Subject: Symmetry: reply to Peter Shenkin

From: "R.G.A. Bone" <[email-masked]>
Date: Thu, 28 Apr 94 11:38:28 -0700

Ok, I'll make this my last statement on this subject:

Peter Shenkin writes in reply to my latest...

>> It is worth restating, though, that there is no theorem which
>> allows you claim that a minimum-energy structure (in fact, any
>> stationary point) must have any point group symmetry at all, in
>> the absence of other information, e.g., a force law. (The
>> converse is open to serious discussion, nevertheless: - does a
>> structure of given symmetry have to be a stationary point?)
>
> Excuse me;  I'm not an expert on this sort of thing, so I may
> be missing something, but consider ethane in the staggered
> conformation.  This conformation is an energetic minimum, and
> has point-group symmetry.  If I simply stretch or compress the
> C-C bond, I do not change the point-group, but I am no longer
> at a minimum, nor at any other stationary point.
> 
> Doesn't this demonstrate that symmetric structures need not
> be stationary points?
> 
> 	-P.
> 

Yes, he is right, I was being a little sloppy; in my first posting on this
subject, I amplified the same point ...

>                                                     ... 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 distortion that Shenkin mentions is, of course, totally symmetric (A_1g in
D3d point group).   I guess I shouldn't have made the unqualified statement,
assuming that people would remember a detail of an earlier posting.

The remaining missing qualifier from the basic statement is that one assumes 
that one's "symmetric structure" has been energy-minimized.

Perhaps one can remove potential confusion by stating that: if one has a 
structure of given symmetry, then one can probably find a stationary point
by energy-minimization, constraining that symmetry.  The energy has changed
only w.r.t. totally-symmetric distortions of the initial structure.  This is
not the same as suggesting that symmetric structures are often stationary-
points, which I accept is rather a bold statement, given that any structure
can be subjected to an infinitesimal distortion.  The main point of practicality
is that the energy of a symmetric structure will be stationary w.r.t. non-
totally-symmetric distortions; in most instances a 'guessed' structure of a
given symmetry (particularly high symmetry) will be close to a stationary
point (requiring small changes only in the totally symmetric degrees of freedom
to minimize to it). And in that sense  one may loosely state that symmetric
structures commonly turn out to be stationary points.

As we can all see - symmetry is a complicated subject and one which is difficult
to write about.  Apologies if I clouded the issue.

Till the next 'virtual interaction' ...

Richard Bone

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R. G. A. Bone.
Molecular Research Institute,
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