Loewdin alpha function and its application to the multicenter molecular integral problem over Slatertype orbitals
Abstract
In this paper we trace the evolution of the Lowdin alphafunction method in its application to multicenter molecular integrals over Slatertype orbitals (STOs). As is wellknown, any STO displaced from the origin can be expanded in an infinite series of spherical harmonics; the functional coefficients have been designated as Lowdin alpha functions. These alpha functions can be represented as exponentials multiplied by polynomials in the displacement distance and the radial distance. The polynomials are used to construct a C matrix with integer elements. To avoid cancellation errors in some cases, the exponentials are expanded to obtain E matrices for interior regions and F matrices for exterior regions. We believe that this careful approach to molecular integrals will succeed in producing accurate and rapid evaluation of the integrals needed in STO basisset methods for quantum chemistry.
 Authors:
 Publication Date:
 Research Org.:
 Florida Agricultural and Mechanical Univ., Tallahassee, FL (USA). Dept. of Physics
 OSTI Identifier:
 7003034
 Alternate Identifier(s):
 OSTI ID: 7003034
 Report Number(s):
 ADA218065/1/XAB
 Resource Type:
 Technical Report
 Resource Relation:
 Other Information: Pub. in Jnl. of Molecular Structure (Theochem), Vol. 199, 233243(1989)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 74 ATOMIC AND MOLECULAR PHYSICS; MOLECULAR STRUCTURE; CALCULATION METHODS; ACCURACY; CANCELLATION; DISTANCE; ERRORS; EVALUATION; INTEGRALS; MATRICES; MOLECULAR ORBITAL METHOD; MOLECULES; NUMERICAL SOLUTION; POLYNOMIALS; SLATER METHOD; SPHERICAL HARMONICS; FUNCTIONS 640302*  Atomic, Molecular & Chemical Physics Atomic & Molecular Properties & Theory
Citation Formats
Jones, H.W., and Weatherford, C.A.. Loewdin alpha function and its application to the multicenter molecular integral problem over Slatertype orbitals. United States: N. p., 1989.
Web.
Jones, H.W., & Weatherford, C.A.. Loewdin alpha function and its application to the multicenter molecular integral problem over Slatertype orbitals. United States.
Jones, H.W., and Weatherford, C.A.. Sun .
"Loewdin alpha function and its application to the multicenter molecular integral problem over Slatertype orbitals". United States.
doi:.
@article{osti_7003034,
title = {Loewdin alpha function and its application to the multicenter molecular integral problem over Slatertype orbitals},
author = {Jones, H.W. and Weatherford, C.A.},
abstractNote = {In this paper we trace the evolution of the Lowdin alphafunction method in its application to multicenter molecular integrals over Slatertype orbitals (STOs). As is wellknown, any STO displaced from the origin can be expanded in an infinite series of spherical harmonics; the functional coefficients have been designated as Lowdin alpha functions. These alpha functions can be represented as exponentials multiplied by polynomials in the displacement distance and the radial distance. The polynomials are used to construct a C matrix with integer elements. To avoid cancellation errors in some cases, the exponentials are expanded to obtain E matrices for interior regions and F matrices for exterior regions. We believe that this careful approach to molecular integrals will succeed in producing accurate and rapid evaluation of the integrals needed in STO basisset methods for quantum chemistry.},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Sun Jan 01 00:00:00 EST 1989},
month = {Sun Jan 01 00:00:00 EST 1989}
}

The Loewdin alpha functions, which are the functions associated with the spherical harmonic expansion of a displaced Slatertype orbital, are expressed using C matrices to represent the polynomials in terms of the displacement distance a and the radial distance r. These polynomials are multiplied by the sum and difference of exponentials. The expansion of the exponentials leads to the use of E and F matrices. By keeping only the r variable identifiable, further simplifications of the alpha functions are possible, which makes for easy programming of all multicenter integrals. Also, no singularities appear in these developments. Everything is demonstrated bymore »

E Matrix' for the Loewdin alpha function, expanded in a Taylor series: an analytic treatment of molecular charge density near the origin
A displaced STO can be expanded in spherical harmonics with the coefficient function or Loewdin or functions characterized by a C matrix. These or functions themselves may be expanded in a Taylor series that is characterized by its own E Matrix. This expansion is necessary for the representation of the or function by a power series and for its evaluation about the origin. As an application, the power series for the molecular charge density in the vicinity of the center of a model diatomic molecule. The analytic approach is general and yields excellent results. 
Numerical evaluation of twocenter integrals over Slater type orbitals
Slater Type Orbitals (STOs) which one of the types of exponential type orbitals (ETOs) are used usually as basis functions in the multicenter molecular integrals to better understand physical and chemical properties of matter. In this work, we develop algorithms for twocenter overlap and twocenter twoelectron hybrid and Coulomb integrals which are calculated with help of translation method for STOs and some auxiliary functions by V. Magnasco’s group. We use Mathematica programming language to produce algorithms for these calculations. Numerical results for some quantum numbers are presented in the tables. Consequently, we compare our obtained numerical results with the othermore »