Evaluation of the alpha-function for large parameter values

Jones, Herbert W.; Jain, J. L.

doi:10.1002/(sici)1097-461x(1996)60:7<1257::aid-qua6>3.0.co;2-y

Cited by 14 publications

(4 citation statements)

References 3 publications

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“…Another general method for the derivation of addition theorems is the alpha function method, which was introduced by Per-Olof Löwdin in his seminal paper on the quantum theory of cohesive properties of solids [33], and which was later used and extended by numerous other authors [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50].…”

Section: Introductionmentioning

confidence: 99%

Addition theorems as three‐dimensional Taylor expansions. II.Bfunctions and other exponentially decaying functions

Weniger

2002

Int J of Quantum Chemistry

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Addition theorems can be constructed by doing three-dimensional Taylor expansions according to f (r + r ′ ) = exp(r ′ · ∇)f (r). Since, however, one is normally interested in addition theorems of irreducible spherical tensors, the application of the translation operator in its Cartesian form exp(x ′ ∂/∂x) exp(y ′ ∂/∂y) exp(z ′ ∂/∂z) would lead to enormous technical problems. A better alternative consists in using a series expansion for the translation operator exp(r ′ · ∇) involving powers of the Laplacian ∇ 2 and spherical tensor gradient operators Y (2000)]. The application of the translation operator in its spherical form is particularly simple in the case of B functions and leads to an addition theorem with a comparatively compact structure. Since other exponentially decaying functions like Slatertype functions, bound-state hydrogenic eigenfunctions, and other functions based on generalized Laguerre polynomials can be expressed by simple finite sums of B functions, the addition theorems for these functions can be written down immediately.

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Section: Introductionmentioning

confidence: 99%

Addition theorems as three‐dimensional Taylor expansions. II.Bfunctions and other exponentially decaying functions

Weniger

2002

Int J of Quantum Chemistry

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“…From a computational standpoint, u max can, as a first guess, be approximated by the positive root of either the lower or the upper bracketing functions (44). Refining such an extremum using a standard method for root finding is only costly when ran for the first time, i.e.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Later, Rashid [36] and Suzuki [46][47][48][49][50][51][52] have independently simplified Sharma's formulation making it more tractable for further mathematical analysis and for numerical experiments. In the meantime, starting with Sharma's expansion, Jones and co-workers [38][39][40][41][42][43][44] have led to another form of BCLFs which is well adapted to integer arithmetic widely available nowadays through many well-known systems, e.g. mathematica, maple and the like.…”

Section: General Propertiesmentioning

confidence: 99%

Addition theorem of Slater type orbitals: a numerical evaluation of Barnett–Coulson/Löwdin functions

Bouferguène¹

2005

J. Phys. A: Math. Gen.

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When using the one-centre two-range expansion method to evaluate multicentre integrals over Slater type orbitals (STOs), it may become necessary to compute numerical values of the corresponding Fourier coefficients, also known as Barnett-Coulson/Löwdin Functions (BCLFs) (Bouferguene and Jones 1998 J. Chem. Phys. 109 5718). To carry out this task, it is crucial to not only have a stable numerical procedure but also a fast algorithm. In previous work (Bouferguene and Rinaldi 1994 Int. J. Quantum Chem. 50 21), BCLFs were represented by a double integral which led to a numerically stable algorithm but this turned out to be disappointingly time consuming. The present work aims at exploring another path in which BCLFs are represented either by an infinite series involving modified Bessel functions K ν (√ a 2 + r 2) or by an integral whose integrand is a smooth function. Both of these representations have the advantage of being symmetrical with respect to the cusp parameter a and the radial variable r. As a consequence, it is no longer necessary to split the integrals over r ∈ [0, +∞) into several components with a different analytical form in each of these. A numerical study is also carried out to help select the most appropriate method to be used in practice.

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“…[15,16]). For the calculation of S nlλ,n l λ (ζ, ζ ; R) we use the Löwdin α radial function [10] in the following form [17][18][19][20][21][22]:…”

Section: Evaluation Of Overlap Integrals Over Nistosmentioning

confidence: 99%

Use of Guseinov's One-Center Expansion Formulae and Löwdin α Radial Function in Calculation of Two-Center Overlap Integrals over Slater Type Orbitals with Noninteger Principal Quantum Numbers

Мамедов

Çopuroğlu

2011

Acta Phys. Pol. A

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An efficient calculation algorithm is presented for two-center overlap integrals over noninteger n * Slater type orbitals in molecular coordinate system based on the use of Guseinov's one-center expansion formulae and Löwdin α radial function. These integrals are expressed in terms of overlap integrals of integer n Slater type orbitals. The analytical formulae offer the advantage of direct and efficient calculation of the two-center overlap integrals over noninteger n * Slater type orbitals without the use of numerical methods. Several numerical results obtained are presented to demonstrate the improvements in convergence rates.PACS: 31 15.-p, 31 15.ae

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Evaluation of the alpha-function for large parameter values

Cited by 14 publications

References 3 publications

Addition theorems as three‐dimensional Taylor expansions. II.Bfunctions and other exponentially decaying functions

Addition theorems as three‐dimensional Taylor expansions. II.Bfunctions and other exponentially decaying functions

Addition theorem of Slater type orbitals: a numerical evaluation of Barnett–Coulson/Löwdin functions

Use of Guseinov's One-Center Expansion Formulae and Löwdin α Radial Function in Calculation of Two-Center Overlap Integrals over Slater Type Orbitals with Noninteger Principal Quantum Numbers

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