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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 CCL-subscribers... 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 point-group, 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 CPU-time 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 E-mail 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 Z-matrix 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 symmetry-optimisation 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 zero-symmery. 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, |-----------------------------------------------------------------------| | C-C Patrick Bultinck | | / \ Dept. Physical & Inorganic Chemistry | | C-O O-C Section Quantum Chemistry | | | | University of Ghent | | C-O O-C Krijgslaan 281 (S-3) | | \ / 9000 Gent | | C-C Belgium | | Tel. Int'l code/32/9/264.44.44 | | Macrocycles Fax. Int'l code/32/9/264.49.83 | | Quantum Chemical E-mail : 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 > symmetry-optimisation 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 symmetry-breaking directions. No gradient can ever (except for numerical reasons, of course) lead to a lower-symmetry 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 > zero-symmery. 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@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 Z-matrix > 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 > symmetry-optimisation 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 > zero-symmery. > > 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 Salter-Duke (Brian Duke) School of Chemistry and Earth Sciences, Northern Territory University, Box 40146, Casuarina, NT 0811, Australia. Phone 089-466702 e-mail: 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 > symmetry-optimisation 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 > zero-symmery. > 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 non-symmetric 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 e-mail: 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 Z-matrix >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 >symmetry-optimisation 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 >zero-symmery. > >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 NO-NO 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)537-4028 culmer-: at :-uoft02.utoledo.edu fax: (419)537-4033 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 GAY-LUSSAC 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 symmetry-unique 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 'closed-shell' 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 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 matter in 'open-shell' systems is more complicated. Imposing high symmetry on a Jahn-Teller 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 open-shell systems, the use of symmetry in UHF-type 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. spin-contamination, 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 high-symmetry species is all-but totally dispelled at this time. Basically, for minima, anything goes, but there is no guarantee that "nature" favours symmetric structures over their lower-symmetry counterparts. (One only has to look at van der Waals molecules to see evidence of that.) For transition states, all that the supposedly-useful symmetry theorems tell you is an upper-limit 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 non-vanishing term in a locally- expanded Taylor series. Thus almost all stationary points that we, as "chemists" meet, are second-order points, because they have non-vanishing second derivatives. Third order points have zero-second derivatives, and are characterised by cubic terms in the potential, e.g., "monkey-saddles". 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 second-order 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 non-zero 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 94304-1011, U.S.A. Tel. +1 (415) 424 9924 x110 FAX +1 (415) 424 9501 E-mail rgab-0at0-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 'closed-shell' 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 non-planar 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 Z-matrix and enforcing certain types of symmetry (like requiring selected C-H 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 C-H 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 non-C1 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 non-C1 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. Joerg-R. 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 Z-matrix 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 'closed-shell' 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 non-planar 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 non-planar 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 N-atom 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 'out-of-plane' atoms in the molecule which are not related to one another by reflection in the plane. (Remember those non-bonded interactions ...). If the _whole_ molecule is planar or has a plane of symmetry then (if closed-shell, 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 reference-list 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 94304-1011, U.S.A. Tel. +1 (415) 424 9924 x110 FAX +1 (415) 424 9501 E-mail rgab -AatT- 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*(n-1)/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 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?) 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 non-totally symmetric distortions (i.e., the oft-desired result that "symmetric structures are (frequently) stationary points") then there are two papers which address the issue (for a non-degenerate 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 perturbation-theory 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 normal-mode expansions. A. Rodger and P. E. Schipper, Chem. Phys., 107, 329, (1986) A *landmark* paper by Pechukas describes the properties of steepest-descent 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 oft-cited example is that of a (putative) monkey-saddle (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 4th-order 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 "valley-ridge 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 Jahn-Teller systems. A. Ceulemans, D. Beyens and L. G. Vanquickenbourne, J. Am. Chem. Soc., 106, 5824, (1984) Symmetry-breaking in Open-Shell 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 Self-Consistent 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 MC-SCF/CAS-SCF 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 Coupled-Cluster 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 saddle-points, 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 symmetry-constraint. 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 low-symmetry than high-symmetry, but a bit of thought shows that two low-symmetry structures are often related to one another by a symmetry operation in a higher-symmetry '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 94304-1011, U.S.A. Tel. +1 (415) 424 9924 x110 FAX +1 (415) 424 9501 E-mail 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 '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 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 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. -- ********************** "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) 854-5143 ***************************************************************************** Peter Shenkin wites: "...consider ethane in the staggered conformation. This conf is an energetic minimum, and 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?" 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 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 ================================================================================ R. G. A. Bone. Molecular Research Institute, 845 Page Mill Road, Palo Alto, CA 94304-1011, U.S.A. Tel. +1 (415) 424 9924 x110 FAX +1 (415) 424 9501 E-mail 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 (E-mail address, address, fax, ... in the original message (see above)). Similar Messages 04/26/1994: Re: CCL:Use of symmetry in optimisations 04/26/1994: symmetry in electronic structure computations 04/28/1994: Symmetry: Definitive References 05/20/1994: Summary on symmetry, part two 04/28/1994: Symmetry: reply to Peter Shenkin 04/27/1994: Symmetry: In Reply to Dan Severence 04/26/1994: Re: Symmetry 04/26/1994: Re: CCL:symmetry in electronic structure computations 10/16/1992: Re: Spin density calculations for benzene (fwd) 04/28/1994: Symmetry again Raw Message Text |