Subject: Symmetry and Stationary Points

From: [email-masked] (Dr. James Stewart)
Date: Thu, 5 May 94 09:28:21 PDT

Apropos the recent discussion on symmetry and stationary
points:

With the exception of Jahn-Teller systems, if the symmetry
of the wavefunction is the same as that of the nuclear
coordinates, then, for an appropriate set of coordinates,
there will always exist a stationary point.  That is easy
to prove from symmetry theory.

Jahn-Teller systems are special - for such systems there
cannot be a stationary point, and a distorting force will
exist which will lower the symmetry.  If a center of symmetry
exists, that will be preserved under the distortion.

Of more interest are systems for which the wavefunction does
NOT have the symmetry of the nuclear coordinates, particularly
in low-symmetry systems (to avoid confusion due to Jahn-Teller
considerations). 

An example of such a system is fluoroethylene, forced into a 
twisted Cs symmetry (the plane of the CH2 being perpendicular 
to that of the CHF).  If the symmetry of the wavefunction
is forced to be Cs, then the energy of the system will be much
higher than if the wavefunction is unconstrained.  Conventionally,
by a self-consistent field, we mean that the electronic energy
is an irreducible minimum.  Given this, it follows that the
forces due to the electronic structure at self-consistency
will not reflect Cs symmetry (Obviously they will be C1).
In that case, there cannot exist a stationary point in the
Cs symmetry.

This does not invalidate the Jahn-Teller effect.  It is only
a logical extension.  To see how, consider a Jahn-Teller system
such as singlet D2d ethylene.  Under the Jahn-Teller theorem,
a distortion of the wave-function from D2d symmetry lower the energy.  
The extension consists of simply making the minimal disturbance 
to the system so as to destroy the high symmetry, but not 
enough to affect the energetics too much.  In such a case 
(C2H3F being an example), the distorting force will still exist.

Jimmy Stewart

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From: [email-masked] (Bersuker)
Date: Thu, 5 May 94 13:57:53 CDT

Dear Dr. Stewart: It is somewhat strange to read your message about stationary  points and the Jahn-Teller effect, as if the problem has not been studied for many years by many authors and several books were not published. You may not be aware of all these publications and, apparently you are not alone; this is why I  refer you to the book of R.Englman and to my book with V.Polinger (Springer, N.Y., 1989) where this and many other related problems are considered in detail.   
Regards,  Isaac-- 
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Isaac B. Bersuker		 | E-mail: 
Dept. of Chemistry	         | [email-masked]
Univeristy of Texas at Austin    | Vox: (512) 471-4671
Austin, TX 78712 	         | Fax: (512) 471-8696
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