Hybrid orbitals evolution and reaction path in addition reactions

Journal of Molecular Structure (Theochem), 260 (1992) 24%251 Elsevier Science Publishers B.V., Amsterdam

Hybrid orbitals evolution addition reactions*

249

and reaction path in

F. Zuccarello” and G. Del Reb “Dipartimento

di Scienze Chimiche,

(Italy) bDipartimento

di Chimica, Universith

Universith di Catania, Viale A. Doria, 6, 95125 Catania di Napoli,

Via Mezzocannone,

4, 80134 Napoli (Italy)

(Received 4 November 1991)

Abstract As a sequel to previous work, the maximum localization procedure for constructing optimum hybrids has been tested with a view to setting up a general algorithm permitting the determination of a “hybridization controlled” reaction path (HCRP) for bimolecular reactions. The test has concerned the “bond straightening” condition used to predict the evolution of an initial conformation of the reactants. The reaction studied has been the Diels-Alder addition of ethylene to cis-butadiene. The trend of the results is encouraging, Work in progress on the overlapbond-length relation should allow a complete description of the HCRP of many reactions.

INTRODUCTION

As interest in quantum chemical studies shifts towards larger and larger molecules, the need to recover simple criteria as guidelines in treating problems such as conformation and reactivity is becoming more and more pressing. In this paper, we attempt a brief analysis of a problem that is central in that context, namely the problem of reaction paths. We try to tackle the problem in the spirit of C.A. Coulson, who always tried to apply simple models with great physical content. It is well known that a general recipe for finding the optimal reaction path for a given reaction requires work with a complicated multidimensional potential energy surface and with a multiplicity of initial conditions. The resulting tendency of computational studies is to try to determine a Correspondence to: F. Zuccarello, Dipartimento di Scienze Chimiche, Catania, Viale A. Doria, 6, 95125 Catania, Italy. *Dedicated to the memory of Professor Charles A. Coulson.

0166-1280/92/$05.00 0

1992 Elsevier Science Publishers

Universita

B.V. All rights reserved.

di

250

minimum energy path (MEP). Even so, there appears to be a need for some quicker procedure for determining a MEP, based on a model enabling the evolution of the structure of a reacting system to be followed step by step, from the initial configuration through the possible transition state to the final product or products. A few attempts in this direction have already been made, using the hybridization procedure proposed by one of us in 1963 [l] and later applied or discussed in several papers [2-91. This procedure is based on the requirement that the A0 basis should be hybridized so as to produce as localized an overlap matrix as possible, under the constraint that the hybridized orbitals (called MLHOs) form an orthonormal set; furthermore (for given bond lengths), the geometry is the more stable the closer the valence angles (as predicted by the hybrids) are to the actual bond angles. Briefly, we recall the essential steps of that procedure. (i) For a given molecular geometry, calculate the overlap matrix on the basis of canonical atomic orbitals. (ii) Diagonalize the overlap matrices associated with the various pairs related to each atom, and select the eigenvectors corresponding to the largest diagonal element. (iii) Orthogonalize the set of the hybrids obtained in step (ii), after having assigned to each a weight equal to the square of the pertinent overlap. In this paper, we shall briefly discuss the scope and prospects of the “hybridization controlled reaction path (HCRP)” thus obtained, on the basis of new conceptual and numerical results. GENERAL DESCRIPTION OF THE PROCEDURE

The conceptual support of the procedure leading to an HCRP can be traced back to the Hellmann-Feynman theorem. For our purposes, that theorem can be stated as follows: in the equilibrium configuration of a molecule, the forces acting on a nucleus must cancel. This is so because “once we have calculated P = ‘3”the forces which act on the nuclei can be computed classically; the forces which operate are exactly those which would act if the positive nuclei were embedded in a distribution of negative charge of density P electron/unit volume” [lo], and in a stable configuration the nuclei are at equilibrium. This theorem can be used to justify the principle of “straight bonds”. For an arbitrary geometrical configuration of the nuclei, the MLHOs are not in general directed along the internuclear axes, and the bonds are “curved’. (Of course, hybrids pointing towards the appropriate nuclei could always be constructed, bu they would not satisfy intra-atomic orthogonality, so they would not describe individual electron pairs as the theory of valence requires.) Now, the negative-charge density associated with a

251

linear combination of MLHOs occupied by two electrons may be approximately treated as if it were concentrated in the centres of the hybrids or in a single centre (depending on the internuclear distance). It follows that if the bonds are bent, the local arrangement of electrostatic forces is largely that of two positive charges A and B held together by one or two negative charges located at a point close to the centre of the segment AB, but displaced perpendicularly to it by some length r. This situation is possible if the molecule is not at liberty to rearrange itself without increasing its energy for other reasons (as in a triangular cycle). Otherwise, the molecule will move so as to bring the negative cloud between the two atoms A and B onto the line between them because this arrangement is the one which corresponds to the lowest electrostatic energy. The straight bond criterion suggests that, if a molecule or a supermolecule turns out to have MLHOs not directed along the pertinent internuclear axes for a given geometry, it will tend to change its geometry so as to align its atoms in the directions of the hybrids. In addition to the angles, the internuclear distances will also change towards optimal values. As soon as the molecular geometry changes, however, different MLHOs will apply. Therefore, unless the new hybrids already obey the straight bond criterion, the molecule or complex under study will not relax completely towards the geometry predicted by the initial distribution of MLHOs, but will gradually change until it comes to a geometry in which the bonds are indeed straight. This iterative process (which we shall call “bond straightening”) can be considered as an algorithm for determining the equilibrium geometry of a group of atoms; indeed, a study in that sense was published a long time ago [4]. However, its best use could be the determination of a reaction path without the computation of a potential energy surface. It has the advantage of corresponding to a clear-cut physical picture of the evolution of the given system under the action of the electrostatic forces to which the Hellmann-Feynman theorem applies. The above idea requires careful exploration before becoming a standard procedure. It has been applied to the opening of the cyclobutene ring [3] and to the reaction of H, with CO adsorbed on an iron surface [8]. The results have been satisfactory, even though the computational process has been simplified by allowing the geometry to relax completely at each step. In this paper, which presents a further step towards the establishment of the procedure, we report results for a bimolecular addition reaction leading to two possible products. The points to be clarified are as follows. (i) Is hybridization really sufficiently sensitive to geometric details to allow determination of a reaction path, even in the case of two unsaturated polyatomic molecules a significant distance apart? (ii) Does the result depend on the initial relative positions of the reactants and, if so, how?

(iii) Is the formation of strained cyclic systems allowed? As is pointed out in the discussion, the answer also throws light on the dependence of the reaction effectiveness on the relative positions of the two partners. CHOICE OF THE SYSTEM AND RESULTS

In order to answer the above questions as simply as possible, we have chosen a reaction only slightly more complicated than the isomerization of cyclobutene: the Diels-Alder addition of ethylene to cis-butadiene. Our procedure should be able to describe the evolution of hybrids from sp3to sp2, with formation of K bonds, and vice-versa. The mechanism of the reaction under consideration is still somewhat obscure, because there is no general consensus about the alternatives: concerted or stepwise. The former mechanism passes through the formation of a symmetric complex (Fig. l(a)); the latter involves a biradical intermediate (Fig. l(b)) for which three conformations (one trans and two gauche) are possible. Recent MC SCF computations [ll] and an analysis of the density Laplacian distribution [12] favour the stepwise mechanism. We think that the two mechanisms are not exclusive, but depend on the initial collision complex formed by the two molecules. For the exploratory purposes mentioned above and for further clarifying the chemical problem itself, even prior to the realization of a complete HCRP programme (which would include bond energy and bond length calculations), we have applied the straight-bond criterion to a butadiene molecule in the presence of an ethylene molecule, in the relative positions shown in Figs. l(a) and l(b). In Fig. l(a) the ethylene molecule is supposed to hover above the butadiene plane parallel to it, with its centre 2 A along the line vertical to the CH,-CH, line; Fig. l(b) is obtained from Fig. l(a) by shifting the ethylene molecule parallel to itself until one of its carbon atoms is outside the butadiene region and the other hovers above carbon atom 4, still at a height of 2A. The choice of 2A as the intermolecular distance is dictated by the necessity of starting with reasonably high overlap values. In the initial step the geometrical parameters of interacting molecules were the standard ones [13];the overlap matrix from which we obtain MLHOs was calculated over STO-3G orbitals. Table 1 allows a comparison of the isolated-molecule results, initial results and final results for the degrees of hybridization and the corresponding overlap values. In both initial geometries the p character of the hybrids involved in CC and CH bonds increases at the expense of the px orbital(s) involved in the new bond(s), which acquire a measure of s character and point away from the plane of the remainder of the molecule. The hybrids

253

Fig. 1. Conformation and atom numbering of ethylene&-butadiene complexes: (a) symmetric complex (concerted mechanism); (b) asymmetric system (biradical mechanism). The geometrical characteristics of the complexes after some cycles of angular modification to give the best matching possible between bonds and corresponding hybrids, following the straightbond criterion, are shown. In the first step, the single interacting molecules were assigned the standard geometrical parameters [13].

involved in the CH bonds now point in the direction opposed to that of the emerging bonds. The C-C bond of ethylene appears somewhat weakened, more because of the variation in the hybridization of the carbon atoms than because of the strain, as one would expect in the symmetric complex. Concerning the final geometries and hybrids, it is important to keep in mind that we have not optimized the bond lengths but have left them unchanged; the bond angles have been changed to the values obtained for the corresponding hybrids, until convergence was reached, if possible. In the case of the biradical complex resulting from an initial asymmetric arrangement, convergence was actually reached and the final complex obeyed the straight-bond criterion; in contrast, the cyclohexene ring appeared to be strained. This strain is certainly due to the fact that the distances are not those of the equilibrium geometry, and therefore the angles at Cl and C2 are much

254 TABLE

1

Typology

of hybrids

Hybrid”

orbitals

Ethylene

and their overlap

cis-Butadiene

Degree of hybridizationd 1.65gd W2)

integrals

Symm. complex

Asymm.

Init. geom.b

Init. geom.b

Fin. geom.”

1.664

1.734

169.178

28.721

2.205

2.233

2.205

2.233

h2(1)

1.659

h2(3) h2(9) h2(10)

2.205 2.205

hl(3) hl(7) hl(8)

h3(1)

complex Fin. geom.”

1.659

1.687

2.330

2.205

2.184

2.330

2.205

2.184

1.664 169.178

1.734 28.721

1.691 79.755

1.821 16.809

2.233

2.330

2.247

2.233

2.330

2.247

2.394 2.394

198.808

28.818 79.567

17.575

1.773 2.143

1.818

1.788

1.839

2.191 2.354

2.180 2.180

2.369

h3(2) h3(4)

1.755

h3(11) h3(12)

2.140 2.140

h4(3) h4(5)

1.693 2.209

1.693 2.209

1.662

1.693

2.311

2.209

1.663 2.310

h4(13)

2.155

2.155

2.103

2.155

2.104

0.750499 0.665710

2.162

2.369

Overlap values” h1(2kh2(1)

0.754033”

0.753855

0.750860

0.752269

h1(7)-H7 hl(8)-HS

0.665284 0.665284

0.663609 0.663609

0663328 0.663328

0.665284 0.665284

0.173213

0.220233

h1(3)h3(1) h2(3)-h3(2) h2(9)H9 h3(4)h4(3)

0.748822

0.747462

0.748146

h3(lltHll h3(12&H12

0.668694 0.668694

0.668639 0.667143

0.667785 0.667785

h4(5)-h5(4) h4(13)H13

0.653624 0.668386

0.653623 0.668386

0.650721 0.669333

“hi(j) denotes

the hybrid

bInitial geometry

orbital

described

of atom i directed

by ethylene

parameters are equal to isolated molecules [13]. ‘Geometries obtained after angular deformations pertinent hybrids. dExponent of hybrid in expression sp”. “Overlap

integrals

between

towards

and cis-butadiene

0.251899 0.663364 0.747123 0.666739 0.666739 0.653624 0.668386

0.665710 0.291800 0.662377 0.747809 0.665232 0.665232 0.650728 0.669318

atom j.

planes at 2.0 A, whose geometrical

that minimize

angles

between

bonds

and

hybrids.

too small to be matched by angles between hybrids: indeed, if the hybrids are computed for cyclohexene in the experimental equilibrium geometry, the bonds appear to be straight. An estimate of the effect of “straightening” the bonds by the procedure

described above can be obtained by computing the corresponding molecular energies. To that purpose STO-3G energies have been computed for the initial and final configurations of the two complexes of Fig. 1; these are given in Table 2, where the charge bond order analysis is also reported. It appears that there is a decrease in energy of about 20 kcal mall* per new bond between the initial and final forms. This value is quite significant because we are only dealing with incipient formation of the bonds in question, the distances have been left unaltered. It is significant that the whole gain in stability is due to the nuclear energy: the electronic energy increases because the very weak bonds formed cannot compensate for the loss in strength of the bonds in the isolated molecules. Details of the latter effect are provided by the population analysis shown in Table 2. CONCLUSIONS

Together with the simpler applications in previous papers [3,4,8], the results reported here confirm the two main assumptions made in the procedure proposed above: “bond straightening” does correspond to a decrease in energy even if it involves weakening of existing bonds and indeed points to the final equilibrium geometry of a reaction complex; the iterative scheme proposed converges within the limits imposed by other geometrical constraints (ring closure). The question of bond lengths (solved by an ad hoc recipe in ref. 8) remains open. Of course, the procedure here proposed can only be autonomous with respect to other computations if it includes a rule for determining the bond lengths that correspond to a given angle distribution. Adaptation of existing optimization programs is one way of providing a general answer, but we are working on a simpler overlap dependent energy-distance relation. Why there should be two different end products of the reaction under consideration, depending on the initial arrangement of the complex is a different sort of question. The answer is that this is largely an artefact of the computation, which preserves the initial symmetry, because, if the open biradical is allowed to come slightly out of the plane, the ring compound is again the end product. Nevertheless, it can be given a physical interpretation by recalling that in any bimolecular reaction different geometries of the collision complex have different probabilities of proceeding towards the final product. Indeed, by following the HCRPs beyond the initial conditions, different activation energies might be found, thus allowing a choice between the concerted and stepwise mechanisms. Finally, the point might be raised that, even in the case of a MEP, the energy will in general increase from the situation when the two reactants are isolated, reach a maximum and then decrease. How do things stand for

256 TABLE 2 Net charges, bond populations and energies (a.u.) Asymm. complex

Symm. complex Initial ge0m.b Net charge 1 Cl c2 c3 c4 c5 C6 H7 H8 H9 HlO Hll H12 H13 H14 H15 H16 Bond population Cl-C2 Cl-C3 C2-C3 c3c4 c4c5 C&C6 Cl-H7 Cl-H8 C%H9 C2-HlO C3Hll C3H12 C4H13 CbH14 C&H15 C&H16 Energies Nucl. energy Electr. energy Total energy

- 0.148 -

0.148 0.103 0.060 0.060 0.103 0.056 0.051 0.056 0.051 0.071 0.075 0.058 0.058 0.071 0.075

Final geom.’ -

0.136 0.136 0.106 0.068 0.068 0.106 0.056 0.058 0.056 0.058 0.069 0.072 0.055 0.055 0.069 0.072

0.602 - 0.002

0.560 0.063

0.586 0.441 0.586 0.406 0.395 0.406 0.395 0.395 0.399 0.395 0.395 0.395 0.399

0.553 0.461 0.553 0.398 0.392 0.398 0.392 0.392 0.393 0.394 0.394 0.392 0.393

236.633677535 - 466.572295601 - 229.938618067

232.728635215 - 462.728109307 - 229.999474092

Initial geom.b -

0.157 0.097 0.093 0.098 0.060 0.145 0.053 0.052 0.087 0.073 0.088 0.077 0.051 0.059 0.058 0.051

Final geom. -

0.141 0.107 0.104 0.081 0.060 0.141 0.054 0.053 0.076 0.073 0.073 0.078 0.054 0.060 0.057 0.055

0.598

0.579

- 0.007 0.587 0.414 0.590 0.394 0.394 0.403 0.396 0.405 0.398 0.396 0.393 0.395 0.394

0.033 0.573 0.413 0.592 0.394 0.394 0.397 0.393 0.397 0.396 0.395 0.394 0.396 0.395

222.460382962 - .452.436437682 - .229.976054720

218.548882622 - 448.558622014 - 230.009739392

“Figures come from STO-3G calculations. bInitial geometry described by ethylene and cis-butadiene planes at 2.0 A, whose geometrical parameters are equal to isolated molecules [13]. ‘Geometries obtained after angular deformations that minimize angles between bonds and pertinent hybrids.

257

the HCRP? The answer is that (as can be seen from Table 2) the isolated molecules are in fact more stable than the initial complexes; the stability gain is only between the complex geometries before and after bond straightening. ACKNOWLEDGMENTS

This paper reports work performed by the authors within their respective projects supported by the Italian Ministry of Universities and Research and by the Italian National Council of Research. REFERENCES 1 2 3 4 5 6 7 8 9 10 11

G. Del Re, Theor. Chim. Acta, 1 (1963) 188. G. Del Re, U. Esposito and M. Carpentieri, Theor. Chim. Acta, 6 (1966) 36. A. Rastelli, A.S. Pozzoli and G. Del Re, J. Chem. Sot., Perkin Trans. 2, (1972) 1571. A.S. Pozzoli, A. Rastelli and M. Tedeschi, J. Chem. Sot., Faraday Trans. 2,69 (1973) 256. G. Del Re and C. Barbier, Croat. Chim. Acta, 57 (1984) 787. V. Barone, G. Del Re, C. Barbier and G. Villani, Gazz. Chim. Ital., 118 (1988) 347. G. Del Re, J. Mol. Struct. (Theochem), 169 (1988) 487. G. Del Re, C. Barbier and G. Villani, Nuovo Cimento D, 12 (1990) 103. G. Del Re and F. Zuccarello, Croat. Chim. Acta, 64 (1991) 449. R. McWeeny, Coulson’s Valence, Oxford University Press, Oxford, 1979. F. Bernardi, A. Bottoni, M.J. Field, N.F. Guest, I.H. Hillier, M.A. Robb and A. Venturini, J. Am. Chem. Sot., 110 (1988) 3050. 12 G. Gatti, M. Barzaghi, L. Bonati and D. Pitea, Studies in Physical and Theoretical Chemistry, Vol. 62, Elsevier, Amsterdam, 1989, p. 401. 13 L.E. Sutton, Interatomic distances, Special Publication No. 18, The Chemical Society, London, 1965.