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## Open Physics

2003 | 1 | 3 | 432-439
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### Two-electron coulomb integrals with minimal group theory. Direct analytic solution and computer program

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EN
A general closed-form analytic solution is developed for two-electron Coulomb integrals such as those that appear in perturbative treatments of multielectron atoms. In developing this solution, no recourse is necessary to group theory, Clebsch-Gordan coefficients, or specialized coordinate transformations: only familiar properties of hydrogenic wavefunctions and standard integrals are employed. One constraint from group theory is adopted for the purpose of establishing an upper limit on an index of summation for computational purposes. An accurate and efficient computer program for evaluating such integrals is made available.
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432-439
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published
1 - 9 - 2003
online
1 - 9 - 2003
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• Department of Physics, Alma College, 48801, Alma, MI, reed@alma.edu
References
• [1] I.N. Levine: Quantum Chemistry, 2nd ed., Ch. 9, Allyn and Bacon, Boston, 1974.
• [2] G. Arfken: Mathematical Methods for Physicists, 3rd ed, Ch. 12, Academic Press, Orlando, 1985.
• [3] J.C. Slater: “The Theory of Complex Spectra”, Phys. Rev., Vol. 34, (1929), pp. 1293–1322. http://dx.doi.org/10.1103/PhysRev.34.1293[Crossref]
• [4] J.J.C. Mulder: “Closed-Form Spherical harmonics: Explicit Polynomial Expression for the Associated Legendre Functions”, J. Chem. Educ., Vol. 77, (2000), pp. 244. Note that Mulder's result is in error in that his coefficients b lmk carry an extra multiplicative factor of (1)l. http://dx.doi.org/10.1021/ed077p244[Crossref]
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