Recurrence relations are derived for constructing rotation matrices between complex spherical harmonics directly as polynomials of the elements of the generating rotation matrix, bypassing the intermediary of any parameters such as Euler angles. The connection to the rotation matrices for real spherical harmonics is made explicit. The recurrence formulas furnish a simple, efficient, and numerically stable evaluation procedure for the real and complex representations of the rotation group. The advantages over the Wigner formulas are documented. The results are relevant for directing atomic orbitals as well as multipoles.
REFERENCES
1.
L. F. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems (MIT, Cambridge, MA, 1987).
2.
E. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren (Vieweg, Wiesbaden, Germany, 1931).
3.
4.
A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton University Press, Princeton, NJ, 1960).
5.
6.
E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, Cambridge, 1951)
7.
H. Bethe, in Handbuch der Physik, edited by H. Geiger and K. Scheel (Springer-Verlag, Berlin, 1933), Vol. 24/1, Chap. 3, Eq. 65.21
8.
D. A. Varshalovich, A. N. Moskaley, and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
9.
See e.g., L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics (McGraw-Hill, New York, 1935).
10.
S. Wolfram, Mathematica, A System for Doing Mathematics by Computer (Addison-Wesley, New York, 1988).
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© 1999 American Institute of Physics.
1999
American Institute of Physics
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