From:	"R.G.A. Bone" <rgab (+ at +) purisima.molres.org>
Date:	Tue, 26 Apr 94 11:27:44 -0700
Subject:	symmetry in electronic structure computations

To add a few remarks to Theresa Windus's comments, which are basically
all correct.

It is true that symmetry is handled differently by different programs.

"Gaussian" finds symmetry present in the input nuclear coordinates and
applies it unless you explicitly switch it off.  But geometry optimizations
will typically, irritatingly, quit as soon as a change in point group occurs.

It is actually easier to deal with symmetry in programs like "cadpac" where
you just input the symmetry-unique atoms and/or specify internal coordinates
which are constrained to be related to one another by symmetry.

The gradient of the energy w.r.t. nuclear displacements should transform as
the totally symmetric 'irrep' in whatever point group you happen to be in.
For 'closed-shell' systems (no electronic degeneracies) this means in practice
that 'symmetric' structures are usually stationary points.  The only distortions
to which the energy gradient could be non-zero are totally symmetric.  Thus one
should be cautious with, say, van der Waals clusters, in which a totally-
symmetric distortion which corresponds to dissociation might be favourable if
the high-symmetry structure contains repulsions.

The matter in 'open-shell' systems is more complicated.  Imposing high symmetry
on a Jahn-Teller system, for example, will obviously lead to problems.
In other systems in which the electronic state does not transform as the totally
symmetric irrep, one should be careful.

Finally, also in open-shell systems, the use of symmetry in UHF-type
calculations contains many 'hidden' pitfalls, most of which are documented.
It is critical to examine the wavefunction of whatever state you converge to.
One can carry out the calculation in a symmetric nuclear configuration both
with and without the constraining the symmetry of the wavefunction.  This
can lead to massive energetic differences, and totally different behaviour
w.r.t. spin-contamination, etc. All results are sensitive to starting guess,
etc., and the "solution" can change during the course of an optimization.
Advice: examine your *whole* computer output very thoroughly!
 
The belief  that 'transition states' are high-symmetry species is all-but
totally dispelled at this time.  Basically, for minima, anything goes, but
there is no guarantee that "nature" favours symmetric structures over their
lower-symmetry counterparts.  (One only has to look at van der Waals molecules
to see evidence of that.)  For transition states, all that the supposedly-useful
symmetry theorems tell you is an upper-limit on the point group symmetry but
even that is almost always a lower symmetry than either of the pertinant minima.
In general, surfaces are sufficiently complicated that transition states end
up having little symmetry at all.  If you somehow converged to one during
a _minimization_ then you either made a shrewd guess or were very lucky.

Finally, there is an unfortunate tendency, which is widespread, to refer to
stationary points at which more than one Hessian eigenvalue is negative as
"higher _order_ saddles". This nomenclature is incorrect.  The 'order' of
a stationary point specifies the lowest non-vanishing term in a locally-
expanded Taylor series.  Thus almost all stationary points that we, as
"chemists" meet, are second-order points, because they have non-vanishing
second derivatives.  Third order points have zero-second derivatives, and
are characterised by cubic terms in the potential, e.g., "monkey-saddles".
As has been discussed many times, these points are extremely rare and, to
my knowledge, a 'real' one has yet to be conclusively identified on a molecular
PES.  The characterisation of second-order stationary points is achieved by
stating the Hessian "index" - the number of negative eigenvalues.
Thus a transition state has a Hessian index of 1.  Structures with indexes>1
are 'maxima' in a subspace.

The term "rank" also has a distinct meaning - it is the number of non-zero
Hessian eigenvalues - and rarely finds application in chemistry.
All these terms are described more fully in P.G.Mezey's book, "Potential
Energy Hypersurfaces", Elsevier.

This point might seem to be pedantic, but there are a number of terms to be
used, and each has a specific and DISTINCT meaning: 'order', 'index', 'rank'
and also 'signature'.  (For the meaning of the last of these, see R.F.W. Bader,
"The Theory of Atoms in Molecules", OUP.)

Richard Bone

================================================================================

R. G. A. Bone.
Molecular Research Institute,
845 Page Mill Road,
Palo Alto,
CA 94304-1011,
U.S.A.

Tel. +1 (415) 424 9924 x110
FAX  +1 (415) 424 9501

E-mail  rgab -AatT- purisima.molres.org



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