Re: CCL:G:Damped Coulomb Potential Summary
Fernando,
Explicit calculation of matrix element of the 'dumped Coulomb potential'
(error function/R) with gaussians is in the paper of G. Lippert, J. Hutter
and M. Parrinello " A hybrid Gaussian and plain wave denisty functional
scheme", Mol Phys 92 (1997) pp 477-487.
They have error function in the local part of the pseudopotential. On page
481 they write: "
erf(a|r-R|)
V(r) = -------------
|r-R|
can be regarded as the Coulomb potential originating from a Gaussian
charge centered at R".
Formula (34) immediately gives the answer.
Ivan Oleinik
------------------------------------------------------------------------
Ivan I. Oleinik E-mail : ivan.oleinik : at : materials.ox.ac.uk
Department of Materials
University of Oxford
Parks Road
Oxford OX1 3PH Tel : +44 (0)1865 283325
United Kingdom Fax : +44 (0)1865 273764
------------------------------------------------------------------------
> ------------------------------------------------------------------------
> > On Wed, 10 Mar 1999, Fernando Vila wrote:
> Dear Fellows;
>
> I would like to thank all the people that replied to me request. Right
> after I sent my e-mail to the list I decided to give it a new try. At the
> very beginning of my equations I found an error that was complicating
> everything. After I fixed that the integrals turned out to be
> embarrassingly easy, easier than those including only the Coulomb
> potential. The references you sent will be useful to double check my
> results.
>
> My original question was:
>
> Dear Fellows,
>
> For the past weeks I have been trying to calculate the matrix elements
> for a damped Coulomb potential to no avail. The potential has the form:
>
> erf(a|r-R|)
> V(r) = -------------
> |r-R|
>
> where r and R are position vectors and erf(t) is the error function:
>
> 2 t
> erf(t) = -------- int exp(-x^2) dx.
> PI^(1/2) 0
>
> The matrix elements then are of the form:
>
> erf(a|r-R|) gij
> <gi|V|gj> = int gi V(r) gj dr = int ----------------- dr
> |r-R|
>
> Here gi are spherical Gaussians and gij is the resulting Gaussian from the
> Gaussian Product Theorem.
>
> I have tried using techniques similar to those used for the calculation of
> the pure Coulomb potential (based on Fourier or Laplace transforms) but
> the usual cancellation is not complete.
>
> Any pointers to papers, books or web pages are welcome.. The complete
> solution is also very much welcome.. :-)
>
> Thanks to all in advance, Fer.
>
> Here is a summary of the replies I received:
>
> ==========================================================================
> From: koelmel <koelmel : at : inetworld.net>
>
> Hi,
>
> Lecture Notes in Chemistry, vol. 48, authored by C. Pisani, R. Dovesi
> and C. Roetti, titled
> "Hartree-Fock Ab Initio Treatment of Crystalline Systems"
> (1988, springer)
>
> may be a useful reference. What you're looking for is mentioned
> on page 60.
>
> One way is to convert 1/r into an integral using the Laplace transform
> trick, then change the order of integration, I think.
>
> Hope this helps somewhat
>
> Cheers,
> Christoph
>
> ==========================================================================
> From: Frank Jensen <frj : at : dou.dk>
>
> Try looking up some of Peter Gill's work in
> recent years. They have played with screened
> Coulomb potentials.
> Frank
>
>
> ==========================================================================
> From: Norge Cruz Hernandez <norge : at : helios.us.es>
>
> Hi ,
>
> Try to see :
>
> S. Obara and A. Saika.
> J. Chem. Phys. 84(7), 3963
>
> best regard
> Norge
>
> ==========================================================================
> From: "Jeremy P. Dombroski" <jack : at :
jubilee.q-chem.com>
>
> Dear Fernando:
>
> This is work that myself, Professor Peter Gill and Dr Ross Adamson have
> spent considerable time on (in particular, the latter). We have recently
> published an article in J. Comp. Chem. and if you contact Ross at
> ross : at : theor.ch.cam.ac.uk he may provide you with a preprint should
the
> article not yet be in print.
>
> Q-Chem 1.2 performs O(N) damped Coulomb SCF calculations, including
> energies, optimizations and frequencies using what we have called the
> CASE approximation where we replace all occurrences of the Coulomb
> operator with the so-called Damped Coulomb potential.
>
> May I ask where your research interest lies in this damped Coulomb
> potential?
>
> Regards,
> Jeremy Dombroski
>
>
> ==========================================================================
> From: "Jan H. Jensen" <jhjensen : at : blue.weeg.uiowa.edu>
>
> Hello,
> I believe Peter Gill at Cambridge has worked on something quite
> similar. I don't have any references on hand.
>
> Best regards,
>
> Jan Jensen
>
>
*******************************************************************************
> Fernando D. Vila
> Department of Chemistry Voice (412)624-8691
> University of Pittsburgh Fax (412)624-8552
> Box 90 Chevron Science Center E-mail fer : at :
theory6.chem.pitt.edu
> Pittsburgh, PA 15260, USA WWW http://www.pitt.edu/~fer
>
*******************************************************************************
>
>
>
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