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Application of hyperspherical coordinates to the correlation problem
Author(s) -
Avery John,
ChristensenDalsgaard Birte L.,
Larsen Peter Sommer,
Hengyi Shen
Publication year - 1984
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560260830
Subject(s) - eigenfunction , hamiltonian (control theory) , physics , wave function , basis (linear algebra) , harmonic oscillator , classical mechanics , basis function , hamiltonian matrix , harmonics , log polar coordinates , identical particles , clebsch–gordan coefficients , schrödinger equation , quantum mechanics , coordinate system , eigenvalues and eigenvectors , orthogonal coordinates , quantum , mathematics , geometry , symmetric matrix , irreducible representation , mathematical optimization , voltage
Techniques for applying hyperspherical coordinates to the quantum‐mechanical many‐body problem are reviewed. An improved method is presented for evaluating matrix elements of the Hamiltonian of a system of particles. This method involves a rotation in the many‐dimensional coordinate space of the system, and it can be applied not only to Coulomb potentials, but also to potentials of other types, such as, for example, the Lennard–Jones potential. It is shown that symmetry‐adapted hyperspherical harmonics in the m = 3 N ‐dimensional coordinate space of an N ‐particle system form a convenient basis set for the solution of the hyperangular part of the many‐particle Schrödinger equation. Methods are presented for constructing hyperspherical harmonics of a type which are simultaneous eigenfunctions of Λ 2 , L 2 , and L z , as well as being basis functions for the group of permutations of identical particles. The method presented here for coupling angular momenta by harmonic projection (without the use of Clebsch‐Gordan coefficients) has broad applicability.