Group theory and reaction mechanisms: Permutation theoretic prediction and computational support for pseudorotation modes in C2H and C5H rearrangements

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XI, 399-413 (1977)

Group Theory and Reaction Mechanisms: Permutation Theoretic Prediction and Computational Support for Pseudorctation Modes in C,Hl and C,H: Rearrange-

ments

THOMAS D. BOUMAN*, CHARLES D. DUNCAN, AND CARL TRINDLE

Department of Chemistry, University of Virginia, Charlottesuille, Virginia 22901, USA

Abstracts

The McIver-Stanton rules concerning the symmetry of transition states have a counterpart in rules concerning the permutation symmetry of single steps in degenerate rearrangements, derivable with the aid of Longuet-Higgins group theory. The generalized rules are illustrated by the widely studied PXs polytopal rearrangements. The analysis leads to prediction of hitherto unexplored “pseudorotation” pathways for rearrangements in ethyl and homotetrahedryl cations. CNDO computations of system energies, gradients, and curvatures at critical points on the C2H; and C5H; surfaces indicate that symmetry-breaking in keeping with the permutation-theoretic predictions is a key feature of the low-energy rearrangements of these systems. In particular, computation indicates that the C,, “classical” homotetrahedral cation corresponds to an energy maximum rather than an energy minimum, or a transition state.

Les regles de McIver-Stanton sur la symetrie des etats de transition correspondent a des regles sur la symetrie de permutation d’etapes simples dans des rkarrangements degenerks, qui peuvent Stre obtenues a l’aide de la thtorie des groupes de Longuet-Higgins. Les regles genkraliskes sont illustrees par des rearrangements polytopes PXs. L’analyse correspondante mkne a des prkdictions de sentiers de “pseudorotation” jusqu’ici non examinks pour des rearrangements dans les cations d’ethyl et de homotetrahedryl. Des calculs de type CNDO d’knergies, de gradients et de courbures des points critiques sur les surfaces des ions C,H: et C5H; indiquent que la symktrie brisee de concert avec les predictions de la thtorie des permutations est un fait saillant des rearrangements B basse tnergie de ces bystemes. En particulier les calculs indiquent que le cation homotetrahedral “classique” C2” correspond a un maximum de I’knergie plut8t qu’a un minimum ou un &tat de transition.

Den McIver-Stanton’ schen Regeln fur die Symmetrie von Ubergangszustanden entsprechen Regeln iiber die Permutationssymmetrie von einzelnen Schritten in entarteten Umordnungen, die mit Hilfe der Longuet-Higgins’ schen Gruppentheorie hergeleitet werden konnen. Die verallgemeinerten Regeln werden mit den polytopen Umordnungen von PXs illustriert. Die Analyse fuhrt zu Vor- hersagungen von bisher nochnicht untersuchten “Pseudodrehungswegen” fur Umordnungen in Athyl- und Homotetrahedrylkationen. CNDo-Berechnungen von Systemenergien Gradienten und Kriimmungen an kritischen Punkten auf den Oberflachen fur C,H; und C,H: deuten an, dass Symrnetriebrechung in Ubereinstimmung mit den Permutationstheoretischen Vorhersagungen ein Haptzug der Tiefenergieumordnungen dieser Systeme ist. Im besonderen weisen Berechnungen darauf hin, dass das “klassische” homotetrahedrale Kation mit C,,-Symmetrie einem Energiemax- imum eher als einem Minimum oder einem Ubergangszustand entspricht.

* On leave 1974-1975 from the Department of Chemistry, Southern Illinois University, Edwardsville, Illinois 62025.

399 @ 1977 by John Wiley & Sons, Inc.

400 BOUMAN, DUNCAN, AND TRINDLE

Introduction

Symmetry arguments of various kinds have found great utility in classifying spectra of stable species, and it is of course appealing to try to apply similar criteria to chemical reactions as well, i.e., to attempt to characterize the shape of the potential energy surface in terms of “allowed” or “forbidden” pathways. The well-known Woodward-Hoffmann rules [ 11 exemplify one such use of symmetry, in the sense of preserving nodal properties of molecular orbitals. Klemperer [2] has used permutation algebra to enumerate possible distinguisha- ble degenerate rearrangement mechanisms, without attempting to characterize their pathwaysin detail. Metiu et al. [3], and McIver and Stanton [4] have deduced theorems on the shape of potential energy surfaces based on the symmetry of the force constant matrix for a given nuclear configuration.

In this report we show that the McIver-Stanton theorems on the symmetry of the transition vector[4] have analogues within the framework of permutation group theory which allow identification of easy single-step processes in degenerate rearrangements [S]. We apply our criteria to a number of proposed mechanisms for the PX5 polytopal rearrangement, and degenerate rearrangements of carbonium ions and fluxional organometallics. Newly discerned paths of rearrangement for C2H: and C5Hl are discussed with the aid of CNDO computations of forces and force constants.

Theory

McIver and Stanton [4] propose that a transition state which may be slightly distorted toward reactant or product can be described by the point group of the undistorted transition state. Operations in this group either may alter the distortion so that a species distorted toward products is transformed to a species distorted toward reactants, or they may leave a distortion unaffected. For the special class of reactions known as degenerate rearrangements “reactants” and “products” are identical save for a permutation of the nuclei. Longuet-Higgins [6] has developed a form of group theory suited to such systems, in which the group includes all “feasible” permutations of the nuclei. “Feasible” permutations actually occur on the time scale of the experiments, and may include the passage from a defined reactant to the defined product. That is, the Longuet-Higgins (LH) group includes operations having the same effect on a static species as the operations of the group of the transition state discussed by McIver.

Consider Figure 1. In McIver and Stanton’s analysis each operation in the point group of the transition state (TS) is applied to the vectors representing small distortions of the transition state geometry along the reaction coordinate toward reactants or products (“transition vectors”). The TS force constant matrix can have only one negative eigenvalue, since otherwise the TS geometry would correspond to a “mountaintop” rather than a pass on the potential energy surface and a lower energy pathway could always be found. Hence the irreducible representation (rep) of the transition vector must be nondegenerate. Further, if the group operation is such as to invert a distortion from reactant side to product

GROUP THEORY AND REACTION MECHANISMS 40 1

McIVER

P‘ PRESENT B / - B 3 ”

C A-B A

B n

$ - I WB

A

x = - 1

0

X = + l X = + l

Figure 1. Illustration of McIver’s rules for the symmetry of the transition vector, and of the present extensions to the concept. (See text).

side, that operation must have a negative character in the rep of the transition vector. All operations which leave the distortion undisturbed must have a positive character in the same rep.

Let us now assume, for the moment, that the “reaction” for which the TS is defined consists of a perhaps finite distortion to one side of the TS geometry being converted to a distortion to the other side. Then the operations of the TS point group on the transition vector in fact interconvert “reactants” and “products”. This is precisely the action of operations in an LH group for a common reference structure visited by all reactants on their way to products.

The above argument requires that Longuet-Higgins group operations correspond to TS point group operations and is no more than a restatement of McIver and Stanton’s analysis. However, the Longuet-Higgins analog of the McIver- Stanton analysis is in fact more general than we have so far made clear.

To make more explicit the similarities and differences between the McIver- Stanton (MS) analysis and our own, we note that MS consider departures from a single reference structure which may be a transition state in a reaction. Their group operations transform internal coordinates of a distorted species, the coordinates measured relative to the possible transition state. We consider the invariance of isomer energy to degenerate rearrangement.

BOUMAN, DUNCAN, A N D TRINDLE 402

The objects of Longuet-Higgins group operators are internal coordinates of a set of degenerate isomers. The actions of the operators are to permute/invert the internal coordinates of the reference set of isomers so that they become identical with those of another set of isomers. Both an operation in the Longuet-Higgins group and the recovered set of isomers may be specified by a single label in permutation notation. Furthermore, the Permutation notation label may specify a path from a reference set of isomers to the set of isomers referred to by that label. In terms of the objects on which the permutation operation acts, the path could be represented by a vector passing from a point in a configuration product space corresponding to the reference set of isomers to another point corresponding to the product set of isomers. Since this vector is a function dependent on sets of internal coordinates, it may be classed in the Longuet-Higgins group relevant to those sets. This vector is notintended to represent details of the dynamics, but rather represents progress from reactants to products. This vector is reversed by exchange of the reactant and product sets, and must be antisymmetric with respect to all operators which exchange reactants and products in a single step. The analogy with the antisymmetry requirement on a wave function describing fermions is useful here. If the system is describable by an 9, Longuet-Higgins group, the nondegenerate totally antisymmetric representation must be spanned by the reactive motion; this is the representation relevant to quantities reversed in sign by pairwise exchange of objects. In systems describable by subgroups of Y,,, we would seek the irreducible representation correlated to the totally antisymmetric representation in Y,,. The sought irreducible representation in the sub- group must be nondegenerate and contain (- 1) characters for paths sharing the label of operations in the group. The operations found in this way, which appear in classes which display a (- 1) character in a nondegenerate irreducible representation of the Longuet-Higgins group, are called proper generators and correspond to single steps in the degenerate rearrangement.

Operators which have positive characters in the irreducible representation in which the proper generators have negative characters may correspond to operations maintaining the system in a given configuration, or may refer to overall reactions consisting of even numbers of direct single steps in sequence. Opera- tions having negative characters may correspond to odd numbers of feasible steps. In Figure 1, P, P', pl'may be feasible steps, but Q might not be an easy single step. The rejection of operations with positive characters is still valid.

To illustrate the symmetry conditions we have adduced, we consider the familiar and much-discussed PX5 system, which is known to undergo sequences of polytopal rearrangements [7]. Refer to Figure 2 in the subsequent discussion. The structure shown on the left-hand side of any of the reactions gives the numbering of the ligands in their defined reference position. A rearrangement is described by specifying the new locations of the numbered ligands, keeping the skeleton fixed. A bodily rotation is performed on the new structure, if necessary, to bring it to maximum coincidence with the old. (See Longuet-Higgins [6] for discussion of the propriety of incorporating this bodily rotation into the group operation.) The reaction labeled M3, for example, scrambles one apical and two equatorial ligands.

GROUP THEORY AND REACTlON MECHANISMS 403

M 1

5

4

M 2 3h;

5

4

M 3

5

4

M 5 3

5

4

M 6 3%;

5

M 7 2A: 5

Figure 2. The mechanisms for PX, polytopal isomerization described by Muetterties [ S ] are sketched, with the class of Y5 associated with each mechanism indicated. The turnstile rotation [7], M7, is included as well.

In permutation notation, it is indicated as (235), a “cycle” of length three; this is read, “2 moves to position 3 ,3 moves to 5 , 5 moves to 2 , l and 4 are unchanged”. Since any apical position can interchange with any equitorial position, ultimately all possible permutations of ligands among skeletal positions will occur, and the relevant permutation group is (the symmetric group on five objects, with 5 ! = 120 elements.) [7] The various mechanistic proposals for the scrambling then amount to the choice of feasible single-step processes, i.e., the choice of generators for the group.

Can our criterion lead to a simplification in the set of possible mechanisms? Muetterties’ mechanisms [8] (Figure 2) can be assigned to classes of Y5 according to the cycle structure of the permutation representing the feasible step [9]. For M3, the permutation consists of one cycle of length 3, and (implicitly) two of

404 BOUIVIAN, DUNCAN, AND TRINDLE

length one. The corresponding class is labeled (12, 3) in a straightforward notation. Thus we have: M1 and M2 in class (13, 2) M3 and M4 in (12, 3), M5 in (1, 22), and M6 (the Berry pseudorotation [lo]) in (1, 4). Ugi [ll] adds M7, the “turnstile” rotation, in class (2,3). Not all members in the classes need be proper generators (they may be odd powers of proper generators), but we require that there be in Y5 a nondegenerate rep which has -1 character for the classes containing proper generators and their odd powers. Examination of the character table shows that in 9’’ only the totally symmetric (labeled [5] ) and the totally antisymmetric [l’] reps are nondegenerate, and in [1’] only the classes (2,3), (1, 4), and (13, 2) have negative characters. Therefore M1 and M2, the Berry pseudorotation, and the turnstile rotation can be proper generators. Muetterties mechanisms of type (12, 3) can be represented as even numbers of turnstile rotations, hence their positive character in [l’]. Mechanisms of type (13, 2) could be represented by an odd number of turnstile motions: [(14) (253)13=[(14) (253)] = (14); hence their negative sign.” Mechanism M5 can be written in several ways as even powers of proper generators, such as two steps of (13, 2) type. Hence the positive sign for (1, 2*) in [15].

Our choice is thus narrowed to “Berry”, “turnstile”, or pairwise proper generators, and in fact our symmetry criterion provides no further choice among them, since each can be formed from powers or products of the other generators. Pairwise interchanges will occur whenever the LH group is 9”. It is interesting to note that experimental evidence [l 11 seems to rule out all mechanisms other than the Berry and turnstile modes and the pairwise permutation exchanging an apical with an equatorial position.

Each of the transition states proposed for these stepwise processes obeys the McIver-Stanton criteria also. In our analysis, however, we make no reference to a postulated transition state, but look only at the overall process. We can also recognize mechanisms which are merely even numbers of steps, each of which occurs by other mechanisms, and we can recognize nonequivalent mechanisms.

Elsewhere we have described LH groups for a number of fluxional carbonium ions [5, 12, 131 and organometallics [5, 141. We quote the principal features of their symmetry analyses here, to indicate that our new criterion gives acceptable mechanistic choices. Figure 3 shows the structures and numbering of the compounds we discuss. (C5H5)benzylidicarbonyl molybdenum (I) has an LH group isomorphic with C2, [14], and the Bl representation has signs consistent with the choice of a single step from minimum to minimum as the proper generator. The same can be said for pi-ally1 tetracarbonyl iron (11) [14]. Pi-cyclooctatetraene tricarbonyl iron’s (111) LH group is isomorphic with CB,, [14], and inspection of a character table shows that the B1 and B2 representations have characters consistent with the choice of C, (Cotton’s “piano stool” path) or a, (1-2 shifts) [15] as the proper generator.

* Multiplication of permutations takes place from right to left. Thus, for example, (123) x (243) = (124); i.e., “1 is unchanged, then replaced by 2 ,2 is replaced b y 4 which is then left unchanged, etc”.

GROUP THEORY AND REACTION MECHANISMS 405

I

--Fe (CO),

II

m

9

P

b

m Figure 3. Structures and numbering for compounds referred to in text

The C6H: cation (IV) circumambulates and produces five-fold equivalences in the ring; [ 12,161 the LH group is isomorphic with Csv. Although an obvious choice for the proper generator might be the five-fold rotation (abcde), our criteria suggest that the ‘‘cT~” operation (cd) (be) is the proper generator. Only the cr, gnerator accounts for inversion at the migrating center and reflects the proposed reaction path [16].

The “barbaralyl” (C9H;) cation (V) exhibits complete positional degeneracy in the NMR, but isotopic distribution studies appear to imply that six of the nine CH positions may interconvert more readily than the remaining three [16]. Elsewhere [13] we analyze the two most probable mechanisms: a concerted rearrangement giving an LH group isomorphic with C3u and mixing six of the nine positions, and a sequence of 1,2-vinyl shifts leading to complete equivalence and described by the symmetric group Y9. In the latter, a proper generator must belong to [19], and the permutation representing ‘a 1,2-vinyl shift, (173) (28)

BOUMAN, DUNCAN, AND TRINDLE 406

(496), indeed has the appropriate negative character. A transition vector for the C3, mechanism must belong to the A2 representation of that group; the permutations describing the concerted process and a degenerate Cope rearrangement both qualify as proper generators by our criteria. As we also deduce elsewhere on nuclear spin symmetry grounds [ 131, however, the D3h symmetric intermediate proposed by Hoffmann et al. [18], and Yoneda et al. [19], is rejected here as well, as the mediator of the six-fold scrambling; its symmetric structure implies that all permutations in the LH group of the six structures are proper generators. This is impossible for any LH group, since no irreducible representation has all negative characters. Note that the McIver-Stanton rules themselves exclude this intermediate, since the transition vector is part of a degenerate pair. As before, we defer the question of whether the D3h structure is in fact the most stable form of barbaralyl in solution.

In Goldstein and Kline’s CIIH:I cation (VI) [12,20] the LH group is isomorphic with the direct product C20C40C,, ; if an irreducible representation in the direct product group is to be nondegenerate, the irreducible representation in each of the factor groups must be nondegenerate, and the McIver criteria must hold in each factor. We can choose generators of form B in C2, B in C4, and A2 in C,,, and our choice of proper generators is limited to (afcd) (b) (gkjih)”. (The * refers to an inversion of all coordinates through a point.) This generator agrees precisely with the path predicted by a Salem-Wright analysis [21] of the HOMO- LUMO transition density. It is worth noting that just as in “C50” C6H: the proper generator for CllHT1 cannot correspond to a simple five-fold rotation of the five-membered ring. The five-fold rotation has a negative character only in degenerate representations.

Of course the five-fold rotations in C6H: and CI1HT1 were never seriously considered contenders for the rearrangement mechanism. On the whole, chemical intuition precedes and is merely reinforced by the extension of the McIver- Stanton rules to fluxional systems. However, we wish to describe two systems for which our extended symmetry rules produce surprising results.

CSH: Cation

The tricyclo [2.1.0.02*s], or homotetrahedryl cation can attain complete degeneracy of CH moieties via a sequence of 1,2 shifts [17]. One evident choice of generators of the LH group is a “shift” permutation and the “rotation” corresponding to the static-symmetry C2 axis. When all powers and products of these permutations are evaluated, one finds that the LH group is identical with that of the PX5 pseudorotations, that is, 9’,, with 120 elements. According to our previous argument, therefore, (13, 2), (1,4), and (2,3) are the classes which may contain proper generators. The structure of C5H: has been studied by ab initio and semiempirical computations, [22,23] and a static symmetric C4, species (Fig. 4a) has been shown to occupy a relative minimum on the potential surface. The operation (2345), i.e., the C4 operation on the square pyramid with CH fragment 1 at the apex, is a symmetry operation corresponding not to a reaction but to a


U

.-

<I

Figure 4. ORTEP plots of (a) CNDO optimized structure for C,, square pyramidal C,H:, (b) direct collapse path such that the classical cationic center becomes the apex of a square pyramid, (c) the favored Berry rotation of homotetrahedryl, and (d) the favored motion for the bicyclopentenyl structure, a turnstile rotation.

bodily rotation. According to our rules, if this motion is nonreactive it must have a positive character in the antisymmetric representation in Ys.

Since this is not the case, we are led to interpret the (1,4) operation as a reactive motion, which in turn forces us to reject the C,, static species as a participant in the processes producing five-fold scrambling in CSH:. Indeed, all computations show that there are more stable “cyclopentadienyl” forms of C5H:, which are not four-fold symmetric. This rejection does not imply that the C4u species is short lived or unstable; it means that it cannot be used to explain rapid five-fold degenerate rearrangements.

The motion explored by Stohrer and Hoffmann [23] is represented as (134) and has a positive character in the anti-symmetric irreducible representation of Y5. It passes through the classical structure shown in Figure 4b,c,d as a transition state, according to their computations. If a single equatorial fragment (say # 2) is susceptible to replacement, and the LH group for the dynamics becomes isomorphic with C2u [operators E, (142), (35)*, and (142) (35)*], only then is the motion proposed by Stohrer and Hoffmann a proper single-step interconversion, having a correct character in the B1 representation. But in this event, not all five positions can become equivalent.

To describe details of rearrangements of the C5H: system, we turn from group theoretic predictions of the permutation symmetry of the easy reaction to approximate computation of the reaction path. In general, for a ten atom system


3N - 6 = 24 independent internal coordinates would be required to describe fully the reactive motion of the system. We can avoid much wasteful exploration of high energy reaches of this potential surface by concentrating on the low energy paths connecting minima on the surface. Chemical intuition and previous computation help us to locate these minima approximately. To define more exactly these minima, we have constructed a computer program which computes not only the molecular energy within the CNDO approximation but also the analytic gradient of the energy, a 3N-component vector. If an initially chosen geometry is in the vicinity of an extremum in the total energy, the gradient points in the direction of steepest descent toward the extremum. Energy gradient computations are familiar from the work of McIver [4], but one warning should be noted at this point; the gradient has the symmetry of the initial nuclear framework so that symmetry-breaking motions will be overlooked. Furthermore, the gradient does not necessarily lead to an energy minimum; it leads to a position where all first derivatives of the energy are zero (a BGV) [3]. This position may be a relative minimum, a saddle point or pass, a shelf or planar region, or conceivably a relative maximum. One can diagnose the nature of the extremum by reference to the second derivative of the energy, d2E/dq, dq] . We construct this force constant matrix by numerical differentiation of the force vector:

Klf = a2E/aq1 = cfi (4, + 4,) - f i (qf )I/dq, Z= (qt + d q ~ ) -6 (41 ) I / ~ Z a2E/aqlaq, = afi/aq, = af&,

The degree to which the latter equality is met, i.e., the degree to which the force constant matrix is symmetric, is a measure of the adequacy of the numerical manipulation. We choose dq, = 0.03 Bohrs. Generally the force constant matrix is symmetric to within 1% to 5% in routine calculations. Diagonalization of a symmetrized constant matrix K, = (K, +Kf,)/2 produces as roots “canoni- cal” force constants Fkk and as associated eigenvectors normal coordinates Qk which allow the energy to be written in the compact form E - Eo + C k d E / d Q k d ~ k + 4 C k F k k ( d Q k ) 2 . If no force constants are negative when the forces d E / d Q k are zero, the energy is at a minimum. If one and only one force constant is negative, the system is at pass, that is a suitable candidate for a transition state. If two or more force constants are negative, the system is at a relative maximum, and a reactive motion will bypass the point in question since a lower energy route can always be found.

We turn our attention first to the C,, structure which Stohrer and Hoffmann find to be the most stable C5H: structure. In agreement with previous CNDO, MINDO, and ab initiu results [22, 231, our computations predict the CdU structure symmetry to occupy a minimum in the surface. It is interesting to note that distortions away from perfect C4U symmetry are very easy: a large number of force constants cluster near zero, including motions producing a pseudo- bicyclopentyl system and also the motion proposed by Stohrer and Hoffmann to effect the scrambling via a CZv homotetrahedryl species.


In our study of Stohrer and Hoffmann’s homotetrahedryl “transition state” [23], we found that, indeed, the motion continuing the (134) rotation directly toward the C40 species possessed a negative force constant, consistent with their result. We were interested to find two other negative force constants, one of which corresponds to direct collapse of the bicyclobutyl fragment to a square pyramid and positioning of the classical cationic center at the apex of the square pyramid (Fig. 4d). This result means that the C2u homotetrahedryl species is a relative maximum and that rearrangements must bypass this structure. Presumably the curvature in the vicinity of this hilltop provides a clue to the path by which interconversions can occur. The most negative force constant is associated with an interesting motion corresponding to a pseudorotation. A view of this motion is given in the ORTEP plot (Fig. 4c) [26]. In this diagram we might first consider CH fragments 2 and 5 to be the apices of a trigonal bipyramid and positions 1,3, and 4 to make up an isosceles-triangular equator, with 3 , 4 being the short side. As the motion proceeds, sites 1 and 3 become the new apical fragments, while 2,4, and 5 assume the equatorial position. Where C1 was once the classical cationic center, Cs becomes the new charged center. Note that this Berry-like pseudorotation mechanism, (4) (1235) in permutation language, provides a means by which a Cqa molecule with C1 as the unique carbon could become a C4u molecule with C4 as the unique carbon. This is to be contrasted with Stohrer and Hoffmann’s proposal, that either C3 or C, was the original unique carbon in a C4u structure and that C4 or C3 respectively must be the unique carbon in the C4v structure resulting from the rearrangement.

Setting aside the direct collapse mode of homotetrahedryl to the CLV species, let us consider the two competitive paths from the C3-apical to the C4-apical C4” species. The homotetrahedryl structure is a maximum in the space defined by motion along the Berry pseudorotation, BPR, and along the Stohrer-Hoff mann path, SH. It is conceivable for a system to pass from reactant to product indepen- dently by BPR or SH over this maximum. The paths are independent and orthogonal, since they correspond to independent nondegenerate roots of the force constant matrix. Of course, we expect that in an attempt to bypass the maximum, the system will mix these distinct paths. The product may be produced in various forms distorted from C4” depending on the proportions of mixing of the two paths.

Because published computations [23, 241 indicate that “bicyclopentenyl” or “cyclopentadienyl” structures may well be more stable than the C4V species, we have computed an optimized structure starting with Dewar’s proposed geometry [22] (Fig. 4d). M I N D O / ~ shortens the “bridge” bond C2-C3 by about 5% relative to Dewar’s M I N D O / ~ value of 2.1 A, while CNDO shortens the c2-c3 distance considerably, to 1.64 A. Force constant computations on the cwo-opt imum structure produce a negative force constant corresponding to the motion drawn in Figure 4d. This motion in which the C1C2C3 triangle rotates relative to the C4-C5 line is very similar to the turnstile motion proposed by Gillespie et al. [8] for PX5 systems. Its permutation representation (123) (45) is consistent with the group theoretic requirement of an odd permutation in Y5. Following this turnstile


motion leads to a variety of highly condensed species: we did not consider that CNDO can describe such systems realistically and did not trace out a reaction path. It was interesting to note that a step along the direct path from the “bicyclopentenyl” structure toward the C4, species does not stabilize the system; there is no tendency to pass spontaneously to the C,, species from the bicyclopentenyl structure.

Ethyl Cation

C2H: systems scramble deuterium labels rapidly under solvolytic conditions. The symmetry arguments are almost identical for the ethyl cation as for the other AB5 systems discussed above. If the framework is fixed except for the migrating proton-that is, no internal rotation is permitted- a relevant LH symmetry group contains the identity and a permutatioh corresponding to the transfer, and is isomorphic with Y2 (or Cs). The direct 1,2-shift then is a proper generator since the (12) (or A’) irreducible representation attaches a negative character to the permutation corresponding to the transfer, (ab) (14) (32).* However, if the methyl group rotates more easily than the transfer occurs, which is the case according to minimal basis set computation [25], the molecular symmetry group is -‘p, the direct transfer is in class (1, 22), and is not a proper generator of Y5, though it was a proper generator of C,. The transfer can occur if attended by a simultaneous internal rotation, producing an operator in class (2, 3). This possibility seems to have been overlooked in reported computations on ethyl cation [25], partly because of the great interest in the stability of the nonclassical ion, and partly because the dimensionality of the surface to be computed is larger for the simultaneous rotation and proton shift than for the direct inversion. We have made preliminary tests of the proposed path by CNDO computations on the potential, including energy gradients and the curvature of the surface. It becomes evident that the C2, bridged structure is not the transition state for the lowest energy proton transfer. CNDO and M I N D O / ~ computations predict that the CZv structure is much more stable than any ‘‘classical” open form of the carbenium ion. This prediction might be dismissed as an expression of the prejudice of ZDO

methods in favor of compact systems; however, the best computations on ethyl cation [26], including estimates of correlation energy, also predict that the C2, structure is a relative minimum within C, symmetry. If the C2= structure is a minimum, it is by definition not a transition state but perhaps an intermediate. CNDO and M I N D O / ~ estimates of force constants for optimized CZu structures indicate that symmetry-breaking (to C2) is very easy; M I N D O / ~ predicts that the most stable bridged structures are of C2 symmetry, while CNDO predicts that the ClU structure occupies a broad valley (Fig. 5). The possibility that the C, plane corresponds to a slight ridge in a wide valley (BGV) cannot be dismissed [3]. While the indication of the ridge for the C2= structure is ambiguous, force constant

* A “proper generator” is defined as having the following properties: (1) element of Longuet- Higgins (LH group of system; (2) interchanges reactant and product in a single step; (3) must have one character in some nondegenerate irreducible representation of LH group.

GROUP THEORY AND REACTION MECHANISMS 41 1

A

Figure 5. Schematic representation of the energy hypersurface for the ethyl cation bridging and symmetry-breaking motions.

computations on a zero-gradient open “classical” ion indicate a negative force constant for a motion combining progress toward the bridged region with internal rotation. Following this motion for a short step and following the gradient toward the bridged region leads us to a zero-gradient region where the C-C-H bridge triangle is isosceles but where the CH2 planes are not symmetrically disposed. Apparently the CH2 plane motion is not strongly enough coupled with the proton bridging to be concerted with it. At all stages in the reactions, as described by CNDO, symmetry breaking by internal rotation is very easy, costing perhaps less than 1 kcal even for substantial motions. On the basis of these computations we cannot exclude the C, symmetry preserving interconversion passing through an intermediate CZv structure, but must recognize the possibility of internal rotation and symmetry breaking accompanying rearrangement perhaps occurring prior to formation of the fully bridged species.

In summary, we have shown that an extension of the McIver-Stanton rules on the symmetry of transition vectors linking products with reactants to systems with ornately branched reaction graphs is made possible by use of LH group theory. Although intuitively proposed reaction mechanisms are often consistent with the group theoretic paths, with the aid of this method and energy gradient and curvature computations some new mechanistic possibilities are recognized for the


interesting (CH): system and the Czv homotetrahedryl species previously proposed as a transition state for interconversion of square pyramidal CSH: species is excluded as a transition state.

Acknowledgment

C. T. gratefully acknowledges the support of the University of Virginia Chemistry Executive Committee and the Division of Academic Computing for a generous grant of computer time. T. D. B. thanks his Virginia hosts for their continued hospitality.

[51 1 1 1 1 1 1 1 ~411 4 2 1 0 0 -1 -1 ~321 5 1 -1 -1 1 1 0 [3i21 6 0 0 0 -2 0 1 [2211 5 -1 -1 1 1 -1 0 ~ 2 1 ~ 1 4 -2 1 0 0 1 -1 US] 1 -1 1 -1 1 -1 1

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Received February 13, 1976

Documents

Group theory and reaction mechanisms: Permutation theoretic prediction and computational support for pseudorotation modes in C2H and C5H rearrangements