Subject: Symmetry again

From: "Sergei Vyboishchikov" <[email-masked]>
Date: 28 Apr 94 13:43:46 MDT

Dear Netters,

I have some comments concerning the discussion about symmetry.

R.G.A.Bone wrote:

>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.

E.Lewars wrote:

>Does anybody know of a formal theorem that says non-C1 structures are
>always stationary points?

Yes, I do. There is the rigorous theorem that states that any symmetrical
structure is a stationary point with respect to NOT totally symmetric nuc-
lear displacements. I know some references to this statement but only in
Russian textbooks (sorry). This is a purely geometrical statement and does
not depend upon the symmetry of electronic state and whether or not the
system has closed shells only. The molecules with Jahn-Teller effect are
not exclusions. Their symmetric structures are also stationary points, but
not minima. But the term "stationary point" is defined as a point where all
partial derivatives (consequently, gradient) is equal to zero. It need not
be a maximum or a minimum, but can be a saddle point.

If one optimizes geometry within symmetry constraints, it guarantees that
the found stricture is a minimum with respect to totally symmetric nuclear
displacements and hence a stationary point with respect to all possible
displacements.

Note that one starts optimizing a structure from a symmetrical geometry, it
is impossible to come to a structure with a lower symmetry, no matter this
symmetry is imposed explicitly (using variables with the same names in
Z-matrix) or simply actually present in the starting geometry. This is due
to the fact that "the gradient of the energy w.r.t. nuclear displacements
should transform" as R.Bone writes. (however, not "should transform" but
"does transform"). The error messages such as GAUSSIAN's "change of point
group or standard orientation" arises either from numerical errors in
transforming Cartesian coordinates into internal or from "false" Z-matrix
(in which some parameters are denoted by the same variables though they are
not symmetrically equivalent).

                            Sergei F.Vyboishchikov
                            Universitaet Marburg, Germany
                            E-Mail: [email-masked]