A Theoretical Study on the Mechanism of the Superacid-Catalyzed

Glen A. Ferguson , Lei Cheng , Lintao Bu , Seonah Kim , David J. Robichaud , Mark R. Nimlos , Larry A. Curtiss , and Gregg T. Beckham. The Journal of ...
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16514

J. Phys. Chem. 1996, 100, 16514-16521

A Theoretical Study on the Mechanism of the Superacid-Catalyzed Unimolecular Isomerization of n-Alkanes and n-Alkenes. Comparison between ab Initio and Density Functional Results 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: April 24, 1996; In Final Form: July 28, 1996X

The mechanism of the branching rearrangement of the 2-pentyl cation has been studied theoretically using both ab initio and density functional theory based methods which include electron correlation and extended basis sets. Two different reaction paths have been considered. In one of them the secondary linear cation is converted through a protonated cyclopropane ring into the secondary branched cation, which is converted into the tertiary cation by a 1,2-H shift. In the other one the secondary linear cation is directly converted into the tertiary cation through the primary cation. Comparison of the calculated activation energies for both reaction paths with the experimental value indicates that the studied reaction occurs via the first mechanism. It has also been found that the protonated 1,2-dimethylcyclopropane ring is not a common intermediate for the reaction, because it is a transition state and not a minimum on the potential energy surface of C5H11+. In relation with the performance of DFT methods it has been found that the local SVWN and nonlocal BP86 and B3P86 optimized geometries are in excellent agreement with the MP2 structures, while the BLYP method tends to reproduce the HF results. From an energetic point of view, the DFT-calculated barrier heights are in better agreement with experiment than the ab initio values.

Introduction The acid-catalyzed transformations of hydrocarbons such as isomerization, alkylation, and cracking play an important role in the industrial processes of petroleum chemistry.1 The interaction of hydrocarbons with solid acids results in the formation of carbenium ions2 and consequently it has been generally assumed that the mechanism of heterogeneous reactions on solid acids is similar to that of homogeneous reactions in superacid media. The mechanism of homogeneous hydrocarbon isomerization reactions in superacid media involves formation and subsequent rearrangement of the carbenium ions. The question of possible participation of primary carbenium ions and of protonated cyclopropanes as intermediates in these rearrangements is a subject of considerable interest. Carbenium ion rearrangements may be classified as branching and nonbranching. The classical mechanism for nonbranching rearrangements, in which the degree of chain branching remains the same, supposes them to proceed by a succession of 1,2hydrogen and alkyl shifts via secondary ions as intermediates. The branching rearrangements, which are about 1000 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 and consequently the mechanism that is currently accepted involves the intermediacy of a protonated cyclopropane ring.3 According to this mechanism (Scheme 1), the C2+ carbon atom attacks the C4 carbon atom and a protonated cyclopropane ring is formed. The opening of the cyclic intermediate at one of the other two sides of the ring results in the formation of a secondary monobranched * Author to whom correspondence should be addressed. † Universidad Polite ´ cnica de Valencia. ‡ Universitat de Vale ` ncia. X Abstract published in AdVance ACS Abstracts, September 1, 1996.

S0022-3654(96)01179-3 CCC: $12.00

carbenium ion if R is an alkyl group. If R is an hydrogen atom, the opening of the cyclic intermediate at side a leads to a n-butyl cation in which a terminal and a nonterminal carbon atoms have changed positions, but the opening of the intermediate at side b becomes more difficult because it would lead to a primary carbenium ion. This mechanism is consistent with the experimental fact that n-butane does not isomerize at an observable rate to isobutane under conditions where n-pentane and n-hexane are rapidly converted into their branched isomers4 and with the finding that the rate of scrambling or isomerization of n-butane1-13C to n-butane-2-13C is comparable to that of isomerization of n-pentane to isopentane.5 In a recent paper,6 a complete theoretical study of the potential energy surface of C4H9+ cation at the Hartree-Fock and MP2 levels was carried out in order to establish the mechanism of the scrambling and branching isomerization reactions of the n-butyl cation. The results obtained indicate that the protonated methylcyclopropane ring is not an intermediate for these two rearrangements, but only the transition state for the carbon scrambling reaction. In the branching isomerization the secondary n-butyl cation is converted into the tertiary isobutyl cation through the primary isobutyl cation, which is the transition state for this reaction. The activation energies calculated assuming this mechanism are in very good agreement with those obtained experimentally. However, since the effect of alkyl substituents on the relative energies and nature of the different carbenium ions involved in the isomerization mechanism is not known, it is not possible to extrapolate the conclusions obtained for C4H9+ cation to higher aliphatic carbenium ions such as C5H11+ or C6H13+. We present in this paper a theoretical study of the mechanism of the branching rearrangement of 2-pentyl cation. This study has been carried out using both traditional ab initio molecular orbital theory7 and density functional theory,8 which is becoming a practical tool for chemical studies due to the fact that it is © 1996 American Chemical Society

Branching Rearrangement of 2-Pentyl Cation

J. Phys. Chem., Vol. 100, No. 41, 1996 16515

SCHEME 1

SCHEME 2

computationally less demanding than ab initio methods for inclusion of electron correlation. DFT-based methods have been successfully applied to predict equilibrium structure properties of molecular systems such as geometries,9-12 thermochemical data10-13 dipole moments,11,14 vibrational frequencies and IR intensities10-12,15-17 interaction energies,18 relative energies of conformational isomers,19 and electronic20 and nonlinear optical21 properties. In general, the results obtained using nonlocal exchange-correlation functionals are in better agreement with experiment than those obtained with post-HF correlated methods. Recent work has shown that partial inclusion of exact exchange significantly improves the DFT results.12,13b,22 Density functional theory has also been used to study transition state structures and activation energies of chemical reactions. Fan and Ziegler found that for isomerization reactions of small systems such as HCN, CH3CN, N2H2, or H2CO,23 both local and nonlocal approximations yield geometries and reaction barriers similar to those obtained from correlated post-HF calculations. However, for processes involving bond breaking or formation such as unimolecular decompositions of small systems,23c,24,25 ring openings,25,26 or Diels-Alder cycloadditions,24,25,27 the LDA underestimates the barrier heights and nonlocal corrections are necessary to remove this error. The activation energies of proton transfer reactions28 are also underestimated by LDA, and although nonlocal corrections yield significant improvements, only hybrid methods which include exact exchange are able to provide results comparable to those obtained at correlated ab initio levels. The present work has two objectives. The first one is to carry out a systematic study of C5H11+ potential energy surface in order to stablish the mechanism of branching rearrangement of 2-pentyl cation. The second one is to test the validity of density functional theory for predicting transition state properties and activation energies of branching isomerization reactions of carbenium ions, which has not previously been done. Scheme 2 shows the two possible reaction paths that have been considered in this work. In both cases, the positive charge on C2 carbon atom in the secondary 2-pentyl cation attacks the

C4 carbon atom. According to Brouwer’s mechanism 1 this attack leads to formation of an edge-protonated 1,2-dimethylcyclopropane ring,3 whose opening results in the formation of the secondary 3-methyl-2-butyl cation. A 1,2-hydrogen shift converts this secondary monobranched cation in the tertiary 2-methyl-2-butyl (or tert-pentyl) cation. The other proposed mechanism 2, is equivalent to that calculated for the branching isomerization of the n-butyl cation.6 A simultaneous strengthening of the C2-C4 bond and breaking of the C3-C4 bond in the secondary 2-pentyl cation would lead to a primary monobranched cation, and from this, a shift of an hydrogen atom from C2 to C3 would lead to the tert-pentyl cation. Experimental data reported by Brouwer3 give an activation energy value of 18.3 kcal/mol for the rearrangement of the tertpentyl cation to the 2-pentyl cation, and from PMR spectroscopic measurements on the tert-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 process.29 This last value was obtained from the study of the interchange of the methyl and methylene protons in the tert-pentyl cation, which occurs via the reversible rearrangement of tert-pentyl to 2-pentyl and again to tert-pentyl cation. The isomerization of the secondary 3-methyl-2-butyl cation into the tertiary 2-methyl2-butyl (or tert-pentyl) cation in the gas phase has been investigated by Collin and Herman.30 From the study of the influence of temperature on the radiolysis of a mixture of xenon, methane, 3-methyl-2-butene, and nitric oxide, an activation energy of 2.1 kcal/mol has been determined for this isomerization process. Computational Details All calculations in this work were performed on an IBM 9021/ 500-2VF computer and on IBM RS/6000 workstations of the University of Vale`ncia using the Gaussian 9231 and Gaussian 9432 computer programs. In the ab initio molecular orbital study, the geometry of the stationary points of C5H11+ potential energy surface has been first fully optimized by using the Hartree-Fock (HF) procedure

16516 J. Phys. Chem., Vol. 100, No. 41, 1996 and the 6-31G* basis set7,33 which has polarization functions (d-type) on non-hydrogen atoms. Afterwards, electron correlation has been included by means of the second-order MoellerPlesset perturbation theory34 that takes into account the core electrons. Two types of calculations have been carried out: single-point calculations on the HF geometries using the MP2 treatment and the 6-31G* basis set (MP2//HF) and a complete geometry optimization of all stationary points at this correlated theoretical level (MP2). The Berny analytical gradient35 and the eigenvalue following36 methods have been used for the minima and transition states geometry optimizations, respectively. All HF and MP2 stationary points have been characterized by calculating the Hessian matrix and analyzing the vibrational normal modes. The relative energies at the MP2 level have been corrected by the zero-point energy (ZPE) obtained from frequency calculations. In the density functional study the calculations have been carried out at two different levels of theory. The first level is the local density approximation (LDA) in which the exchange and correlation parts of the functional are those dervied by Slater37 and Volk, Wilk, and Nusair,38 respectively (SVWN). The second level includes gradient-based corrections to the LDA expressions. We have used the nonlocal exchange correction due to Becke39 combined with either the Perdew40 or the Lee, Yang, and Parr41 nonlocal correlation corrections (BP86 and BLYP, respectively), and also the Becke’s hybrid threeparameter exchange functional13b which includes the HartreeFock exchange with the Perdew nonlocal correlation corrections (B3P86). The 6-31G* basis set has also been used in these calculations in order to facilitate the comparison between DFT and ab initio results. Single-point calculations using the MP2 treatment have also been carried out on all DFT-optimized geometries. Results and Discussion The six structures that according to Scheme 2 have to be localized and characterized at ab initio and DFT levels on C5H11+ potential energy surface are depicted in Figure 1. The optimized geometries of the secondary 2-pentyl cation A, the protonated 1,2-dimethylcyclopropane ring B, the secondary 3-methyl-2-butyl cation C, the transition state for the hydrogen shift that converts the secondary branched cation into the tertiary one D, the tertiary 2-methyl-2-butyl or tert-pentyl cation E, and the primary branched pentyl cation F are summarized in Tables 1, 2, 3, 4, 5, and 6, respectively. Table 7 shows the relative energies of all structures and Table 8 shows the calculated barrier heights for the different processes involved in the mechanism together with available experimental data. The energy profile for the two reaction paths considered in this work is represented in Figure 2. The discussion of the results has been structured into three parts. In section A we report the ab initio study of the potential energy surface of C5H11+ cation, paying a special attention to the influence that inclusion of electron correlation has on the geometry, energy, and nature of the stationary points. In section B we discuss the performance of different DFT functionals by comparing the calculated geometries and energies with the ab initio results. Finally, in section C, the mechanism of the branching isomerization of the 2-pentyl cation is determined by comparison of the calculated activation energies for the two possible reaction paths with the experimental value. A. Ab Initio Results. The tertiary 2-methyl-2-butyl (tertpentyl) cation E is the most stable minimum on the potential energy surface of C5H11+ both at the HF and at the MP2 levels. The optimized geometries summarized in Table 5 indicate that

Boronat et al.

Figure 1. MP2/6-31G* optimized structures of C5H11+ cation.

TABLE 1: Optimized Geometry of Structure Aa parameter

SVWN

BLYP

BP86

B3P86

HF

MP2

r(C2-C1) r(C3-C2) r(C4-C3) r(C4-C2) r(C5-C4) ∠(C3-C2-C1) ∠(C4-C3-C2) ∠(C5-C4-C3) average devb

1.462 1.419 1.588 1.898 1.506 123.9 78.1 111.9 0.018

1.464 1.430 1.674 2.365 1.544 126.0 98.9 106.7 0.014

1.473 1.426 1.654 2.098 1.534 125.4 85.5 109.8 0.011

1.468 1.413 1.632 2.025 1.522 125.1 83.0 110.4 0.005

1.461 1.441 1.599 2.366 1.531 125.4 102.1 108.8 0.015

1.472 1.402 1.656 1.941 1.524 124.4 78.3 108.8

a Distances in Å and angles in deg. All calculations were carried out using a 6-31G* basis set. b Average deviations in absolute value from MP2/6-31G* results, calculated including all C-C and C-H bond lengths of the system.

the tert-pentyl cation is not fully classical. A partial bridging between the C4-C5 bond and the positive charge on C2+ cation atom can be deduced from the lengthening of the C4-C5 bond to 1.565 Å at the HF level and 1.581 Å at the MP2 level, and also from the calculated C2+C4C5 angle values of 106.3° and 101.5° at the HF and MP2 levels, respectively. The structure of the tert-pentyl cation has already been studied by comparing the 13C chemical shifts of the C+ carbon calculated by IGLO using ab initio geometries with the experimental values. Our results are in complete agreement with those previously reported.42 Two other minima have been found on the potential energy surface of C5H11+: the linear A and the branched C secondary pentyl cations. As can be seen in Table 1, the lengthening at the HF level of the C3-C4 bond in the linear pentyl cation A to 1.599 Å together with the calculated value of 102.1° for the C2+C3C4 angle indicates that there is a partial bridging between the C3-C4 bond and the positive charge on C2+ carbon atom, similar to that previously reported for the tertiary pentyl cation.

Branching Rearrangement of 2-Pentyl Cation

J. Phys. Chem., Vol. 100, No. 41, 1996 16517

TABLE 2: Optimized Geometry of Structure Ba

TABLE 5: Optimized Geometry of Structure Ea

parameter

SVWN

B3P86

HF

MP2

parameter

SVWN

BLYP

BP86

B3P86

HF

MP2

r(C2-C1) r(C3-C2) r(C4-C3) r(C4-C2) r(C5-C4) r(H-C3) r(H-C4) ∠(C3-C2-C1) ∠(C4-C3-C2) ∠(C5-C4-C3) ∠(H-C4-C3) average devb

1.494 1.431 1.734 1.578 1.503 1.564 1.166 121.3 58.9 122.0 61.6 0.012

1.511 1.445 1.768 1.567 1.518 1.508 1.175 120.9 57.3 123.6 57.5 0.003

1.519 1.450 1.807 1.553 1.526 1.478 1.176 120.3 55.7 125.6 54.6 0.013

1.510 1.443 1.747 1.562 1.520 1.490 1.174 120.7 57.7 123.1 57.4

r(C2-C1) r(C3-C2) r(C4-C2) r(C5-C4) ∠(C3-C2-C1) ∠(C4-C2-C1) ∠(C4-C2-C3) ∠(C5-C4-C2) average devb

1.450 1.449 1.443 1.561 120.2 119.6 120.1 101.1 0.012

1.480 1.480 1.466 1.608 120.0 119.9 120.1 105.2 0.012

1.474 1.473 1.464 1.593 119.8 119.8 120.4 104.8 0.011

1.463 1.463 1.454 1.576 119.8 119.8 120.4 104.1 0.003

1.474 1.474 1.471 1.565 120.0 120.0 120.0 106.3 0.011

1.464 1.464 1.446 1.581 119.9 120.0 120.1 101.5

a Distances in Å and angles in deg. All calculations were carried out using a 6-31G* basis set. b Average deviations in absolute value from MP2/6-31G* results, calculated including all C-C and C-H bond lengths of the system.

TABLE 3: Optimized Geometry of Structure Ca parameter

SVWN

BLYP

BP86

B3P86

HF

MP2

r(C2-C1) r(C3-C2) r(C4-C2) r(C4-C3) r(C5-C4) ∠(C3-C2-C1) ∠(C4-C2-C1) ∠(C4-C2-C3) ∠(C5-C4-C2) average devb

1.488 1.664 1.406 1.813 1.469 114.4 122.6 71.8 124.8 0.016

1.5351 1.682 1.429 2.179 1.471 110.7 119.8 88.5 126.6 0.017

1.5163 1.7085 1.419 1.943 1.485 113.8 122.1 76.2 125.8 0.018

1.505 1.691 1.405 1.896 1.479 113.8 122.4 74.9 125.4 0.006

1.532 1.592 1.444 2.295 1.462 111.1 115.6 98.1 125.8 0.023

1.501 1.733 1.396 1.836 1.489 113.4 123.3 70.9 124.6

a

Distances in Å and angles in deg. All calculations were carried out using a 6-31G* basis set. b Average deviations in absolute value from MP2/6-31G* results, calculated including all C-C and C-H bond lengths of the system.

TABLE 4: Optimized Geometry of Structure Da parameter

SVWN

B3P86

HF

MP2

r(C2-C1) r(C3-C2) r(C4-C2) r(C5-C4) r(H-C2) r(H-C4) ∠(C3-C2-C1) ∠(C4-C2-C1) ∠(C4-C2-C3) ∠(C5-C4-C2) ∠(H-C2-C4) average devb

1.503 1.560 1.427 1.427 1.128 1.990 114.0 118.2 108.0 126.7 101.6 0.011

1.519 1.571 1.439 1.441 1.118 1.976 113.9 118.2 109.1 128.0 100.5 0.002

1.526 1.522 1.411 1.479 1.180 1.582 115.4 118.1 121.7 128.4 74.6 0.020

1.519 1.563 1.437 1.441 1.118 1.961 113.8 117.7 109.6 127.7 99.5

a

Distances in Å and angles in deg. All calculations were carried out using a 6-31G* basis set. b Average deviations in absolute value from MP2/6-31G* results, calculated including all C-C and C-H bond lengths of the system.

The inclusion of the electron correlation at the MP2 level lengthens the C3-C4 bond to 1.656 Å and closes the C2+C3C4 angle to 78.3°, that is to say, increases the degree of methylbridging. This is reflected in the important shortening of the C2+-C4 bond length from 2.366 Å at the HF level to 1.941 Å at the MP2 level, while the C3-C5 bond length remains nearly constant (2.545 and 2.587 Å). The changes in geometry produced when electron correlation is included are more important in the branched cation C, as can be seen in Table 3. At the HF level the C2-C3 bond length value of 1.592 Å and the C3C2C4+ angle value of 98.1° are indicative of a partial bridging between the C2-C3 bond and the positive charge on C4+ carbon atom. At the MP2 level, however, the C2-C3 bond is lengthened to 1.733 Å, the C3C2C4+ angle is closed to 70.9°, and the C3-C4 bond is shortened from 2.295 Å at the

a Distances in Å and angles in deg. All calculations were carried out using a 6-31G* basis set. b Average deviations in absolute value from MP2/6-31G* results, calculated including all C-C and C-H bond lengths of the system.

TABLE 6: Optimized Geometry of Structurea parameter

SVWN

B3P86

HF

MP2

r(C2-C1) r(C3-C2) r(C4-C2) r(C5-C4) r(H-C2) r(H-C3) ∠(C3-C2-C1) ∠(C4-C2-C1) ∠(C4-C2-C3) ∠(C5-C4-C2) ∠(H-C2-C3) average devb

1.498 1.391 1.644 1.498 1.131 1.967 120.5 114.2 110.1 109.9 102.0 0.013

1.516 1.402 1.631 1.517 1.125 1.931 119.1 115.4 111.7 111.5 99.0 0.002

1.529 1.435 1.589 1.5281 1.103 1.974 115.6 115.5 108.9 114.2 101.3 0.013

1.516 1.398 1.618 1.521 1.127 1.921 118.7 115.5 111.1 111.5 98.5

a Distance in Å and angles in deg. All calculations were carried out using a 6-31G* basis set. b Average deviations in absolute value from MP2/6-31G* results, calculated including all C-C and C-H bond lengths of the system.

TABLE 7: Calculated Relative Energies (kal/mol) of the Stationary Points Found on the C5H11+ Potential Energy Surface method

A

B

C

D

E

F

SVWN BLYP BP86 B3P86 HF MP2 MP2+ZPE MP2//SVWN MP2//BLYP MP2//BP86 MP2//B3P86 MP2//HF

11.08 11.81 11.98 12.31 13.60 10.33 11.67 10.83 13.05 10.93 10.63 13.63

17.19

8.73 11.29 10.57 10.50 12.85 7.24 8.04 7.55 9.98 7.60 7.42 11.75

14.55

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

36.62

20.42 29.45 17.05 17.81 19.44 19.68 18.49

14.68 15.62 14.84 14.40 14.84 14.77 12.04

35.35 34.18 34.48 34.09 34.32 34.33 35.02

TABLE 8: Calculated and Experimental Activation Energies (kcal/mol) for the Processes Depicted in Figure 2 method

Ea1

Ea2

Ea3

SVWN B3P86 HF MP2 MP2+ZPE MP2//SVWN MP2//B3P86 MP2//HF exp

17.19 20.42 29.45 17.05 17.81 19.44 19.68 18.49 18.8

5.82 4.18 2.77 7.6 6.36 7.29 7.35 0.29 2.1

36.62 35.35 34.18 34.48 34.09 34.32 34.33 35.02

uncorrelated level to 1.836 Å. The degree of methyl-bridging becomes so important that this structure could be best described as an unsymmetrical corner-protonated 1,2-dimethylcyclopropane ring. All these geometric changes are reflected in the relative energies of the secondary cations A and C with respect to the tertiary cation E summarized in Table 7. The HF and MP2//

16518 J. Phys. Chem., Vol. 100, No. 41, 1996

Figure 2. Energy profile corresponding to the branching isomerization of the secondary 2-pentyl cation. The tertiary ion has been taken as the origin of energies.

HF calculated relative energies of A are equivalent. The increase in the degree of bridging resulting from the MP2 optimization stabilizes this structure and yields a relative energy value of 10.33 kcal/mol, and 11.67 kcal/mol with the ZPE correction. The HF calculated relative energy of C, 12.85 kcal/ mol, is about 2 kcal/mol too high in relation to the experimental value of 11 kcal/mol reported by Collin and Herman30 for the energy difference between the secondary 3-methyl-2-butyl and the tertiary 2-methyl-2-butyl cation. Since there is a partial bridging in the HF optimized geometry of C, the MP2//HF calculation slightly stabilizes it and yields a relative energy value of 11.75 kcal/mol. But the MP2 and MP2+ZPE calculated values, 7.24 and 8.04 kcal/mol, respectively, are more than 3 kcal/mol too low in relation to the experimental value. The overstabilization of structure C at this level of theory can be explained if we take into account the inclusion of the electron correlation by means of the Moeller-Plesset treatment preferentially stabilizes nonclassical bridged structures43 and, as can be observed in Table 3, the degree of bridging in C is more important than in any of the other structures. According to Brouwer’s mechanism (1 in Scheme 2), the isomerization of the linear A to the branched C secondary cations passes through an edge-protonated 1,2-dimethylcyclopropane ring B. The structure of this cyclic species was very difficult to obtain. Taking as a starting point the optimized geometry of the linear cation A, two variables were simultaneously controlled: the C2+C3C4 angle and the C3-H bond length. In each calculation these two variables were fixed and all other parameters were allowed to fully optimize. When the edge-protonated cyclic structure was obtained, its geometry was completely reoptimized at the HF and MP2 levels using the eigenvalue following transition state search technique. The two stationary points obtained were characterized by force constant calculations and they were found to be transition states, with only one imaginary vibration frequency clearly associated to the movement of the bridged hydrogen atom either to C3 or to C4. The HF and MP2 optimized geometries of B are nearly equivalent. The C2-C3, C2-C4, and C3-C4 bond lengths in the ring are 1.450, 1.553, and 1.807 Å, respectively, at the HF level, while at the MP2 level the calculated values are 1.443, 1.562, and 1.747 Å (see Table 2). The hydrogen bridge is markedly unsymmetrical with C-H bond lengths of 1.176 and 1.478 Å at the HF level and 1.174 and 1.490 Å at the MP2 level. Despite this geometric similarity between the correlated and uncorrelated optimized structures, the stability of B in relation to the tert-pentyl cation is strongly dependent on the theoretical level considered. At the HF level the calculated relative energy is 29.45 kcal/mol. The inclusion of electron correlation at the

Boronat et al. MP2//HF level stabilizes structure B in nearly 11 kcal/mol, being the calculated relative energy at this level 18.49 kcal/mol. When the geometry is optimized at the MP2 level the relative energy obtained is slightly lower, 17.05 kcal/mol, and rises to 17.81 kcal/mol at the MP2+ZPE level. These energetic changes are easily explained by the stabilization that the Moeller-Plesset treatment introduces on nonclassical bridged structures.47 Since the HF optimized geometry of B corresponds to a nonclassical bridged structure, the single-point calculation at the MP2//HF level is able to introduce all the stabilization due to electron correlation. The MP2 geometry optimization cannot increase the degree of bridging in this structure and consequently no significant stabilization with respect to the single-point calculation is observed at this level of theory. The last step in Brouwer’s mechanism is the hydrogen shift that converts the secondary branched cation C into the tertiary cation E. Starting from the optimized geometry of the tertiary cation, the distance between C2 and the hydrogen atom that is going to migrate has been shortened from the optimized value in E (2.115 Å at the HF level and 2.125 Å at the MP2 level) to a value of 1.10 Å. Then, the geometry of the obtained structure was completely reoptimized at both theoretical levels using the eigenvalue following transition state search method. The force constant calculations indicate that both structures are transition states on their respective potential energy surfaces, showing only one imaginary vibration frequency associated to the movement of the migrating hydrogen atom toward C2 or C4. At the HF level the relative energy calculated for D in relation to the tertpentyl cation is 15.62 kcal/mol. Since D is a nonclassical hydrogen-bridged structure the single-point calculation at the MP2//HF level stabilizes it and yields a relative energy value of 12.04 kcal/mol. However, the value obtained from the MP2 optimization is higher, 14.84 kcal/mol, and 14.40 kcal/mol when the ZPE correction is included. The reason for this behavior can be deduced from the different geometries reported in Table 4. At the HF level the optimized geometry of D corresponds to an unsymmetrically hydrogen-bridged structure with a C-C bond length of 1.411 Å and two C-H bond lengths of 1.189 and 1.582 Å. At the MP2 level the optimized C-C bond length is 1.437 Å and the C-H bond lengths are 1.118 and 1.961 Å. These values are indicative of a lesser degree of hydrogenbridging and consequently the structure is slightly destabilized. Finally, in order to complete the mechanism 2 depicted in Scheme 2, the structure of the primary pentyl cation F has been calculated. Starting again from the optimized geometry of the tertiary pentyl cation E, the distance between C2 and one of the hydrogen atoms attached to C3 was slowly shortened from its optimized value in E to 1.10 Å and then, using the eigenvalue following transition state search technique, its geometry was completely reoptimized at the HF and MP2 levels. Characterization of the stationary points by force constant calculations indicates that at both theoretical levels the primary pentyl cation is a transition state on the C5H11+ potential energy surface. If the eigenvector associated to the imaginary vibration frequency is followed in one direction the tertiary cation E is reached, while following it in the opposite direction leads to the secondary linear cation A. As can be observed in Table 6, the correlated and uncorrelated optimized geometries of F are very similar. The lengthening of the C2-C4 bond to 1.589 Å at the HF level and 1.618 Å at the MP2 level could be indicative of partial C-C bridging, but the calculated C3+C2C4 angle values of 109.0 and 111.1° at the HF and MP2 levels, respectively, eliminate this possibility. Since the optimized geometry of the primary cation F at both theoretical levels is fully classical, the inclusion of electron

Branching Rearrangement of 2-Pentyl Cation correlation produces no stabilization of this structure in relation to the tert-pentyl cation. Thus, as can be observed in Table 7, the four ab initio calculated values for the relative energy of F are very similar, 34.18, 35.02, 34.48, and 34.09 kcal/mol, at the HF, MP2//HF, MP2, and MP2+ZPE levels, respectively. B. Density Functional Results. The geometry of the three minima found on the ab initio potential energy surfaces of C5H11+, the linear A and branched C secondary cations and the tertiary cation E, has been fully optimized using the local SVWN, the nonlocal BP86 and BLYP and the hybrid B3P86 methods. Since these methods take into account most of the electron correlation effects, the MP2-optimized geometries obtained in section A have been used as starting point for the DFT calculations. It can be seen in Table 5 that there is an excellent agreement between MP2 and all DFT optimized geometries of the tertiary cation E. The average deviations in absolute value between the MP2 and all other methods calculated C-C and C-H bond lengths, given at the end of the table, are less than 0.012 Å in all cases, corresponding the lowest value to the B3P86 results. More specifically, it can be observed that the local SVWN method slightly underestimates the C-C bond lengths while the nonlocal BP86 and BLYP methods slightly overestimate them. Although no information about the C-H bond lengths is given in the tables, it has been found that, for the six structures considered, all DFT methods yield values which are about 0.01 Å too long. The different performance of the various DFT methods used in this work can be seen in the optimized geometries of the secondary cations A and C summarized in Tables 1 and 3, respectively. The degree of methyl-bridging in the linear cation A, which can be measured from the C4-C3-C2+ angle or from the C4-C2+ bond length shows a strong dependence on the functional used. The local SVWN angle value of 78.1° is equivalent to the MP2 value of 78.3°. When the nonlocal corrections are included the angle value increases. This increase is small when the Perdew correlation functional is used, but at the BLYP level the optimized angle value of 98.8° is nearly equivalent to the HF-optimized value of 102.1°. The same tendency is observed in the C4-C2+ bond lengths. The BLYPand HF-optimized values, 2.365 and 2.366 Å, are equivalent. The BP86- and B3P86-calculated bond lengths are 0.156 and 0.084 Å, respectively, longer than the MP2 value, while the SVWN bond length is 0.043 Å too short. Although it could seem from these data that SVWN yields the best geometry in relation to MP2, there are other parameters as for example the C3-C4 bond length for which the biggest difference between the MP2 and DFT optimized values is obtained at the SVWN level. The average deviations given at the end of the table, in which all C-C and C-H bond lengths are considered, indicate that the best geometry in relation to MP2 is that calculated with the B3P86 method. It has been seen in section A that the degree of methylbridging in the branched secondary cation C, given by the C2C3C4+ angle and by the C2-C3 and C3-C4+ bond lengths, is considerably increased by the inclusion of the electron correlation. At the DFT level this degree of methyl-bridging strongly depends on the functional used. As in the case of the linear cation A, the SVWN- and MP2-optimized angle values, 71.8 and 70.9°, respectively, are equivalent. The SVWN C3C4+ bond length is only 0.023 Å too short, but the calculated deviation for the C2-C3 bond length, 0.069 Å, is more important. At the B3P86 level the increase of the angle to 74.9°, the lengthening of the C3-C4+ bond length to 1.896 Å, and the shortening of the C2-C3 bond length to 1.691 Å are indicative

J. Phys. Chem., Vol. 100, No. 41, 1996 16519 of a slight decrease in the degree of methyl-bridging, a decrease which is accentuated at the BP86 level. The previously suggested tendency of the BLYP method to reproduce the HF results can be observed in the optimized angle value of 88.5°, and more clearly in the C3-C4+ bond length value of 2.179 Å, nearly equivalent to the HF value of 2.295 Å and very different from the MP2 value of 1.836 Å. Again, it can be seen that the lowest average deviation from the MP2 results has been obtained with the B3P86 method. The relative energies summarized in Table 7 indicate that the tertiary cation E is the most stable structure on C5H11+ potential energy surface at all theoretical levels. The calculated DFT relative energies of the secondary cations A and C are in all cases lying between the MP2 and the HF levels. For structure A the BLYP and BP86 calculated energies are equivalent to the MP2+ZPE value, the SVWN and B3P86 results being only 0.6 kcal/mol too low and too high, respectively. The SVWN-calculated relative energy of structure C, 8.73 kcal/mol, is the most similar to the MP2+ZPE value of 8.04 kcal/mol and consequently it is too low in relation to the experimental value of 11 kcal/mol. On the contrary, the BLYP-, BP86-, and B3P86-calculated energies, 11.29, 10.57, and 10.50 kcal/mol, respectively, are between 2.5 and 3.3 kcal/mol higher than the MP2+ZPE result but are in excellent agreement with the value obtained experimentally. These results suggest that the local SVWN method reproduces the tendency of MP2 treatment to overestimate the stability of bridged structures and that the nonlocal corrections remove this tendency. Table 7 also shows the results of MP2 single-point calculations carried out on the DFT- and HF-optimized geometries. For both secondary cations A and C the MP2//SVWN, MP2//BP86, and MP2//B3P86 relative energies are less than 0.6 kcal/mol higher than the MP2 values, indicating that the geometries yielded by these three methods are very similar to the MP2-optimized structures. The MP2//BLYP energies are, however, more similar to the MP2//HF values, confirming the previously suggested tendency of the BLYP method to reproduce the HF geometries. Since the best geometries and energies in relation to the MP2 results are those provided by the SVWN and B3P86 methods, the geometry of structures B, D, and F has only been optimized at these two theoretical levels. The optimizations have been carried out using the eigenvalue following transition state search technique, and the stationary points have been characterized by force constant calculations in order to make sure that they have only one imaginary vibration frequency. The most important optimized parameters of structure B, D, and F are summarized in Tables 2, 4, and 6, respectively. It has been seen in section A that although the HF- and MP2optimized geometries of the cyclopropane ring B are nearly equivalent, with an average deviation between HF and MP2 bond lengths of only 0.013 Å, the relative energies are strongly dependent on the level of theory considered. Something similar happens at the DFT level. The average deviations between DFT and MP2 bond lengths are really low, 0.012 Å for SVWN and 0.003 Å for B3P86 results. The SVWN-calculated relative energy of structure B, 17.19 kcal/mol, is in excellent agreement with the MP2 value of 17.05 kcal/mol, while the B3P86 value is slightly higher, 20.42 kcal/mol. The MP2//SVWN and MP2/ /B3P86 relative energies, however, do not reflect the similarity between the DFT- and MP2-optimized geometries; they are 2.39 and 2.63 kcal/mol, respectively, too high in relation to the MP2 value. For structure D it has been found in section A that the C2-H-C4 bridge is more unsymmetrical at the MP2 than at the HF level. Both DFT-optimized geometries reproduce the

16520 J. Phys. Chem., Vol. 100, No. 41, 1996 MP2 structure with average deivations of 0.011 Å for the SVWN results and only 0.002 Å for the B3P86 results. This excellent geometrical agreement is reflected in the relative energies summarized in Table 7. The SVWN, B3P86, MP2//SVWN, MP2//B3P86, MP2, and MP2+ZPE calculated relative energies are all equivalent, the differences between them being less than 0.5 kcal/mol. The DFT- and MP2-optimized geometries of the primary pentyl cation F are, as can be seen in Table 6, nearly equivalent with average deviations of 0.013 Å for the SVWN results and only 0.002 Å for the B3P86 results. The local and nonlocal relative energies of F are 2.53 and 1.26 kcal/mol, respectively, too high in relation to the MP2+ZPE value of 34.09 kcal/mol. However, the MP2//SVWN and MP2//B3P86 values of 34.32 and 34.33 kcal/mol, respectively, reproduce the excellent agreement existent between the DFT and MP2 structures of F. C. Mechanism. Figure 2 shows the calculated energetic profile for the branching isomerization of the 2-pentyl cation. As mentioned in the Introduction, two different reaction paths (1 and 2 in Scheme 2) have been studied. In both cases the secondary linear A and the tertiary E minima are the starting point and the product of the isomerization reactions, respectively. According to mechanism 1, strengthening of the C2-C4 bond in A together with weakening of the C3-C4 bond and migration of one of the hydrogen atoms attached to C4 to a bridged position between C3 and C4 leads to transition state B. From this, breaking of the C3-C4 bond together with migration of the bridged hydrogen atom to C3 leads to minimum C. Then, the hydrogen atom attached to C2 migrates to C4 through transition state D, and minimum E is reached. According to mechanism 2, a simultaneous strengthening of the C2-C4 bond and breaking of the C3-C4 bond in A directly leads to transition state F and, from this, the migration of the hydrogen atom from C2 to C3 leads to minimum E. The activation energy for the rearangement of the tert-pentyl cation to the 2-pentyl cation can be calculated as the energy difference between the primary transition state F and the tertiary minimum E if reaction path 2 is followed (Ea3 in Figure 2). In 1, the rate-determining step is the conversion of the branched C into the linear A secondary cations, and consequently the activation energy for the global process is the energy difference between transition state B and minimum E (Ea1 in Figure 2). The calculated activation energies together with available experimental data are summarized in Table 8. At the HF level the calculated activation energies for the two reaction paths are quite similar, 29.45 kcal/mol for 1 and 34.18 kcal/mol for 2, and too high as compared with the experimental value of 18.8 kcal/mol. When electron correlation is included either at the MP2 or at the DFT levels, the energy of the primary cation experiences no changes and the calculated activation energies for mechanism 2, Ea3, lie between the 34.09 kcal/mol obtained at the MP2+ZPE level and the 36.62 kcal/mol provided by the SVWN method. Electron correlation, however, highly stabilizes the edge-protonated cyclopropane ring B and consequently the calculated activation energies for mechanism 1 at correlated levels are lowered to values that differ in less than 1.75 kcal/ mol from the experimental data. Comparison of the calculated activation energies Ea1 and Ea3 with the experimental value of 18.8 kcal/mol clearly indicates that the branching isomerization of the 2-pentyl cation does not follow reaction path 2 as was the case for n-butyl cation but occurs via mechanism 1. As already mentioned, mechanism 1 consists of two steps. The rate-determining one is the rearrangement that converts the linear cation A into the branched cation C discussed above. The second step is the hydrogen shift that converts the branched

Boronat et al. secondary cation C into the tertiary cation E. The activation energy for this process (Ea2 in Figure 2) can be calculated as the energy difference between transition state D and minimum C. The Ea2 calculated value at the HF level, 2.77 kcal/mol, compares well with the experimental one of 2.1 kcal/mol reported by Collin and Herman.30 At the MP2//HF level the obtained value is too low, 0.29 kcal/mol, while the MP2 optimization yields a too high energy barrier for this process, 7.61 kcal/mol, and 6.36 kcal/mol when the ZPE correction is included. The B3P86 and SVWN values, 4.18 and 5.82 kcal/ mol, respectively, are also too high and the MP2 single-point calculations on the DFT geometries overestimate the barrier height by more than 5 kcal/mol. The reported experimental energy difference between the secondary C and the tertiary E cations of 11 kcal/mol30 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 E into the secondary C cation of 13.1 kcal/mol, which can be calculated as the energy difference between transition state D and minimum E. It can be seen in Table 7 that, except for the HF result, the calculated relative energies of D are between 12.04 and 14.84 kcal/mol, that is to say, they are basically correct. Consequently, the reason for the observed difference between the calculated and experimental values of Ea2 must be in the relative energy of the secondary branched cation C. As has been previously discussed, the HFcalculated relative energy of C is too high, but the DFT (exceptuating SVWN) calculated values are in very good agreement with the experimental energy difference between the secondary C and the tertiary E cations. At the B3P86 level, for example, the relative energy of structure C is 10.5 kcal/ mol. The 0.5 kcal/mol of difference with respect to the experimental value added to the 1.5 kcal/mol of difference reported for transition state D yields an overestimation of the barrier height for the hydrogen shift process of only 2.0 kcal/ mol. The MP2, MP2+ZPE, and MP2//DFT calculated relative energies of C, however, are more than 3 kcal/mol lower than the experimental value. This more than 3 kcal/mol of discrepancy between the calculated and the experimental relative energies of C added to the 1.50-1.75 kcal/mol reported before for transition state D is the reason that explains the too high energy barrier obtained when the MP2 treatment is used. Conclusions A theoretical study of the branching rearrangement of the 2-pentyl cation including geometry optimization and characterization of the stationary points at correlated levels has been carried out. Two different reaction paths have been considered (Scheme 2). If reaction path 1 is followed the secondary linear cation A is converted into the secondary branched cation C through the edge-protonated cyclopropane ring B. A 1,2hydrogen shift converts then the secondary branched cation C into the tertiary cation E. According to reaction path 2, the secondary linear cation A is directly converted into the tertiary cation E through the primary cation F. Comparison of the calculated activation energies for reaction paths 1 and 2 with the experimental value of 18.8 kcal/mol indicates that the branching isomerization of the 2-pentyl cation occurs via mechanism 1 and not through the primary cation as was the case for the n-butyl branching isomerization. This conclusion can be extrapolated to higher aliphatic carbenium ions because, as can be observed in Scheme 2, 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 in the

Branching Rearrangement of 2-Pentyl Cation isomerization mechanism. It is also important to note that the calculated mechanism 1 is not equivalent to that empirically proposed by Brouwer, the main difference being the nature of the protonated cyclopropane ring. Our results indicate that this cyclopropane ring cannot be an intermediate as affirmed by Brouwer, because it is a transition state and not a minimum on the potential energy surface of C5H11+ cation. From the results obtained at the different ab initio levels of calculation it can be concluded that the inclusion of the electron correlation in the geometry optimizations and in the characterization of the stationary points is essential in the study of this kind of reaction mechanisms. The performance of several density functional theory based methods, which include electron correlation, has been studied by comparing the DFT results with those obtained from the ab initio calculations and from experiment. Excepting the BLYP method that tends to reproduce the HF results, an excellent agreement has been obtained between MP2- and DFT-optimized geometries, while the DFTcalculated barrier heights have been found to be in better agreement with experiment than the HF and also the MP2 values. Among all ab initio and DFT methods studied in this work, the hybrid B3P86 gives the best overall performance. These results demonstrate that density functional theory is an adequate tool for studying the mechanism of acid-catalyzed organic reactions. It is important to note that all ab initio and DFT calculations have been carried out using the Hartree-Fockoptimized 6-31G* basis set in order to facilitate the comparison between the two methodologies, and consequently it is to be expected that the use of DFT optimized basis sets will improve the already very good DFT results. Acknowledgment. The authors thank the Centre de Informa`tica and Departament de Quı´mica Fı´sica of the University of Valencia for computing facilities. They thank C.I.C.Y.T. (Project MAT 94-0359) and Conselleria de Cultura, Educacio´ i Cie`ncia de la Generalitat Valenciana, for financial support. M.B. thanks the Conselleria de Cultura, Educacio´ i Cie`ncia de la Generalitat Valenciana, for a personal grant. References and Notes (1) (a) Pines, H. In The Chemistry of Catalytic Hydrocarbon ConVersion; Academic Press: New York, 1981. (b) Jenkins, J. H.; Stephens, T. W. Hydrocarbon Process. 1980, 60, 163. (c) Condon, F. E. In Catalysis; Emmet, P. H., Ed.; Reinhold Publishing Corp.: New York, 1958; Vol. VI, Chapter 2. (2) (a) Gates, B. C.; Katzer, J. R.; Schuit, G. C. A. In Chemistry of Catalytic Processes: McGraw-Hill: New York, 1979; Chapter 1. (b) Voge, H. H. In Catalysis; Emmet, P. H., Ed.; Reinhold Publishing Corp.: New York, 1958; Vol. VI, Chapter 5. (3) (a) Brouwer, D. M.; Hogeveen, H. Prog. Phys. Org. Chem. 1972, 9, 179. (b) Brouwer, D. M. In Chemistry and Chemical Engineering of Catalytic Processes; Prins, R., Schuit, G. C. A., Eds.; Sijthoff and Noordhoff: Alphen aan Rijn, The Netherlands, 1980; p 137. (4) Brouwer, D. M.; Oelderik, J. M. Recl. TraV. Chim. 1968, 87, 721. (5) Brouwer, D. M. Recl. TraV. Chim. 1968, 87, 1435. (6) (a) Boronat, M.; Viruela, P.; Corma, A. J. Phys. Chem. 1996, 100, 633. (b) Sieber, S.; Buzek, P.; Schleyer, P. v. R.; Koch, W.; Carneiro, J. W. de M. J. Am. Chem. Soc. 1993, 115, 259. (7) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab initio Molecular Orbital Theory; Wiley-Interscience: New York, 1986. (8) (a) Parr, R. G.; Yang, W. In Density Functional Theory of Atoms and Molecules; Oxford University Press: New York, 1989. (b) Density Functional Methods in Chemistry; Labanowski, J. R., Andzelm, J., Eds.; Springer-Verlag: New York, 1991. (c) Ziegler, T. Chem. ReV. 1991, 91, 651. (9) (a) Versluis, L.; Ziegler, T. J. Chem. Phys. 1988, 88, 322. (b) Fan, L.; Ziegler, T. J. Chem. Phys. 1991, 95, 7401. (10) Andzelm, J.; Wimmer, E. J. Chem. Phys. 1992, 96, 1280. (11) Johnson, B. G.; Gill, P. M. W.; Pople, J. A. J. Chem. Phys. 1993, 98, 5612. (12) C. W. Bauschlicher, C. W., Jr. Chem. Phys. Lett. 1995, 246, 40.

J. Phys. Chem., Vol. 100, No. 41, 1996 16521 (13) (a) Becke, A. D. J. Chem. Phys. 1992, 96, 2155; 1992, 97, 9173. (b) Becke, A. D. J. Chem. Phys. 1993, 98, 1372, 5648. (14) Rashin, A. A.; Young, L.; Topol, I. A.; Burt, S. K. Chem. Phys. Lett. 1994, 230, 182. (15) (a) Fan, L.; Ziegler, T. J. Phys. Chem. 1992, 96, 6937, 9005. (b) Berces, A.; Ziegler, T. J. Chem. Phys. 1993, 98, 4793; J. Phys. Chem. 1995, 99, 11417. (16) Dobbs, K. D.; Dixon, D. A. J. Phys. Chem. 1994, 98, 4498. (17) (a) El-Azhary, A. A.; Suter, H. U. J. Phys. Chem. 1995, 99, 12751. (b) Wheeles, C. J. M.; Zhou, X.; Liu, R. J. Phys. Chem. 1995, 99, 12488. (c) Ho, Y. P.; Yang, Y. C.; Klippenstein, S. K.; Dunbar, R. C. J. Phys. Chem. 1995, 99, 12115. (d) Kozlowski, P. M.; Rauhut, G.; Pulay, P. J. Chem. Phys. 1995, 103, 5650. (18) (a) Sim, F.; St-Amant, A.; Papai, I.; Salahub, D. R. J. Am. Chem. Soc. 1992, 114, 4391. (b) Branchadell, V.; Sbai, A.; Oliva, A. J. Phys. Chem. 1995, 99, 6472. (c) Novoa, J. J.; Sosa, C. J. Phys. Chem. 1995, 99, 15837. (19) (a) Oie, T.; Topol, I. A.; Burt, S. K. J. Phys. Chem. 1994, 98, 1121. (b) Collins, S. J.; O’Malley, P. J. Chem. Phys. Lett. 1994, 228, 246. (c) Tsuzuki, S.; Uchimaru, T.; Tanabe, K. Chem. Phys. Lett. 1995, 246, 9. (20) (a) Ziegler, T.; Nagle, J. K.; Snijders, J. G.; Baerends, E. J. J. Am. Chem. Soc. 1989, 111, 5631. (b) Eriksson, L. A.; Lunell, S.; Boyd, R. J. J. Am. Chem. Soc. 1993, 115, 6896. (c) Arduengo III, A. J.; Bock, H.; Chen, H.; Denk, M.; Dixon, D. A.; Green, J. C.; Herrmann, W. A.; Jones, N. L.; Wagner, M.; West, R. J. Am. Chem. Soc. 1994, 116, 6641. (d) Doublet, M. L.; Kroes, G. J.; Baerends, E. J.; Rosa, A. J. Chem. Phys. 1993, 103, 2538. (21) (a) Guan, J.; Duffy, P.; Carter, J. T.; Chong, D. P.; Casida, K. C.; Casida, M. E.; Wrinn, M. J. Chem. Phys. 1993, 98, 4753. (b) Matsuzawa, N.; Dixon, D. A. J. Phys. Chem. 1994, 98, 2545, 3967, 11677. (c) Matsuzawa, N.; Ata, M.; Dixon, D. A. J. Phys. Chem. 1995, 99, 7698. (22) (a) Lejl, F.; Adamo, C.; Barone, V. Chem. Phys. Lett. 1994, 230, 189. (b) Baker, J.; Andzelm, J.; Muir, M.; Taylor, P. R. Chem. Phys. Lett. 1995, 237, 53. (23) Fan, L.; Ziegler, T. J. Chem. Phys. 1990, 92, 3645. (b) Fan, L.; Ziegler, T. J. Am. Chem. Soc. 1992, 114, 10890. (c) Deng, L.; Ziegler, T.; Fan, L. J. Chem. Phys. 1993, 99, 3823. (24) Stanton, R. V.; Merz, K. M., Jr. J. Chem. Phys. 1994, 100, 434. (25) Baker, J.; Muir, M.; Andzelm, J. J. Chem. Phys. 1995, 102, 2063. (26) Deng, L.; Ziegler, T. J. Phys. Chem. 1995, 99, 612. (27) Jursic, B.; Zdravkovski, Z. J. Chem. Soc., Perkin Trans. 2 1995, 1223. (28) (a) Zhang, Q.; Bell, R.; Truong, T. N. J. Phys. Chem. 1995, 99, 592. (b) Latajka, Z.; Bouteiller, Y.; Scheiner, S. Chem. Phys. Lett. 1995, 234, 159. (29) Saunders, M.; Rosenfeld, J. J. Am. Chem. Soc. 1969, 91, 7756. (30) Collin, G. C.; Herman, J. A. J. Chem. Soc., Faraday Trans. 1978, 74, 1939. (31) Frisch, M. J.; Trucks, G. W.; Head-Gordon, M.; Gill, P. M. W.; Wong, M. W.; Foresman, J. B.; Johnson, B. G.; Schlegel, H. B.; Robb, M. A.; Replogle, E. S.; Gomperts, R.; Anches, J. L.; Raghavachari, K.; Binkley, J. S.; Gonzalez, C.; Martin, R. L.; Fox, D. J.; DeFrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. Gaussian 92; Gaussian: Pittsburgh, PA, 1992. (32) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Cioslowski, J.; Stefanov, B. B.; Nanayakkara, A.; Challacombe, M.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; DeFrees, D. J.; Baker, J.; Stewart, J. P.; HeadGordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 94, Revision B.1; Gaussian: Pittsburgh, PA, 1995. (33) Hariharan, P. C.; Pople, J. A. Chem. Phys. Lett. 1972, 16, 217. (34) (a) Moeller, C.; Plesset, M. S. Phys. ReV. 1934, 46, 618. (b) Binkley, J. S.; Pople, J. A. Int. J. Quantum Chem. 1975, 9, 229. (35) Schlegel, H. B. J. Comput. Chem. 1982, 3, 214. (36) Baker, J. J. Comput. Chem. 1986, 7, 385; 1987, 8, 536. (37) Slater, J. C. Phys. ReV. 1951, 81, 385. (38) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (39) Becke, A. D. Phys. ReV. A 1988, 38, 3098. (40) Perdew, J. P. Phys. ReV. B 1986, 33, 8822. (41) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. B 1988, 37, 785. (42) Schleyer, P. v. R.; Carneiro, J. W. de M.; Koch, W.; Forsyth, D. A. J. Am. Chem. Soc. 1991, 113, 3990. (43) Raghavachari, K.; Whiteside, R. A.; Pople, J. A.; Schleyer, P. v. R. J. Am. Chem. Soc. 1981, 103, 5649.

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