Subject: Finite Fields in Gaussian

From: "Fernando D. Vila" <fer(at)freyr.chem.washington.edu>
Date: Thu, 15 Apr 2004 15:21:50 -0700 (PDT)

Hello everybody..

I have pretty much exhausted all other alternatives, so here I'm asking 
the CCL community...

I have been using the Field keyword in Gaussian for many years, mostly to 
calculate multipole moments and dipole and quadrupole polarizabilities. 
Now I wanted to calculate some hyperpolarizabilities using the 
following formulas:

E    = E0 - mu0_i F_i - 1/2 alp_ij F_i F_j - 1/6 bet_ijk F_i F_j F_k

mu_i = m0_i + alp_ij F_j + 1/2 bet_ijk F_j F_k

where F_i is the perturbing electric field in the i direction, E is the
total energy including the perturbation, E0 is the energy without the 
perturbation, mu, alp and bet are the dipole, polarizability and 
hyperpolarizability. The summation over the i, j, k indices is implied.

To verify that the finite differences were giving me a small error I
decided to check by calculating NH3 at HF level (Gaussian produces
analytic values at this level). Here are the results:

Dipole=0.,0.,-0.7000795
Polar=9.4875931,0.,9.4875931,0.,0.,9.799189
HyperPolar=0.,-14.2904879,0.,14.2904879,-3.6541021,0.,-3.6541021,0.,0.,-19.6122974

If I now apply a finite electric field along the z axis and fit the 
results to a polymomial I get:

For the energy:
   -56.200911807600
    -0.700079533332 * F
    -4.899299996216 * F^2
     3.333331240659 * F^3
  -400.002401571933 * F^4

For the dipole moment:
      -0.700079500000
      -9.799183333333 * F
       9.783333333291 * F^2
    -466.666666569064 * F^3
   66666.666631576154 * F^4

If you compare the coefficients of this polymomials with the coefficients
in the formulas above you will see that there are many sign differences
with respect to the analytic results I showed above. This problem is not
new to me and the Gaussian manual very helpfully :-P informs us that "the
coefficients are those of the Cartesian operator matrices; be careful of
the choice of sign convention when interpreting the results".

Before I had always assumed that although Gaussian says that is adding an
electric field, what is actually doing is adding a potential gradient (
dV/di = V_i = -F_i ). This assumption always gave me correct results, but
now it is totally irreconciliable with the sign I get for the
hyperpolarizability.

So finally, my question: what is the CORRECT FORM of the perturbation 
introduced by the Gaussian keyword Field.

Cheers and I will really appreciate any comments (Doug Fox, are you 
there?? :-)

Fer.

"The biggest cause of trouble in the world today is that the stupid
people are so sure about things and the intelligent folks are so
full of doubts." Bertrand Russell
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Fernando D. Vila                Voice    (206)616-3207
Department of Chemistry         Fax      (206)685-8665
University of Washington        E-mail   fdv(at)u.washington.edu
Seattle, WA 98195, USA          WWW      http://faculty.washington.edu/fdv
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