J. Phys. Chem. 1987, 91, 2233-2234
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Permutation-Inversion Symmetry Considerations on the Allowed Geometry of a Transition-State Complex Aristophanes Metropoulos Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation,t Athens I 1635, Greece (Received: February 2, 1987)
A recently developed selection rule scheme concerning the allowed geometry of a transition-state complex is expressed in the language of permutation-inversion groups. The superiority of these groups for treating chemical reactions is discussed.
Introduction Very recently, a symmetry selection scheme developed by Schipper et al.’ for spectroscopic processes has been extended to reaction mechanisms by the same authors.2 This scheme, called generalized selection rule (GSR) scheme, is a two-stage procedure. The first stage is a “classical selection rule” (CSR) procedure based on the fact that the quantum mechanical operators corresponding to natural observables must be always totally symmetric. The second stage is the usual state selection rules (SSR) procedure applied on the states allowed in the first stage. For reaction mechanisms the natural observable is the geometry of the molecules and the geometry changes effected during a reaction. At the CSR stage one tries to arrive a t the allowed geometry of the transition complex (T) once the geometries of the reactants (R) and products (P) are known. To do this Schipper et al. have exploited the general features of the potential energy surfaces (PES)3*4limiting themselves to concerted reactions for which a well-behaved path from R to P through a well-defined T exists.2 However, their results for the CSR stage are independent of the quantum nature of the system. The arguments employed in the actual derivation of their selection rules are based on point group symmetry considerations intermixed with permutations of identical nuclei. Their method represents a significant advance in our ability to predict the allowed structure of the transition-state complex. The purpose of this brief note is to point out that their excellent results can be recast entirely in the language of the permutation-inversion (PI) groups of Longuett-Higgins5 and that the PI symmetries are the “natural” symmetries to be exploited in chemical reaction processes. PI vs. Point Group Concepts in Chemical Reactions A point group treatment of a reaction process involves the concepts of creation and destruction of symmetry elements as the system moves along the reaction path from R to T to P. This destruction and creation of symmetry elements is a mathematical device necessitated by the fact that point groups are geometry oriented. When the molecular or system geometry changes as a result of a reaction, the corresponding point group also changes. Since a symmetry operation on a system commutes with the Hamiltonian of this system, a change of the applicable point group in different regions of the reaction path implies the existence of different Hamiltonians along this path. Their difference is of course hidden in the potential function which is implicitly assumed different for every limiting case along the path. That is, each rovibrational Hamiltonian corresponds to a solution of the electronic part of the Schroedinger equation at the different geometries of R, T, and P as well as at the intermediate regions denoted here by I. It is conceptually more satisfying and probably closer to reality if the entire reaction system is considered as one molecule (supermolecule) controlled by a single Hamiltonian. Under this picture, a large amplitude vibration leads from R to T to P and symmetry is conserved during the entire process, as it should be. In this way, the bothersome concept of symmetry breaking and ‘48 Vas Constantinou Avenue.
0022-3654/87/2091-2233$01.50/0
creating is replaced by the notion of feasible and unfeasible operations (see below). To achieve this, one has to abandon all point group considerations in favor of the perhaps less familiar PI group concepts. The PI groups are widely used in the spectroscopy of nonrigid molecules and they are more appropriate for the description of a system, which by its nature undergoes large amplitude distortions. A particularly illuminating discussion on the differences and uses of point, PI, and the so-called extended PI6 groups has been given by H ~ u g e n . In ~ short, one may say that the point groups exploit the geometry (shape) of a molecule while the PI groups rely on the indistinguishability of identical nuceli regardless of where they happen to be located. All small-amplitude normal coordinates in a reaction system (nonrigid supermolecule) are measured with respect to the reference configuration, which is the analogue of the equilibrium configuration in rigid molecules (point groups). A nonrigid system is in its reference configuration when “all the small amplitude vibrational displacements are set to zero but when the Euler angles and the large amplitude coordinate(s) are a r b i t r a r ~ ” . ~ The use of PI symmetries necessitates the consistent labeling of the permutable nuclei and the introduction of the concept of a f r a m e w ~ r k . ~A, ~framework may be thought of as the original labeled system before a PI operation is performed on it. Two labeled structures which can be superimposed by a sequence of rotations around their structure-fixed axes so that their nuclear labels coincide are said to belong to the same framework. Since in a system with many identical nuclei the number of PI operations may be extremely large, giving rise to a large number of accidentally degenerate states which are not observable within experimental times, the concept of feasibility or unfeasibility of a PI operation has been introduced as an integral part of PI group treatmentQ5s7If a PI operation leaves the system in a framework consistent with the resolution of the experimental observation which one tries to symmetry analyze, the operation is feasible; otherwise it is unfeasible. Usually, an unfeasible operation corresponds to breaking and/or creating a chemical bond when no such a change is observable during the time scale of the experiment. The existence of unfeasible operations gives rise to the so-called “distinctly different nuclear labeled forms” (DDNLF)? which may be comprised of one or more frameworks. These forms do not interconvert spontaneously within the experimental time scale and the PI operations that interconvert such forms may be ignored except when a reaction takes place (see below). Each DDNLF may be a rigid (single framework) or a nonrigid (two or more frameworks) system and belongs to a subgroup of the full PI group of this system. When a system undergoes a change that brings (1) Schipper, P. E.; Rodger, A. Chem. Phys. 1985, 98, 29. See also: Schipper, P. E. J. Am. Chem. SOC.1978, 100, 3658. (2) Rodger, A.; Schipper, P. G. Chem. Phys. 1986, 107, 329. (3) Murrell, J. N.; Laidler, K. J. Trans. Faraday SOC.1968, 64, 371. (4) Pearson, R. G. Symmefry Selection Rules for Chemical Reactions; Wiley: New York, 1976. ( 5 ) Bunker, P. R. Molecular Symmetry and Spectroscopy; Academic: New York, 1979. (6) Bunker, P. R.; Papousek, D. J. Mol. Spectrosc. 1969, 32, 419. (7) Hougen, J. T. J . Phys. Chem. 1986, 90, 562.
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The Journal of Physical Chemistry, Vol. 91, No. 9, 1987
it from one DDNLF to another, we say that the system has undergone a chemical change or that a chemical reaction has occurred. The original DDNLF is identified with R while the new DDNLF is identified with P. We may view both of these DDNLF’s as a single DDNLF which may exist under the conditions that induce the reaction and which is identified with the nonrigid supermolecule mentioned above. This “composite” DDNLF belongs to a subgroup of the full PI group of this supermolecule which contains all the feasible operations of the supermolecule and which is larger than the subgroups of each of the component DDNLF‘s. We refer to this larger subgroup as the reaction symmetry (RS) group in analogy to the molecular symmetry group of ~pectroscopy.~
CSR Procedure in PI Symmetry The main results of Schipper et aL2 regarding their CSR procedure are their theorem 2, the principle of maximum symmetry (PMS), and their corollaries 1 and 2. These results are valid only for concerted reactions and they may be stated as follows: (1) When a chemical reaction occurs along a well-behaved reaction path all the common feasible operations between R and P must remain feasible throughout the path and, moreover, they must be the only common operations between R, T, and P. (2) If there is a framework conserving feasible operation in T that is unfeasible in R and P, it must be an antisymmetry operation for the reaction coordinate (Q) and it must interconvert R and P. (3) If there is a PI operation that interconverts R and P, it must be a framework conserving feasible operation in T and it must be an antisymmetry operation for Q. The basic premise in deriving the above rules is that the geometry of a system (natural observable) must always be totally symmetric under both the point and the PI groups of the system. The essence of the CSR procedure is that one must now ensure that a proposed geometry of T is consistent with the above rules. Let us now take a look at how the intermediate steps that lead to the above results may be expressed in the PI group language. The principle of continuity, the definition of a well-behaved path, and the definition of a stationary point on a PES are transferred intact into the PI group treatment since they are group-independent concepts. Everything else needs to be expressed in terms of feasible and unfeasible operations, frameworks, and DDNLF interconversions. The major intermediate step that we will focus on is the treatment of the infinitesimal motions along both a totally and a nontotally symmetric reaction coordinate. Thus, under the operations of the pertinent R S group, a reaction system will maintain its symmetry during a sequence of infinitesimal motions along a symmetry-adapted reaction coordinate Q, be it totally symmetric or not. However, if Q is antisymmetric under some operations, these operations will gradually become unfeasible as the motion of the nuclei progresses. The operations that remain feasible for all geometries during the motion of the nuclei must form a subgroup of the RS group under which Q must be totally symmetric (symmetry reduction). We refer to this subgroup as the “common feasible operations” (CFO) subgroup of the RS group. Notice that Q will remain antisymmetric within a portion of the I region in which the antisymmetry causing operations are feasible but not framework conserving. If in the new geometry one considers only the CFO subgroup of the R S group and the sign of the now totally symmetric Q is reversed, the operations that had become unfeasible now become gradually feasible again leading back to the full R S group (symmetry increase). On the other hand, if Q is totally symmetric under the full R S group, no operations are added or removed in any manner during the
Letters motion (original symmetry retained). Here, we have stressed the gradual nature of a PI “symmetry change” as opposed to the sudden change that necessarily occurs under a point group analysis. A contradiction may seem to arise at this point. An antisymmetric Q in the I region implies a zero potential energy gradient with respect to this Q in some portion of the I region (that is aE/aQ = 0). This contradicts the notion that aE/aQ > 0 everywhere in I and therefore, that Q must be totally symmetric everywhere in Is4 The contradiction is lifted if one recognizes that the positive gradient rule is valid only for operations that allow the system to maintain its instantaneous position along the barrier. These are the operations that operate on a single DDNLF Under the constraints of ref 2, it is shown there that the operations for which Q is antisymmetric interconvert R and P. That is, these operations convert one DDNLF into an other. This conversion corresponds to a barrier penetration for which the positive gradient rule is not valid. Hence, Q may be antisymmetric in I but only for operators that interconvert the two DDNLF‘s that correspond to R and P. Notice that Schipper at al. have actually used PI operations at some points of their treatment without explicitly mentioning so. Thus, their stationary operators correspond to PI framework conserving feasible operators and their nonstationary operators correspond to PI unfeasible or framework nonconserving feasible operators. The correspondences between point and PI group treatments of the remaining intermediate steps are now fairly obvious and need not be elaborated upon any further. Discussion From the above considerations it is obvious that one may ignore the apparent symmetry changes during the motion of the nuclei and use the full RS group throughout the reaction path. If such a choice is made, theorem 2 of ref 2 and the PMS become in a sense redundant and the corollaries 1 and 2 may be expressed as follows: (1) Some of the operators of the RS group of a reaction system that are antisymmetric for Q interconvert R and P; (2) All the operators of the RS group of a reaction system that interconvert R and P are antisymmetric for Q; (3) Q must belong to an irreducible representation (ir) of the RS group that correlates to the totally symmetric representation of its CFO subgroup. This last rule severely limits the allowed symmetry of the reaction coordinate and it is an underlying concept in ref 2. It is possible that the totally symmetric ir of the CFO subgroup correlates to a nontotally symmetric ir of the subgroup of R or P. The corresponding antisymmetric operators will not inerconvert R and P; hence, the “some” in rule 1 above is meant to exclude these operators if they happen to be present in a particular system. The CSR rules in ref 2 are most appropriate for rearrangement reactions. In this note we have only recast the symmetry analysis therein in the language of what we feel is a more appropriate group. We would like now to pose a few questions that we feel could be answered much more efficiently with a PI rather than a point group analysis. (1) What would be the allowed geometry of T when (a) R and P are comprised of more than one fragment? (b) R, P or both contain nonrigid fragments? (2) How could the CSR/SSR approach be enhanced to allow for the prediction of the possible rovibronic states of P once the rovibronic states of R are known? We will attempt to answer some of these questions in future work. For now we mention that rearrangement reactions need only one large-amplitude vibrational coordinate and they can be treated relatively e a ~ i l y .In ~ reactions where more than one fragment (rigid and/or nonrigid) is involved, at least two and possibly more such coordinates must be considered simultaneously, making the treatment of such reactions substantially more difficult.
