Subject: DFT gradients in a Gaussian Basis

From: Ed Brothers <[email-masked]>
Date: Thu, 20 Jun 2002 17:50:26 -0400

Hi.
    I am trying to implement DFT gradients in a gaussian basis as per
Johnson, Gill, and Pople, J. Chem Phys, 98 (7) 1993.  I am having a
great deal of difficulty understanding some of the equations and
methodology.  Any help would be hugely appreciated.

    Specifically, I am having trouble with the equation for the
derivative of exchange/correlation energy with respect to nuclear
displacement (the last term in eq. 9 in the paper).  None of the dels
inside the integral are subscripted, which appears to mean that in
cartesian coordinates the gradient of the functional would be the same
in all  directions for any pair of basis functions.  Is this correct, or
am I misreading it?

    I also do not understand the X(mu,nu) matrix equation (eq. 10),
which shows up in the gradient equation as a multiplier to the nonlocal
portion of the functional, specifially what the gradient of the
transpose of the gradient of a basis function means.  Is it just the
second derivative of the basis function, or is the math more
complicated?

    Finally, I am having a bit of trouble with the statement, "The
atomic grids are defined such that an atoms grid 'moves with' a
discpacement of its nucleus, exactly as do the basis functions situated
on the atom. As a result, there is a contribution on A's grid to the
gradient with respect ot A from all basis functions except those
centered on A."  Does this mean that when both phi(mu) and Phi(nu) are
centered on A the gradient of the functional is zero or is there more to
it?

    I realize this sounds like I'm fairly in the dark, which I am.  :)
I understand the need for the weight derivatives, and how to use
translational invariance to get the gradient of a weight with respect to
the displacement of it's parent atom, but tI'm missing something quite
basic in the actual gradient of the functional.  Any help you can render
would be appreciated!

        Ed Brothers.
        Merz Group
        Penn State.