Manuscripts - Further Links

Well, what's been happening in the world of molecular integrals since my researches wound down in 1976? Quite a lot, it would appear...

A great many papers about the Barnett-Coulson / Lowdin expansions and their application to multi-centre integrals over Slater-type atomic orbitals have in fact appeared since that time. Whether everybody has been reinventing the wheel I cannot say; perhaps genuinely new and improved formulations have been found, but my instinct (as stated earlier) is that efficient structuring of the computer implementation, plus effective use of symmetry, plus parallel processing, are probably of equal or greater importance.

Being no longer affiliated to an academic organisation, I can only establish the existence of relevant research papers by means of (say) Google in the first instance. Their titles and abstracts are accessible this way, but not their detailed content. For that I must either pay quite a substantial online viewing fee, or else visit the local University library, which doesn't really carry any hard-copy journals in this specialist area. So to be honest, I'm not really much the wiser at the moment - but I'd be very interested to receive feedback from any of you out there who are actively researching this area.

It should be mentioned at this point that the Barnett-Coulson expansion coefficients were traditionally called **zeta** functions, whereas in the Lowdin formalism they are referred to as **alpha** functions. Clearly the two must be essentially equivalent, but the latter terminology now seems to be the norm.

The names of A Bouferguene, H W Jones, R R Sharma and N Suzuki seem to be particularly prominent. I have itemised some of their publications below, and in the case of Suzuki whose abstracts are particularly detailed I have included these as well, to give a flavour of his investigations.

(Disclaimer: In some of the following mathematical expressions, the builder of this web page has been uncertain which symbol correctly renders certain characters and has tentatively replaced them with a hash symbol (#). Readers are urged to study the original abstracts directly from the publishers. If you can ascertain what the characters should be, please send an email and this page will be updated accordingly.)

Bouferguene A, Jones H W,

J. Chem. Phys. 109, pp 5718-5729, 1998

"Convergence analysis of the addition theorem of Slater orbitals and its application to three-center nuclear attraction integrals"

Bouferguene A,

J Physics A, Vol. 38, No. 13, pp 2899-2916, 2005

"Addition theorem of Slater type orbitals: a numerical evaluation of Barnett-Coulson / Lowdin functions"

Bouferguene A, Safouhi H,

J Physics A, Vol. 39, No. 3, pp 499-511, 2006

"A complexity analysis of the Gauss-Bessel quadrature as applied to the evaluation of multi-centre integrals over STFs"

Bouferguene A, Safouhi H,

Int J Quant Chem, Vol. 106, Issue 11, pp 2398-2407, 2006

"Gauss-Bessel Quadrature: A tool for the evaluation of Barnett-Coulson / Lowdin functions"

Duret S, Bouferguene A, Safouhi H,

J Comput Chem, 29(6), pp 934-944, 2008

"Strategies for an efficient implementation of the Gauss-Bessel quadrature for evaluation of multicenter integrals over STFs"

Jones H,

Int J Quant Chem, Vol. 41, pp 749-754, 1992

"Lowdin Alpha Function, Overlap Integral, and Computer Algebra"

Jones H, Weatherford C,

J Mol Struct, Vol. 199, pp 233-243, 1989

"The Lowdin alpha function and its application to the multi-center molecular integral problem over Slater-type orbitals"

Macdonald, J.R., Dolding, R.M.

Molecular Physics, Vol. 31, Number 1, pp 289-293, 1976

"Analytic expressions for multi-centre integrals using Slater-type orbitals"

J. Math. Phys. 25, 1133 (1984); doi:10.1063/1.526256 (

(Received 27 July 1983; accepted 30 September 1983)

A function of the type *f(R)Y ^{M}_{L}*(Θ,Φ) around a specific center

J. Math. Phys. 26, 3193 (1985); doi:10.1063/1.526648 (

(Received 3 April, 1985; accepted 25 June 1985)

This paper subsequent to the one [J. Math. Phys. 25, 1133 (1984)] (referred to as Part I) presents the following new results: It is found out that for *M=L* and *L-1* the coefficients *b _{K k}*(

J. Math. Phys. 31, 2314 (1990); doi:10.1063/1.528639 (

(Received 30 October 1989; accepted 9 May 1990)

When *r/a≊0*, Löwdin's α function (1/*r)α _{#}(fLM ‖ a,r)* is expressed as (1/

The closed form of the coefficients *h _{n,2n-i}*(

Introducing into the expression for #* _{n,2n-i}*(

J. Math. Phys. 33, 4288 (1992); doi:10.1063/1.529831 (

(Received 17 December 1991; accepted 10 July 1992)

By using the explicit expression for the coefficients *b _{K k}*(

Perhaps we can finish where we began, with the maestro himself - several of his recent publications, such as this one, seem to have revisited the topic of molecular integrals and his eponymous expansion:

Journal of Symbolic Computation

Volume 42, Issue 3, March 2007, Pages 265-289

Abstract

I have used MATHEMATICA to solve several problems that relate to the symbolic calculation of the 'molecular integrals' that are a mainstay of computational chemistry.

This work has provided many new results of chemical and mathematical interest, and it has led to a powerful programming methodology that I call MATHSCAPE that uses a novel open ended set of macros. Some further work on molecular integrals is presented here, largely as an introduction to MATHSCAPE. I discuss (1) the immediate mathematical problem of the '*J*' integrals, (2) the key features of MATHSCAPE, (3) a novel reduction of the *J* integrals, (4) the chemical context of this work, and (5) the computer science context.

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