E. Otto Steinborn

Translations of fields represented by spherical-harmonic expansions for molecular calculations: II. Translations of powers of the length of the local vector

AbstractFor an arbitrary integerN, the expansion theorem

Analytische Lösung des Molekülintegrals vom Drei-Zentren-Ein-Elektron-Typ über Slater-Atomorbitale und eine reguläre räumliche Kugelfunktion@@@Analytical solution of molecular three-center one-electron integrals over slater-type atomic orbitals and regular solid spherical harmonics

|r_ > - r_< |^N = r_ > ^N \Sigma _l \Sigma _k (2l + 1)T_{l,k}^N (r_< /r_ > )^k P_l (\cos \theta )

Fourier transform of a two-center product of exponential-type orbitals. Application to one- and two-electron multicenter integrals

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.

Translations of fields represented by spherical-harmonic expansions for molecular calculations: III. Translations of reduced Bessel functions, Slater-types-orbitals, and other functions

Three-dimensional functions f(r) = g(r) · Yml(θ, ø), which transform like an irreducible tensor, are transformed simultaneously under rotations and translations. The relationships governing the transformation reveal some general properties. If the addition theorem of a function f(r) can be represented by a one-center expansion in terms of surface spherical harmonics Yml, each expansion coefficient is given by a Clebsch-Gordan coefficient and a radial function.Because of these properties, addition theorems are especially helpful for the simplification and evaluation of quantum-mechanical matrix elements and multi-center energy integrals in molecular LCAO calculations. The application of addition theorems has two major advantages: First, because addition theorems are equivalent to translation formulas, the number of centers of an integral can be reduced by translation of orbitals and operators. Second, due to the typical analytical structure of the series expansion representing the addition theorem, the dimensionality of a molecular integral can be reduced, because the integration over the angular variables can be executed. Then, a molecular multi-center integral is represented by a series of one-center integrals over functions of the radial variable only.

Translations of Fields Represented by Spherical-Harmonic Expansions for Molecular Calculations

In calculations of molecular properties matrix elements of solid spherical harmonics are often needed. For the molecular three-center one-electron integral over two Slater-type orbitals of different atoms and a regular solid spherical harmonic, which is defined with respect to a third center, a completely analytical solution is given that contains only finite sums. Due to the strict observation of the concept of separate computation of the charge density distribution, and since elliptic coordinates are used, expressions can be utilized which occur in calculations of diatomic molecules. This is advantageous for practical applications.