Harmonic analysis generalizes Fourier expansions and transforms, and we develop it in order to provide efficient computational implementation to the particularly demanding cases of atomic and molecular physics where the density of states is high. Physically, wavefunctions for the angular part of the kinetic energy operator are spherical harmonics, i.e. eigensolutions to Laplace operator on the ordinary sphere: they add and multiply according to the rotation group operations, so providing the framework for the quantum theory of angular momentum (Clebsch-Gordan Series, sum rules for 3-j and 6-j coefficients...).
The mathematical tools are the classical orthogonal polynomials as well as their discrete analogues (hyperquantization algorithm). The computational advantages are due to the existence of closed form expressions, three-term recurrences, analytical integrals. Work is presented here on multidimensional extensions, to treat exactly chemical reactions including spin and electronic angular momenta [(J. Chem. Phys., 108, in press, 1998) and references therein].
For many-body problems, the kinetic energy operators, when conveniently
written in hyperspherical coordinates, have as eigensolutions hyperspherical
harmonics, which can be interpreted as wavefunctions of hyperangular momenta,
whose algebra, which is within the boundaries of Gauss' hypergeometric
function theory, is being developed.
Hyperharmonics can also be exploited as atomic and molecular orbitals,
extending Fock projection into momentum space for the hydrogen atom to
the n-dimensional case, and introducing alternative expansions for a multidimensional
plane wave, of use for generalized Fourier transforms. This method allows
us to work both in configuration space (on eigenfunctions expanded on a
Sturmian basis) and in momentum space (on a (n + 1) dimensional hyperspherical
harmonics expansions) [( Phys. Rev. Letters, 80, 3209, 1998) and
references therein].