8028
J. Phys. Chem. A 2007, 111, 8028-8035
Hyperconjugative Stabilization in Alkyl Carbocations: Direct Estimate of the β-Effect of Group-14 Elements Israel Ferna´ ndez and Gernot Frenking* Fachbereich Chemie, Philipps-UniVersita¨t Marburg, Hans-Meerwein Strasse, 35043 Marburg, Germany ReceiVed: May 15, 2007; In Final Form: May 26, 2007
DFT calculations at the BP86/TZ2P level have been carried out for the primary, secondary, and tertiary carbenium ions [H2C-CH(EH3)2]+ (1a-e), [HC{CH(EH3)2}2]+, (2a-e), and [C{CH(EH3)2}3]+ (3a-e) for E ) C, Si, Ge, Sn, Pb. The nature of the interaction between the carbenium center H2-nC+ and the substituents {CH(EH3)2}m has been investigated with an energy decomposition analysis (EDA) aiming at estimating the strength of the π hyperconjugation which electronically stabilizes the carbenium ions. The results of the EDA show that the calculated ∆Eπ values can be used as a measure for the strength of hyperconjugation in carbenium ions arising from the interactions of saturated groups possessing π orbitals. The theoretical data suggest that the ability of σ C-E bonds to stabilize positive charges by hyperconjugation follow the order C , Si < Ge < Sn < Pb. Hyperconjugation of C-Si bonds is much stronger than hyperconjugation of C-C bonds while the further rising from silicon to lead is smaller and has about the same step size for each element. The strength of the hyperconjugation in primary, secondary, and tertiary alkyl carbenium ions does not increase linearly with the number of hyperconjugating groups; the incremental stabilization becomes smaller from primary to secondary to tertiary cations. The effect of hyperconjugation is reflected in the shortening of the C-C bond distances and in the lengthening of the C-E bonds, which exhibits a highly linear relationship between the calculated C-C and C-E distances in carbocations 1-3 and the hyperconjugation estimated by the ∆Eπ values.
Introduction Carbocations are arguably the most important reactive intermediates in organic chemistry.1 Meerwein suggested in 1922 for the first time that positively charged alkyl cations might be involved in a chemical reaction.2 He recognized that the rearrangement reactions of camphene hydrochloride showed similarities with the behavior of triarylhalomethanes, which could be understood if ionization takes place during the reaction. Meerwein’s proposal was a breakthrough in physical organic chemistry3 because later research revealed that carbocations are the key for understanding not only rearrangement reactions but for many other organic processes like substitution and elimination reactions.1 Carbocations may be distinguished into fivecoordinated species (carbonium ions) and three-coordinated species (carbenium ions). A particular class of carbenium ions which is termed nonclassical ions has hydrogen atoms or other groups in a bridging position to the electron deficient positively charged carbon atom. Nonclassical cations have been the topic of intensive theoretical and experimental research.4 The parent carbenium ion CH3+ is not stable in a condensed phase. Substitution of hydrogen by substituents which are capable of donating electronic charge into the formally empty p(π) AO stabilizes the cations which may eventually become isolated and characterized with spectroscopic methods. Groups which carry lone-pair electrons such as NR2, OR, or halides are particularly suitable for stabilizing carbenium ions.1 Another possibility for yielding stable carbocations is through conjugation with a π-electron systems, yielding conjugated species such as * To whom correspondence should be addressed. Phone: +49(0)6421-282 5563. Fax: +49(0)6421-282 5566. E-mail: frenking@ chemie.uni-marburg.de.
allyl or benzyl cations which are notably stable species. Saturated groups are generally less suitable for stabilizing carbenium ions because hyperconjugation (Figure 1) is less efficient than conjugation. Recent theoretical studies by us and others have shown, however, that stabilization due to hyperconjugation is not negligible and that it may significantly contribute to the structure and stability of molecules.5,6 Hyperconjugative stabilization in a carbenium ion CR3+ involves the donation of electronic charge from a substituent R, which has no multiple bonds into the formally empty p(π) orbital of the carbenium carbon atom. Figure 1a depicts a situation where the hyperconjugative stabilization in an alkyl carbenium ion consists of genuine π interactions such as conjugative interactions because the cation possesses an appropriate mirror plane. The difference between hyperconjugation and conjugation is that in the latter interaction the substituent has a multiple bond consisting of σ and π bonding and that the charge donation comes from the π component. The charge donation from an electronically saturated substituent comes from an orbital which has local σ symmetry. This is the reason that hyperconjugation is often referred to as donation from an occupied σ orbital into an empty π orbital. Figure 1b depicts a situation where the methyl group is rotated about the C-C bond by 90°. The hyperconjugation now takes place between an occupied orbital of the CH3 group and the vacant p orbital of C(+) atom, which in the strictest sense have σ symmetry because they are symmetric with respect to the mirror plane of that conformation. However, these orbitals have π character because the sign of the orbital changes in the local H2C(+)-C plane which is not a mirror plane of the ethyl cation conformation depicted in Figure 1b. The resulting orbital is therefore
10.1021/jp073737k CCC: $37.00 © 2007 American Chemical Society Published on Web 07/14/2007
Stabilization in Alkyl Carbocations
Figure 1. Schematic representation of hyperconjugation in an alkyl carbenium cation. (a) Hyperconjugative charge donation from an occupied π orbital of the alkyl substituent. (b) Hyperconjugative charge donation from an occupied σ orbital (pseudo-π orbital) of the alkyl substituent.
said to possess pseudo-π-symmetry.7 In this paper we only investigate systems which correspond to the situation which is sketched in Figure 1a. It is well-known that alkyl-substituted carbenium ions CR3+ exhibit the stability order tertiary > secondary > primary > CH3+. Until very recently, the only aliphatic cations which could become structurally characterized by X-ray analysis were tertiary carbocations like the adamantyl cation8 and the tert-butyl cation.9 Bochman and co-workers now report about the first isolation and X-ray structure determination of a secondary alkyl carbenium ion.10 The remarkably stable crystal compound [HC{CH(SiMe3)(SnMe3)}2]+Zr2Cl9- melts at 109 °C while the analogous species with Hf2Cl9- as counterion melts at 120 °C. The X-ray structure analysis of the former cation shows that there are no close contacts between the cation and the anion.10 The authors reported also about DFT calculations of the secondary carbenium ion and related all-silicon model compounds. It was suggested that the stability of the cation comes from hyperconjugation, where the C-Sn bonds which are in a β position should be much more effective than the C-Si bonds. Evidence for the suggestion comes from analysis of the charge distribution, the geometry and from the NMR chemical shifts.10 The same conclusion was achieved experimentally by Lambert11 through comparison of the extrusion reaction of Me3MX from Me3MCH2CH2X (M ) Si to Sn; X ) nucleofuge) in solvents of different nucleophilicities and theoretically by Gordon12 through comparison of the isodesmic reactions of H3MCH2CH2+ and CH3+. The stabilizing β-effect of group-14 elements which is widely employed in synthetic transformations13 was the topic of extensive experimental and theoretical studies with the goal to elucidate the origin of the stabilization.14 A pivotal question concerns the strength of the hyperconjugative stabilization which is the driving force of the β-effect. It would be helpful if a direct estimate of the strength of the hyperconjugation could be made based on a well-defined quantum chemical partitioning of the interaction energy which does not need an external reference system. A method which uses only the π orbitals of the interacting fragments in the geometry of the molecule for estimating π interactions is the energy decomposition analysis (EDA).15 We recently reported that the calculated ∆Eπ values given by the EDA can be used as a probe to estimate the relative contributions of π interactions which come from conjugation between multiple bonds or from hyperconjugation which arise from the interactions of saturated groups possessing π orbitals.5f,5g,16 We successfully applied this method in order to compare the calculated strength of the π conjugation in metaand para-substituted benzylic cations and anions with Hammett substituent constants.16a The advantage of the EDA calculations for estimating the strength of π interactions was revealed by the excellent correlation between the calculated ∆Eπ values and experimental data such as the 13C NMR chemical shifts in large π conjugated systems.16b The good performance of the EDA encouraged us to employ the method for a theoretical investigation of the β-effect in
J. Phys. Chem. A, Vol. 111, No. 32, 2007 8029
Figure 2. Schematic representation of the primary (1a-e), secondary (2a-e), and tertiary (3a-e) alkyl carbenium ions investigated in this work.
primary, secondary and tertiary alkyl carbocations which carry C-E bonds (E ) C- Pb). We calculated the geometries and analyzed the bonding situation in the carbenium ions [H2CCH(EH3)2]+ (1a-e), [HC{CH(EH3)2}2]+, (2a-e), and [C{CH(EH3)2}3]+ (3a-e) (E ) C- Pb), which are shown in Figure 2. The calculated data give detailed information about the interatomic interactions in the cations. The theoretical work should be helpful for further experimental work which aims at the synthesis of carbenium ions. Methods The geometries of the molecules have been optimized at the nonlocal DFT level of theory using the exchange functional of Becke17 in conjunction with the correlation functional of Perdew18 (BP86). Uncontracted Slater-type orbitals (STOs) were employed as basis functions for the SCF calculations.19 The basis sets have triple-ζ quality augmented by two sets of polarization functions, that is, p and d functions for the hydrogen atoms and d and f functions for the other atoms. This level of theory is denoted as BP86/TZ2P. An auxiliary set of s, p, d, f, and g STOs was used to fit the molecular densities and to represent the Coulomb and exchange potentials accurately in each SCF cycle.20 Scalar relativistic effects have been considered using the zero-order regular approximation (ZORA).21 The vibrational frequencies of the optimized structures have been calculated in order to investigate the nature of the stationary points. The Hessian matrices of the optimized geometries have in all cases only positive eigenvalues which means that the fully optimized structures reported here are minima on the potential energy surface. The atomic partial charges were calculated by using the Hirshfeld partitioning scheme.22 The calculations were carried out with the program package ADF 2003.23 The interactions were analyzed by means of the energy decomposition analysis (EDA) of ADF,15 which was developed by Ziegler and Rauk24 following a similar procedure suggested by Morokuma.25 EDA has proven to give important information about the nature of the bonding in main-group compounds and transition-metal complexes.26 The focus of the bonding analysis is the instantaneous interaction energy, ∆Eint, of the bond, which is the energy difference between the molecule and the fragments in the electronic reference state and frozen geometry of the compound. The interaction energy can be divided into three main components:
∆Eint ) ∆Eelstat + ∆EPauli + ∆Eorb
(1)
∆Eelstat gives the electrostatic interaction energy between the fragments, which are calculated using the frozen electron density distribution of the fragments in the geometry of the molecules. The second term in eq 1, ∆EPauli, refers to the repulsive interactions between the fragments, which are caused by the fact that two electrons with the same spin cannot occupy the same region in space. ∆EPauli is calculated by enforcing the Kohn-Sham determinant on the superimposed fragments to obey the Pauli principle by antisymmetrization and renormal-
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Ferna´ndez and Frenking
Figure 3. (a) Fully optimized geometries at BP86/TZ2P of (a) 1a-e. (b) Partially optimized geometries at BP86/TZ2P of 1b′-e′ where the C-C distance is kept frozen at the value of 1a (1.405 Å). Bond distances are given in Å and angles in deg.
ization. The stabilizing orbital interaction term, ∆Eorb, is calculated in the final step of the energy partitioning analysis when the Kohn-Sham orbitals relax to their optimal form. This term can be further partitioned into contributions by the orbitals belonging to different irreducible representations of the point group of the interacting system. The interaction energy, ∆Eint, can be used to calculate the bond dissociation energy, De, by adding ∆Eprep, which is the energy necessary to promote the fragments from their equilibrium geometry to the geometry in the compounds (eq 2). The advantage of using ∆Eint instead of De is that the instantaneous electronic interaction of the fragments becomes analyzed which yields a direct estimate of the energy components. Further details of the energy partitioning analysis can be found in the literature:15
-De ) ∆Eprep + ∆Eint
(2)
The interacting fragments in the EDA calculations are the open-shell species which come from breaking the σ bond(s) between them. The calculations of the open shell fragments for the EDA can only be carried out in the ADF program using the restricted formalism while for the optimization of the fragments the unrestricted formalism is used. The energy differences
between the restricted and unrestricted calculations were always
