J. Phys. Chem. B 1997, 101, 10069-10074
10069
Theoretical Study on the Mechanism of the Hydride Transfer Reaction between Alkanes and Alkylcarbenium Ions M. Boronat,† P. Viruela,‡ and A. Corma*,† Instituto de Tecnologı´a Quı´mica UPV-CSIC, UniVersidad Polite´ cnica de Valencia, aV/ dels Tarongers s/n, 46022 Valencia, Spain, and Departament de Quı´mica Fı´sica, UniVersitat de Vale` ncia, c/Dr. Moliner 50, 46100 Burjassot (Valencia), Spain ReceiVed: May 29, 1997; In Final Form: July 26, 1997X
Several density functional theory based methods and the ab initio MP2 method were used to investigate the mechanism of secondary-secondary, tertiary-tertiary, and secondary-tertiary hydride-transfer reactions between alkanes and alkylcarbenium ions. All these reactions were found to proceed through a mechanism consisting of two steps: formation of a stable tight intermediate complex and decomposition of the complex to the products without activation energy. The optimized geometry of all the intermediate complexes is linear, with an open three-center two-electron C-H-C bond that is more or less symmetrical depending on the nature of the involved carbon atoms. By comparison of the results obtained at different levels of theory, it was shown that DFT-based methods provide optimized geometries in very good agreement with those obtained at the MP2 level, but the former fail in describing the energetics of the complex formation reactions studied. While DFT underestimates in some cases the stability of the intermediate complexes, the MP2 results are in very good agreement with the experimental data, indicating the adequacy of this method for studying this type of reaction mechanism. It was also shown that when large systems are involved and geometry optimization and characterization of the stationary points at the MP2 level become computationally too expensive, MP2 single-point calculations on DFT-optimized geometries yield results of comparable quality. Finally, energy profiles along the reaction coordinate were calculated for the three types of hydride-transfer processes considered, and neither a loose complex nor an internal potential barrier was found for any of them.
Introduction A large number of important commercial catalytic processes are catalyzed by acid catalysts. Among them, solid acids such as zeolites account for the largest amount of any type of catalysts used today, with the largest economical impact. From a mechanistic point of view, the acid-catalyzed reactions occur through the formation of carbenium and carbonium ions, which can be formed on Brønsted sites either by hydride abstraction or proton addition. In many of those processes, the reaction proceeds by means of a chain-propagation mechanism where a carbenium ion generated in a product abstracts a hydride ion from a reactant molecule to form the corresponding carbenium ion which will evolve, forming, for instance, branched, alkylated, or cracked products.1 If hydride transfer is important for chain propagation in many acid-catalyzed reactions, this is also of vital importance during secondary reactions occurring during the cracking of gasoil, which is responsible for more than 30% of the total gasoline produced in the world. In this case, the process is called hydrogen transfer and involves the saturation of olefins. This occurs when they adsorb on Brønsted acid sites, forming carbenium ions, and in a following step abstract hydride ions from cycloalkanes to yield paraffins and aromatics. In this case, hydrogen transfer is responsible for changes in the gasoline octane number, as well as for the amount and nature of the coke formed.1 It is then important to understand in depth the detailed mechanism through which the hydride transfer between saturated molecules and carbenium ions occurs. * To whom correspondence should be addressed. † Universidad Polite ´ cnica de Valencia. ‡ Universitat de Vale ` ncia. X Abstract published in AdVance ACS Abstracts, October 15, 1997.
S1089-5647(97)01735-5 CCC: $14.00
Today there is a considerable amount of experimental information on hydride-transfer reactions and other reactions between ionic and neutral molecules in the gas phase.2 By using the technique of radiation chemistry, Ausloos and Lias3 studied the hydride-transfer reactions between tert-butyl ions and 22 larger alkanes having tertiary carbon atoms. They found very low rate constants and small activation energies which were inversely proportional to the exothermicity of the reaction. MeotNer and Field4 investigated the kinetics of the hydride-transfer reactions between C2H5+, C3H7+, t-C4H9+, and t-C5H11+ ions and larger tertiary hydrocarbons using the pulsed high-pressure mass spectrometric technique. These reactions were found to be fast at low temperatures, but above a given transition temperature, which is characteristic of each system, they become slow in agreement with the results of Ausloos and Lias. To explain this behavior, they proposed a consecutive mechanism proceeding through two reaction steps. The first step is formation of a loose complex by the collision of an ion and a neutral molecule. It is assumed that in this loose complex, the ion and the molecule preserve their rotational degrees of freedom and are bonded to each other through ion-induced dipole and dispersion forces. The second step involves the formation of a tight complex having a bridging hydride between the two fragments and in which the relative orientation of the ion and the molecule is fixed. The results obtained for the reaction C3H7+ + i-C4H10 ) C3H8 + t-C4H9+ were later reproduced by Sunner et al.,5 who also tried to clarify whether there is an internal energy barrier separating the loose and the tight complexes involved in the mechanism proposed by Meot-Ner and Field. They could not detect experimentally the intermediate complex C3H7+-C4H10, as was their aim, but observed the formation of other complexes, © 1997 American Chemical Society
10070 J. Phys. Chem. B, Vol. 101, No. 48, 1997
Boronat et al.
C3H7+-C3H8, t-C4H9+-i-C4H10, and t-C4H9+-C3H8, and measured their rate constants and thermochemistry. They also calculated several points on the potential energy surface along the reaction coordinate for the C3H7+ + i-C4H10 ) C3H8 + t-C4H9+ process, and they found neither an internal potential barrier nor a C3H7+-C4H10 loose complex. However, those calculations were carried out using the STO-3G basis set and MNDO-optimized geometries, and as the authors themselves pointed out, the results obtained cannot be very reliable. Other theoretical studies concerning the mechanism of hydride-transfer reactions between alkanes and alkylcarbenium ions found in the literature6,7 employed semiempirical methods such as MINDO/3 or minimal basis sets, and therefore, they are also inconclusive. More recently, an analysis of the hydride transfer in the (CH3-H-CH3)+ system8 and an extensive study of the C2H7+ potential energy surface9 using high-level ab initio methods were reported. We have now carried out a detailed theoretical study of the hydride-transfer reactions
C3H7+ + C3H8 ) C3H8 + C3H7+
(1)
t-C4H9+ + i-C4H10 ) i-C4H10 + t-C4H9+
(2)
In the first step, all geometries have been fully optimized using the B3PW91 functional and the 6-31G* basis set which has polarization functions of the d type on non-hydrogen atoms. Taking the obtained structures as the starting point, all geometries have been completely reoptimized using the B3PW91, B3LYP, and MP2(fu) methods and the 6-31G** basis set which also includes polarization functions of the p type on all hydrogen atoms. To test the effects of the basis set and of the methodology on the relative energies, single-point calculations at the B3PW91/ 6-311G**, MP2(fu)/6-31G*, MP2(fu)/6-31G**, and MP2(fu)/ 6-311G** levels have been performed on the B3PW91/6-31G*optimized structures. The basis set superposition error (BSSE) has been calculated for reaction 3 using the counterpoise (CP)19 technique. Geometry optimizations have been carried out using the Berny analytical gradient method.20 The nature of each stationary point was characterized at the B3PW91/6-31G*, B3PW91/6-31G**, B3LYP/6-31G**, and MP2(fu)/6-31G** theoretical levels by calculating the Hessian matrix and analyzing the vibrational normal modes. Zero-point vibrational corrections to the total energies and standard state enthalpies for comparison with experimental data were also obtained from the frequency calculations.
C3H7+ + i-C4H10 ) C3H8 + t-C4H9+
(3)
Results and Discussion
using both traditional ab initio molecular orbital theory10 and density functional theory11,12 based methods that include electron correlation and extended basis sets. Reactions 1-3 have been chosen to model secondary-secondary, tertiary-tertiary and secondary-tertiary hydride-transfer reactions, respectively. Hydride transfer between alkanes and alkylcarbenium ions is the elementary step responsible for chain propagation of many hydrocarbon transformations catalyzed by solid acids such as zeolites. The main objective of this work is to localize the stable intermediates, that is to say, the tight complexes, through which the hydride-transfer reaction occurs in order to, in a future work, extend this study to larger systems in which the presence of the solid acid catalyst is explicitly taken into account. Therefore, it is also of great importance to test the validity of density functional theory, which includes electron correlation at a much lower computational cost than post-HF ab initio methods, for studying this type of reaction mechanism. This has been done by comparing the results obtained at the different levels of theory with the experimental data previously reported. Finally, we have tried to localize the loose complexes postulated in refs 4 and 5. Computational Details All calculations in this work were performed by means of the Gaussian 9413 computer program on the IBM RS/6000 and SGI Power Challenge L workstations of the University of Valencia. Since the systems under study have nonclassical twoelectron three-center bonds, only methods that include electron correlation have been used in the calculations. In the density functional study, the methods used combine Becke’s hybrid three-parameter exchange functional14 with either the Perdew and Wang15 or the Lee, Yang, and Parr16 gradient-corrected correlation functionals (B3PW91 and B3LYP, respectively), because it has been recently found that hybrid methods that include the Hartree-Fock exact exchange yield results that are comparable to those obtained with post-HF correlated methods.17 In the ab initio molecular orbital study, the electron correlation has been included by means of the second-order Møller-Plesset perturbation theory18 that takes into account the core electrons (MP2(fu)).
This section is structured into two parts. First, the optimized geometries of the intermediate complexes through which reactions 1-3 proceed calculated at different levels of theory are given and compared (Table 1), and the nature of the structures obtained is discussed. Second, the energy and enthalpy changes of the complex formation reactions obtained from the optimizations and from single-point calculations are compared with the experimental data (Table 2) in order to test the quality of the different theoretical levels employed. The influence of BSSE on the energies is discussed, and the energy profiles along the reaction coordinate are reported for the three hydride-transfer processes. Geometries. Although there is no direct experimental information about the structure of the intermediate complexes through which reactions 1-3 are supposed to proceed in the gas phase, an open linear structure for the intermediate (D3CH-CH3)+ was found to be consistent with the experimental data obtained in a crossed beam scattering study of the hydride transfer between CD3+ and methane,21 and therefore, open linear structures were assumed for the intermediate complexes studied in this work. Two possible conformations were initially considered for the (C3H7-H-C3H7)+ complex. In one of them, the two hydrogen atoms attached to the secondary carbon atoms were in a cis conformation, and in the other one, these two hydrogen atoms were in the trans conformation. Their geometries were optimized at the B3PW91/6-31G* level by imposing C2V and C2h symmetry constraints, respectively, and they were characterized by frequency calculations. Both structures were found to be transition states, showing only one imaginary vibration mode associated with rotation of one C3H7 fragment with respect to the other. Complete optimization of the geometry of the (C3H7H-C3H7)+ complex without symmetry restrictions leads to the gauche structure 1 in Figure 1, which was characterized as a true minimum on the B3PW91/6-31G* potential energy surface. In this structure, the hydrogen atom of the bridge is equidistant and quite strongly bonded to the two secondary carbon atoms of the C3H7 fragments, and consequently, it can be considered to be a tight complex. It can be seen in Table 1 that the C--H--C bridge is completely linear at this level of theory and
Alkane and Alkylcarbenium Ion Reactions
J. Phys. Chem. B, Vol. 101, No. 48, 1997 10071
TABLE 1: Optimized Values of the Most Important Geometric Parameters of the (C3H7-H-C3H7)+ (1), (t-C4H9-H-t-C4H9)+ (2), and t-C4H9+-C3H8 (3) Complexes Shown in Figure 1 (Distances in Å and Angles in deg) parameter
B3PW91/6-31G*
B3PW91/6-31G**
B3LYP/6-31G**
MP2(fu)/6-31G**
r(Csec-H) r(Csec-Csec) r(Csec-Cmethyl) ∠(Csec-H-Csec)
1.276 2.544 1.496-1.497 180.0
(C3H7-H-C3H7) 1.272 2.536 1.495-1.496 170.3
1.277 2.547 1.500-1.501 171.0
1.236 2.436 1.497-1.499 160.0
r(Ctert-H) r(Ctert-Ctert) r(Ctert-Cmethyl) ∠(Ctert-H-Ctert)
1.295 2.590 1.506 179.8
(t-C4H9-H-t-C4H9)+ 1.292 2.585 1.505 179.9
1.297-1.296 2.593 1.511 179.9
1.252-1.251 2.504 1.503 179.8
r(Csec-H) r(Ctert-H) r(Csec-Ctert) r(Csec-Cmethyl) r(Ctert-Cmethyl) ∠(Csec-H-Ctert)
1.214 1.391 2.602 1.505 1.498 174.7
t-C4H9+-C3H8 1.215 1.378 2.590 1.504 1.498 174.9
1.204 1.417 2.619 1.511 1.501 175.4
1.207 1.289 2.493 1.500 1.500 173.8
+
TABLE 2: Calculated Relative Energies and Enthalpies with Respect to Separated Fragments of the (C3H7-H-C3H7)+ (1), (t-C4H9-H-t-C4H9)+ (2), and t-C4H9+-C3H8 (3) Complexes (in kcal/mol) theoretical level B3PW91/6-31G* B3PW91/6-31G** B3LYP/6-31G** MP2(fu)/6-31G**b B3PW91/6-311G**//B3PW91/6-31G* MP2(fu)/6-31G*//B3PW91/6-31G* MP2(fu)/6-31G**//B3PW91/6-31G* MP2(fu)/6-311G**//B3PW91/6-31G*
∆E ∆E + ZPE ∆H ∆E ∆E + ZPE ∆H ∆E ∆E + ZPE ∆H ∆E ∆E + ZPE ∆H ∆E ∆E + ZPE ∆H ∆E ∆E + ZPE ∆H ∆E ∆E + ZPE ∆H ∆E ∆E + ZPE ∆H
exptla
1
2
3
-14.5 -13.6 -13.7 -14.7 -13.7 -13.9 -13.6 -12.7 -12.9 -17.4 -16.1 -16.1 -13.6 -12.7 -12.8 -15.4 -14.0 -14.0 -17.1 -15.7 (-16.2) -15.7 (-16.3) -17.8 -16.4 -16.5 -13.6
-8.0 -7.3 -7.4 -8.1 -7.6 -7.7 -7.1 -6.5 -6.7 -15.4 (-14.8) (-15.0) -7.2 -6.6 -6.8 -13.0 (-12.5) (-12.2) -14.9 (-14.4) (-14.5) -16.0 (-15.5) (-15.6) -12.1
-3.4 -2.7 -2.9 -3.5 -2.9 -3.1 -2.7 -2.1 -2.3 -8.9 -7.8 -8.0 -2.7 -2.1 -2.3 -6.5 -5.4 -5.6 -8.1 -7.1 (-7.5) -7.3 (-7.6) -9.2 -8.1 -8.3 -6.6
a Standard state enthalpies for the complex formation reactions. Data from ref 5. b The values in parentheses were calculated using zero-point and thermal corrections obtained from B3PW91/6-31G** frequency calculations.
that the Csec-Cmethyl bond lengths are intermediate between the 1.438-Å calculated for the propyl cation and the 1.527-Å value calculated for propane. The gauche structure 1 is the most stable, but the cis and trans transition states are only 0.5 and 0.2 kcal/mol, respectively, higher in energy, indicating that rotation around the C--H--C axis is nearly free. For the (t-C4H9-H-t-C4H9)+ complex, two different conformations were initially considered, one with the methyl groups in cis and one with the methyl groups in trans. The geometry of both structures was fully optimized without symmetry restrictions at the B3PW91/6-31G* level, and the two stationary points were characterized by force-constant calculations. The trans conformer (structure 2 in Figure 1) is a minimum on the potential energy surface, while the cis structure is the transition state for rotation of one t-C4H9 fragment with respect to the other. The calculated B3PW91/6-31G* activation energy for rotation around the C- -H- -C axis in the (t-C4H9-H-t-C4H9)+ complex is 1.0 kcal/mol. This value is higher than that calculated for the (C3H7-H-C3H7)+ complex, because the steric repulsions between the methyl groups of the two t-C4H9
fragments are more important than those between two C3H7 fragments. The C- -H- -C bridge in the (t-C4H9-H-t-C4H9)+ complex is also linear (see Table 1), and the Ctert-H and CtertCtert bond lengths in it are slightly longer than in the (C3H7H-C3H7)+ complex because the t-C4H9 fragments are bulkier than the C3H7 ones. The calculated Ctert-Cmethyl bond lengths are intermediate between 1.461 Å calculated for the tert-butyl cation and 1.527 Å calculated for isobutane. The secondary-tertiary hydride-transfer process was modeled by the reaction of the sec-propyl cation with isobutane to give propane and the tert-butyl cation. In relation to the reaction mechanism, Sunner et al.5 did not detect in their mass spectrometric study the C3H7+-i-C4H10 intermediate complex through which reaction 3 is supposed to proceed but detected only a relatively loose t-C4H9+-C3H8 complex. Structure 3 in Figure 1 corresponds to this t-C4H9+-C3H8 complex, which was obtained from full optimization of all the geometric parameters of the system at the B3PW91/6-31G* level and was characterized as a true minimum on the potential energy surface. The C- -H- -C bridge of this complex is almost linear (see Table
10072 J. Phys. Chem. B, Vol. 101, No. 48, 1997
Figure 1. Structures of the (C3H7-H-C3H7)+ (1), (t-C4H9-H-tC4H9)+ (2), and t-C4H9+-C3H8 (3) complexes.
1) and markedly unsymmetrical, due to the different natures of the two carbon atoms involved in it. The distance from the central hydrogen atom to the less stable secondary carbon atom is shorter than that to the tertiary carbon atom, indicating that in this complex the C4H9 fragment has more ionic character than the C3H7 one. This in turn affects the Ctert-Cmethyl bond lengths, which are slightly shorter than in the (t-C4H9-H-tC4H9)+ complex, and the Csec-Cmethyl distances, which are elongated in relation to the (C3H7-H-C3H7)+ complex. Although the hydrogen atom of the bridge in this structure is less strongly bonded to the tertiary carbon atom than to the secondary one, the bonding between propane and the tert-butyl cation is site specific. That is to say, propane and the tert-butyl cation cannot undergo free rotation as would happen in a loose complex in which the two fragments are only bonded by ion-induced dipole and dispersion forces. Their relative orientation is fixed, and consequently, the t-C4H9+-C3H8 complex can be considered to be relatively tight. To obtain a C3H7+-i-C4H10 complex, a geometry optimization at the B3PW91/6-31G* level was performed in which the Ctert-H distance was held fixed at a value of 1.1 Å (the C-Hoptimized bond-length value in isolated isobutane) and all other parameters were allowed to fully optimize. The Csec-Ctert obtained distance, 2.463 Å, is shorter than in the t-C4H9+-C3H8 complex. The Ctert-Cmethyl bond lengths, 1.517 Å, are similar to those calculated for isobutane at the same level of theory, 1.527 Å. However, the Csec-Cmethyl bond lengths, 1.487 Å,
Boronat et al. are not as short as in the propyl cation, 1.438 Å, because there is a strong interaction between the C3H7+ fragment and the hydrogen atom of the bridge, reflected in the Csec-H bond length value of 1.365 Å. The restricted C3H7+-i-C4H10 complex was characterized by frequency calculations and was found to be a hilltop showing two imaginary vibration frequencies associated with the movement of the central hydrogen atom out of the linear bridge, and when its geometry was fully optimized without any restriction, minimum 3 was reached. Since we are mainly interested in the intermediate complexes involved in the hydride-transfer reactions, only the three minima shown in Figure 1 have been studied at higher levels of theory. Their geometries have been reoptimized without symmetry constraints at the B3PW91/6-31G**, B3LYP/6-31G**, and MP2(fu)/6-31G** levels, and in all cases, the same conformations as at the B3PW91/6-31G* level have been obtained. It can be seen in Table 1 that all C- -H and C- -C distances of the bridge obtained using the DFT methods are similar. The differences between them are less than 0.01 Å except for the B3LYP/6-31G**-calculated Ctert-H distance in complex 3 which is 0.04 Å longer than the other DFT values. The MP2(fu)/6-31G**-optimized C- -H and C- -C distances of the bridge are slightly shorter than the DFT values, but the differences are in general less than 0.1 Å. The Cbridge-Cmethyl bond lengths are equivalent at all theoretical levels. In relation to the basis set, inclusion of polarization functions on hydrogen atoms has no influence on the calculated bond lengths, and only in the case of the (C3H7-H-C3H7)+ complex has some effect on the Csec-H-Csec angle. This is completely linear at the B3PW91/6-31G* level, close to 170° and 171° at the B3PW91/ 6-31G** and B3LYP/6-31G** levels and has a value of 160° at the MP2(fu)/6-31G** level. Energetics. The calculated relative energies and enthalpies with respect to separated fragments of the (C3H7-H-C3H7)+, (t-C4H9-H-t-C4H9)+, and t-C4H9+-C3H8 complexes are summarized in Table 2 together with experimental enthalpy data reported by Sunner et al.5 The zero-point vibrational corrections to the total energy (ZPE) and the thermal corrections to the standard state enthalpies were obtained from frequency calculations at the B3PW91/6-31G*, B3PW91/6-31G**, B3LYP/631G**, and MP2(fu)/6-31G** levels of theory. In the analysis of the vibrational normal modes, it was found that for the three complexes, there exists one very low frequency (lower than 50 cm-1 in all cases) related to rotation of the alkyl fragments around the C- -H- -C axis. The contribution of this lowfrequency mode was subtracted from the zero-point vibrational correction, and it was substituted in the thermal correction to the enthalpy by the contribution of an unhindered internal rotation, which is RT/2.22 The B3PW91/6-31G**- and MP2(fu)/ 6-31G**-modified zero-point and thermal corrections were used to correct the B3PW91/6-311G**//B3PW91/6-31G* and the MP2(fu)/6-31G*//B3PW91/6-31G*, MP2(fu)/6-31G**//B3PW91/ 6-31G*, and MP2(fu)/6-311G**//B3PW91/6-31G* relative energies, respectively. The calculated energy and enthalpy changes for the reaction of formation of 1 are quite similar at all theoretical levels and in good agreement with the experimental value of -13.6 kcal/ mol, although the MP2(fu)/6-31G** method tends to overestimate the complex formation energy. Inclusion of the zero-point energy correction destabilizes the complex in relation to separated fragments in about 1 kcal/mol at all theoretical levels, and the same occurs with the calculated enthalpy changes, which are nearly identical with the zero-point-corrected energy changes. For the reactions of formation of 2 and 3, the differences between the DFT- and MP2-calculated energies are more important. All DFT relative energies of complex 2 with respect
Alkane and Alkylcarbenium Ion Reactions to separated reactants are equivalent, as in the case of the (C3H7-H-C3H7)+ complex, but are about 4-5 kcal/mol too high in relation to the experimental value of -12.1 kcal/mol. The same is observed for the calculated relative energies of the t-C4H9+-C3H8 complex. The DFT values are similar and about 3-4 kcal/mol higher than the “experimental value” of -6.6 kcal/ mol. On the other hand, the MP2(fu)/6-31G** energy changes calculated for formation of complexes 2 and 3 are slightly overestimated in relation to the experimental values. Inclusion of the zero-point and thermal corrections destabilizes the complexes in ∼ 0.5 kcal/mol at the DFT levels and in ∼1 kcal/ mol at the MP2 level, what implies worse agreement between the DFT results and the experimental results and an improvement of the already better MP2 results. The calculated relative energies and enthalpies of complex 3 have been compared to the -6.6 kcal/mol value reported by Sunner et al. for a t-C4H9+C3H8 complex that they characterize as relatively loose, although in the discussion of the geometric results we have considered this structure to be relatively tight. We think that this comparison is valid in view of the agreement existing between the MP2(fu)/6-31G**-calculated values and the experimental data reported for complexes 1 and 2, indicative of the coherence of the theoretical results. What could be questioned is the experimental consideration of this complex as relatively loose, not only because the theoretical results indicate that it has a relatively tight structure but also because -6.6 kcal/mol is a high value for a complex that is only stabilized by ion-induced dipole and dispersion forces. From these data, it seems that density functional theory based methods are not able to correctly evaluate the energetics of hydride-transfer reactions between alkanes and alkylcarbenium ions, but keeping in mind that our final objective is to include explicitly the presence of the solid acid catalyst in the theoretical investigation of the mechanism of these reactions, and that geometry optimization and characterization of stationary points using the ab initio MP2 method will be prohibitively expensive for such extended systems, we have carried out a series of singlepoint calculations on the B3PW91/6-31G*-optimized geometries using both the B3PW91 and the MP2(fu) methods and different basis sets. The calculated values are listed in Table 2. It can be observed that the B3PW91/6-311G**//B3PW91/6-31G* results are similar to those obtained at the other DFT levels. The energy and enthalpy changes calculated for the formation of the (C3H7-H-C3H7)+ complex are in good agreement with the experimental value, but the stabilities of the (t-C4H9-Ht-C4H9)+ and t-C4H9+-C3H8 complexes are greatly underestimated. On the other hand, the results provided by the MP2 single-point calculations on the DFT-optimized geometries are comparable to those obtained from full optimization at the correlated ab initio level. It can then be concluded that while density functional theory based methods are not able to accurately evaluate the energetics of formation of µ-hydridobridged complexes, the MP2(fu)/6-31G** level of theory is adequate to investigate this kind of reaction. Nevertheless, when geometry optimization and characterization of stationary points at this ab initio correlated level become too expensive, MP2(fu)/ 6-31G** single-point calculations on B3PW91/6-31G*-optimized geometries yield results of comparable quality. It is important to note that the differences between our calculated values and the experimental values are, both for DFT and MP2 results, within the mean absolute theory vs experiment errors reported by Scheiner et al.23 in a comparative study of more than 300 chemical reactions. But while the MP2calculated energies are lower than the experimental values due to the known fact that inclusion of the electron correlation by means of Møller-Plesset treatment overestabilizes nonclassical
J. Phys. Chem. B, Vol. 101, No. 48, 1997 10073 bridged structures,24 Density functional theory seems to provide too high energies for these complexes having nonclassical twoelectron three-center bonds. Another reason that could explain the bad performance of DFT methods in this type of reactions is the previously reported failure of density functional theory to properly evaluate weak van der Waals interactions such as dispersion energies25 and steric repulsions. This effect seems to be the cause of the large relative conformational energies of some n-alkane rotamers calculated using DFT methods26and for the failure in estimating the rotational barrier of conjugated systems such as 2,2′-bithiophene and biisothianaphthene, for which planar conformers are overstabilized.27 These effects might also be important in the complexes studied in this work, because the atoms of the methyl groups of the two fragments of every complex are situated at distances close to their van der Waals radii, this effect being more important in complexes 2 and 3 than in the gauche structure 1. Anyway, a more complete study of the interactions existing in analogous complexes having nonclassical bonds should be performed in order to obtain definitive conclusions. Another question that has to be taken into account when calculating the energy of a complex formation reaction as the difference between the total energy of the complex and the sum of the energies of the separated fragments is the basis set superposition error (BSSE). This error is associated with the fact that the basis sets commonly used to calculate the energies of both the complex and the separated fragments are far from being saturated. Therefore, in any complex, each fragment tends to use the basis functions of the other fragment to lower its energy, giving nonphysical stabilizing contributions to the energy of the complex. The most widely used method to estimate the importance of this error is the CP19 method, which defines it as
BSSE ) ∆ECP - ∆E ∆E ) E(AB)AB - E(A)A - E(B)B ∆ECP ) E(AB)AB - E(A)AB - E(B)AB where E(X)XY is the total energy of fragment X at its equilibrium geometry in the complex computed with a basis set formed by adding to the basis set of fragment X all basis functions of the other fragment without its electrons and nuclei (ghost functions). It is important to remark that it is implicitly assumed in this method that the geometry of the fragments does not change significatively by formation of the complex. Otherwise, the calculated BSSE values can only be indicative of the existence or not of this error and of the quality and balance of the basis set used.28 Since in the systems studied in this work the geometry of both the alkane and the carbenium ion changes significatively by formation of the complex, the CP method can only give us an indication of the existence or not of the BSSE error, but not a quantitative estimation of its importance. Therefore, the CP correction has been calculated only in one case, the formation of the t-C4H9+-C3H8 complex, the only one that can be considered to be formed by two different fragments. The calculated BSSE values at all DFT levels are similar and destabilize the t-C4H9+-C3H8 complex in about 1 kcal/mol, while the BSSE values calculated for the MP2 method are larger, as previously observed.29 These results indicate that there is a basis set superposition error that overestabilizes the complexes formed and that correction of this error would bring the MP2(fu)/ 6-31G** results still more in line with the experimental values. To clarify whether the loose complexes postulated in refs 4 and 5 exist and if there is an internal energy barrier separating
10074 J. Phys. Chem. B, Vol. 101, No. 48, 1997 them from the tight intermediates, energy profiles along the reaction coordinate have been calculated for the three hydridetransfer reactions considered in this work. In view of the big computational effort necessary to study the energy profile of a reaction, the cheapest B3PW91/6-31G* method has been initially used. The energy profiles of reactions 1 and 2 were calculated in the same way. Starting from the optimized geometry of the intermediate complex, the Csec-Csec distance in 1 and the Ctert-Ctert distance in 2 were taken as the independent variable, and the rest of the parameters were completely optimized. As the C-C distance increases, the central hydrogen atom moves toward one of the carbon atoms of the bridge, the resulting molecule and cation separate, and the energy rises in a continuous way until the calculated value for separated fragments is reached. The energy profile for reaction 3 was calculated in two steps. First, taking the CtertHbridge distance as the reaction coordinate, three points corresponding to fixed Ctert-Hbridge distances of 1.3, 1.2, and 1.1 Å were localized on the potential energy surface. As the CtertHbridge distance decreases, the Csec-Hbridge and the Csec-Ctert bond lengths increase, the rest of the parameters change little and the energy of the system rises. Afterward, two other points were obtained in which the Csec-Htert distance was held fixed at 3.0 Å and the central hydrogen atom was bonded to the C3H7 fragment in one case and to the t-C4H9 fragment in the other. No internal potential barrier was found for any of the hydridetransfer reactions studied at this level of theory. Several attempts have also been done to localize the loose (t-C4H9-H-t-C4H9)+ complex using the MP2(fu)/6-31G** method, but none of them has succeeded and the only minimum obtained corresponds to the tight structure 2. Conclusions The mechanism of secondary-secondary, tertiary-tertiary, and secondary-tertiary hydride-transfer reactions between alkanes and alkylcarbenium ions was studied theoretically by means of several density functional theory based methods and the correlated ab initio MP2 method. It was found that these reactions proceed through formation of tight intermediate complexes having open linear C- -H- -C bonds, which are symmetrical in the (C3H7-H-C3H7)+ and (t-C4H9-H-tC4H9)+ complexes and markedly unsymmetrical in the t-C4H9+C3H8 complex due to the different natures of the two carbon atoms involved in it. The optimized geometries of the three intermediate complexes are equivalent at all considered theoretical levels, the differences between the MP2- and DFT-calculated bond lengths being less than 0.1 Å. The MP2(fu)/6-31G**-calculated enthalpy changes for the reactions of formation of the three complexes are in good agreement with the experimental data, indicating that the method is adequate to investigate this type of mechanism. However, all DFT methods underestimate the stability of complexes 2 and 3. When the systems under study are large, the MP2(fu)/ 6-31G** method becomes too expensive for carrying out geometry optimizations and characterization of stationary points, but it has been shown that MP2(fu)/6-31G** single-point calculations on the B3PW91/6-31G*-optimized geometries of these complexes yield results of comparable quality. The three calculated energy profiles indicate that in all cases, as the cation and the neutral molecule approach, the hydrogen atom is transferred and the system is stabilized, corresponding the minimum of the curve to the intermediate tight complex. No internal potential barrier has been found, and all our attempts to localize a loose (t-C4H9-H-t-C4H9)+ complex using the MP2(fu)/6-31G** method have failed.
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