Finite Field PT in Gaussian

   I don't send questions to the CCL very often, but it looks like this
 week I'm going to send a couple.. And this one is looong.. :-)
   I have been trying to calculate the dipole-quadrupole (A) and
 quadrupole-quadrupole (C) polarizabilities of water by means of Finite
 Field Perturbation Theory (FFPT) using Gaussian 98 Rev A.7. I have
 previous experience with this method, I've used it to calculate dipole
 moments and dipole polarizabilities. Now I've run into some trouble
 because I need to transform the Cartesian-based properties that Gaussian
 generates into the traceless ones. Right now I'm pretty confused, so I'll
 try to explain the problem as clearly as I can.
 I think the main question is: what is the expression of the perturbation
 Hamiltonian used by Gaussian??
 I have found that when a dipole perturbation is applied (lets say, in the
 z direction), the perturbed energy can be written as:
 E = E0 + Mz * Vz
 where E0 is the unperturbed energy, Mz is the Cartesian z component of the
 dipole moment and Vz is the magnitude of the perturbation. I have checked
 and the Mz moment is identical to the one reported at the end by Gaussian.
 This is reasonable, the problem comes when you check that for a quadrupole
 perturbation the SAME formula is valid. Now, if the standard Cartesian
 perturbation Hamiltonian was used, the formula should be
 E = E0 + 1/2 * Mzz * Vzz
 I did the same for the octupole and the hexadecapole and the apparent
 Hamiltonian used by Gaussian is
 H' = Mi * Vi + Mij * Vij + Mijk * Vijk + Mijkl * Vijkl
 where the repeated indices are summed. This series is missing the 1/n!
 coefficients in front of each term.
 I tried checking if these coefficients are folded into the Cartesian
 moments, but this is not the case: when the Cartesian moments are used to
 calculate the traceless moments, the results are exactly correct, so no
 coefficients are included (the moments are simple the Cartesian
 Since I don't know what perturbation is being used, I can't derive the
 transformation relation that would take the Cartesian components of, say,
 the dipole-quadrupole polarizability and give me the ones associated with
 the traceless operators.
 What is really frustrating is that the Gaussian manual is extremely terse
 in the explanation of what is being done. Moreover, I've found that the
 warning in the manual regarding the sign ("be careful of the choice of
 sign convention when interpreting the results") is quite misleading:
 according to usual standards, what Gaussian is using are not electric
 fields but potential gradients.
 Let me say, finally, that I have actually gone into the code to see if
 there were any more explanatory comments, without success..
 Hope anybody can help, thanks in advance.. Fer.
 Fernando D. Vila                Voice    (206)616-3207
 Department of Chemistry         Fax      (206)685-8665
 University of Washington        E-mail   fdv;at;
 Seattle, WA 98195, USA          WWW