j APPLIED CATALYSIS A: GENERAL

ELSEVIER Applied Catalysis A: General 146 (1996) 207-223

Theoretical study of the mechanism of branching rearrangement of carbenium ions

M. Boronat a p. Viruela b A. Corma a,*

a lnstituto de Tecnolog& Qulmica UPV-CSIC, Universidad Polit~cnica de Valencia, c~ Camino de Vera s~ n, 46071 Valencia, Spain

b Departament de Qulmica F&ica, Universitat de Valencia, c~ Dr. Moliner 50, 46100 Burjassot (Valencia), Spain

Abstract

Owing to the practical interest of the acid catalyzed isomerization reactions of hydrocarbons, the mechanism of the branching rearrangements of C4H ~- and C5H+1 carbenium ions has been studied theoretically using ab initio methods which include electron correlation and extended basis sets. It has been found that the protonated cyclopropane-type species does not appear as a common intermediate for these reactions, since it is a transition state and not a minimum on the potential energy surfaces studied. In the case of C4H ~- cation, the protonated methyl-cyclopropane ring is the transition state for the carbon scrambling reaction in the secondary n-butyl cation, while the isomerization of n-butyl cation into t-butyl cation occurs via a primary cation. The activation energies calculated assuming this mechanism are in very good agreement with those obtained experimentally. For the branching rearrangement of n-pentyl cation two reaction paths have been considered. In the first one the secondary n-pentyl cation is converted through the 1,2-dimethyl- cyclopropane ring into the secondary 3-methyl-2-butyl cation, which is converted into the t-pentyl cation by a 1,2-hydrogen shift. In the second one the secondary n-pentyl cation is directly converted into the t-pentyl cation through a primary monobranched cation. Comparison of the calculated activation energies for both mechanisms with the experimental value indicate that this reaction does not occur via the primary cation as was the case for n-butyl cation, but occurs via the protonated 1,2-dimethyl-cyclopropane ring.

Keywords." Carbenium ions; Cracking; Isomerization; Mechanism; Rearrangement

1. Introduct ion

Catalytic transformations of hydrocarbons such as isomerizat ion, alkylat ion

and cracking are processes of great importance for the chemistry of petro leum

* Corresponding author.

0926-860X/96/$15.00 Copyright 1996 Elsevier Science B.V. All rights reserved. PII SO926-860X(96)O0160-3

208 M. Boronat et al. / Applied Catalysis A: General 146 (1996) 207-223

[1-3]. The most active and widely used solid acid catalysts for these reactions are silica-alumina and zeolites either exchanged with transition metals [4-6] or in their pure acid forms [7-9]. It is generally accepted that the interaction of hydrocarbons with solid acids results in the formation of carbenium ions [10,11 ]. Consequently, it has been assumed that the mechanism of heterogeneous reactions on solid acids is similar to that of homogeneous reactions in superacid media, although the influence of the solid acid catalyst on the formation and reactivity of the carbenium ion is not explicitly considered in this formal mechanism.

The mechanism of homogeneous isomerization reactions in superacid media involves three steps: formation of the carbenium ion either by protonation of an alkene or by hydride transfer from an alkane, rearrangement of this carbenium ion, and deprotonation or hydride ion abstraction to give the rearranged hydro- carbon. The carbenium ion rearrangements involved in the second step may be classified as branching and non-branching. The classical mechanism for the non-branching rearrangements, in which the degree of chain branching remains the same, supposes them to proceed by a succession of 1,2-hydrogen and alkyl shifts via secondary ions as intermediates. The branching rearrangements, which are about a thousand times slower, involve a decrease or an increase in the degree of chain branching. For this type of rearrangements, a mechanism with only 1,2-hydrogen and alkyl shifts would necessarily include primary carbenium ions as intermediates. This is not consistent neither with the experimental fact that n-butane is not isomerized at an observable rate by HF/SbF 5 to isobutane under conditions where n-pentane and n-hexane are rapidly converted into their branched isomers [12], nor with the finding that the rate of scrambling or isomerization of n-butane-l-13C to n-butane-2-13C is comparable to that of isomerization of n-pentane to isopentane [13]. Consequently, the mechanism proposed by Brouwer [14,15] which includes a protonated cyclopropane ring as intermediate has been accepted.

According to this mechanism (see Scheme 1), the positive charge on C 2 carbon atom attacks the C 4 carbon atom and a protonated cyclopropane ring is formed. The opening of the cyclic intermediate at one of the other two sides of

\ I \ / \1 \ / Y --C 1 \+ 1c3\ / q,, ~/c,3,,) /

C2 C 4 ,,/C2 "--~--- C4 ~. R I I "R H +

,.I I.. ..I cl/_ - - --C 1 ci\+ /c3- \c + /

/C2- -C4QR ~ . i I2--C4...R

>cN3 ,,I __~Cl [C2__~4j --C3~+ I / / [ ~'R ~ f2 --C4~. R

Scheme 1.

M. Boronat et al./ Applied Catalysis A: General 146 (1996) 207-223 209

the ring results in the formation of a secondary monobranched carbenium ion if R is an alkyl group. If R is an hydrogen atom, i.e., for C4H ~- cation, the opening of the cyclic intermediate at side a leads to a n-butyl cation in which a terminal and a non-terminal carbon atoms have changed positions (C scrambling reaction), but the opening of the intermediate at side b becomes more difficult because it would lead to a primary carbenium ion.

The accepted values for the activation energies for the carbon scrambling process in the n-butyl cation and for the conversion of n-butyl into t-butyl cation are 7.5 and 18.0 kcal/mol respectively [16,17]. The similarity of the last value to the activation energy for the carbon scrambling in the isopropyl cation, 15.7 kcal/mol [18], is consistent with the supposition that the branching isomeriza- tion reaction of n-butyl cation passes through the high-energy primary isobutyl cation. The low barrier obtained for the scrambling process, however, suggests that this reaction passes through a protonated methyl-cyclopropane species, more stable than the primary isobutyl cation. For the rearrangement of the t-pentyl cation to the n-pentyl cation an experimental activation energy value of 18.3 kcal/mol is reported by Brouwer [14,15]. This reversible rearrangement leads to interchange of the methyl and methylene carbon atoms and, conse- quently, to scrambling of the methyl and methylene protons in the observable t-pentyl cation. From PMR spectroscopic measurements on the t-pentyl cation at high temperatures Saunders and Rosenfeld have obtained a more precise value of 18.8 kcal/mol for the activation energy of this proton scrambling process [19]. The isomerization of the secondary 3-methyl-2-butyl cation into the tertiary 2-methyl-2-butyl or t-pentyl cation (the last step in Scheme 1) has been found to have an activation energy value of 2.1 kcal/mol [20].

In order to establish the mechanism of the branching rearrangements of carbenium ions, we present in this paper a complete theoretical study of the potential energy surfaces of C4H ~- and C5H/1 cations which include geometry optimization and characterization of the stationary points at correlated levels.

2. Computational details

All ab initio molecular orbital calculations in this work were performed on an IBM 9021/500-2VF computer and on IBM RS/6000 workstations of the University of Valencia using the Gaussian 88 [21] and Gaussian 92 [22] computer programs.

The geometry of the stationary points on C4H ~- and C5H/1 potential energy surfaces was first fully optimized by using the Hartree-Fock procedure and the 6-31G * basis set [23,24] (HF/6-31G* ) which has polarization functions (d-type) on non-hydrogen atoms. Afterwards, electron correlation was included by means of the second-order Moeller-Plesset perturbation theory that takes into account the core electrons. Two types of calculations were carried out: single point

210 M. Boronat et al./Applied Catalysis A: General 146 (1996) 207-223

calculations on the HF optimized geometries using the MP2 treatment and the 6-31G * basis set (MP2/6-31G * / /HF /6 -31G * ), and complete geometry opti- mization of all stationary points at this correlated theoretical level (MP2/6- 31G * ). The Berny analytical gradient [25] and the eigenvalue following [26,27] methods were used for the minima and transition states geometry optimizations, respectively. All HF/6-31G * and MP2/6-31G * stationary points were charac- terized by calculating the Hessian matrix and analyzing the vibrational normal modes. The relative energies at these two levels were corrected by the zero point energy (ZPE) obtained from frequency calculations.

3. Results and discussion

3.1. C 4 H9 +

According to Scheme 1, there are four structures involved in the mechanism of branching rearrangement of C4H ~- cation: the secondary n-butyl cation, the protonated methyl-cyclopropane ring, the primary isobutyl cation and the ter- tiary isobutyl cation. The geometry of these four structures has been completely optimized without any symmetry restriction.

The tertiary isobutyl cation D is the most stable minimum on C4H ~- potential energy surface both at the HF/6-31G * and MP2/6-31G * theoretical levels. As can be seen in Fig. 1, the conformation adopted minimizes the repulsions between the hydrogen atoms and also between the C -H and C-C bonds.

.479) 1.491

1.447 (1.403),f \2.364 (1.923)

\~ "~1.288 A

1.518(1.S11) 1.494 ) 1.496

B

~1.517) ~1.473(1.459)

C D

Fig. 1. Structures of the C4H ff cation. The HF/6-31G* bond lengths are given first, followed by the MP2/6-31G* values in parentheses.

M. Boronat et al. /Appl ied Catalysis A: General 146 (1996) 207-223 211

The structure of the secondary n-butyl cation has already been studied at many standard theoretical levels by Schleyer et al. [28]. They found that both the partially methyl-bridged form A (shown in Fig. 1) and a trans-hydrogen bridged form (not shown) are minima on the potential energy surface, structure A being slightly more stable than the hydrogen-bridged one at every considered theoreti- cal level. Since apart from being the most stable, structure A shows a strong positive charge on a secondary carbon atom, it seems more suitable to induce the cycle formation process and consequently we have taken it as the starting point for the carbon scrambling and branching isomerization reactions.

In the present work the geometry of structure A has been completely optimized at the HF/6-31G* and MP2/631G* levels and, as was previously reported by Schleyer, both stationary points have been found to be minima on the potential energy surface. The bond lengths depicted in Fig. 1 show the changes in geometry due to the inclusion of electron correlation. The degree of methyl-bridging becomes more important when electron correlation is included, as can be observed from the !engthening of the C3-C 4 bond from 1.584 ,~ at the HF/6-31G * level to 1.653 A at the MP2/6-31G * level, and also from the fact that the C2C3C 4 angle is closed from 102.4 at the HF/6-31G * level to 77.5 at the MP2/6-31G * level.o This implies an important ShOortening of the C2-C 4 bond length from 2.364 A at the HF/631G * level to 1.923 A at the MP2/631G * level, while the C 1-C3 bond length remains the same (2.582 and 2.549 ,~). The calculated energy differences between the secondary A and the tertiary D butyl cations (or relative energies) summarized in Table 1 are in good agreement with the experimental values of the enthalpy of rearrangement of n-butyl to t-butyl cation, which are 15-17 [29-31] and 14-15 [32,33] kcal/mol in gas phase and in solution, respectively.

Three different structures were proposed by Wiberg and Kass [34] for the protonated methyl-cyclopropane ring: a corner-protonated species formed by adding a proton to the methine carbon atom, an edge-protonated species in which the proton is shared by two neighboring methylene carbon atoms, and a structure formed by adding a proton to a methylene carbon atom which resulted to be the open form of the secondary n-butyl cation A. Although at the Hartree-Fock level the corner-protonated species was found to be more stable than the edge-protonated one, the inclusion of the electron correlation reversed

Table 1 Calculated relative energies (kcal/mol) of the stationary points found on C4H ~- potential energy surface

Method A B C D

HF/6-31G * 13.6 31.5 32.9 0.0 HF/6-31G * +ZPE 14.7 32.8 33.3 0.0 MP2/6-31G * / /HF /6 -31G * 13.9 20.2 34.2 0.0 MP2/6-31G * 11.0 19.9 33.6 0.0 MP2/6-31G * +ZPE 12.9 21.5 34.1 0.0

212 M. Boronat et al. / Applied Catalysis A: General 146 (1996) 207-223

the relative stability of these two forms. Besides, the optimized bond lengths reported by Wiberg and Kass for the corner-protonated species suggest that it could be best described as a complex between CH3CH ~- cation and CH 2 =CH 2 molecule, without any chemical meaning in the rearrangements studied. Conse- quently, we have only considered in this work the edge-protonated form of the methyl-cyclopropane ring (structure B in Fig. 1).

The HF/6-31G* and MP2/631G* optimized geometries of B depicted in Fig. 1 are nearly identical, and exhibit a partially distorted C S symmetry. The C2-C 3 and C2-C 4 bond lengths in the ring and the C-H bond lengths of the almost symmetrical hydrogen bridge are equivalent at both theoretical levels. The only noticeable change that electron correlation introduces in the geometry of B is a slight shorteningoOf the C3-C 4 bond length from 1.763 A at the HF/6-31G * level to 1.717 A at the MP2/6-31G * level. Despite this similarity between the correlated and uncorrelated optimized geometries of B, the calcu- lated relative energies summarized in Table 1 are strongly dependent on the theoretical level used. At the HF/6-31G * level the relative energy of B is 31.5 kcal/mol, and 32.8 kcal/mol with the ZPE correction. The inclusion of the electron correlation at the MP2/6 -31G* / /HF /6 -31G* level stabilizes this structure and yields a relative energy value of 20.2 kcal/mol, very similar to the values obtained at the MP2/6-31G * and MP2/6-31G * + ZPE levels, 19.9 and 21.5 kcal/mol, respectively.

These energetic changes are not surprising if we take into account that the inclusion of the electron correlation by the Moeller-Plesset perturbation treat- ment preferentially stabilizes the non-classical bridged structures [35]. Since the HF/6-31G* optimized geometry of B corresponds to a non-classical bridged structure, the single point calculation at the MP2/6-31G * / /HF /6 -31G * level is able to introduce all the stabilization due to electron correlation. The MP2/6-31G * geometry optimization cannot increase the degree of bridging in B and therefore there is no stabilization with respect to the single point calculation observed at this level of theory.

What is more surprising is the result obtained from the analysis of the vibrational normal modes. This analysis indicates that structure B is a transition state on C4H ~- potential energy surface both at the HF/6-31G* and MP2/6- 31G* levels of theory. The negative frequency is clearly associated to the movement of the bridged hydrogen atom towards one of the methylene carbon atoms to give the secondary n-butyl cation A. The fact that structure B is not a minimum but a transition state on the potential energy surface means that the protonated meth2)l-cyclopropane ring cannot be a true intermediate of the scrambling and branching isomerization reactions of n-butyl cation as suggested by Brouwer, but only a species through which the scrambling reaction passes.

Finally, the structure of the primary isobutyl cation C has been calculated. Taking as a starting point the optimized geometry of the tertiary isobutyl cation D, the distance between one of the hydrogen atoms attached to C 3 and C2 has

M. Boronat et al. / Applied Catalysis A: General 146 (1996) 207-223 213

been slowly shortened from 2.037 ,~ to 1.075 A, already corresponding to a primary cation. In each calculation the C2-H distance has been fixed and all other parameters have been allowed to fully optimize. Then, the geometry of the primary cation obtained has been completely reoptimized using the eigenvector following transition state search technique. The HF/6-31G * and MP2/6-31G * optimized geometries of C are very similar, and the relative energies summa- rized in Table 1 are all of them about 33 or 34 kcal/mol. The force constant calculations indicate that at both theoretical levels structure C is a transition state with only one imaginary vibration frequency.

Taking into account all these data and the idea that a transition state must be connecting two minima on a potential energy surface, a new mechanism, shown in Scheme 2, is proposed for the carbon scrambling and branching isomerization reactions of n-butyl cation. According to this new mechanism the secondary n-butyl cation is the starting point for the two reactions that, from the first moment, follow different reaction paths. When the positive charge on C 2 carbon atom in minimum A attacks C 4 carbon atom two different processes can take place: (a) a simultaneous strengthening of the C2-C 4 bond and breaking of the C3-C 4 bond in A directly leads to transition state C and from this, a shift of the hydrogen atom from C 2 to C 3 leads to minimum D. The primary cation is the transition state for the conversion of n-butyl into t-butyl cation; (b) strengthen- ing of the C2-C 4 bond in A together with an elongation of a C4-H bond length leads to transition state B. If the C2-C 3 bond is then broken and the hydrogen atom of the bridge moves to C 3, the minimum A' (equivalent to A) is reached and the carbon scrambling reaction has occurred. The protonated cyclopropane ring is, at least in the case of C4H ~- cation, not an intermediate but only the transition state for the carbon scrambling reaction.

Fig. 2 shows the energetic profile for the two reactions studied and Table 2 summarizes the calculated activation energies together with experimental data. The activation energy for the carbon scrambling process, Eal, has been calcu-

\1 \ i --CiN+ i c3 \ /

7 2 74 .

/ L A j i< B A'

X [-\1 I+ -It \1 I/ I -q ~c3- -q\+/c3-- / ~ J ~ 12 L " "7< i7'\

C D

Scheme 2.

214 M. Boronat et al. /Applied Catalysis A: General 146 (1996) 207-223

AE (kcal/mol)

35.0 -

30 .0 -

25 .0 -

20 .0 -

15 .0 -

10 .0 -

5 .0 -

0 .0 -

B /--'~\ / / ~ Eal \N

/ \ / r.-J ~ J A' A

I I

I I

i I E~a I

I I

c

k

N

Fig. 2. Energy profile corresponding to the scrambling and isomerization reactions of the secondary n-butyl cation. The tertiary ion has been taken as the origin of energies.

lated as the energy difference between the protonated methyl-cyclopropane ring B and the secondary n-butyl cation A. The HF/6-31G * and HF/6-31G * + ZPE values, 17.9 and 18.1 kcal/mol respectively, are too high as compared with the experimental value of 7.5 kcal/mol. As we have seen before, electron correla- tion stabilizes structure B and the calculated activation energies at the MP2/6- 31G * / /HF /6 -31G *, MP2/6-31G * and MP2/6-31G * + ZPE levels, 6.3, 8.9 and 8.6 kcal/mol respectively, are much closer to the experimental value. These results indicate that the Hartree-Fock level is not adequate to study mechanisms in which non-classical species are involved, and that inclusion of electron correlation is essential to treat these species correctly. The activation energies for the direct (n-butyl ~ t-butyl), Ea2, and reverse (t-butyl ~ n-butyl), Ea3, isomerization reactions have been calculated as the energy differences between the primary cation C and structures A and D, respectively. The calculated activation energies are quite similar at all theoretical levels, between 18.6 and 22.6 kcal/mol for the direct reaction and between 32.9 and 34.2 kcal/mol for the reverse reaction, and in good agreement with the experimental values.

Table 2 2 + 13Calculated and experimental activation energies (kcal/mol) for the processes depicted in Fig. 2

Method gal Ea2 Ea3 HF/6-31G * 17.9 19.3 32.9 HF/6-31G* +ZPE 18.1 18.6 33.3 MP2/6-31G * / /HF /6 -31G * 6.3 20.3 34.2 MP2/6-31G * 8.9 22.6 33.6 MP2/6-31G * +ZPE 8.6 21.2 34.1 Exp. 7.5 18.0 32.0

M. Boronat et al. / Applied Catalysis A: General 146 (1996) 207-223 215

3.2. C5H1~

According to Brouwer's mechanism (see Scheme 1), the following structures have to be localized and characterized on CsH~- ~ potential energy surface: the secondary n-pentyl cation, the protonated 1,2-dimethyl-cyclopropane ring, the secondary 3-methyl-2-butyl cation, the tertiary 2-methyl-2-butyl or t-pentyl cation and the transition state for the hydrogen shift that converts the secondary branched cation into the tertiary one.

Both at the HF/6-31G * and MP2/6-31G * levels the tertiary t-pentyl cation I is the most stable minimum on CsH~- ~ potential energy surface. The optimized bond lengths depicted in Fig. 3 indicate that the t-pentyl cation is not fully classical. A partial bridging between the C4-C 5 bond and the positive charge on

I .S53

1 .526~'~ 'q

~ 74)

~.519(1.510)

747)

1.462 (I .489)

1.59Z

.836) ~"~0

G H

_ 1.471 ~x "474 (1.464)

I

1.589~ 1"529 (I"516)

J

Fig. 3. Structures of C5H~1 cation. The HF/6-31G* bond lengths are given first, followed by the MP2/6-31G* values in parentheses.

216 M. Boronat et al. /Appl ied Catalysis A: General 146 (1996) 207-223

Table 3 Calculated relative energies (kcal/mol) of the stationary points found on CsHll + potential energy surface

Method E F G H I J

HF/6-31G * 13.6 29.5 12.9 15.6 0.0 34.2 HF/6-31G * +ZPE 14.2 30.2 13.0 14.5 0.0 34.9 MP2/6-31G * / /HF /6 -31G * 13.6 18.5 11.8 12.0 0.0 35.0 MP2/6-31G * 10.3 17.1 7.2 14.8 0.0 34.5 MP2/6-31G * +ZPE 11.7 17.8 8.0 14.4 0.0 34.1

C 2 carbon atom can be deduced from the lengthening of the C4-C 5 bond to 1.565 ,~ at the HF/6-31G* level and 1.581 A at the MP2/6-31G * level, and also from the calculated C2C4C 5 angle values of 106.3 and 101.5 at the HF/6-31G * and MP2/6-31G * levels, respectively. The structure of the t-pentyl cation has already been studied by comparing the ~3C chemical shifts of the C + carbon atom calculated by IGLO using ab initio geometries with the experimen- tal values, and our results are in complete agreement with those previously reported [36].

The secondary n-pentyl cation E has also been found to be a minimum on the potential energy surface at the HF/6-31G* and MP2/6-31G* theoretical levels. The optimized bond lengths depicted in Fig. 3 show the same tendencies that were observed in the secondary n-butyl cation A. Inclusion of electron correlation at the MP2/6-31G * level lengthens the C3-C 4 bond from 1.599 to 1.656 ~, and closes the C2C3C 4 angle from 102.1 to 78.3 , i.e., increases the degree of methyl-bridging. This is reflected in the important shortenin~ of the Ca-C 4 bond length from 2.366 ,~ at the HF/6-31G* level to 1.941 A at the MP2/6-31G* level, while the C3-C 5 bond length remains nearly constant (2.545 and 2.587 A at the HF/6-31G * and MP2/6-31G * levels respectively). These geometric changes are reflected in the relative energies summarized in Table 3. The HF/6-31G*, HF/6-31G* + ZPE and MP2/6 -31G* / /HF /6 - 31G* calculated energies, 13.6, 14.2 and 13.6 kcal/mol respectively, are very similar. The increase in the degree of methyl-bridging produced by the inclusion of the electron correlation in the MP2/6-31G * optimization stabilizes structure E and at the MP2/6-31G* and MP2/6 -31G*+ ZPE levels the calculated relative energies are 10.3 and 11.7 kcal/mol respectively.

The third minimum found on CsH~] potential energy surface is the secondary 3-methyl-2-butyl cation, structure G in Fig. 3. At the HF/6-31G* level the C2-C 3 bond length value of 1.592 A and the C3C2C 4 angle value of 98.1 are indicative of a partial bridging between the C 2-C3 bond and the positive charge on C 4 carbon atom, similar to that previously reported for the t-pentyl cation. At the MP2/6-31G* level, however, the C2-C 3 bond is lengthened to 1.733 ~,, the C3C2C 4 angle is closed to 70.9 and the C3-C 4 bond is shortened from 2.295 A at the uncorrelated level to 1.836 .A. The degree of methyl-bridging becomes so important that this structure could be best described as an unsym-

M. Boronat et al./Applied Catalysis A: General 146 (1996) 207-223 217

metrical comer-protonated 1,2-dimethyl-cyclopropane ring. The relative ener- gies summarized in Table 3 reflect the importance of the geometric changes produced when electron correlation is included in the calculations. The HF/6- 31G* and HF/6-31G* +ZPE calculated relative energies, 12.9 and 13.0 kcal/mol respectively, are about 2 kcal/mol too high in relation to the experimental value of 11 kcal/mol reported by Collin and Herman [20] for the energy difference between the secondary 3-methyl-2-butyl and the tertiary 2-methyl-2-butyl cations. The MP2/6-31G*/ /HF/6-31G* calculated value, 11.8 kcal/mol, is slightly lower than the two uncorrelated values, probably due to the fact that there is a partial bridging in the HF/6-31G * optimized geometry of structure G. But the MP2/6-31G * and MP2/6-31G * + ZPE values are 3.8 and 3 kcal/mol respectively too low in relation to the experimental value. This overestabilization of structure G at the best levels of theory used in this work can be explained if we take into account that the inclusion of the electron correlation using the Moeller-Plesset treatment preferentially stabilizes non- classical bridged structures [35] and, as can be seen in Fig. 3, the degree of bridging in G is more important than in any of the other structures.

According to Brouwer's mechanism (Scheme 1), the isomerization of the linear E to the branched G secondary cations passes through an edge-protonated cyclopropane ring F. Taking as a starting point the optimized geometry of the linear cation E, two variables have been simultaneously controlled in order to obtain structure F: the C2C3C 4 angle and the C3-H bond length. In each calculation these two variables have been fixed and all other parameters have been allowed to fully optimize. The geometry of the cyclic structure obtained has been completely reoptimized at the HF/6-31G * and MP2/6-31G * levels using the eigenvalue following transition state search technique and the two stationary points have been characterized by force constant calculations. As in the case of C4H ~- cation, they have been found to be transition states, with only one imaginary vibration frequency associated to the movement of the bridged hydrogen atom.

The two optimized geometries of structure F depicted in Fig. 3 are ne~ly equivalent, with C-C bond lengths in the ring of 1.450, 1.553 and 1.807 A at the HF/6-31G * level and 1.443, 1.562 and 1.747 A at the MP2/6-31G * level. The effect of alkyl substitution on the structure of the protonated cyclopropane ring can be clearly observed in the different symmetry exhibited by the hydrogen bridge in B (protonated methyl-cyclopropane ring) and F (protonated 1,2-dimethyl-cyclopropane ring). While the two C-H bond lengths in structure B are equivalent, the hydrogen bridge in F is markedly unsymmetrical, with C-H bond lengths of 1.176 and 1.478 A at the HF/6-31G * level and 1.174 and 1.490 ~, at the MP2/6-31G* level.

The relative energies of structure F reproduce the tendencies previously observed for structure B. The HF/6-31G* and HF/6-31G * + ZPE calculated values are high, 29.5 and 30.2 kcal/mol respectively. The single point calcula-

218 M. Boronat et al. / Applied Catalysis A: General 146 (1996) 207-223

tion at the MP2/6-31G*//HF/6-31G* level strongly stabilizes this cyclic species, yielding a value of 18.5 kcal/mol, while the MP2/6-31G* and MP2/6-31G* + ZPE calculated values are only slightly lower, 17.1 and 17.8 kcal/mol respectively.

The last step in Brouwer's mechanism is the hydrogen shift that converts the secondary branched cation into the tertiary one. Starting from the optimized geometry of I, the distance between C 2 and the hydrogen atom that is going to migrate has been slowly shortened from 2.12 A to 1.10 A and the geometry of the obtained structure has been then completely reoptimized at the HF/6-31G * and MP2/6-31G* levels using the eigenvector following the transition state search technique. The two stationary points have been characterized by force constant calculations and they have been found to be transition states on their respective potential energy surfaces, showing only one imaginary vibration frequency. At the HF/6-31G* level the relative energy of structure H is 15.6 kcal/mol, and 14.5 kcal/mol with the ZPE correction. The single point calculation at the MP2/6-31G * / /HF/6-31G * level stabilizes this hydrogen- bridged structure and yields a relative energy value of 12.0 kcal/mol. However, the value obtained from the MP2/6-31G * optimization is higher, 14.8 kcal/mol, and 14.4 kcal/mol with the ZPE correction. The HF/6-31G * and MP2/6-31G * optimized bond lengths depicted in Fig. 3 explain this energetic values. At the HF/6-31G * level, the optimized geometry of H corresponds to an unsymmetri- cally hydrogen-bridged structure with a C-C bond length of 1.411 A and two

o

C-H bond lengths of 1.189 and 1.582 A. At the MP2/6-31G* level the optimized C-C bond length value of 1.437 A and the C-H bond length values of 1.118 and 1.961 A are indicative of a lesser degree of hydrogen bridging and consequently the structure is slightly destabilized at this level of theory.

Taking into account all these results, the mechanism depicted in Scheme 3 is proposed for the branching rearrangement of n-pentyl cation. According to it, strengthening of the C2-C 4 bond in minimum E together with weakening of the C3-C 4 bond and migration of one of the hydrogen atoms attached to C 4 to a bridged position between C 3 and C a leads to transition state F. From this, braking of the C 3-C4 bond together with migration of the bridged hydrogen atom to C 3 leads to minimum G. Then, the hydrogen atom attached to C 2 migrates to C 4 through transition state H and minimum I is reached. It is important to note that this calculated mechanism is not equivalent to that empirically proposed by Brouwer, the main difference being the nature of the

,,t L \ i \ ] I t , , , / F , , I / lt,.i - -c \ c ( c / [\~1 ~'- - / c3 / c, 3 | --c3

E F (3 H I

Scheme 3.

M. Boronat et al. / Applied Catalysis A: General 146 (1996) 207-223 219

\1 \ / I / --ctx, +/c3 \ / c5 - -

7 /k

\+ / I t c3

\ I / \ l _ _ _ _ _

/ CI 7 2 C4"~5_._ j ~7 "C5--i\ J I

Scheme 4.

protonated 1,2-dimethyl-cyclopropane ring. Our results indicate that this cyclic species cannot be an intermediate as affirmed by Brouwer, because it is a transition state and not a minimum on CsH~- 1 potential energy surface.

A second mechanism, equivalent to that calculated for the branching isomer- ization of n-butyl cation, has been considered in this work. As can be seen in Scheme 4, a simultaneous strengthening of the C2-C 4 bond and breaking of the C3-C 4 bond in the secondary n-pentyl cation leads to a primary monobranched cation J and from this, a shift of an hydrogen atom from C 2 to C 3 leads to the t-pentyl cation. The structure of the primary cation has also been calculated. Starting again from the optimized geometry of the t-pentyl cation I, the distance between C 2 and one of the hydrogen atoms attached to C 3 has been slowly shortened from 2.12 ,~ to 1.10 A and then, using the eigenvalue following method, its geometry has been completely reoptimized at the HF/6-31G * and MP2/6-31G * levels. Characterization of the two stationary points by force constant calculations indicate that the primary pentyl cation J is a transition state at both theoretical levels. Since the HF/6-31G* and MP2/6-31G* optimized geometries of structure .1 depicted in Fig. 3 are very similar and fully classical, the inclusion of the electron correlation in the calculations produces no stabiliza- tion of this structure in relation to the t-pentyl cation and the calculated relative energies of J summarized in Table 3 are all of them similar, between 34 and 35 kcal/mol.

AE (kca l /mo l )

35.0-

30.0-

25.0-

20.0 -

15.0-

10.0-

5.0-

0.0-

F

l/if I ~\ E Ea4 H

G \ \

J / \

I I

/ I

I /

I I I

I I I

I /

I \\ t I

\ \ \ \ \ \ \

~a6 \ \ \

E

Fig. 4. Energy profile corresponding to the branching isomerization of the secondary n-pentyl cation. The tertiary ion has been taken as the origin of energies.

220 M. Boronat et al. /Appl ied Catalysis A: General 146 (1996) 207-223

Fig. 4 shows the energetic profile for the branching isomerization of the n-pentyl cation. The activation energy for the rearrangement of the t-pentyl cation to the n-pentyl cation, for which experimental data are available, can be calculated as the energy difference between the primary transition state J and the tertiary minimum I if the reaction path depicted in Scheme 4 is followed. In the mechanism of Scheme 3, the rate determining step is the conversion of the branched G into the linear E secondary pentyl cations, and consequently the activation energy for the global process is the energy difference between transition state F and minimum I. The calculated activation energies together with available experimental data are summarized in Table 4. At the HF/6-31G * and HF/6-31G * + ZPE levels the calculated activation energies for the two mechanisms are similar, 29.5 and 30.2 kcal/mol for Ea4 and 34.2 and 34.9 kcal/mol for E,6, and too high as compared with the experimental value of 18.8 kcal/mol. When electron correlation is included the energy of the primary cation J experiments no changes, and the calculated values for Ea6 are again between 34 and 35 kcal/mol. However, the transition state F is highly stabilized by inclusion of electron correlation and consequently the calculated values for Ea4 at the MP2/6-31G*//HF/6-31G*, MP2/6-31G* and MP2/6-31G* + ZPE levels are lowered to 18.5, 17.1 and 17.8 kcal/mol respectively. Compari- son of the calculated activation energies for the two mechanisms considered with the experimental value of 18.8 kcal/mol indicates that the branching isomerization of the n-pentyl cation does not occur through the primary cation, as was the case for the n-butyl cation, but occurs via the protonated 1,2-di- methyl-cyclopropane ring, following the reaction path shown in Scheme 3.

As already told, this mechanism consists of two steps: the conversion of the secondary linear cation E into the secondary branched cation G discussed above, and the conversion of G into the tertiary cation I. The activation energy for this process, Ea5 in Fig. 4 and Table 4, can be calculated as the energy difference between transition state H and minimum G. The Ea5 calculated values at the HF/6-31G * and HF/6-31G * + ZPE levels, 2.7 and 1.5 kcal/mol respectively, compare well with the experimental value of 2.1 kcal/mol. At the MP2/6- 31G */ /HF/6-31G* level the obtained value is too low, 0.2 kcal/mol, while the MP2/6-31G* optimization yields a too high barrier for this process, 7.6 kcal/mol, and 6.4 kcal/mol with the ZPE correction. The reported energy

Table 4 Calculated and experimental activation energies (kcal/mol) for the processes depicted in Fig. 4

Method Ea4 Ea5 Ea6

HF/6-31G * 29.5 2.7 34.2 HF/6-31G * + ZPE 30.2 1.5 34.9 MP2/6-31G * / /HF /6 -31G * 18.5 0.2 35.0 MP2/6-31G * 17.1 7.6 34.5 MP2/6-31G * +ZPE 17.8 6.4 34.1 Exp. 18.8 2.1 -

M. Boronat et al. /Appl ied Catalysis A: General 146 (1996) 207-223 221

difference between the secondary 3-methyl-2-butyl and the t-pentyl cations, 11 kcal/mol, together with the activation energy value for the hydrogen shift that converts this secondary branched cation into the tertiary one yields an energy barrier for the conversion of the tertiary I into the secondary G cations of 13.1 kcal/mol, which corresponds to the energy difference between transition state H and minimum I. Exceptuating the HF/6-31G * result, which is 2.5 kcal/mol too high, the calculated relative energies of structure H are between 12.0 and 14.8 kcal/mol, i.e., they are basically correct. Consequently, the source of the difference between the calculated and the experimental values of Ea5 must be in the calculated energy of the secondary branched cation G. As has been previously discussed, the Moeller-Plesset perturbation treatment overestabilizes non-classical bridged structures such as G. The 3 or more kcal/mol of discrepancy between the calculated and the experimental relative energies of G reported before added to the 1-2 kcal/mol found for transition state H are the reason that explains the too high Ea5 values obtained when the MP2 treatment is used.

4. Conclusions

A theoretical study of the potential energy surfaces of C4H ~- and C5H~1 cations including geometry optimization and characterization of the stationary points at correlated levels has been carried out in order to establish the mechanism of branching rearrangement of carbenium ions. The results obtained suggest alternative mechanisms to that empirically proposed by Brouwer for these reactions, the main difference being the nature of the protonated cyclo- propane-type species. According to Brouwer's mechanism, this cyclic type of structure is an intermediate, from which two different routes lead to two different products. However, both the protonated methyl-cyclopropane ring and the protonated 1,2-dimethyl-cyclopropane ring have been found to be transition states and not minima on Call ~- and CsH~- 1 potential energy surfaces respec- tively, and consequently they cannot be true intermediates but only species through which the reactions pass.

The new mechanism proposed for the carbon scrambling and branching isomerization reactions of the n-butyl cation (Scheme 2) supposes that, from the first moment, the two reactions follow different reaction paths. The protonated methyl-cyclopropane ring is the transition state for the carbon scrambling reaction, and the isomerization of the linear n-butyl cation into the branched t-butyl cation occurs through a primary cation. However, the branching rear- rangement of n-pentyl cation does not occur through a primary cation as shown in Scheme 4, but it follows the reaction path depicted in Scheme 3. The secondary n-pentyl cation is converted through the protonated 1,2-dimethyl- cyclopropane ring into the secondary 3-methyl-2-butyl cation and then, an

222 M. Boronat et al. / Applied Catalysis A: General 146 (1996) 207-223

hydrogen shift converts this branched secondary cation into the tertiary one. This mechanism can be extrapolated to higher aliphatic carbenium ions because, as can be observed in Scheme 3, it does not seem probable that addition of a methyl (or alkyl) group to the C5 carbon atom introduces any important change in the relative energy or nature of the different species involved.

From the results obtained at the different levels of calculation it can also be concluded that the inclusion of the electron correlation in the geometry optimiza- tions and in the characterization of the stationary points is essential for the study of this type of reaction mechanisms, in which non-classical bridged structures are involved. Thus, the MP2/6-31G* would be the lowest acceptable level of calculation.

Acknowledgements

The authors thank the Centre de Informhtica and Departament de Qufmica Ffsica of the University of Valencia for computing facilities. They thank C.I.C.Y.T. (Project MAT 94-0359) and Conselleria de Cultura, Educaci6 i Cibncia de la Generalitat Valenciana for financial support. MB thanks the Conselleria de Cultura, Educaci6 i Ci~ncia de la Generalitat Valenciana for a personal grant.

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