Research: Science and Education
Closed-Form Spherical Harmonics: Explicit Polynomial Expression for the Associated Legendre Functions J. J. C. Mulder Gorlaeus Laboratories, Leiden University, P.O. Box 9502, 2300 RA Leiden, The Netherlands;
[email protected] The solutions of the nonrelativistic Schrödinger equation for the hydrogen atom figure prominently in all physical chemistry textbooks. Their presentation can be improved upon, however. In general a small number of functions for low values of the quantum numbers n, l, and m is given, together with some rather cryptic comments on spherical harmonics and (associated) Legendre and Laguerre functions. For the student with some interest in the subject, this is an unsatisfactory situation. The general expression as a function of the polar coordinates r, θ, and ϕ, should be available in such a way that only a choice of the quantum numbers is necessary to generate a particular solution simply and directly. This has the additional advantage that the structure of the problem (i.e., the relation between the number and type of nodes and the degree in θ or r, or the connection between the three quantum numbers and the three variables) is shown in a transparent fashion. The expression sought for is: φn,l,m = R n,l (r)Yl,m(θ,ϕ) = R n,l (r)Θl,m(θ)Fm(ϕ)
A quick search through the literature (1–5) leads to the surprising conclusion that, whereas the general solutions for r and ϕ are readily available, this apparently is not the case for the associated Legendre function. Typically one finds:
R n,l r =
2Z ; β = na o
βr n–l–1 N R e 2 p=0
Σ
1
p
n +l p
n +l ! n +l = ; p p! n + l – p !
Θl, m θ = N Θ
1
l
l
2 l+ m !
sin m θ
βr n –l–p–1 ! n–p–1
F m ϕ = 1 e imϕ 2π d l+ m sin2l θ d cos θ
l+ m
The normalization constants are: NR =
n – l – 1 ! β3 ; NΘ = 2n n + 1 !
2l + 1 l – m ! l + m ! 2 l! l!
To obtain the explicit polynomial form for Θl,m, the m = 0 case is considered first (6 ). The result can thereafter be differentiated |m| times and multiplied with sin|m| θ to arrive at the final expression. It is only by virtue of the possibility of
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using cos θ and sin θ simultaneously as variables, however, that one is able to get the desired result:
Θl, m θ = NΘ
sin
m
θ
l
2
l– m 2
Σ q=0
1
l+q
l q
2l – 2q cosl– l+ m
m –2q
θ
In the summation q is incremented with a step size of one, and the upper limit is the largest integer ≤(l – |m|)/2. The independence of the degree of Θl,m with respect to |m| is clearly demonstrated in the q = 0 lead term of the polynomial. When preparing this manuscript I searched the literature more exhaustively, as it seemed unlikely that this result would not have been discovered before. It turned out that it had (L. J. Oosterhoff, unpublished notes, found in his personal copy of ref 5; personal communication from M. C. van Hemert). It seems desirable, though, to bring it to the attention of the chemistry and physics community. Possibly the textbooks will follow. Acknowledgment I wish to thank an anonymous referee, who directed my attention to J. C. Slater’s The Quantum Theory of Matter, 2nd ed., p 116, where an expression for Θl,m(θ) is to be found that is essentially the same as the one derived above. It is probably the series notation used, instead of the closed-form polynomial, that has prevented it from becoming current. Literature Cited 1. Abramowitz, M.; Segun, I. A. Handbook of Mathematical Functions; Dover: Mineola, NY, 1968; p 331. 2. Magnus, W.; Oberhettinger, F. Formeln und Sätze für die speziellen Funktionen der mathematischen Physik; Springer: Heidelberg, 1948; p 73. 3. Karplus, M.; Porter, R. N. Atoms and Molecules; Benjamin/ Cummings: Menlo Park, CA, 1970; p 164. 4. Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: New York, 1994; pp 426, A26. 5. Pauling, L.; Wilson, E. B. Introduction to Quantum Mechanics; McGraw-Hill: New York, 1935; p 125. 6. Handbook of Applicable Mathematics, Vol. IV; Ledermann, W., Ed.; Wiley: New York, 1982; p 462.
Journal of Chemical Education • Vol. 77 No. 2 February 2000 • JChemEd.chem.wisc.edu
