Symmetry again

["Sergei Vyboishchikov" <SERGEI-: at :-ps1515.chemie.uni-marburg.de>]

 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: sergei-: at :-ps1515.chemie.uni-marburg.de