well as of an internal hyperangular momentum operator, L =−iប͑‫ץ‬ / ‫,͒␦ץ‬…”

Fundamental Approaches to Nonadiabaticity: Toward a Chemical Theory beyond the Born–Oppenheimer Paradigm

“…However, it was Mead who first studied and pointed out the theoretical similarity of the Longuet-Higgins phase to the Aharonov-Bohm effect. , Later, Mukunda and Simons established its common mathematical foundation for a geometric phase in terms of anholonomy in fiber bundle theory . The effects of the Longuet-Higgins phase on chemical dynamics have been discussed extensively in conjunction with quantum scattering phenomena ,,,,− and vibronic spectroscopy in a Na 3 system, which was examined theoretically by Kendrick. − In a control theory of chemical reactions, Domcke et al examined the effect of phase on a photodissociation branching ratio of phenol . The geometric phase is critically important, not only in molecular science, but also in solid state physics; for instance, it is claimed that the quantum hole effect in quantum transport is related to the geometric phase. , …”