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Journal of Computational and Applied Mathematics Harmonic Polynomials Journal of Computational and Applied Mathematics 233 (2010) 1366–1379 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Harmonic polynomials, hyperspherical harmonics, and atomic spectra John Scales Avery H.C. Ørsted Institute, University of Copenhagen, Copenhagen, Denmark article info a b s t r a c t Article history: The properties of monomials, homogeneous polynomials and harmonic polynomials in Received 12 October 2007 d-dimensional spaces are discussed. The properties are shown to lead to formulas for the canonical decomposition of homogeneous polynomials and formulas for harmonic Dedicated to Professor Jesús S. Dehesa on projection. Many important properties of spherical harmonics, Gegenbauer polynomials the occasion of his 60th birthday. and hyperspherical harmonics follow from these formulas. Harmonic projection also provides alternative ways of treating angular momentum and generalised angular Keywords: momentum. Several powerful theorems for angular integration and hyperangular Harmonic polynomials integration can be derived in this way. These purely mathematical considerations have Hyperangular integration Hyperspherical harmonics important physical applications because hyperspherical harmonics are related to Coulomb Sturmians Sturmians through the Fock projection, and because both Sturmians and generalised Atomic spectra 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 A monomial of degree n in d coordinates is a product of the form D n1 n2 n3 ··· nd mn x1 x2 x3 xd (1) where the nj's are positive integers or zero and where their sum is equal to n. n1 C n2 C···C nd D n: (2) 3 2 For example, x1, x1x2 and x1x2x3 are all monomials of degree 3. Since @m n D −1 njxj mn; (3) @xj it follows that d X @mn xj D nmn: (4) x jD1 @ j A homogeneous polynomial of degree n (which we will denote by the symbol fn) is a series consisting of one or more D 3 C 2 − monomials, all of which have degree n. For example, f3 x1 x1x2 x1x2x3 is a homogeneous polynomial of degree 3. Since each of the monomials in such a series obeys (4), it follows that d X @fn xj D nfn: (5) x jD1 @ j 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 Xd @ 2 ≡ (6) 1 2 jD1 @xj and the hyperradius defined by d 2 ≡ X 2 r xj (7) jD1 we can show (with a certain amount of effort!) that β β−2 β 1 r fα D β(β C d C 2α − 2/r fα C r 1fα (8) where α and β are positive integers or zero, β being even. We next define an harmonic polynomial of degree n to be a homogeneous polynomial of degree n which also satisfies the generalised Laplace equation: 1hn D 0: (9) D 2 − 2 C For example, h3 x1x2 x3x2 x1x2x3 is a harmonic polynomial of degree 3. Combining (8) and (9), we obtain β β−2 1 r hα D β(β C d C 2α − 2/r hα: (10) 2. The canonical decomposition of a homogeneous polynomial Every homogeneous polynomial fn can be decomposed into a sum of harmonic polynomials multiplied by powers of the hyperradius [2]. This decomposition, which is called the canonical decomposition of a homogeneous polynomial, has the form 2 4 fn D hn C r hn−2 C r hn−4 C··· : (11) To see how the decomposition may be performed, we can act on both sides of Eq. (11) with the generalised Laplacian operator 1. If we do this several times, making use of (10), we obtain 2 1fn D 2.d C 2n − 4/hn−2 C 4.d C 2n − 6/r hn−4 C··· 2 1 fn D 8.d C 2n − 6/.d C 2n − 8/hn−4 C··· (12) 3 1 fn D 48.d − 2n − 8/.d − 2n − 10/.d − 2n − 12/hn−6 C··· ; and in general b n c 2 WW C − − WW ν X .2k/ .d 2n 2k 2/ 2k−2ν 1 fn D r hn−2k (13) .2k − 2ν/WW .d C 2n − 2k − 2ν − 2/WW kDν where j.j − 2/.j − 4/ ··· 4 × 2 j D even jWW ≡ (14) j.j − 2/.j − 4/ ··· 3 × 1 j D odd: An important special case occurs when ν D n=2. In that case, (13) becomes WW C − WW n=2 n .d n 2/ 1 fn D h0 (15) .d − 2/WW or − WW .d 2/ n=2 h0 D 1 fn: (16) nWW.d C n − 2/WW We will see below that this result leads to powerful angular and hyperangular integration theorems. 3. Harmonic projection Eqs. (12) or (13) form a set of simultaneous equations that can be solved to yield expressions for the various harmonic polynomials that occur in the canonical decomposition of a homogeneous polynomial fn. In this way we obtain the general result [3–5]: b n −νc C − − WW 2 − j C − − − WW .d 2n 4ν 2/ X . 1/ .d 2n 4ν 2j 4/ 2j jCν hn−2ν D r 1 fn: (17) 2 WW d C 2n − 2 − 2 WW 2j WW d C 2n − 4 − 4 WW . ν/ . ν / jD0 . / . ν / 1368 J.S. Avery / Journal of Computational and Applied Mathematics 233 (2010) 1366–1379 If we let n − 2ν D λ, this becomes C − WW bλ/2c − j C − − WW .d 2λ 2/ X . 1/ .d 2λ 2j 4/ 2j jC 1 .n−λ) OλTfnUD hλ D r 1 2 fn: (18) n − WW d C n C − 2 WW 2j WW d C 2 − 4 WW . λ) . λ / jD0 . / . λ / Here Oλ can be thought of as a projection operator that projects out the harmonic polynomial of degree λ from the canonical decomposition of the homogeneous polynomial fn. The projection is of course taken to be zero if λ and n have different parities. 4. Generalised angular momentum The generalised angular momentum operator Λ2 is defined as [3–5] d d 2 2 X X @ @ Λ ≡ − xi − xj : (19) @x @x i>j jD1 j i When d D 3 it reduces to the familiar orbital angular momentum operator 2 D 2 C 2 C 2 L L1 L2 L3 (20) where 1 @ @ L1 D x2 − x3 (21) i @x3 @x2 and where L2 and L3 are given by similar expressions with cyclic permutations of the coordinates. If we expand the expression in (19), we obtain d 2 d 2 2 X @ X @ Λ D −r 1 C xixj C .d − 1/ xi : (22) @x @x @x i;jD1 i j iD1 i 2 We next allow Λ to act on a homogeneous polynomial fn, and make use of (5). This gives us d 2 2 2 X @ fn Λ fn D −r 1fn C n.d − 1/fn C xixj : (23) @x @x i;jD1 i j The relationship d 2 X @ fn xixj D n.n − 1/fn (24) @x @x i;jD1 i j can be derived in a manner similar to the derivation of (5). Substituting this into (24), we have 2 2 Λ fn D −r 1fn C n.n C d − 2/fn: (25) From (25), it follows that a harmonic polynomial of degree λ is an eigenfunction of the generalised angular momentum operator with the eigenvalue λ(λ C d − 2/, i.e., 2 Λ hλ D λ(λ C d − 2/hλ: (26) When d D 3, this reduces to 2 L hl D l.l C 1/hl: (27) We can conclude from this discussion that the canonical decomposition of a homogeneous polynomial can be viewed as a decomposition into eigenfunctions of generalised angular momentum. 5. Angular and hyperangular integration 2 In a 3-dimensional space the volume element is given by dx1dx2dx3 in Cartesian coordinates or by r dr dΩ in spherical polar coordinates. Thus we can write 2 dx1dx2dx3 D r dr dΩ (28) where dΩ is the element of solid angle. Similarly [3,6,4,5], in a d-dimensional space we can write d−1 dx1dx2 ··· dxd D r dr dΩ (29) J.S. Avery / Journal of Computational and Applied Mathematics 233 (2010) 1366–1379 1369 where r is the hyperradius and where dΩ is the element of generalised solid angle. From the Hermiticity of the generalised angular momentum operator Λ2, one can show [6,4,5] that its eigenfunctions corresponding to different eigenvalues are orthogonal with respect to hyperangular integration. Thus from (26) it follows that Z ∗ D 0 6D dΩ hλ0 hλ 0 if λ λ. (30) In the particular case where λ0 D 0, this becomes Z Z ∗ D ∗ D 6D dΩ h0hλ h0 dΩ hλ 0 if λ 0 (31) ∗ since the constant, h0, can be factored out of the integration over generalised solid angle. Thus we obtain the important result: Z dΩ hλ D 0 if λ 6D 0: (32) Let us now combine this result with Eq. (11), which shows the form of the canonical decomposition of a homogeneous polynomial fn. From (11) and (32) it follows that if a homogeneous polynomial is integrated over a generalised solid angle, the only term that will survive is the constant term in the canonical decomposition, i.e., h0.
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