Stabilization of carbocations CH₃⁺, C₂H₅⁺, i-C₃H₇⁺, tert-Bu⁺, and cyclo-pentyl⁺ in solid phases: experimental data versus calculations

Abstract

Comparison of experimental infrared (IR) spectra of the simplest carbocations (with the weakest carborane counterions in terms of basicity, CHB₁₁Hal₁₁⁻, Hal = F, Cl) with their calculated IR spectra revealed that they are completely inconsistent, as previously reported for the t-Bu⁺ cation [Stoyanov E. S., et al. J. Phys. Chem. A, 2015, 119, 8619]. This means that the generally accepted explanation of hyperconjugative stabilization of the carbocations should be revised. According to the theory, one CH bond (denoted as ) from each CH₃/CH₂ group transfers its σ-electron density to the empty 2p_z orbital of the sp² C atom, whereas the σ-electron density on the other CH bonds of the CH₃/CH₂ group slightly increases. From experimental IR spectra it follows that donation of the σ-electrons from the bond to the 2p_z C-orbital is accompanied by equal withdrawal of the electron density from other CH bonds, that is, the electrons are supplied from each CH bond of the CH₃/CH₂ group. As a result, all CH stretches of the group are red shifted, and IR spectra show typical CH₃/CH₂ group vibrations. Experimental findings provided another clue to the electron distribution in the hydrocarbon cations and showed that the standard computational techniques do not allow researchers to explain a number of recently established features of the molecular state of hydrocarbon cations.

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Introduction

The research on carbocation stabilization is one of the most important trends in carbocation chemistry. Experimental studies mainly deal with gas phase infrared (IR) mass spectroscopy,^1–4 IR and NMR characterization in superacidic media,^5–12 and X-ray structural analysis.^13–17 As shown by ab initio calculations, the isolated carbocations are stabilized due to the intramolecular charge distribution under the influence of hyperconjugation^1,18–22 and polarization.²³ The simplest carbocation, CH₃⁺, is subject to only the polarization effect.²³ For the gaseous C₂H₅⁺ cation, the theory predicts and experiments confirm that the hyperconjugative interaction between the CH₃ group and the empty 2p_z orbital in the carbon atom of the CH₂ group is so strong that a bridged-proton symmetrical structure is formed.³ The isolated i-propyl (i-Pr⁺) cation has been the subject of extensive ab initio calculations.^24–27 In its lowest-energy C₂ conformer, two C–H bonds are aligned with the up and down lobes of the 2p_z orbital of the sp² C-atom, suggesting that these C–H bonds are involved in strong hyperconjugation.

Most attention has been given to the tert-butyl cation, whose hyperconjugative delocalization of the positive charge has been used as a textbook explanation of its stability. As predicted by quantum chemical calculations, the most energetically stable t-Bu⁺ cation (C_s symmetry) has three CH bonds, one from each CH₃ group, which are aligned in parallel with the empty 2p_z orbital of the central sp² carbon atom.^1,18–20 This situation makes donation of its σ-electron density to this orbital possible, resulting in σ-p_z hyperconjugation (Scheme 1). These three C–H bonds (one of them is marked with in Scheme 1) are weakened and their νCH frequencies are significantly decreased, whereas the remaining C–H bonds (CH′ and CH′′ in Scheme 1) are slightly strengthened and their νCH increases.

Scheme 1 Schematic representation of the electron density redistribution in accordance with the classical understanding of the mechanism of hyperconjugation stabilization in t-Bu⁺.

Thus, ab initio calculations predict a coherent picture of the mechanism underlying positive charge dispersion in the naked cations C₂H₅⁺, (CH₃)₂CH⁺ (i-Pr⁺), and (CH₃)₃C⁺ (t-Bu⁺) primarily via the hyperconjugative effect. It is clear that the polarization effect must play an important role in cation stabilization although convincing data based on calculations are actually absent.

Our recent experimental study of the t-Bu⁺ cation in condensed phases, together with interpreted IR spectra of gaseous t-Bu⁺, showed²⁸ that empirical findings contradict the generally accepted hyperconjugation mechanism (Scheme 1). It follows from the experiments that all C–H bonds of the two CH₃ groups of t-Bu⁺ donate σ-electrons to the carbon's 2p_z orbital (Scheme 2), and the third CH₃ group is mostly affected by polarization.

Scheme 2 Schematic representation of electron density redistribution in t-Bu⁺ according to empirical data.²⁸

It should be noted that in condensed phases, there is an additional intermolecular effect of carbocation stabilization: hydrogen bonding with the immediate surroundings. It leads to the strengthening of hyperconjugation and contributes to the scattering of the positive charge to the environment.^17,28

In the present work we expanded the comparison of the computational results explaining stabilization of carbocations (structurally optimized and having the lowest energy) with empirical findings in IR spectra of carbocations in the series CH₃⁺, C₂H₅⁺, i-C₃H₇⁺, t-Bu⁺, and cyclo-C₅H₉⁺, with the carborane counterions (CHB₁₁Hal₁₁⁻, Hal = F, Cl; hereinafter abbreviated as {Hal₁₁⁻}, see Fig. 1), whose conjugated acids are the strongest pure Brønsted superacids available today.^29,30 We established that IR spectra of {F₁₁⁻} or {Cl₁₁⁻} in ionic carbocation salts are not dependent on the nature of the cation and therefore are not considered here. The aims of the present study were (i) to determine whether t-Bu⁺ is a special case of a cation or a common representative of saturated carbocations and (ii) to verify the compatibility of the theoretical data with experimental results for a range of simplest carbocations. We focused on the C–H stretch vibrations because they are mostly sensitive to hyperconjugation and polarization effects. Despite the importance of intermolecular hydrogen bonding for carbocation stabilization in the condensed phase, this effect will not be examined here in detail because it does not change the hyperconjugation mechanism when going from vacuum to a condensed phase as shown previously.^17,28

Fig. 1 Icosahedral carborane anions of CHB₁₁Hal₁₁⁻ type (Hal = F, Cl) abbreviated as {F₁₁⁻} or {Cl₁₁⁻}. Halogen atoms are depicted in green.

Experimental

All sample handling was carried out in an inert atmosphere (H₂O, O₂ < 1 ppm) in a dry box. Carborane acids H{Hal₁₁} and their CH₃{Hal₁₁} salts were prepared as previously described.^30–33 The salts of C₂H₅⁺ and C₃H₇⁺ cations with the {F₁₁⁻} anion were obtained by reacting CH₃F with H{F₁₁} in an IR cell-reactor with simultaneous recording of the IR spectra of the formed cations.³³

The spectrum of the C₃H₇⁺{F₁₁⁻} salt was also obtained in another way: instead of the H{F₁₁} acid, Et₃Si{F₁₁} was used, synthesized as described in ref. 30. A small drop of liquid 2-Cl-propane was added to an excess of solid Et₃Si{F₁₁} placed on the surface of the diamond attenuated total reflectance (ATR) accessory. A rapid reaction takes place

Et₃Si{F₁₁} + C₃H₇Cl → C₃H₇⁺{F₁₁⁻} + Et₃SiCl↑, (1)

and gaseous Et₃SiCl is quickly removed. An ATR spectrum of the formed solid revealed strong absorption of the i-C₃H₇⁺{F₁₁⁻} salt and weak absorption of unreacted (excess) Et₃Si{F₁₁}. Computer subtraction of the weak spectrum of residual Et₃Si{F₁₁} allowed us to obtain a high quality spectrum of the i-C₃H₇⁺{F₁₁⁻} salt.

The cyclo-C₅H₉⁺{F₁₁⁻} salt was obtained by means of direct interaction of liquid cyclo-pentane with a powder of H{F₁₁} in accordance with eqn (2):

cyclo-C₅H₁₀ + H{F₁₁} → C₅H₉⁺{F₁₁⁻} + H₂ (2)

It was reported³⁰ that all alkanes, C_nH_m (n ≥ 4), react with the H{F₁₁} acid in the same way. Reaction (2) proceeded under ambient conditions for 4 hours. A white powder of the C₅H₉⁺{F₁₁⁻} salt was obtained from the solution after vacuum removal of the solvent, cyclo-pentane.

IR spectra of isopropyl and cyclo-pentyl carbocations with the {Cl₁₁⁻} counterion were obtained as follows. A grain of the H{Cl₁₁} acid was placed on the top face of the diamond crystal of the attenuated total reflectance (ATR) IR accessory and was wetted with a small drop of liquid 2-chloropropane or chloro-cyclo-pentane; this amount was not sufficient for full acid utilization. This approach ensures stoppage of the reaction at the first-stage, as illustrated in eqn (3) (an example of preparation of the i-Pr⁺ salt):

C₃H₇Cl + H{Cl₁₁} → C₃H₇⁺{Cl₁₁⁻} + HCl. (3)

Recording the spectrum without applied pressure allowed us to obtain the spectrum of the oily byproduct covering the diamond surface as a thin film. This byproduct is formed when an excess of the chlorine hydrocarbon interacts with an acid. Recording the spectrum of the solid particles crushed under pressure, which contained mainly the carbocation salt and an excess of unreacted acid, enabled us to obtain their spectra with a minor overlap with the spectrum of byproducts. Computer-based subtraction of the spectrum of the byproducts and excess acid allowed for isolation of the carbocation spectra.

IR spectra were recorded on a Perkin Elmer Spectrum-100 spectrometer in the 4000–400 cm⁻¹ frequency range in transmission or ATR mode. The spectrometer was installed inside the dry box. The spectra were manipulated using GRAMMS/A1 (7.00) software from Thermo Scientific.

Computational details

Geometric parameters of the species under study were optimized at the B3LYP-D3/def2-TZVPD level of theory^34–37 using ultrafine grids. Equilibrium structures of compounds R{F₁₁} and t-Bu{Cl₁₁} were also calculated in a dichloroethane (DCE) solution (ε = 10.125) using the SMD solvation model.³⁸ All stationary points were characterized as minima by a vibrational analysis (the number of imaginary frequencies [NImag] was equal to zero), unless stated otherwise. Zero-point energies (ZPEs) were computed from the corresponding vibrational frequencies without scaling factors. (SMD-)B3LYP-D3/def2-TZVPD optimized structures were used in all subsequent computations.

To compare calculated and experimental vibrational frequencies, (SMD-)B3LYP-D3/def2-TZVPD harmonic frequencies were scaled by a factor of 0.9674 as recommended by Kesharwani et al.³⁹ To obtain more accurate relative energies of some isomers, single-point high-level CCSD(T)/def2-TZVPD coupled-cluster computations⁴⁰ within a frozen core approximation were additionally performed.

The energy decomposition analysis (EDA)⁴¹ was carried out for the most stable compounds at the B3LYP-D3/TZ2P level of theory using the scalar relativistic (SR) zeroth-order regular approximation (ZORA) Hamiltonian (core potentials were not used, and the quality of the Becke numerical integration grid was set to the keyword good).⁴²

SR-ZORA-B3LYP-D3/TZ2P computations were performed using the ADF2016 software,^43–45 while the remaining gas-phase calculations in the gas phase and in a DCE solution were conducted using the Gaussian09 software.⁴⁶ The natural bond orbital (NBO) analysis^47,48 was performed for the species of interest as implemented in Gaussian09. def2-TZVPD basis sets were taken from the EMSL database.^49,50

Density functional theory (DFT) calculations with periodic boundary conditions were performed using the PBE functional⁵¹ with D3 dispersion correction³⁶ using the VASP software package.^52,53 The projector augmented wave (PAW) method⁵⁴ was used to analyze the core states along with a plane-wave energy cutoff of 400 eV. Only the Γ point was used for sampling the Brillouin zone.

The convergence criteria for electronic and structural optimization were set to 10⁻⁶ eV and 10⁻⁵ eV Å⁻¹, respectively. Harmonic vibrational frequencies were calculated by the finite difference method using the central difference with a step size of 0.015 Å.

All compounds were considered in their ground states. The spin-restricted formalism was employed for both DFT and CCSD(T) computations.

Results and discussion

Characterization of carbocations on the basis of calculations

To obtain a unified computational view, we considered all carbocations (R⁺) under study and their complexes with Ar and {F₁₁⁻} in the gas phase and in solution at the same level of theory. The lowest-energy structures were chosen for subsequent analysis (see Fig. S1–S4 in ESI† for a complete list of isomers). The results obtained for the R⁺, R⁺⋯Ar and R{F₁₁} species are given in Tables S1–S5 in ESI.† The stretching vibrations and some valent angles for naked and solvated carbocations are compared in Table 1.

Table 1 CH stretch frequencies (scaled by a factor of 0.9674) and their IR intensities (km mol⁻¹, in parentheses) calculated at the B3LYP-D3/def2-TZVPD level of theory for the most energetically stable naked and Ar-solvated carbocations. Relevant structural parameters are also given

Compound	C2H5^+⋯Ar	i-Pr^+		i-Pr^+⋯Ar		t-Bu^+		t-Bu^+⋯Ar		c-C5H9^+	c-C5H9^+Ar
a E(CCSD(T)/def2-TZVPD//B3LYP-D3/def2-TZVPD) + ZPE(B3LYP-D3/def2-TZVPD). b The θ angle is shown in Scheme 3. Averaged values of angle θ are presented. c The characteristic frequencies of bonds mostly affected by hyperconjugation are boldfaced.
Symmetry	C s	C 2	C 2v	C 1	C s	C 1	C s	C 1	C 1	C 2	C 1
NImag	0	0	1	0	1	0	1	0	0	0	0
Relative energy,^a kcal mol^−1	0	0	0.2	0	0.1	0	0	0	0.1	0	0
Dihedral angle θ, degrees^b	0	17	36	13	20	21	2	1	14	13	12
angle, degrees	92.6	102	107.8	101.4	104.6	104.9	103.3	103.2	103.9	98.1	98.1
		102	107.8	102.8	104.6	105.9	104.1	104.1	104.3	98.1	98.6
						106.8	104.1	104.1	105.2
Scaled CH stretches,^c cm^−1	2612 (154)	2764 (224)	2841 (239)	2766 (145)	2803 (141)	2853 (91)	2835 (109)	2840 (78)	2842 (93)	2704 (130)	2711 (83)
		2777 (7)	2855 (1)	2799 (59)	2815 (54)	2864 (64)	2838 (85)	2843 (104)	2851 (53)	2712 (15)	2735 (45)
						2881 (34)	2852 (19)	2855 (9)	2871 (33)
	2985 (17)	2942 (54)	2856 (0)	2938 (28)	2921 (53)	2931 (16)	2974 (5)	2976 (3)	2952 (12)	2961 (0)	2962 (0)
	3024 (8)	2944 (0)	2856 (64)	2960 (16)	2923 (11)	2944 (15)	2975 (2)	2977 (5)	2970 (8)	2969 (8)	2970 (9)
	3071 (11)	3022 (0)	3022 (1)	3032 (0)	3032 (1)	2953 (10)	2977 (8)	2981 (5)	2972 (5)	2984 (22)	2982 (12)
	3128 (28)	3053 (7)	3062 (9)	3053 (7)	3059 (8)	3050 (2)	3044 (3)	3045 (2)	3048 (2)	2986 (0)	2994 (8)
		3058 (9)	3068 (5)	3060 (8)	3067 (5)	3052 (3)	3046 (1)	3049 (2)	3049 (2)	3037 (0)	3036 (1)
										3042 (1)	3042 (1)

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The hyperconjugation effect is assumed to be the case starting from naked C₂H₅⁺. In this cation, the hyperconjugation strength is so significant that the proton is converted to the bridge state leading to a symmetrical structure³ (II.1 in Fig. S1 in ESI†), which does exist in the condensed phase. Thus, we excluded the bridged structure from further analyses.

The C₂H₅⁺ structure becomes asymmetrical after solvation by the Ar atom via the C center. The bond is oriented strictly parallel to the 2p_z orbital of the sp² C atom (θ = 0, see Scheme 3) and the angle is reduced (92.6°, Table 1); these data indicate a strong hyperconjugative interaction. The similar orientation of the bond from each CH₃ group relative to the 2p_z orbital of the sp² C atom has only naked t-Bu⁺ (C_s, θ = 2°) and t-Bu⁺⋯Ar solvate (C₁, θ = 1°).This finding corresponds to the generally accepted mechanism of carbocation hyperconjugative stabilization (Scheme 1). Nevertheless, the θ angles of t-Bu⁺ (C₁), despite almost the same relative energy as in t-Bu⁺ (C_s), are increased by ca. 20°. The θ angle reaches values of 36° and 20° for the low-lying isomers of i-Pr⁺ (C_2v) and i-Pr⁺⋯Ar (C_s) respectively (Fig. S2 and S3 in ESI†), which means that one C–H bond from each CH₃ group is perpendicular to the vacant 2p_z orbital, and two other C–H bonds are equivalently affected by hyperconjugation (Scheme 4). An intermediate case takes place in c-C₅H₉⁺ and c-C₅H₉⁺⋯Ar species, with θ ranging from 12° to 13°. Thus, rotation of the CH₃ group by angle θ has a weak influence on the carbocation hyperconjugation stabilization.

Scheme 3 A schematic representation of C₂H₅⁺–Ar (C_3v, Ar is omitted) along the C–C bond. Dihedral angle θ shows deviation of the plane from the direction of carbon's 2p_z orbital.

Scheme 4 Schematic representation of i-Pr⁺ (C_2v). The views of the two CH₃ groups are overlapping.

The strong hyperconjugation influence on the CH₃ group in asymmetric C₂H₅⁺⋯Ar (Scheme 3) leads to a strong decrease in the stretch to 2612 cm⁻¹, whereas frequencies of the other two CH′ and CH′′ bonds (νCH₂) are changed much less as compared to neutral hydrocarbons (Table 1). The and νCH₂ frequencies are highly characteristic and are not mixed (Fig. S5b in ESI†). Unlike C₂H₅⁺, solvation of other carbocations with Ar via the sp² C atom has a slight effect on the calculated IR spectra (Table 1). The spectra of all carbocations under study (both naked and solvated with Ar) show similar features: (i) the bands of C–H stretches affected by hyperconjugation are significantly red shifted and have increased intensity (bold faced in Table 1); (ii) frequencies of the other C–H bonds of CH₃/CH₂ groups are rather increased or are in the frequency region common for neutral hydrocarbons; (iii) the normal vibrations of these bonds or groups have a highly characteristic nature.

To gain insights into the bonding situation for the representative R⁺⋯Ar and R{F₁₁} complexes, we performed the EDA (Table S1 in ESI†) and NBO analyses (Table S2 in ESI†). The results of our gas-phase calculations for the carbocations paired with the {F₁₁⁻} anion indicate the following: (a) in the case of R = C₂H₅, the hyperconjugation effect is the weakest one because of a covalent bond between ethyl and {F₁₁} leading to a modest energy gain corresponding to hyperconjugative stabilization (ΔE⁽²⁾_ij = 9.2 kcal mol⁻¹; Table S2, ESI†) and rather weak IR intensity of the corresponding CH stretch vibration (Table S4, ESI†); (b) the spectrum of i-Pr⁺ (Table S4 in ESI†) shows only some signs of weak hyperconjugation (two slightly decreased frequencies at 2923 and 2916 cm⁻¹ with somewhat increased intensity), suggesting that the bonding between i-Pr and {F₁₁} moieties is still covalent (ΔE⁽²⁾_ij = 11.7 kcal mol⁻¹; Table S2, ESI†); (c) the spectra of cyclo-pentyl⁺ and t-Bu⁺ (Table S4 in ESI†) show greater resemblance to those of the corresponding Ar-solvated cations, as compared to the lighter carbocations.

NBO analysis revealed a polar covalent two-center two-electron C–F bond populated by 2.0 |e| at R = CH₃, C₂H₅, and i-Pr. Indeed, according to the EDA (Table S1 in ESI†), the electrostatic interaction (ΔE_elstat, 52–79%) between R⁺ and {F₁₁⁻} moieties dominates over orbital (ΔE_orb) and dispersion (ΔE_disp) interactions in all compounds. On the other hand, the energy of orbital interactions makes a noticeable contribution to the total attractive interactions when R = CH₃ (47%), C₂H₅ (42%), and i-Pr (34%), in line with the calculated IR spectra. Thus, a covalent contribution to the C–F bond is present in all cases and gradually decreases in the order R = CH₃ < C₂H₅ < i-Pr < cyclo-pentyl ≤ t-Bu simultaneously with the increase of hyperconjugative stabilization in the same order as can be seen from the corresponding ΔE⁽²⁾_ij values (Table S2, ESI†).

However, as will be illustrated below, all solid R{F₁₁} compounds, except for CH₃{F₁₁}, are characterized by ionic bonding. The reason is the bulk effect, which most clearly manifests itself in the example of alkali halides. They are ionic in the solid state, but their neutral diatomics are linked covalently in the gas phase showing stretch vibration (for example, νNa–Cl at 364.6 cm⁻¹).⁵⁵ So, the crystal lattice promotes ionic binding.

To model the effect of the environment taking place in the condensed phase, we performed NBO and EDA analyses of the R{F₁₁} species when they are transferred from the gas phase to a DCE solution. This change of the phase is accompanied by an increase of the R–{F₁₁} distance (Tables S1 and S2, ESI†), which is relatively small for R = CH₃ (Δr = 0.033 Å) and pronounced when R = C₂H₅ (Δr = 0.120 Å), making the C–F bond more polar but retaining its covalent nature (ΔE_orb, 38–48%). In the case of i-C₃H₇{F₁₁}, the lengthening of C–F is the most pronounced (Δr = 0.604 Å), indicating that the character of the bond changes to presumably ionic (ΔE_orb, 16%; ΔE_elstat, 78%). Replacing i-C₃H₇⁺ with c-C₅H₉⁺ further increased the C–F separation to 2.584 Å, increasing the ionicity of the bond (ΔE_orb, 12%). Finally, the t-C₄H₉{F₁₁} compound has the C–F distance typical for ionic species (>2.6 Å, ΔE_orb, 11%; Table S4, ESI†). Similar results were obtained in SMD-B3LYP-D3/def2-TZVPD calculations for DCE solutions: the dielectric medium increased the separation between R and {Hal₁₁} fragments (Fig. S4, ESI†), which made C–Hal bonds more ionic. The C–Cl distance in t-Bu{Cl₁₁} became very close to that in an ionic salt (3.545 Å, Fig. S6, ESI†), resulting in a small covalent contribution—to the bonding between the ions (ΔE_orb, 14%; Table S1, ESI†)—approaching that in the solid phase. The calculated CH stretch frequencies for both phases also correlate well (Table S5, ESI†).

Increasing the ionicity of the R{F₁₁} compounds in the series R = C₂H₅ < i-Pr < cyclo-pentyl < t-Bu strengthens the effect of hyperconjugation, as followed from the corresponding ΔE⁽²⁾_ij values (Table S2, ESI†) and from the increase in the red-shift and intensity of the bands of CH stretches involved in hyperconjugation (Table S4, ESI†). It should be noted that the condensed phase enhances both ionicity and the effect of hyperconjugation in this series of R{F₁₁} compounds gradually; this result contradicts experimental findings.

We have proved previously⁵⁶ that “effective” basicity of {F₁₁⁻} in the solid salt, (HCO)⁺{F₁₁⁻}, is close to the basicity of the Ar atom, and the calculated spectra of the Ar⋯H⁺–CO and CO–H⁺⋯Ar solvates agree well with the experimental spectra of ionic compound (CO)H⁺{F₁₁⁻}. Therefore, the experimental IR spectra of the ionic bulk salts under study are expected to better match the calculated IR spectra of weakly bound R⁺–Ar solvates rather than R⁺–{F₁₁⁻} species with a notable covalent contribution (Table S1 in ESI†). The same conclusion follows from DFT calculations for t-Bu{Cl₁₁}: the structure of t-Bu⁺ in the t-Bu{Cl₁₁} crystal (Fig. S7 in ESI†) is similar to that in the naked or Ar-solvated carbocation rather than to the structure of molecular compound t-Bu{Cl₁₁} with a distorted t-Bu⁺ moiety (Fig. S6a in ESI†). To sum up, computational results obtained for R⁺–Ar (or naked R⁺) should be used for comparison with the presented experimental findings.

Experimental IR spectra and properties of the carbocations

The CH₃⁺ cation has an extremely high ability to bind to the bases. Even in the CH₃{F₁₁} compound with the weakest basic ion {F₁₁⁻}, the methyl group forms a highly polar covalent bond.³³

The CH₃⁺ cation is the most convenient object for evaluating the influence of the polarization effect on charge scattering over the methyl group in the absence of hyperconjugation.

Polarization is responsible for the transfer of some electron density from hydrogens to the central carbon. This effect leads to an increase of the CH stretch frequencies and strengthening of the force constants of CH bonds.^23,28,33 For methyl carboranes, CH₃{Hal₁₁}, the dependence of ν_asCH₃ on anion basicity is linear and extrapolates to zero basicity (neat CH₃⁺) at 3103 cm⁻¹.³³ This value corresponds to the true CH₃⁺ cation in the condensed phase, and is very close to that obtained with the B3LYP-D3/def2-TZVPD approximation (3108 cm⁻¹, see Table S3 in ESI†).

C₂H₅⁺ and i-C₃H₇⁺ cations. An experimental IR spectrum of the C₂H₅⁺ cation in vacuum, when targeted with Ar at H⁺, shows three CH stretch vibrations (3114, 3032, and 2158 cm⁻¹) well-fitting those calculated for the non-classic bridged-proton symmetrical structure.³ Thus, the hyperconjugative interaction in the bare cation is so strong that H⁺ is equally shared between the two carbon atoms.

The IR spectrum of the ethyl carborane, C₂H₅{Cl₁₁}, revealed another extreme case:³² all C–H stretch frequencies are in the region 2933–3057 cm⁻¹ indicating covalent binding between C₂H₅ and {Cl₁₁} moieties, which prevents the hyperconjugation effect. The frequencies of the CH₂ group of C₂H₅{Cl₁₁} (3057 and 2976 cm⁻¹) are slightly higher than those of the chloronium cation (C₂H₅)₂Cl⁺ (3038 and 2974 cm⁻¹), pointing to the more polar C–Cl bond in the case of C₂H₅{Cl₁₁}.

The C₂H₅⁺ cation was obtained in the form of ionic salt C₂H₅⁺{F₁₁⁻}. Its formation through an unstable salt of fluoronium cation, (CH₃)₂F⁺, is accompanied by the formation of i-Pr⁺.³³ IR spectra showed the bands of both cations, whose intensities vary depending on the ratio of cations (Fig. 2a). Obtaining the difference between the spectra of the samples with a low and high portion of i-Pr⁺ (Fig. 2, red and black)—up to full compensation of the absorption from i-C₃H₇⁺—allowed us to obtain a spectrum of the C₂H₅⁺ cation (Fig. 2b). Reverse subtraction of the spectra (“black” minus “red”) did not fully remove the spectrum of the C₂H₅⁺ cation but allowed us to see the main features of the i-C₃H₇⁺ spectrum (Fig. 2c). The absorption contour of CH stretches affected by hyperconjugation is asymmetric for both cations and can be broken down into at least two components (Fig. 2b and c).

Fig. 2 IR spectra of (a) a mixture of C₂H₅⁺{F₁₁⁻} and i-Pr⁺{F₁₁⁻} salts,³³ showing the transformation of C₂H₅⁺ into i-Pr⁺ (red-blue-black); (b) the C₂H₅⁺ cation spectrum obtained by subtracting the black spectrum (a) from the red spectrum (a) until full compensation of the band at 2831 cm⁻¹ of i-Pr⁺; (c) the i-C₃H₇⁺ spectrum isolated by reverse subtraction of the spectra: black (a) minus red (a). An uncompensated remnant from C₂H₅⁺ absorption is marked by an asterisk. The bands 3078 and 2960 cm⁻¹ belong to the methyl group of the unreacted CH₃{F₁₁} salt. The strong band, marked with **, belongs to the {F₁₁⁻} counterion.

The IR spectrum of the i-C₃H₇⁺{F₁₁⁻} salt obtained in reaction (1) differs from the spectrum shown in Fig. 2c. This means that two isomers of i-C₃H₇⁺ can form. This issue will be discussed below.

We did not succeed in obtaining a high-quality spectrum of the i-C₃H₇⁺{Cl₁₁⁻} salt according to eqn (3) because formation of this salt is accompanied by formation of the chloronium cation (C₃H₇)₂Cl⁺. Superimposition of the two spectra allowed us to determine only the maximum of the absorption from the C–H stretches of i-C₃H₇⁺ disturbed by hyperconjugation (Fig. S8 in ESI†).

The cyclo-pentyl⁺ cation. A white powder of the cyclo-C₅H₉⁺{F₁₁⁻} salt obtained via direct interaction of acid H{F₁₁} with cyclo-pentane (eqn (2)) yielded a high-quality IR spectrum (Fig. 3). Nevertheless, the cyclo-C₅H₉⁺{Cl₁₁⁻} salt did not form in a similar manner. It was obtained in the reaction of H{Cl₁₁} with chloro-cyclo-pentane; this reaction was accompanied by formation of oligomeric carbocations, which are degradation products of the intermediate chloronium cations decomposing at room temperature.³³ In Fig. 4, the IR bands from these byproducts are marked with asterisks. The important region of CH stretches of the cyclo-C₅H₉⁺ cation, which was disturbed by hyperconjugation, was not distorted by impurities, and separation of their bands into individual components became possible.

Fig. 3 The IR spectrum of the cyclo-C₅H₉⁺{F₁₁⁻} salt (red; CH stretches of the anion are marked with asterisks) versus the calculated spectrum of naked cyclo-C₅H₉⁺ at the B3LYP-D3/def2-TZVPD level of theory (blue curve; frequencies were scaled by a factor of 0.9674; band positions are indicated by black bars).

Fig. 4 The IR spectrum of cyclo-pentyl⁺{Cl₁₁⁻}, complicated by the bands of impurities from oligomeric cations (marked with asterisks).

The patterns observed in the IR spectra of the carbocations

t-Bu⁺ is the most actively studied carbocation. On the basis of experimental data from the gaseous and condensed phase experiments, two CH₃ groups of the t-Bu⁺ cation were proven to be involved in hyperconjugation.²⁸ All three CH bonds of each group donate σ-electrons to the 2p_z orbital of the sp² C atom (Scheme 2) and their CH stretches appear in IR spectra as group vibrations, ν_sCH₃ and ν_asCH₃, with significantly decreased frequencies (at 2867, 2822 and 2771 cm⁻¹ in the case of the salt with {F₁₁⁻} counterion, Fig. 5a). The third CH₃ group (marked as CH₃*) is predominantly polarized and shows three higher-frequency bands in the gas phase. This observation is supported by important experimental evidence: the stretch vibrations of CH₃ and CH₃* groups in t-Bu⁺ show a good correlation (Table 2), which confirms their attribution to group vibrations and indicates that both types of CH₃ groups are isoelectronic.

Fig. 5 IR spectra of cations t-Bu⁺ (a) and i-C₃H₇⁺_as (b) in the salts with the {F₁₁⁻} anion. The spectra are broken down into Gaussian components. The broad absorption band is highlighted in blue. Compensation of the weak absorption of residual Et₃Si{F₁₁}, which was used in a small excess, leads to some remnants at 2900–2800 cm⁻¹ that look like increased noise in spectrum (b).

Table 2 The valence group vibrations (in cm⁻¹) for gaseous t-Bu⁺ with the proposed attribution

Group	ν asCH3		ν sCH3
CH3*	3038	2965	2913
CH3	2880	2839	2793
Ratio	1.055	1.044	1.043

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In addition, the C–H stretch frequencies of t-Bu⁺ involved in hyperconjugation very well correlate with those of the isoelectronic planar trimethyl boron (CH₃)₃B⁵⁷ (Table 3). This observation confirms that the types of local CH₃ normal vibrations for both compounds coincide, and the electron distributions in their methyl groups are identical.

Table 3 Comparison of experimental CH stretches of t-Bu⁺ with those of isoelectronic (CH₃)₃B⁵⁷ (in cm⁻¹)

Group	ν asCH3		ν sCH3
t-Bu^+ gas	2880	2839	2793
(CH3)3B gas	3000	2975	2875
Ratio	0.960	0.954	0.971

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Finally, a good correlation was also observed between the frequencies of the CH₃* group in t-Bu⁺ and those of the strongly polarized methyl group in CH₃{F₁₁}³³ (Table 4), confirming the above-mentioned conclusions.

Table 4 Comparison of experimental CH stretches of the polarized CH₃* group of t-Bu⁺ with those of the CH₃{F₁₁} compound (in cm⁻¹)

Group	ν asCH3		ν sCH3
t-Bu^+ gas	3038	2965	2913
CH3{F11} solid	3098	3086	2975
Ratio	0.981	0.961	0.979

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In the condensed phases, t-Bu⁺ forms noticeable H-bonds with the nearest surroundings favoring involvement of the CH₃* group in hyperconjugation, which is, however, weaker than that formed by its other two CH₃ groups.²⁸ This phenomenon is accompanied by enhanced dynamic properties of the H atoms of CH₃*, resulting in collapse of the three bands of CH₃* vibrations into one broad absorption (with an intensity of ca. 30% of that from all CH stretches) at a lower frequency²⁸ (Fig. 5a).

The spectrum of i-Pr⁺ obtained in reaction (1) differs from that shown in Fig. 2b, but is very similar to the spectrum of t-Bu⁺ (Fig. 5). It shows a broad absorption pattern at 2922 cm⁻¹ with intensity ca. 50% of the total intensity of the CH stretch vibrations of i-Pr⁺. The three bands at a lower frequency belong to the second CH₃ group involved in stronger hyperconjugation (the ratio of their intensity values may change, depending on the conditions of separation of the bands, but the number of bands is always three). Thus, the above-mentioned i-Pr⁺ isomer is an analog of t-Bu⁺, with similar asymmetry, and will be designated as i-Pr_as⁺.

The spectrum of the second i-Pr⁺ isomer contains only two bands of CH₃ groups involved in hyperconjugation (the third band was obviously not resolved) without the broad absorption (Fig. 2c). Thus, both CH₃ groups equally participate in hyperconjugation. Hereinafter, we will refer to this symmetrical isomer as i-Pr_sym⁺.

The frequencies of hyperconjugated CH₃ groups for both i-Pr⁺ isomers are close (Fig. 2c and 5b), which means that i-Pr_as⁺ and i-Pr_sym⁺ have roughly equal energy. Therefore, the efficiency of the positive charge redistribution over the CH₃ and CH₃* groups in carbocations is energetically equivalent.

A charged C₃ core of the cyclo-C₅H₉⁺ cation is similar to that of i-Pr_sym⁺. Its IR spectrum does not contain the broad absorption pattern (Fig. 3), indicating that this cation is symmetric. CH stretches of the two equivalent CH₂^α groups involved in hyperconjugation (Scheme 5) result in two low-frequency bands at 2836 and 2765 cm⁻¹. The remote CH₂^β groups yield the bands ν_as at 2956 and ν_s at 2904 cm⁻¹. Moreover, the ratio of frequencies for these bands coincides (Table 5). This finding implies identical assignment of frequencies for CH₂^α and CH₂^β groups and their isoelectronic character, as established for two types of CH₃ groups of t-Bu⁺ (Table 2). This result leads to a major conclusion: both CH bonds of CH₂^α groups equally donate σ-electrons to the 2p_z orbital of the sp² C atom, in agreement with the case of t-Bu⁺.

Scheme 5 Schematic representation of the cyclo-pentyl cation.

Table 5 CH stretch frequencies (in cm⁻¹) of the cyclo-pentyl cation

Group	ν asCH2	ν sCH2
CH2^α	2836	2765
CH2^β	2956	2904
Ratio	0.959	0.952

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The C₂H₅⁺ spectrum shows two bands from the CH₃ group vibrations, which are similar to those observed in the i-Pr_sym⁺ spectrum (Fig. 2b and c). The broad absorption pattern, of course, is absent. This finding points to some generic relation between C₂H₅⁺ and i-Pr_sym⁺. The overall picture of the generic relations between the analyzed carbocations with hyperconjugative stabilization is shown in Fig. 6.

Fig. 6 The generic relation between the simplest carbocations with hyperconjugative stabilization. The charged core is marked in red; the C–C bond formed by the more strongly polarized CH₃* group is indicated by a bold red line.

One can see that there are two kinds of carbocations: (i) carbocations with one type of hyperconjugated CH₃/CH₂ groups (C₂H₅⁺, i-Pr_sym⁺, and c-C₅H₉⁺), which form a symmetrical C₃ charged core; (ii) carbocations with an asymmetrical C₃ or C₄ charged core formed by two types of CH₃ groups.

The frequencies of the CH₃/CH₂ groups mostly affected by hyperconjugation for all carbocations under study are summarized in Table 6. The decrease in the frequencies from {F₁₁⁻} to {Cl₁₁⁻} salts is due to the formation of H-bonds.¹⁷

Table 6 IR frequencies (in cm⁻¹) of the CH₃/CH₂ groups of carbocations under study affected by hyperconjugation

Cation	Anion	νCH3/νCH2
a The most intense bands are boldfaced. b Not determined.
C2H5^+	{F11^−}	2791	2728
i-Prsym^+	{F11^−}	2874	2831
i-Pras^+	{F11^−}	2866	2825	2770
t-Bu^+	{F11^−}	2867	2822	2771
Cyclo-pentyl^+	{F11^−}		2836	2765
i-Pr^+	{Cl11^−}		2780
t-Bu^+ 28	{Cl11^−}	2830	2791	2746
Cyclo-pentyl^+	{Cl11^−}		2777	2724

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All CH stretch vibrations of C₂H₅⁺, i-Pr⁺, cyclo-pentyl⁺, and t-Bu⁺ cations were observed in IR spectra with the exception of those from CH₂ and CH groups at the sp² C atom in C₂H₅⁺ and i-Pr⁺, respectively, which are weak and located in the frequency region of the intense C–H stretches from {F₁₁⁻} or {Cl₁₁⁻} anions.

Experimental data versus calculations. Experimental and calculated data showed that the impact of the environment on the nature of the R–{F₁₁} bond may be significant. The CH₃ group is linked covalently to {F₁₁⁻}, and the calculated CH₃ frequencies (both for vacuum and for the condensed phase) are in good agreement with the empirical ones for solid CH₃–{F₁₁} (Table S3, ESI†). When R is C₂H₅ and larger, the ionicity of the C–F bond and the hyperconjugated effect in compounds R{F₁₁} gradually increase in the series R = C₂H₅, i-C₃H₇, cyclo-C₅H₉, and t-Bu. Nonetheless, the experiments showed that starting from R = C₂H₅ (and larger), the compounds are purely ionic (Table 7). In the case of the more basic {Cl₁₁⁻} carborane, an abrupt switch from the covalent to ionic state takes place between C₂H₅{Cl₁₁} and i-C₃H₇⁺{Cl₁₁}. For the more basic CHB₁₁Me₅Br₆⁻ carborane, the switch “covalent–ionic state” occurs between i-C₃H₇{Cl₁₁} and t-C₄H₉⁺{Cl₁₁}. Thus, the switch from the covalent to ionic state is abrupt, without intermediates. Possibly, this happened because permittivity of carborane compounds is much higher than that of DCE, used in the calculations.

Table 7 The covalent and ionic R–X compounds (X = carborane) identified on the basis of the presence or absence of the strong hyperconjugation effect during R stabilization

Carborane, X	Covalent compounds, R–X	Ionic salts, R^+X^−
{F11^−}	H3C–X	R = C2H5^+ and larger
{Cl11^−}	H3C–X, C2H5–X	R = i-C3H7^+ and larger
CHB11Me5Br6^−^58	H3C–X, C2H5–X, i-C3H7–X	R = t-C4H9^+ and larger

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As the calculations revealed, the experimental spectra of carbocations in condensed phases must be compared with those calculated for naked carbocations or carbocations solvated by argon, taking into account that the nature of hyperconjugative stabilization of carbocations does not depend on the phase state. This comparison of cations C₂H₅⁺, i-Pr⁺, and t-Bu⁺ is shown in Fig. 7. It uncovered some similarities and some significant discrepancies, already found for t-Bu⁺.²⁸

Fig. 7 Dependence of CH stretch frequencies of CH₃ groups involved in hyperconjugation in ethyl⁺, i-Pr⁺, and t-Bu⁺ on the number of their CH₃ groups. Empirical data (blue) are shown for {F₁₁⁻} salts, and the calculated data (red) are shown for cations with a similar type of normal vibrations (C₂H₅⁺·Ar, C_s; i-Pr⁺, C₂; t-Bu+, C_s). The number of CH frequencies is specified. The bold and thin lines represent the most and least intense vibrations, respectively.

The discrepancies can be summarized as follows: (1) the calculated frequencies of hyperconjugated C–H bonds continuously increase as the number of CH₃ groups in carbocations increases (C₂H₅⁺, i-Pr⁺, and t-Bu⁺; Fig. 7, red), but this is not the case for the empirically determined dependence (Fig. 7, blue). (2) The calculated frequencies are notably lower than empirical ones for C₂H₅⁺ and i-Pr⁺ and actually coincide only for t-Bu⁺, which is mentioned in the literature as proof of a good match between theory and experiment.¹ Judging by the above results, it is obvious that this coincidence is rather accidental. (3) According to calculations, the number of C–H stretches of hyperconjugated CH bonds is equal to the number of CH₃ groups: one for C₂H₅⁺, two for i-Pr⁺, and three for t-Bu⁺. The experiments imply that these numbers are 2(3), 3 and 3 respectively. (4) The calculated splitting of the bands is rather small (13–17 cm⁻¹) and cannot be detected in the condensed-phase IR spectra because the corresponding bands should be merged into one slightly broadened (degenerated) band. In contrast, experimental spectra show three common bands for CH₃ group vibrations separated by 96 cm⁻¹.

A comparison of the calculated frequencies for the cyclo-pentyl⁺ cation (C₂ symmetry) with the experimental spectrum of cyclo-C₅H₉⁺{F₁₁⁻} revealed (Fig. 3) that there is also a significant discrepancy between theory and experiment, as in the case of C₂H₅⁺ and i-Pr⁺ cations. From the experimental spectrum it follows that both CH bonds of the CH₂^α group of cyclo-pentyl⁺ are equally influenced by hyperconjugation, and CH₂^α and CH₂^β are isoelectronic and have the same type of normal vibrations. The same is true for t-Bu⁺ (and i-Pr_as⁺): three C–H bonds of CH₃ are equally involved in hyperconjugation, and both CH₃ and CH₃* groups are isoelectronic, with the same type of local group vibrations. Unfortunately, these results contradict computational data.

To try to reconcile the quantum-chemical calculations with the experiment, we performed PBE-D3 computations with periodic boundary conditions for the t-Bu{Cl₁₁} crystal lattice previously determined by X-ray crystallography.¹⁷ No significant structural changes were found in the fully relaxed unit cell, as compared to the experiment. In contrast, the calculated vibrational frequencies of t-Bu⁺ in the crystal lattice were found to have the same features as those in gaseous t-Bu⁺ (Table S5 in ESI†), which again contradict the experimental findings. Similar results follow from the calculations when experimental atomic coordinates were used without relaxation (Table S5 in ESI†). Thus, the origin of inconsistencies in the mechanisms of carbocation hyperconjugation stabilization that follow from calculations and experiments has yet to be established.

Conclusions

According to the experiment, two sets of carbocations exist: with a symmetric and an asymmetric charged core. Our DFT calculations predict no carbocations with an asymmetrical charged core and their results in general contradict the experimental data.

The standard theoretical representation of the mechanism of hyperconjugation predicts that σ-electron density is transferred to the empty 2p_z orbital of the sp² C atom from one preferred bond of each CH₃ group. At the same time, the σ-electron density on the other CH′ and CH′′ bonds slightly increases (Scheme 1). It looks like σ-electrons of the three C–H bonds of the CH₃ group do not merge into molecular orbitals with the generalized electrons. This phenomenon manifests itself in calculated IR spectra, including high-level CCSD(T) computations,¹⁸ as a strong reduction in the frequency (much stronger than in the experiment) making it highly characteristic. Accordingly, vibrations of C–H′ and C–H′′ bonds also appear to be highly characteristic ones, which are not mixed with those of the bond. To the best of our knowledge, this case is unknown in the practice of vibrational spectroscopy.

From the experimental data it follows that donation of the σ-electrons from the bond to the empty 2p_z orbital of the sp² C atom is accompanied by an equal withdrawal of the electron density from C–H′ and C–H′′ bonds. That is, the electrons are supplied from each CH bond of the CH₃ group, and the bond does not act as the predominant donor. Thus, all three bonds of CH₃ groups are equivalent. As a result, the CH stretches manifest themselves as typical CH₃ group vibrations with a smaller red shift (∼60–100 cm⁻¹) as compared with that predicted by calculations for stretches (100–200 cm⁻¹). The vibrations of the stronger polarized CH₃* group also appear to be common CH₃ group vibrations with equivalent C–H bonds.

Because calculations revealed that the impact of hyperconjugation on the CH₃ group weakly depends on the dihedral θ angle (the CH₃ rotation around the C–C bond), the donation of the σ-electron density from CH₃ to the empty 2p_z orbital of the sp² C atom may take place not only via the bond, but also via CH′ and CH′′ bonds. This mechanism is indicated in Scheme 2 by a dashed arrow.

The aforementioned inconsistencies are valid for all quantum-chemical calculations of naked carbocations known to date. Furthermore, our DFT calculations for the t-Bu⁺{Cl₁₁⁻} crystal and R{Hal₁₁} molecular compounds in a DCE solution turned out to be comparable with those for naked carbocations, leading to the generally accepted mechanism of hyperconjugative stabilization rather than the mechanism deduced from our experiments.

We can hypothesize that fine electronic structure of carbocations that has not yet been taken into account may be responsible for the observed theory-versus-experiment discrepancies. It seems that potential energy surfaces of the carbocations under study are more complicated than expected and cannot be analyzed by the standard quantum-chemical methods such as conventional DFT. This is the subject of our ongoing research.

Undoubtedly, further computational studies in this field, possibly based on the more advanced methods, will result in convergence of the theory with experiment and will allow researchers to explain a number of recently detected features of carbocations, such as the energy equivalence of the CH₃ and CH₃* groups in asymmetric carbocations; why increasing the number of methyl groups from two (i-Pr⁺) to three (t-Bu⁺) does not influence the frequencies of C–H stretches; and the abrupt change in the state of the CH₃* group when t-Bu⁺ is transferred from gaseous to condensed phases.²⁸

Acknowledgements

This work was supported by grant # 16-13-10151 from the Russian Science Foundation and by the Ministry of Education and Science of the Russian Federation within the Project of the joint Laboratories of the Siberian Branch of the Russian Academy of Sciences and National Research Universities. Quantum chemical calculations were supported by the Russian Foundation for Basic Research (grant # 16-03-00357). The Siberian Supercomputer Center is acknowledged for providing computational resources. The authors thank Irina S. Stoyanova for providing the carborane acids and technical support.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6cp06839a

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