| There have been several posts
recently with questions and proposed solutions to the problem of "too many
negative frequencies" for transition-state (TS) structures. First, at the risk of
sounding needlessly didactic, the frequencies are not "negative," they are
imaginary. It is the computed force constant that is negative. In the quantum
mechanical harmonic oscillator (QMHO) approximation, the frequency is related to
the square root of the force constant, and hence it is imaginary. It is a
historical curiosity, probably relating to the difference between FORTRAN
floating point and character variables, that most codes print the specific
frequencies as negative numbers rather than appending Euler's "i" after the
magnitude.
On a more substantial front, several posters have offered suggestions for
removing the "minor" imaginary frequency by perturbing geometries along the
predicted normal mode, reoptimizing, generally jiggling structures, etc. Such
procedures can indeed be effective, but only if the imaginary frequency is
really there... Left unaddressed (at least in the most recent iteration of this
thread -- I have vague memories that others may have posted to CCL on this point
before) is the possibility that the unwanted imaginary frequency is an artifact
of the quadrature grid used in a DFT calculation (recent posters have not
actually specified their level of electronic structure theory, but given the
prevalence of DFT in modern calculations, one suspects this was indeed their
choice).
Most (if not all) modern DFT functionals do NOT permit an analytic
evaluation of the necessary volume integrals of the exchange-correlation
potential. Instead, the integrals are solved via a quadrature procedure over a
3-dimensional grid. The so-called analytic derivatives and second derivatives
are thus NOT analytic derivatives of the correct integrals, they are analytic
derivatives of the quadrature schemes for these integrals. The accuracy of any
quadrature approach depends on the density of the grid points, and, of course,
the cost of the integral evaluation also goes up with the number of grid points.
So, codes like Gaussian, Jaguar, ADF, NWChem, ORCA, etc. (sorry if I left out
your favorite) have come up with default integration grids that represent good
compromises for speed and accuracy. However, the demands on grid size increase
as one becomes interested in not just the value of an integral, but also in the
value of its first and second derivatives with respect to atomic positions.
Thus, it is not at all uncommon with default grids to find a geometry that seems
by all accounts to be, say, a local minimum, but gives a small imaginary
frequency even though it has no symmetry and all attempts to rotate methyl
groups (for instance) fail to eliminate the problem. TS structures can certainly
suffer from exactly the same phenomenon. The problem is that the force constant
is not being computed accurately enough by the quadrature scheme, NOT that there
really is a negative curvature on the potential energy surface. So, what can one do?
Some codes permit one to choose a finer quadrature grid, and this often does
solve the problem. Of course, one can then worry about whether one should go
back and recompute one's full set of stationary points with this finer grid
(it's only computer time...) but at least one knows that the issue is not a
chemical one. Another option is to compute the frequencies by finite difference
of the first derivatives (probably even MORE expensive and not really an ideal
option). Lastly, one can boldly rely on one's chemical intuition to know when
one is being plagued by this problem and attempt to sleep well while blithely
ignoring the issue (after all, the mode in question will have a "true" frequency
that will contribute negligibly to zero-point vibrational energy, and negligibly
to enthalpy, and will be so small that the correct value should not be used on
the QMHO approximation for entropy in any case). For those wishing for a
more complete discussion of these issues, I believe that Fritz Schaefer and
co-workers recently published a paper or two comparing the convergence of
default quadrature grids for various properties in various codes, but I'm too
lazy to do the literature search for these inquiring individuals. Happy
hunting. Chris
Cramer -- Christopher J. Cramer University of Minnesota Department of Chemistry 207 Pleasant St. SE Minneapolis, MN 55455-0431 -------------------------- Phone: (612) 624-0859 || FAX: (612) 626-2006 Mobile: (952) 297-2575 http://pollux.chem.umn.edu/~cramer (website includes information about the textbook "Essentials of Computational Chemistry: Theories and Models, 2nd Edition") |