Hyperspherical Harmonics and Generalized Sturmians

Avery, John

doi:10.1007/0-306-46944-8

Cited by 71 publications

(134 citation statements)

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“…As mentioned above, the Coulomb Sturmians satisfy a partial differential equation [6,7] r 2 r r S ðmÞ ðnlÞ ð;r rÞ ¼ 2 À 2n r ! S ðmÞ ðnlÞ ð;r rÞ:…”

Section: Solution Of Poisson's Equation Using Spectral Formsmentioning

confidence: 98%

Solution of Poisson's equation using spectral forms

Weatherford¹,

Red²,

Hoggan³

2005

Molecular Physics

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A new technique is presented for the solution of Poisson's equation in spherical coordinates. The method employs an expansion of the solution in a new set of functions defined herein for the first time, called 'spectral forms'. The spectral forms have spherical harmonics as their angular part, but use a new set of radial functions that automatically statisfy the boundary conditions, up to a multiplicative constant, on the Poisson solution. The resultant problem reduces to a set of simultaneous equations AC C ¼B B for the expansion coefficientsC C. The matrix A is block diagonal in the spherical harmonic indices l; m and is independent of any parameters. The simultaneous equations may be solved by LU decomposition. The LU decomposition only needs to be done once and multiple right hand sides (B-vectors) can be treated by a matrix-vector multiply. For a parallel computing platform, each such B-vector may be dealt with on a separate processor. Thus the algorithm is highly parallel. This technique may be used to calculate Coulomb energy integrals efficiently on a parallel computer.

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“…As mentioned above, the Coulomb Sturmians satisfy a partial differential equation [6,7] r 2 r r S ðmÞ ðnlÞ ð;r rÞ ¼ 2 À 2n r ! S ðmÞ ðnlÞ ð;r rÞ:…”

Section: Solution Of Poisson's Equation Using Spectral Formsmentioning

confidence: 98%

Solution of Poisson's equation using spectral forms

Weatherford¹,

Red²,

Hoggan³

2005

Molecular Physics

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“…The angular factor is just a spherical harmonic of order l. These functions are ortho-normal. The optimal values of the β parameters may be determined analytically by setting up secular equations which make use of the fact that the Sturmian eigenfunctions also orthogonalise the nuclear attraction potential, as developed by Avery (Avery, 2000).…”

Section: Basis Setsmentioning

confidence: 99%

“…This is complete without continuum states, satisfies the cusp condition as required and orthogonalises the Coulomb attraction. Avery has shown that the latter property can be used to optimize the exponents for single or multiple-zeta basis sets (Avery, 2000). Furthermore, before CS software is completed, it is possible to express any CS as linear combinations of STO and use STOP.…”

Section: Introductionmentioning

confidence: 99%

Suitable basis sets for accurate NMR chemical shift calculations: an application to in vivo studies of benzothiazole metabolites

Hoggan

2009

Interdiscip Sci Comput Life Sci

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This paper describes atomic orbitals which have direct physical interpretation, i.e. Coulomb Sturmians and hydrogen-like orbitals. The radial nodes are shown to be essential in obtaining accurate nuclear shielding tensors for NMR work. The method is applied ab initio to chemical shifts for (15) N NMR, which can also be measured. This procedure is shown to be a useful tool in structure determination for benzothiazoles, some of which are used as pesticides and represent a challenging use of biodegradation. The measurements may be made in biological media, for which a rudimentary simulation is provided by including a few discrete solvent molecules.

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“…Coulomb Sturmians have been used to treat problems involving many centres and many electrons [12,13]; pilot calculations were performed for N-electron molecules using the generalised Sturmian method. Generalised Sturmian basis functions, their connection to hyperspherical harmonics and their application to solve many-electron problems directly, without the use of the self-consistentfield approximation, are explored elsewhere [14].…”

Section: Introductionmentioning

confidence: 99%

Derivation, properties and application of Coulomb Sturmians defined in spheroidal coordinates

et al. 2015

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Coulomb Sturmian amplitude functions are derived in prolate spheroidal coordinates and are presented in a closed algebraic form. Spheroidal Sturnian functions are revealed to be related to the polynomial solutions of Heun's confluent equation. A reduction of symmetry from spherical to axial leads to the coupling of spherical polar orbitals and the formation of hybrid orbitals. The contribution of each spherical orbital into a hybrid orbital depends strongly on distance R from a nucleus to the dummy centre, and substantially alters when R varies. At two limiting cases R = 0 and R → ∞ spheroidal Sturmians are purely atomic orbitals, whereas at intermediate R they contain many features intrinsic to diatomic molecular orbitals. Applications of spheroidal Sturmian basis are discussed; Coulomb spheroidal Sturmians are asserted to be the most appropriate basis functions for diatomic molecular calculations.

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Hyperspherical Harmonics and Generalized Sturmians

Cited by 71 publications

References 0 publications

Solution of Poisson's equation using spectral forms

Solution of Poisson's equation using spectral forms

Suitable basis sets for accurate NMR chemical shift calculations: an application to in vivo studies of benzothiazole metabolites

Derivation, properties and application of Coulomb Sturmians defined in spheroidal coordinates

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