Translational invariance in DFT derivative calculations

[JOHNSON' at \`CMCHEM.CHEM.CMU.EDU (Benny Johnson)]

 Katrina Werpetinski writes:
 > I've got another problem with which I'm hoping someone can help. We're
 > using an Xalpha LDF code to do geometry optimizations.  The gradient
 > calculations wrt Cartesian coordinates are not conserving linear and
 > angular momentum; that is, there are net translational and rotational
 > energies.  This, of course, makes it impossible to find gradients wrt
 > internal coordinates.  Has anyone come up against this problem before
 > and fixed it?
 One possible source of loss of translational invariance in DFT gradients
 is in the way the numerical quadrature weights are handled.  Most DFT codes
 use atomic weighting schemes to partition the molecular XC integrals
 into single-center contributions (the most popular scheme is that of Becke,
 JCP 88, 2547 (1988)).  These weights depend on the nuclear configuration,
 and hence have derivatives with respect to the nuclear coordinates.  Strictly
 speaking, in order to be consistent with the implementation of the XC energy
 (as a numerical sum over points and weights, not an analytic integral), when
 taking gradients one must include the terms involving the weight derivatives.
 However, none of the currently available DFT packages of which I am aware
 include these.
 Translational invariance is strictly not obeyed unless the weight derivatives
 are included.  The magnitude of the loss of translational invariance depends
 on the "completeness" of the quadrature grid used.  I am doing some
 coding of
 DFT methods as part of my Ph.D. work, and have looked at the effect of the
 weight derivatives.  For fairly large grids (say, O(10000) points per atom)
 the effect is insignificant for practical purposes.  For small grids, though
 (O(1000) points per atom), the errors in the nuclear forces from leaving out
 the weight derivatives can be significant (around 10**-4 - 10**-3 a.u.) unless
 the point distribution is very well-chosen.  Some optimizers can indeed get
 confused by this inconsistency.  The effect is more severe for vibrational
 frequencies, where both first and second derivatives of the weights contribute.
 Once the weight derivative terms are put in, however, e.g. the related
 optimization difficulties go away, regardless of the size of the grid, since
 one is now exactly differentiating what has been defined as the energy.  We
 routinely include the weight derivatives in our optimization and frequency
 calculations.
 As far as the loss of rotational invariance is concerned, that is a tougher
 problem, and one which I have not yet considered in detail myself.  However,
 as I implied earlier, in the case of geometry optimization the TI problem is
 more significant.
 Benny Johnson
 Carnegie Mellon University