Translations of fields represented by spherical-harmonic expansions for molecular calculations

AbstractFor an arbitrary integerN, the expansion theorem $$|r_ > - r_< |^N = r_ > ^N \Sigma _l \Sigma _k (2l + 1)T_{l,k}^N (r_< /r_ > )^k P_l (\cos \theta )$$ is derived by an induction method, which yields explicit expressions for the expansion coefficientTl,kN. Such expansions are useful in molecular theory because functions (r′)N withN′=|r> −r<| are contained in many operators. This investigation provides also a basis for the derivations of expansion theorems for more complicated functions which will be dealt with in later articles of this series.

[1]  K. Kay,et al.  Dirac Delta Functions in the Laplace‐Type Expansion of rnYlm(θ, φ) , 1969 .

[2]  T. A. Bromwich.  An Introduction To The Theory Of Infinite Series , 1908 .

[3]  Freeman J. Dyson,et al.  Irreducible Tensorial Sets , 1960 .

[4]  H. Weyl.  Gruppentheorie und Quantenmechanik , 1928 .

[5]  J. F. Perkins.  Atomic Integrals Containing rijp Correlation Factors with Unlinked Indices , 1969 .

[6]  R. B. Zipin,et al.  Principles of Atomic Spectra , 1968 .

[7]  Otto Steinborn.  Poly-polar expansions for regular and irregular spherical harmonics in molecules , 1969 .

[8]  Symmetric Expansion of One‐ and Two‐Center Coulomb Potentials , 1961 .

[9]  E. Hobson.  The Theory of Spherical and Ellipsoidal Harmonics , 1955 .

[10]  R. A. Sack,et al.  Generalization of Laplace's Expansion to Arbitrary Powers and Functions of the Distance between Two Points , 1964 .

[11]  E. Steinborn,et al.  Translations of fields represented by spherical-harmonic expansions for molecular calculations , 1975 .

[12]  K. Ruedenberg,et al.  New Aspects of the Bipolar Expansion and Molecular Multicenter Integrals , 1968 .

[13]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[14]  R. A. Sack,et al.  Three‐Dimensional Addition Theorem for Arbitrary Functions Involving Expansions in Spherical Harmonics , 1964 .

[15]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[16]  K. Ruedenberg,et al.  Expansion of r12 and r 12−1 in terms of analytical functions , 1969 .

[17]  R. P. Soni,et al.  Formulas and Theorems for the Special Functions of Mathematical Physics , 1967 .

[18]  B. L. Waerden.  Die Gruppentheoretische Methode in der Quantenmechanik , 1932 .

[19]  Two‐Center Hybrid Integrals between Slater‐Type Atomic Orbitals , 1967 .

[20]  J. P. Dahl,et al.  Expansion theorems for solid spherical harmonics , 1965 .

[21]  R. Balescu.  Interactions between symmetric polyatomic molecules , 1956 .

[22]  K. Knopp.  Theorie und Anwendung der Unendlichen Reihen , 1924 .

[23]  E. Wigner,et al.  Book Reviews: Group Theory. And Its Application to the Quantum Mechanics of Atomic Spectra , 1959 .

[24]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[25]  Klaus Ruedenberg.  Bipolare Entwicklungen, Fouriertransformation und Molekulare Mehrzentren-Integrale , 1967 .

[26]  A. R. Edmonds.  Angular Momentum in Quantum Mechanics , 1957 .

[27]  G. Weiss,et al.  Differential Equations of Applied Mathematics , 1966 .

[28]  K. C. Daiker,et al.  Natural basis expansion of functions of the distance between two points: Generalization of Laplace's and Neumann's expansions , 1974 .

[29]  Ying‐Nan Chiu.  Irreducible Tensor Expansion of Solid Spherical Harmonic‐Type Operators in Quantum Mechanics , 1964 .

[30]  A. Zemanian,et al.  Distribution theory and transform analysis , 1966 .

[31]  M. E. Rose.  Elementary Theory of Angular Momentum , 1957 .

[32]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[33]  R. P. Soni,et al.  The gamma function and related functions , 1966 .