Carbocations' stabilization by hyperconjugation is one of the cornerstones of chemistry, and has received a considerable attention, particularly in educational, organic, and theoretical literature (Hehre, 1975). They are involved in numerous reactions, whenever an anionic chemical group leaves a carbon atom, as it is the case in S_N1 reaction for instance, or by positively charged species attachment (White et al., 1999). Olah et al. have boosted their experimental study with exceptionally strong acids (Olah, 1993, 2001). However, computational studies are essential for the evaluation of the energetics at work (Lambert and Ciro, 1996; Müller et al., 2005). The recent review by Aue (2011) made a special emphasis on the study of their stability, plus a presentation of carbocations of practical interest, for instance in biological systems.

Carbocations stability is a key in numerous reaction mechanisms, particularly near transition states, where bond breaking/forming processes modify the electronic density of a species. Thus, their stabilization occurs frequently in systems that can be distorted from their equilibrium geometry. Their stability relies particularly on charge delocalization over the whole chemical species, and this can be attained via conjugation and hyperconjugation (Müller et al., 2005; Hadzic et al., 2011; Newhouse and Baran, 2011; Emanuelsson et al., 2013; Zimmerman and Weinhold, 2013). Even when it is small in magnitude, hyperconjugation can determine reactivity, and is of primary importance (Cieplak, 1999; Ingold and DiLabio, 2006; Braïda et al., 2009; Fernandez et al., 2013).

The conformation plays an important role, and sometimes it can be used to switch the delocalization off (deconjugated bond), and evaluate its effects by difference with the conjugated conformation (Wiberg et al., 1990; Gobbi and Frenking, 1994).

Conjugation involves an interaction between π orbitals. It reputedly implies large stabilization energy (or resonance energy) and the effect extends across several bonds (Milian-Medina and Gierschner, 2012). Here, in the carbocation case, the charge delocalization corresponds to the interaction between the empty π orbital of the carbocation center, and at least one filled π orbital expanded on neighboring atoms (Scheme 1A) (Alabugin et al., 2011). We shall use the allyl cation as a model system to evaluate such stabilization. The hyperconjugation (Scheme 1B) involves filled CH orbitals, which are in principle lower in energy. Because orbitals interact better if they are close in energy, the effect is in principle larger for conjugation than for hyperconjugation.

As stated above in the allyl cation, a deconjugation by rotation around a CC bond (Scheme 2) can give an estimation of the resonance energy. However, this gives an underestimation of the energy because hyperconjugative effects lower the rotated structure (Mo, 2004). The value of the resonance energy in allyls has been the subject of some debates, which chiefly concerned the allyl anion (Mo et al., 1996; Mo and Peyerimhoff, 1998; Barbour and Karty, 2004; Linares et al., 2008). As far as the cation is concerned, there are less discrepancies among the authors although electronic correlation is significant and some variations are encountered. We shall retain that, with an Orbital Deletion Procedure (ODP) (Mo, 2006) the resonance energy in the allyl cation was evaluated to 36.6 kcal/mol at the HF level, and to 48.8 with B3LYP/6-311+G(d,p) level using the “Block Localized Wavefunction” (BLW) approach (Mo et al., 2007). These correspond to “Adiabatic Resonance Energies” (ARE), which means that geometrical parameters are relaxed in the localized calculation (Figure 1). The later value is close to the value of 50 kcal/mol, obtained with a corrected Hydride Ion Affinities procedure (HIA) (Aue, 2011). It is also close to the value obtained with our Lewis-based Valence Bond BOND scheme, 55 kcal/mol (Linares et al., 2006), although our value is to be considered as a Vertical Resonance Energy (VRE) because the geometry of the localized structure is constrained to that of the delocalized system.

As reminded above, hyperconjugation is frequently considered as a (small) second order conjugation, which is justified by the lower energy of CH bonding orbitals compared to π (Scheme 1). This is particularly true in neutral systems, and the interaction can be small in these cases. However, there are evidences that hyperconjugation can be large, and even of similar magnitude as conjugation (Daudey et al., 1980; Mullins, 2012; Wu and Schleyer, 2013). Large hyperconjugation effects (in silicon substituted species) are reported to lead to very significant rate enhancements (up to 10¹² times larger) (Lambert and Chelius, 1990; Creary and Kochly, 2009). They also have been isolated and an X-ray structure is even available.¹

Several approaches are being used to describe the conjugation and hyperconjugation effects. We reminded in the introduction different publications using isodesmic reactions, based on hydride abstraction. Hyperconjugation in neutral systems has also been described by a large panel of methods such as the Energy Decomposition Analysis (EDA) (Fernandez and Frenking, 2006; Feixas et al., 2011; Mo et al., 2011), Valence Bond (VB) (Shaik and Hiberty, 2008; Braïda et al., 2009; Havenith and Van Lenthe, 2009) or the already mentioned BLW (Bickelhaupt and Baerends, 2003; Mo et al., 2004a,b; Mo and Schleyer, 2006; Wu et al., 2012), and the well-known Natural Bond Orbital analysis (NBO) (Weinhold, 2003; Glendening et al., 2012).

There is also a rich literature on silicon substituted carbocations, and cross method evaluations have been recently published on these systems (Fernandez and Frenking, 2007; Dabbagh et al., 2012, 2013). Still on carbocations, Schleyer et al. very recently (Wu and Schleyer, 2013), showed large hyperconjugation effects in various strained systems. However, for simple carbocations, which are our subject here, there have been fewer studies. Particularly, the values published by Mo (2006) with the BLW method at the HF level have not been updated with a care for electronic correlation.

Such an evaluation at the correlated level is certainly desired, and this is an objective of the present paper. We evaluated the energetics of hyperconjugation at the B3LYP/6-311G(d) level, which includes some correlation effects. We used the BLW approach in all the cases. Despite some discussions on its apparent basis set dependency (Mo et al., 2010; Zielinski et al., 2010) this type of calculation is becoming a standard for such an evaluation (Steinmann et al., 2011; Wu et al., 2012; Fernandez et al., 2013). Our BLW results shall update and extend the values published previously at the uncorrelated level (Mo, 2006). We expect that, as it was the case for conjugation, correlated value for hyperconjugation will be somehow larger than Hartree-Fock.

The paper is organized as follows. In the computational considerations, we first define our levels of calculations, programs we used, and we write a short memo on the way we used the “BLW” approach in the specific case of carbocations. We then turn our attention to the conformations of the ethyl cation as a model of all the hyperconjugated cations. The results and discussion part is divided into three subsections. The first one deals with the hyperconjugation in the ethyl cation. The second extends to secondary and tertiary carbocations, with methyl substituents and silicon β-effects. In the last part, we added a conjugated link between the C⁺ atom and the substituent (e.g., a C#C triple bond). We evaluated here the incremental stabilization due to hyperconjugation in already conjugated species.

The computations of the ethyl cation displayed in Table 1 were done with Gaussian 03 (Frisch et al., 2004). The three methods (HF, B3LYP, and CCSD) were used for the geometrical optimization with two basis sets, Pople's 6-311G(d) (Krishnan et al., 1980) and Dunning's cc-pvQZ (McLean and Chandler, 1980). DFT calculations are not very sensitive to the size of the basis set, but CCSD is much more basis set dependant. For the B3LYP calculations we used the default implementation of Gamess, with the original VWN5 correlation functional² rather than the defaults Gaussian's implementation (Vosko et al., 1980; Lee et al., 1988; Becke, 1993).³ As it is also the default in Gamess, 6D orbitals were used throughout. As the basis set dependency was small only the 6-311G(d) results are discussed here. The results obtained with the cc-pvQZ basis set are given in the supplementary materials.

For all the BLW calculations, we used a version of Gamess that was modified by Mo to implement the BLW method (Mo and Peyerimhoff, 1998; Mo et al., 2000; Cembran et al., 2009). This implementation permits to re-optimize the geometry of the cations with the localization constraints. This feature was used to compute the geometrical effects of the localization: we obtained the CC⁺ bond lengthening, and its impact on the resonance energy (hence we obtained both VREs and AREs). However, the ARE will be of little use here. We rather discuss the vertical energies because they can concern directly and unequivocally reactive intermediates in their genuine geometry. Geometrical variations upon localization/delocalization (Δd) are discussed though.

With BLW, we used there the standard 6-311G(d) basis set, with no diffuses on heavy atoms and no polarizations on hydrogens. These restrictions intend to preserve the localized calculations' meaning. Diffuse orbitals on first neighbors of the carbocation, as well as π orbitals on H atoms could bring some confusion on the validity of the localization constraints (Galbraith et al., 2013). In these calculations we oriented the systems in such a way that the π system is along the z axis, and two blocks are defined for the localized calculations. One contains zero electrons (it is empty), and is defined over the p_z, d_xz, d_yz atomic orbitals of the C⁺ site. This block ensures an appropriate localization of the positive charge. The other block contains all the electrons and is defined over all the remaining orbitals. For the delocalized calculations, we removed the (empty) block, and added the p_z, d_xz, d_yz atomic orbitals to the other block, so delocalization is now allowed.

The analysis of the BLW results concerns both energies and difference between electronic densities. For these densities we used two “cube” files generated by Gaussian 03. One has densities obtained with the orbitals of the localized calculation. The second uses the delocalized orbitals.⁴ The density differences at each point of the grid defined in the cube files were drawn with the VMD freeware.⁵ We refer to these drawings as Electronic Densities Difference maps (EDD maps). They indicate the flux of electron density (gain/loss) when localization constraints are relaxed.

The results for hyperconjugation in simple carbocations are in Table 2 and Figure 3, while Table 3 and Figure 4 concern the evaluation of the hyperconjugation in conjugated carbocations. In the tables, the two first columns correspond to VRE and ARE as defined in Figure 1. In the three last columns is the CC⁺ bond variation when hyperconjugation is activated. The CC⁺ bond shortens when the delocalization is allowed and Δd is thus always negative. These results were of course expected since the electronic delocalization evidently builds a kind of π bonding between these two atoms. In the discussions that follow, VRE's are used more often than ARE's because their definition is more straightforward.

To have in mind an order of magnitude for our calculations, we shall recall that for the allyl cation, the VRE amounts to 56.0 kcal/mol (Table 2, entry 7). This value is to be considered as large.

Using B3LYP we pinned down resonance energies in a variety of carbocations, with a special attention to hyperconjugative effects. Our discussion focused of vertical resonance energies, and we showed here how hyperconjugation could be of similar magnitude as conjugation, but this evaluation necessitates some correlated methods.

The fact that we considered cationic systems enhanced the delocalization effects. Smaller effects are expected (and reported) for neutral systems (Fernandez and Frenking, 2006). Nevertheless, a dimethyl-silyl substituent (DMS), delocalizes a significant amount of electron density from the CSi bonds onto the neighboring C⁺, and this hyperconjugation corresponds to a stabilization energy as large as 61.8 kcal/mol. This is to be compared to the vinyl delocalization onto the C⁺, in the allyl cation. It amounts to “only” 56.0 kcal/mol of conjugation; hence hyperconjugative effects on energy can be larger than conjugation. The CC⁺ bond distances are accordingly short, e.g., 1.371Å for the DMS-CH₂₊ carbocation.

Long-range hyperconjugative effects travel across an unsaturated linkage (a double or a triple bond here). We show that the energy associated to them can be as large as 25 kcal/mol. They can be logically extended to aryl linkages, for instance in para substituted benzyl cations.

Cartesian coordinates of all the compounds discussed in the text.