
From: 
Patrick Bultinck <Patrick.Bultinck #  at  # rug.ac.be> 
Date: 
Fri, 29 Apr 1994 10:36:52 +0200 (MET DST) 
Subject: 
Summary on symmetry (what a rhyme) 
Hello netters,
Seems I started something...
I am sending you this summary, since people have pointed out to me that
the subjct is possibly interesting for a whole lot of CCLsubscribers...
It seems to me that the discussion will remain open for quite some time,
since both pro and contra messages to my opinion have reached my address
or the CCl.
It seems to me that there are two more or less basic approaches :
 Do not use symmetry, and if you reach a minimum energy structure that
is only a little apart from a certain pointgroup, you may want to do a
symmetry calculation to see if the deviation of the symmetry is relevant
or not. (so you need a frequency analysis...)
 Use symmetry, and do a frequency analysis after you reached a
stationary point. If the F.A. prooves that you have reached a minimum,
cheer... Otherwise you need another calculation in which you lower the
symmetry, and repeat the above...
The prime matter you need to consider is (my opinion) what will be the
time for the calculation of the minimum, and if it is at all possible to
define (guess) a symmetry for the minimum. Off course the lower you take
the symmetry, the higher the time for the calculation, but the lower the
risk you will need to do the calculations for a lower symmetry group...
So in short (still my opinion after what I read, learned ...) :
Symmetry No Symmetry
CPU time depending high
on the start
Symmetry group
(the higher the
symmetry, the
higher the risk
of having to do
more calculations)
a good choice will
make substantial lower
ings of the CPU time
Integral eval. Relatively cheap Expensive
(also determines
CPU time,
off c(o)urse)
Number of Often higher than One
required one
F.A. calc.
It is not clear to me if one can a priori state for a certain molecule
(unless you have certitude of a certain symmetry point group, from
experiment) what will be the most CPUtime friendly way of calculating
stuff.
I hope this was, and is, a useful topic for the CCl, and would very much
appreciate if futher comments are sent to my Email address, if there
would be messages I find very useful, or if I would have received about
five more, I will send them to CCl as a sort of appendix to this summary.
Off course, you can continue the discussion on CCl, I would enjoy the
discussion, since it illustrates that there is still quite something left
to say on a routine matter like optimising geometries.....
THANKS EVERYBODY WHO REACTED, AND GAVE HIS COMMENT TO ME !!!!!!!
Original message :
Netters,
I have just sent a reply to a question concerning the use of a Zmatrix
for a calculation on ethylene, using GAMESS (US).
As a direct consequence I rembered something that I am not very
comfortable with...
In section 4 of the GAMESS manual, I find under a title : ***The role of
symmetry*** that it is best to use symmetry during an optimisation.
However GAMESS never lowers the symmetry during an optimisation. It is a
well known fact that sometimes a minimal energy structure differs (a lot)
from the symmetry you might expect. Isn't it then safer to always do
unconstrained optimisations, and if you think that the result differs so
little from a certain symmetry point group, you can always do a
symmetryoptimisation starting from the geometry with that symmetry, and
compare energies...
The most important disadvantage is off course that it often takes a lot
of time to reach a certain symmetry point group, when you start from
zerosymmery.
Any thoughts on this...
Those interested in a summary of some sort should send me a note, if
enough interest exists, I send one to CCL.
Thanks,

 CC Patrick Bultinck 
 / \ Dept. Physical & Inorganic Chemistry 
 CO OC Section Quantum Chemistry 
   University of Ghent 
 CO OC Krijgslaan 281 (S3) 
 \ / 9000 Gent 
 CC Belgium 
 Tel. Int'l code/32/9/264.44.44 
 Macrocycles Fax. Int'l code/32/9/264.49.83 
 Quantum Chemical Email : Patrick.Bultinck &$at$& rug.ac.be 
 Calculations 

Replies :
> Isn't it then safer to always do
> unconstrained optimisations, and if you think that the result differs so
> little from a certain symmetry point group, you can always do a
> symmetryoptimisation starting from the geometry with that symmetry, and
> compare energies...
If you start with a high symmetry geometry, this structure is always
a stationary point with respect to any symmetrybreaking directions.
No gradient can ever (except for numerical reasons, of course) lead
to a lowersymmetry structure. Therefore, if you are not sure whether
your choice of symmetry is correct for the desired minimum, you could
start the optimisation with a C1 geometry.
> The most important disadvantage is off course that it often takes a lot
> of time to reach a certain symmetry point group, when you start from
> zerosymmery.
Even worse, the calculation of the wave function needs much more time
in C1 than in any other point group. If the minimum has symmetry, you
waste a lot of time optimizing it in C1.
I prefer the optimisation in high symmetry with subsequent calculation
of harmonic frequencies. If one force constant gets negative, then I know
(more or less) precisely which symmetry reduction leads to the minimum.
If the minimum is "nearly symmetric", restarting the optimizer will be
relatively cheap.
Frequencies are, however, quite costly.
Just my thoughts.
Greetings,
Gernot
 Gernot Katzer
How does a system manager change a light bulb?  katzer %! at !%
bkfug.kfunigraz.ac.at
 katzer %! at !%
balu.kfunigraz.ac.at
He doesn't. He just denies access to everyone to 
the area served by the light bulb in question.  NEVER make me sysmgr!

>
> Netters,
>
> I have just sent a reply to a question concerning the use of a Zmatrix
> for a calculation on ethylene, using GAMESS (US).
>
> As a direct consequence I rembered something that I am not very
> comfortable with...
>
> In section 4 of the GAMESS manual, I find under a title : ***The role of
> symmetry*** that it is best to use symmetry during an optimisation.
> However GAMESS never lowers the symmetry during an optimisation. It is a
> well known fact that sometimes a minimal energy structure differs (a lot)
> from the symmetry you might expect. Isn't it then safer to always do
> unconstrained optimisations, and if you think that the result differs so
> little from a certain symmetry point group, you can always do a
> symmetryoptimisation starting from the geometry with that symmetry, and
> compare energies...
>
> The most important disadvantage is off course that it often takes a lot
> of time to reach a certain symmetry point group, when you start from
> zerosymmery.
>
> Any thoughts on this...
>
> Those interested in a summary of some sort should send me a note, if
> enough interest exists, I send one to CCL.
>
Patrick,
You are essentially right, but if you do a constrained optimisation
and then do a frequency calculation, you you will find out whether you have
a local minimum or not. If you do not, that is the time to lower the
symmetry. Hope this helps.
Cheers, Brian,

Associate Professor Brian SalterDuke (Brian Duke)
School of Chemistry and Earth Sciences, Northern Territory University,
Box 40146, Casuarina, NT 0811, Australia. Phone 089466702
email: b_duke ( at ) lacebark.ntu.edu.au or b_duke ( at ) uncl04.ntu.edu.au
Dear Patrick:
It has been our experience that Nature does tend to favor high symmetry
conformations, so much in fact that it is interesting to find cases
where this is not the case. So that is a "philosophical" reason
to optimize with symmetry (this assumes of course that one calculates and
diagonalizes a Hessian to verify the nature of the stationary point).
You mentioned the the most pratical reason yourself: geometry convergence
is greatly accelerated by using symmetry. Should your molecule be C1, it it
still an efficient way to refine your guess at the structure.
Best regards,
Jan Jensen
Iowa State University
In reply to Patrick Bultinck's question about symmetry:
>
> In section 4 of the GAMESS manual, I find under a title : ***The role of
> symmetry*** that it is best to use symmetry during an optimisation.
> However GAMESS never lowers the symmetry during an optimisation. It is a
> well known fact that sometimes a minimal energy structure differs (a lot)
> from the symmetry you might expect. Isn't it then safer to always do
> unconstrained optimisations, and if you think that the result differs so
> little from a certain symmetry point group, you can always do a
> symmetryoptimisation starting from the geometry with that symmetry, and
> compare energies...
>
> The most important disadvantage is off course that it often takes a lot
> of time to reach a certain symmetry point group, when you start from
> zerosymmery.
>
Several quantum chemistry codes do not lower symmetry during an
optimization. As a matter of fact, if the gradient does not reflect
the full symmetry of the point group used, there is a symmetrization
problem.
Concerning using point group symmetry or not: If a molecule has
a possibility of a higher point group symmetry than C1, it is a
good idea to exploit it. Integrals, optimizations, etc. can run
faster if the program is able to use symmetry information. While
the molecule with symmetry may not be the global minimum on the
surface it is certainly some sort of stationary point on the
surface. (Note that it might be a transition state or "higher order
transition state". If the stationary point is a transition state, it
is possible to follow the imaginary frequency leading to a minimum.
You would also have found an important point on the reaction surface!)
Even if you find a nonsymmetric minimum that is lower than the symmetric
molecule, you are not guaranteed that it is the global minimum. Only in
fairly small molecules is it "easy" to verify that the global minimum
has been found.
As was pointed out, it can take time to "find" symmetry if you
don't start out with it. On the other hand, if you have a symmetric
molecule, it is possible to distort it into lower symmetry and
search for a lower symmetry molecule. My point is that even if it
is not necessarily the lowest energy isomer on the surface, a
molecule with symmetry may tell you something about the surface and
it is a good starting point. Of course, "forcing" a molecule into
higher symmetry than is practical is not necessarily a good idea
either.
Hope this helps,
Theresa Windus
Department of Chemistry email: windus <at> chem.nwu.edu
Northwestern University
Patrick Bultinck writes.....
>Netters,
>
>I have just sent a reply to a question concerning the use of a Zmatrix
>for a calculation on ethylene, using GAMESS (US).
>
>As a direct consequence I rembered something that I am not very
>comfortable with...
>
>In section 4 of the GAMESS manual, I find under a title : ***The role of
>symmetry*** that it is best to use symmetry during an optimisation.
>However GAMESS never lowers the symmetry during an optimisation. It is a
>well known fact that sometimes a minimal energy structure differs (a lot)
>from the symmetry you might expect. Isn't it then safer to always do
>unconstrained optimisations, and if you think that the result differs so
>little from a certain symmetry point group, you can always do a
>symmetryoptimisation starting from the geometry with that symmetry, and
>compare energies...
>
>The most important disadvantage is off course that it often takes a lot
>of time to reach a certain symmetry point group, when you start from
>zerosymmery.
>
>Any thoughts on this...
Yes, I would like to see a discussion on this subject opened up. I would
especially be interested in comments from some of the more experienced
computation chemists. Also, those persons who did calculations long enough
ago when the lack of computer power necessitated the use of symmetry could,
I am sure, have some interesting input on this subject.
Up to this point, I thought that the use of symmetry in calculations was
a big NONO because you are arbitrarily constraining the system and "garbage
in is garbage out". But I just ran across a time when we had to use symmetry:
We are investigating a complexation between a small organic molecule and a
fragment containing several metal centers. The whole point is to match the
metal orbitals with those calculated for the organic fragment and determine
why one fragment reacts one way and other fragments react anther way. The
only way that we can make sense out of the orbitals in the organic fragment
is to constrain the system to some point group  must get orbital symmetries.
So, the questions is: If this works, is it by some stroke of luck or does
this reaction ONLY occure if the organic fragment (considering free rotation)
reaches d3h symmetry while in the proximity of the metal centers? (I am being
facetious to some degree).
Don't we, as scientists, have to keep all constraints and limitations of
our model in mind when looking at systems and use our chemical intuition as
a guide? Isn't the use of symmetry just one more constraint to keep in mind
or does the use of symmetry *invalidate* any scientific findings?
Of course, the symmetry argument (to use or not to use) probably goes along
with the "bigger is better" argument and I will probably still stay away
from symmetry as much as possible and run with the largest basis set and
highest level of theory that disk allows, provided OSC permits.
Chuck

Charles W. Ulmer, II
Graduate Student Senior Scientist
D.A.Smith Group DASGroup, Inc.
University of Toledo 3807 Elmhurst Road
Toledo, OH, 43606 Toledo, OH 43613
phone: (419)5374028 culmer "at" uoft02.utoledo.edu
fax: (419)5374033
WE ARE PERHAPS NOT FAR REMOVED FROM THE TIME WHEN WE SHALL BE ABLE TO
SUBMIT THE BULK OF CHEMICAL PHENOMENA TO CALCULATION.
 JOSEPH LOUIS GAYLUSSAC
MEMOIRES DE LA SOCIETE D'ARCUEIL, 2, 207 (1808)
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 symmetryunique 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 'closedshell' 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 nonzero 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 highsymmetry structure contains repulsions.
The matter in 'openshell' systems is more complicated. Imposing high symmetry
on a JahnTeller 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 openshell systems, the use of symmetry in UHFtype
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. spincontamination, 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 highsymmetry species is allbut
totally dispelled at this time. Basically, for minima, anything goes, but
there is no guarantee that "nature" favours symmetric structures over their
lowersymmetry counterparts. (One only has to look at van der Waals molecules
to see evidence of that.) For transition states, all that the supposedlyuseful
symmetry theorems tell you is an upperlimit 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 nonvanishing term in a locally
expanded Taylor series. Thus almost all stationary points that we, as
"chemists" meet, are secondorder points, because they have nonvanishing
second derivatives. Third order points have zerosecond derivatives, and
are characterised by cubic terms in the potential, e.g., "monkeysaddles".
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 secondorder 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 nonzero
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 943041011,
U.S.A.
Tel. +1 (415) 424 9924 x110
FAX +1 (415) 424 9501
Email rgab #  at  # purisima.molres.org
Hi,
Just a few more remarks regarding Richard Bone's comments:
> "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.
If you wish to force the symmetry, this can be averted by using the
same variable name for the symmetric lengths and angles, thus there is
no possibility of a change in point group.
> 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 'closedshell' systems (no electronic degeneracies) this means in practice
> that 'symmetric' structures are usually stationary points. The only
There are actually a number of cases where this is indeed not the
case. Amides, for instance, tend not to be planar but are slightly
puckered at the N. It often doesn't take much energy to force it to
be planar, but nonplanar is the minimum at any levels of theory that
I've seen. Also, quite bulky molecules often avoid a symmetric
form in favor of spreading out the steric interactions.
The only way to be sure anything is really a minimum or a
transition state is to perform a frequency calculation at the end,
whether or not one uses symmetry in the calculation. I had started
the methyl groups in DMA in one orientation and they stayed put, even
though I did not invoke symmetry. The frequency calculation showed
it to be a transition state, and further optimizations using those
forces lead to the minimum structure.
The use of symmetry is still quite useful in speeding
computations, and the freq. calc. will tell you whether you were
correct or incorrect in assuming the particular level of symmetry.
(Use individual variables for each bond and angle if you want to
optimize further, though!)
Dan Severance (APLS  Association for Prevention of Long Sigs)
dan %! at !% omega.chem.yale.edu
Regarding the use of symmetry in quantum calculations. Most of the
stuff I've looked at never was more than C1 to start with so it's not
been a problem. By tinkering with the Zmatrix and enforcing certain
types of symmetry (like requiring selected CH bonds to have the same
length), I've seen between 10 and 300% differences in speed for test
systems I've worked with. For example, in dimethyl terephthalate, I
can require all of the aromatic CH bonds to be of the same length,
and I can require certain other bonds (e.g. both carbonyls) to be the
same length. Any time I'm really interested in the most accurate,
quantitative results, I'll do a symmetry disallowed optimization, or
at least a symmetry disallowed frequency calculation, to see how
far from a true minimum I happen to be.
My opinion (worth what you paid for it ;) ): Symmetry can be used
to advantage, particularly less restrictive symmetry than full point
group symmetry, but you take a risk of getting a misleading answer
when you do that. The classic one (and one that a student of mine
got tripped up on this fall) was keeping the symmetry and getting
a planar biphenyl. If I'm really concerned, I've been known to run
the starting geometry through a few femtoseconds of dynamics to
completely ruin any symmetry/stationary points I've happened to
introduce in constructing the molecule.
Bruce Wilson (bewilson;at;emn.com)
Yes, I agree it is safer never to use symmetry. But sometimes it is
worth the risk. I am willing to use symmetry in the optimization
of, e.g., benzene and ethylene. Sometimes you get burned. Once I
used symmetry to optimize the geometry of sulfuric acid (C2v according
to the JANNAF Tables), and missed the opportunity of discovering that
it should have been C2.
Symmetry can also give lower energies than not using symmetry. Thus,
in benzene the only `error' can be in the two bond lengths, if symmetry
is used. If symmetry is NOT used, errors due to unequal bond lengths and
due to angles differing from 120 degrees will cause the energy to rise.
Jimmy Stewart
Patrick Bultinck started a discussion about whether or not symmetry constraints
should be used in calculations. When we do a geometry optimization we're
seeking a stationary point (which almost alwars means a relative minimum or
a transition state). Since it's sound practice to characterize any stationary
point you have found by calculating its freqs, it is probably a good idea to
routinely impose symmetry (for esthetic reasons and to save time) and let the
number of imaginary vibrations tell you when to relax it to go from a transition
state to a minimum.
Theresa Windus says "while the molece with symmetry may not be the global
minimum...it is certainly some sort of stationary point..." Does anybody
know of a formal theorem that says nonC1 structures are always stationary
points, or some place where this is explicitly stated? It's just that it
would be nice to see some sort of proof; it's not quite obvious that it
MUST be so, altho' I do see that it looks reasonable.
Thanks.
===
E. Lewars wrote:
> [ ... ]
> Theresa Windus says "while the molece with symmetry may not be the global
> minimum...it is certainly some sort of stationary point..." Does anybody
> know of a formal theorem that says nonC1 structures are always stationary
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> points, or some place where this is explicitly stated? It's just that it
^^^^^^
If you OPTIMIZE your molecule under the restriction of a specific point group
and you find a point where all force vanish you have found a stationary point
(that's how stationary points are defined). So it doesn't matter whether the
molecule possesses symmetry or not at this point of its potential surface.
> would be nice to see some sort of proof; it's not quite obvious that it
> MUST be so, altho' I do see that it looks reasonable.
But the probability that you have found a MINIMUM is very small. In most cases
you will found one or even more imaginary frequencies and you have to optimize
further using a subgroup of your current point group (C1 is a always a subgroup)
and starting with a distortion of your current geometry (preferably along a
normal mode with an imaginary frequency), since otherwise you stay at your
current stationary point of the potential surface (there are no force your
optimizer can minimize).
Using symmetry makes sense if you know from experiment that your molecule belongs
to a special point group or your chemical intuition tells you that symmetry may
play a role. Use of symmetry for speeding up or making calculations possible has
always to be considered as giving you wrong answers (no minima). If your
molecules
are models for larger systems and the imaginary frequencies affect only parts
of the molecule which are not present in your larger system, you may use
symmetry,
too.
JoergR. Hill
In reply to Dan Severence's comments:
i) Gaussian calculations.
>> "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.
>
> If you wish to force the symmetry, this can be averted by using the
>same variable name for the symmetric lengths and angles, thus there is
>no possibility of a change in point group.
>
Yes, Dan is correct, provided that you can do it. Constraining the
internal coordinates in the Zmatrix is the way out. But I had one fiendish
example once (in "C_i" symmetry) where it was not possible to do this: two
related angles could not be specified by the same internal coordinate and
the optimization failed as soon as these angles became slightly inequivalent.
ii) I was very careful in my comment about the gradient.
>> 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 'closedshell' systems (no electronic degeneracies) this means in practice
>> that 'symmetric' structures are usually stationary points. The only
>
> There are actually a number of cases where this is indeed not the
> case. Amides, for instance, tend not to be planar but are slightly
> puckered at the N. It often doesn't take much energy to force it to
> be planar, but nonplanar is the minimum at any levels of theory that
> I've seen. Also, quite bulky molecules often avoid a symmetric
> form in favor of spreading out the steric interactions.
>
Note that I said that symmetric structures are usually STATIONARY POINTS,
not 'MINIMA'. Nothing I said contradicted the example of a nonplanar amide.
But, symmetry can not be interpreted to be 'local' either. I was about to
venture that the configuration in which the bonds about the Natom are planar
would probably be a 'transition state' but of course such a structure may not
be a stationary point at all if there are other 'outofplane' atoms in the
molecule which are not related to one another by reflection in the plane.
(Remember those nonbonded interactions ...). If the _whole_ molecule is
planar or has a plane of symmetry then (if closedshell, and in the absence of
totally symmetric distortions) it will be a stationary point but I couldn't
tell you, a priori, what sort.
N.B., Just because symmetric structures are usually stationary, does not mean
the converse, either, viz: stationary points do not have to be 'symmetric'.
There are clearly a number of pitfalls and potential confusions in the
discussion of symmetry. I will shortly post a useful referencelist to CCL
which not only provides background to my own comments but should hopefully
be a useful almanac to anybody with an interest in this field.
Watch this space .........
Richard Bone
================================================================================
R. G. A. Bone.
Molecular Research Institute,
845 Page Mill Road,
Palo Alto,
CA 943041011,
U.S.A.
Tel. +1 (415) 424 9924 x110
FAX +1 (415) 424 9501
Email rgab;at;purisima.molres.org
Dear netters
Here's my two ore contribution to the symmetry discussion.
The goal of a calculation is to get a physically relevant result. Therefore the
use of absolute symmetry restrictions should be scientifically motivated. Of
course, it is usually a good approach to start with a high symmetry and then
step by step relax the restrictions. This will economize resources and may at
the same time give meaningful information about saddle points.
Frequency calculations in principle involve quite a lot  something like
n*(n1)/2 for n degrees of freedom  of lower symmetry calculations, involving
all possible pairs of displacements of individual atoms. That may be too much
effort, since a check for saddle points requires only n such calculations  or
less if bond stretching is avoided.
Thanks for your bandwidth ...
In view of the current comments on the philosophy of the use of symmetry
in the computation of molecular potential energy surfaces, I thought it
would be useful to put out a compendium of 'primary' references in this
field. I revisited my (1992) PhD thesis and culled the following.
It is worth restating, though, that there is no theorem which allows you
claim that a minimumenergy 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?)
Symmetry can be used somewhat heuristically, however, with caution.
And, as has also, already been stated, once one has located a stationary point
(confirmed by checking all gradient components), the only sure way of testing
whether or not it is a local minimum, is to calculate the Hessian.
A general discussion of all basic features of PESs can be found in;

P.G.Mezey, Studies in Physical and Theoretical Chemistry, 53, "Potential
Energy Hypersurfaces", Elsevier, (1987).
Symmetry at Stationary Points

If one is looking for a 'proof' that the energy must be stationary w.r.t.
all nontotally symmetric distortions (i.e., the oftdesired result that
"symmetric structures are (frequently) stationary points") then there are
two papers which address the issue (for a nondegenerate electronic state):
D. J. Wales, J. Am. Chem. Soc., 112, 7908, (1990)
 a paper which considers the balance of forces on the atoms at various points
in a given structure.
H. Metiu, J. Ross, R. Silbey and T. F. George, J. Chem. Phys.,61, 3200, (1974)
This paper observes that the discussion is linked closely with the notion of
expanding the local potential surface in normal coordinates (and their
derivatives, etc.). [Because, of course, normal coordinates are guaranteed
to transform as irreducible representations of the point group.]
This, in turn, leads on to discussions of reaction paths, etc. and an
understanding of how the gradient vector transforms under symmetry operations,
a subject which has been addressed by many authors.
Symmetry of the gradient.

The first of these papers contains a straightforward and rigorous formal proof.
J. W. McIver, Jr. and A. Komornicki, Chem. Phys. Lett., 10, 303, (1971)
The papers by Pearson describe a perturbationtheory approach.
R. G. Pearson, J. Am. Chem. Soc., 91, 1252, (1969)
R. G. Pearson, J. Am. Chem. Soc., 91, 4947, (1969)
R. G. Pearson, Theor. Chim. Acta., 16, 107, (1970)
R. G. Pearson, J. Chem. Phys., 52, 2167, (1970)
R. G. Pearson, Pure Appl. Chem., 27, 45, (1971)
R. G. Pearson, Acc. Chem. Res., 4, 152, (1971)
R. G. Pearson, Symmetry Rules for Chemical Reactions, Wiley Interscience, (1976)
Rodger and Schipper have a formalism based upon normalmode expansions.
A. Rodger and P. E. Schipper, Chem. Phys., 107, 329, (1986)
A *landmark* paper by Pechukas describes the properties of steepestdescent
paths and leads to cornerstone statements of transition state symmetries.
P. Pechukas, J. Chem. Phys., 64, 1516, (1976)
Transition State Symmetry Rules

The believed limitations on the symmetries of transition states have lead to
many analyses, beginning with Stanton and McIver:
J. W. McIver, Jr. and R. E. Stanton, J. Am. Chem. Soc., 94, 8618, (1972)
R. E. Stanton and J. W. McIver, Jr., J. Am. Chem. Soc., 97, 3632, (1975)
J. W. McIver, Jr., Acc. Chem. Res., 7, 72, (1974)
A. Rodger and P. E. Schipper, Chem. Phys., 107, 329, (1986)
A. Rodger and P. E. Schipper, Inorg. Chem., 27, 458, (1988)
A. Rodger and P. E. Schipper, J. Phys. Chem., 91, 189, (1987)
R. G. A. Bone, Chem. Phys. Lett., 193, 557, (1992)
Exceptions and Pitfalls

The application of symmetry and many assumptions about the form of a PES can
lead to problems if unusual features are present. The oftcited example is
that of a (putative) monkeysaddle (3rd order 'degenerate' critical point),
but except where symmetry itself may impose severe restrictions on the form
of the potential function, it is safe to assume that there will always be
minor perturbations which lift the degeneracy. Near 4thorder critical points
can arise on exceptionally flat surfaces and when symmetry demands that
3rd order terms in the potential must vanish. Far more frequent are bifur
cations. Strictly a "bifurcation" is a feature of a path. Valtazanos and
Ruedenberg demonstrated their association with "valleyridge inflection points"
on a PES:
P. Valtazanos and K. Ruedenberg, Theor. Chim. Acta., 69, 281, (1986)
Bifurcations are legion. For description of one on a very simple and surpris
ing surface, see:
R. G. A. Bone, Chem. Phys., 178, 255, (1993).
Other more exotic examples, include:
J. F. Stanton, J. Gauss, R. J. Bartlett, T. Helgaker, P. Jorgensen,
H. J. Aa. Jensen and P. R. Taylor, J. Chem. Phys., 97, 1211, (1992).
T. Taketsugu and T. Hirano, J. Chem. Phys., 99, 9806, (1993)
The main significance of bifurcations in the context oif symmetry is that they
are points at which, symmetry may be 'broken'.
For a qualitative discussion of the plethora of forms of PES and the influence
of point group symmetry, see:
R. G. A. Bone, T. W. Rowlands, N. C. Handy and A. J. Stone,
Molec. Phys., 72, 33, (1991)
And, a paper which describes concepts useful for characterisation the
local symmetry of potential energy surfaces in JahnTeller systems.
A. Ceulemans, D. Beyens and L. G. Vanquickenbourne,
J. Am. Chem. Soc., 106, 5824, (1984)
Symmetrybreaking in OpenShell Systems

This subject has a long history, so I will be selective here:
General considerations/review:
E. R. Davidson, J. Phys. Chem., 87, 4783, (1983)
UHF calculations:
L. Farnell, J. A. Pople and L. Radom, J. Phys. Chem., 87, 79, (1983)
D. B. Cook, Int. J. Quant. Chem., 43, 197, (1992)
D. B. Cook, in "The SelfConsistent Field Approximation", Ed. R. Carbo
D. B. Cook, J. Chem. Soc. Far. Trans. 2, 82, 187, (1986)
P. O. Lowdin, Rev. Mod. Phys., 35, 496, (1963)(
W. D. Allen and H. F. Schaefer, III, Chem. Phys., 133, 11, (1989)
Calculations with MCSCF/CASSCF
P. Pulay and T. P. Hamilton, J. Chem. Phys., 88, 4926, (1988)
J. M. Bofill and P. Pulay, J. Chem. Phys., 90, 3637, (1989)
Calculations with a CoupledCluster Wavefunction
J. D. Watts and R. J. Bartlett, J. Chem. Phys., 95, (1991)
Final Comments:
There is no definitive answer to the question of what role symmetry should
play in a given molecular problem. There will always be a compromise between
(computational) resources available and thoroughness of analysis. Sometimes
it makes sense to start in high symmetry, sometimes not. Sometimes it is only
possible to use symmetry  because of the difference it makes to the size of
the calculation. In calculating a reaction path, it's probably a good idea
to use as little symmetry as possible. When optimizing a geometry, symmetry
will help you get to a stationary point (probably) but only a frequency
calculation can tell you what it is. When searching for saddlepoints, the
theorems tell you that you'll only increase in symmetry (at the T.S.) if you're
finding a T.S. for a 'degenerate' rearrangement. Otherwise, your transition
state cannot be searched for by minimizing in a symmetryconstraint. This
means that the majority of T.S.'s don't have any useful distinguishing
symmetries. Sad, but true.
Despite the zeal with which chemists have pursued those symmetric 'beauties',
the Fullerenes, Nature more often displays lowsymmetry than highsymmetry,
but a bit of thought shows that two lowsymmetry structures are often
related to one another by a symmetry operation in a highersymmetry 'midpoint'
so symmetry of the PES is always conserved.
Richard Bone
================================================================================
R. G. A. Bone.
Molecular Research Institute,
845 Page Mill Road,
Palo Alto,
CA 943041011,
U.S.A.
Tel. +1 (415) 424 9924 x110
FAX +1 (415) 424 9501
Email rgab <at> purisima.molres.org
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 'closedshell' 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 nonC1 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 JahnTeller 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
Zmatrix) 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" Zmatrix
(in which some parameters are denoted by the same variables though they are
not symmetrically equivalent).
Sergei F.Vyboishchikov
Universitaet Marburg, Germany
EMail: sergei \\at// ps1515.chemie.unimarburg.de
On Apr 27, 11:27pm, R.G.A. Bone wrote:
> Subject: CCL:Symmetry: Definitive References
> It is worth restating, though, that there is no theorem which
> allows you claim that a minimumenergy 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 pointgroup symmetry. If I simply stretch or compress the
CC bond, I do not change the pointgroup, 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.

********************** "So much for global warming...." *********************
Peter S. Shenkin, Box 768 Havemeyer Hall, Dept. of Chemistry, Columbia Univ.,
New York, NY 10027; shenkin;at;still3.chem.columbia.edu; (212) 8545143
*****************************************************************************
Peter Shenkin wites: "...consider ethane in the staggered conformation. This
conf is an energetic minimum, and pointgroup symmetry. If I simply stretch
or compress the CC 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?"
Yes, but wer're not really asking if a symmetric structure of *arbitrary*
geometry is a stationary point, but rather if such a structure, optimized
within the limits of its symmetry (keeping the Sch:/nfliess point group, e.g.
C2v, Cs, etc) must go to a stationary point.
===
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 minimumenergy 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 pointgroup symmetry. If I simply stretch or compress the
> CC bond, I do not change the pointgroup, 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 nonzero 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 highsymmetry 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 energyminimized.
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 energyminimization, constraining that symmetry. The energy has changed
only w.r.t. totallysymmetric 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
totallysymmetric 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
================================================================================
R. G. A. Bone.
Molecular Research Institute,
845 Page Mill Road,
Palo Alto,
CA 943041011,
U.S.A.
Tel. +1 (415) 424 9924 x110
FAX +1 (415) 424 9501
Email rgab ^at^ purisima.molres.org
That's all folks, at least upto now.
As mentionned, future mailings will still be very welcome, and I hope
that CCl will be considering this matter for the next few days (weeks) to
come.
Greetings, from Belgium
Patrick (Email address, address, fax, ... in the original message (see
above)).
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