Harmonic polynomials, hyperspherical harmonics, and atomic spectra

The properties of monomials, homogeneous polynomials and harmonic polynomials in d-dimensional spaces are discussed. The properties are shown to lead to formulas for the canonical decomposition of homogeneous polynomials and formulas for harmonic projection. Many important properties of spherical harmonics, Gegenbauer polynomials and hyperspherical harmonics follow from these formulas. Harmonic projection also provides alternative ways of treating angular momentum and generalised angular momentum. Several powerful theorems for angular integration and hyperangular integration can be derived in this way. These purely mathematical considerations have important physical applications because hyperspherical harmonics are related to Coulomb Sturmians through the Fock projection, and because both Sturmians and generalised Sturmians have shown themselves to be extremely useful in the quantum theory of atoms and molecules. © 2009 Elsevier B.V. All rights reserved. 1. Monomials, homogeneous polynomials, and harmonic polynomials Amonomial of degree n in d coordinates is a product of the form mn = x n1 1 x n2 2 x n3 3 · · · x nd d (1) where the nj’s are positive integers or zero and where their sum is equal to n. n1 + n2 + · · · + nd = n. (2) For example, x1, x 2 1x2 and x1x2x3 are all monomials of degree 3. Since ∂mn ∂xj = njx j mn, (3) it follows that d ∑ j=1 xj ∂mn ∂xj = nmn. (4) A homogeneous polynomial of degree n (which we will denote by the symbol fn) is a series consisting of one or more monomials, all of which have degree n. For example, f3 = x1 + x 2 1x2 − x1x2x3 is a homogeneous polynomial of degree 3. Since each of the monomials in such a series obeys (4), it follows that d ∑ j=1 xj ∂ fn ∂xj = nfn. (5) E-mail address: [email protected]. 0377-0427/$ – see front matter© 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2009.02.057 J.S. Avery / Journal of Computational and Applied Mathematics 233 (2010) 1366–1379 1367 This simple relationship has very far-reaching consequences. If we now introduce the generalised Laplacian operator