Associated legendre functions on the cut

Olver, F. W. J.; Smith, John M.

doi:10.1016/0021-9991(83)90166-3

Cited by 26 publications

(12 citation statements)

References 15 publications

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“…The case in which # and x are constant and v runs over a sequence of values is more conveniently treated by an algorithm that will be given in a subsequent paper [9]. Pertinent formulas are supplied by the From the results of [9], it is known that the computation of successive values of P~(x) by means of (1.6) is numerically stable, provided that the sequence of values of # is descending. In other words, /3~(x) is a dominant solution of the difference equation in this direction, and therefore individual rounding errors grow at approximately the same rate as P?(x).…”

mentioning

confidence: 99%

Extended-Range Arithmetic and Normalized Legendre Polynomials

Smith

Olver

Lozier

1981

ACM Trans. Math. Softw.

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An algorithm is presented for the computation of normalrzed Legendre polynomials. In order to permit wide ranges of argument, degree, and order of these functions, an "extended-range" arithmetic is introduced whereby a separate storage location is allocated to the exponent of a floatmg-pomt number. Since this device may have other applications, separate subroutines are developed for addition of extended-range numbers and also for conversion to and from ordinary floating-point form.

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confidence: 99%

Extended-Range Arithmetic and Normalized Legendre Polynomials

Smith

Olver

Lozier

1981

ACM Trans. Math. Softw.

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“…The choice of this coefficient is purely conventional since it works in a tandem with the constants C (i) μ , i = 1,2, from Eq. (29). For instance, one can put C (1) μ equal to zero and then fix the value of c μ μ from the initial conditions.…”

Section: Calculation Of the L μ (α) Functions For Small Values Of αmentioning

confidence: 97%

“…in the double precision arithmetic (around 15 significant figures). New column shows results of calculations with Eqs (29). and(30), also in the double precision arithmetic.…”

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confidence: 98%

Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. II. Neumann expansion of the exchange integrals

Lesiuk

Moszyński

2014

Phys. Rev. E

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In this paper we consider the calculation of two-center exchange integrals over Slater-type orbitals (STOs). We apply the Neumann expansion of the Coulomb interaction potential and consider calculation of all basic quantities which appear in the resulting expression. Analytical closed-form equations for all auxiliary quantities have already been known but they suffer from large digital erosion when some of the parameters are large or small. We derive two differential equations which are obeyed by the most difficult basic integrals. Taking them as a starting point, useful series expansions for small parameter values or asymptotic expansions for large parameter values are systematically derived. The resulting expansions replace the corresponding analytical expressions when the latter introduce significant cancellations. Additionally, we reconsider numerical integration of some necessary quantities and present a new way to calculate the integrand with a controlled precision. All proposed methods are combined to lead to a general, stable algorithm. We perform extensive numerical tests of the introduced expressions to verify their validity and usefulness. Advances reported here provide methodology to compute two-electron exchange integrals over STOs for a broad range of the nonlinear parameters and large angular momenta.

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“…Calculation of the Legendre functions and their derivatives is a standard task as has been discussed many times in the literature [134][135][136]. Let us pass to the second part, i.e., the integral over [1, ∞].…”

Section: A Spin-averaged Potentials Iaaa Typementioning

confidence: 99%

Combining Slater-type orbitals and effective core potentials

2017

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We present a general methodology to evaluate matrix elements of the effective core potentials (ECPs) within one-electron basis set of Slater-type orbitals (STOs). The scheme is based on translation of individual STO distributions in the framework of Barnett-Coulson method. We discuss different types of integrals which naturally appear and reduce them to few basic quantities which can be calculated recursively or purely numerically. Additionally, we consider evaluation of the STOs matrix elements involving the core polarisation potentials (CPP) and effective spin-orbit potentials. Construction of the STOs basis sets designed specifically for use with ECPs is discussed and differences in comparison with all-electron basis sets are briefly summarised. We verify the validity of the present approach by calculating excitation energies, static dipole polarisabilities and valence orbital energies for the alkaline earth metals (Ca, Sr, Ba). Finally, we evaluate interaction energies, permanent dipole moments and ionisation energies for barium and strontium hydrides, and compare them with the best available experimental and theoretical data.

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Associated legendre functions on the cut

Cited by 26 publications

References 15 publications

Extended-Range Arithmetic and Normalized Legendre Polynomials

Extended-Range Arithmetic and Normalized Legendre Polynomials

Reexamination of the calculation of two-center, two-electron integrals over Slater-type orbitals. II. Neumann expansion of the exchange integrals

Combining Slater-type orbitals and effective core potentials

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