Subject: Field gradient integrals

From: Frank Neese <[email-masked]>
Date: Thu, 13 Jan 2000 13:17:53 +0100

Hi all,
I have recently had a little trouble with the implementation of field
gradient integrals over GTOs, i.e. <a|v(i,j)|b> where a and b are GTOs
centered at points A and B respectively and v(i,j) is a component of the
field gradient operator located at center C (= d/dxC d/dyC 1/rC for the x,y
component etc). I have used the McMurchie-Davidson algorithm as described
in the original article or by Helgaker and Taylor in their extensive
review. I noticed that if all centers coincide the results are correct.
However, if a and b are located somewhere else and the product a(r)b(r) has
a significant amplitude at point C, where the operator is located, the
results are incorrect. For example, two unnormalized 1s GTOs (exponent=1.0)
located at +1 bohr and the v(z,z) operator at the origin. In general in
these cases the algorithm returns <a|v(x,x)+v(y,y)+v(z,z)|b> <> 0 but it
must =0  for the correct integrals. Judging from tests with numerical
integration of the same integrals the algorithm appears to produce correct
integrals in the case that the amplitude of a*b at point C is very small. I
know from other integrals that the F[m] generation, E-coefficients,
R-coefficients etc. are all definitively correct and this makes me believe
that I have a conceptional problem with the singularity of the operator at
the origin and not just with the numerics.

I hope that this is a standard problem for anyone who has succesfully
implemented these integrals (and related ones like the one electron part of
the spin-orbit operator) and I would greatly appreciate if some experienced
persons could give me a hint about how to deal with it so I don't have to
reinvent the wheel. 

Your help is much appreciated,
best regards,

Frank
    -----------------------------------------
    ! Dr. Frank Neese                       !
    ! Fakultaet fuer Biologie               !
    ! Universitaet Konstanz                 ! 
    ! D-78457 Konstanz                      !
    ! Germany                               !
    ! e-mail: [email-masked]   !
    ! Tel   : 07531/883205                  !
    ! FAX   : 07531/882966                  !
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