Abstract

Ab initio quantum mechanics has been employed in an attempt to gain some insight into the structure and dynamics of the C₄H₇ cation and radical. The C₄H₇ cation and radical potential energy surfaces, including transition-states, have been calculated using B3LYP/6-31G**. The calculated transition-state energies agree with experimental trends.

Introduction

Over 100 years ago, W. H. Perkin Jr. attempted to synthesize cyclobutanol from cyclobutylamine (Roberts, 1990). He reported the synthesis of "ordinary butyl alcohol." Thirteen years later, N. J. Demjanow reported that Perkin had actually synthesized a mixture of cyclobutanol and cyclopropylmethanol (Roberts, 1990). This was the first indication that C₄H₇⁺ cations rearrange (Roberts, 1990). In later examinations, Mazur showed that cyclopropylmethylamine, cyclobutylamine, or 3-buten-1-ylamine all react with nitrous acid to produce mixtures of cyclopropylmethanol, cyclobutanol, and 3-buten-1-ol (Fig. 1) (Roberts 1990). This indicates that a common C₄H₇⁺ intermediate for all of these reactions.

Figure 1: Reactions examined by Mazur.

Five structures have been proposed for the non-classical C₄H₇⁺ cation (Fig. 2: I-V) (Roberts, 1990). All five of these structures are non-classical in the sense that their structures do not conform to the standard bonding concepts observed for the overwhelming majority of organic compounds. These molecules can react at more than one position to give more than one of the observed products (Roberts, 1990). The evidence from theory and experiment, suggests that structures I, II, and III are the most likely to be the stable structures. Structure IV has been discarded as a possible structure because one of its configurational isomers does not agree with the experimental NMR data in solutions of C₄H₇⁺ in Olah's "magic acid" and its other isomer should be greatly disfavored by molecular orbital theory (Roberts, 1990).

Figure 2: Proposed C4H7+ structures. All have a positive charge.

Low-temperature ¹³C NMR spectra of C₄H₇⁺ in "magic acid" show shifts that vary substantially with temperature (Roberts, 1990). This indicates that there is a mixture of interconverting C₄H₇⁺ isomers whose composition changes with temperature. Debate still exists as to what is actually being observed. Roberts believes that the most stable ion has structure III (Roberts, 1990) while the NMR experiments of Myhre (Myhre, 1990) and the IR spectra of Vancik (Vancik, 1993) seem to indicate that I and II rapidly interconvert even at temperatures as low as 5 K. Theorists agree that I and II are probably interconverting, but they do not agree on which structure is the lowest in energy (Koch, 1988; McKee, 1986; Roberts, 1990; Saunders, 1988). Saunders believes that III is the transition state between I and II (Saunders, 1988).

To help clarify some of these issues, we examined the potential-energy surface for both C₄H₇⁺ and C₄H₇^·, including transition states for interconversion between isomers.

Theory

The potential energy surfaces for C₄H₇⁺ and C₄H₇^· isomers were calculated using Density Functional Theory, including the generalized gradient approximation and hybrid exchange, at the level referred to as B3LYP. We used the 6-31G** basis set. [All calculations used the Jaguar 3.5 code from Schrodinger, Inc (www.schrodinger.com).] The transition states were located using QST-guiding procedures The results of the calculations for C₄H₇⁺ are summarized in figure 3 (with the energies referenced to structure II) while the results for C₄H₇^· are summarized in figure 4 (with the energies referenced to structure VII). The energies are expected to be accurate to ~2 kcal/mol. For the C₄H₇⁺ cation, we could not converge the transition state from V to I or from V to II; in both cases the calculation, converged instead to structure VI (indicated by NC).

Figure 3: C4H7+ isomer and transition state energies calculated using B3LYP/6-31G**. NC indicates that the calculation did not converge to the desired transition state, but instead converged to VI.

Figure 4: B3LYP/6-31G** bond orders and bond lengths for C4H7+ structures. The top number is the bond order, and the bottom number, in parentheses, is the bond length in Angstroms. All structures have a positive charge. The pair of dots (· ·) indicates the electron density (number of electrons/2), from lone pair electrons, at the atom which would carry the positive charge in the classical cation structure.

To interpret the bond orders for these molecules and intermediates, we calculated the natural bond orbitals (NBO) (Foster 1980) using Jaguar. These are shown in Fig. 4 for C₄H₇⁺ and in Fig. 6 for C₄H₇^·.

Experimental evidence indicates that structures I and II of the cation are interconverting (Myhre, 1990; Vancik, 1993) indicating that they have nearly equivalent energies. This is consistent with our calculated energies which find I to be 1.49 kcal/mol more stable than II. Saunders calculated that I is 0.26 kcal/mol more stable than II using MP4SDTQ/6-31G*//MP2(FULL)/6-31G* (Saunders, 1988). The calculated ¹³C NMR shifts agree better with experiment if II is assumed to have slightly lower energy than I (Saunders, 1988). We calculate that the transition barrier from cations II to I is only 0.101 kcal/mol suggesting rapid equilibrium. Myhre was unable to measure the energy barrier using low-temperature NMR (5 K) but believes that it is only a few tenths of a kcal/mol (Myhre, 1990), in agreement with our results. Koch calculated the transition state to be 0.6 kcal/mol using MP2/6-31G** (Koch, 1988). The global minimum for the current cation calculations is structure VI. This agrees with Koch's theoretical work (Koch, 1988) and Vancik (Vancik, 1993) and Schultz's experimental work (Schultz, 1984).

My bond order analysis of the current wavefunctions (Fig. 4) shows that there is no bond across the ring in structure II even though the two carbon atoms are only 1.65 A apart. This agrees with the conclusion of Bader, who critically analyzed calculated C₄H₇⁺ electronic distributions to determine the actual bonding in the isomers (Bader, 1992). The non-classical bonding of the C₄H₇⁺ isomers is indicated by the distances between adjacent carbons, as shown in figure 3. Clearly, there is not always a bond when the CC distances are close. The unusual bonding in C₄H₇⁺ isomers allows I and II to be more stable than structure V whereas the classical description would suggest that they are less stable because of the strain in the small rings.

The complete potential energy surface for C₄H₇^· was successfully calculated (Fig. 5). It is seen that structures VI, VII, and VIII are effectively all of the same energy, and structure IX is the absolute minimum, about 21 kcal/mol lower in energy than the other structures. Of the calculated radical transition-state energies, the VI-to-VII transition barrier is the lowest at 10.0 kcal/mol. The other calculated barriers are 30-50 kcal/mol, which indicates that the reactions associated with them will not be observed within usual radical lifetimes. The conversion from VI to VIII is, in fact, the only rearrangement of the radical observed experimentally (Beckwith, 1994; Engel, 1997; Newcomb, 1993) and has an activation energy of 7.05 kcal/mol (Beckwith, 1994). This compares reasonably well with our calculated value. To verify the radical calculations, the rotational barrier about the exo-CH₂ bond of structure VI was calculated. We found a barrier of 3.0 kcal/mol, which agrees well with the experimental value of 2.7 kcal/mol obtained by NMR (Walton, 1987).

Figure 5: C4H7· isomer and transition state energies calculated using B3LYP/6-31G**.

Figure 6: B3LYP/6-31G** bond orders and bond lengths for C4H7· structures. The top number is the bond order, and the bottom number, in parentheses, is the bond length in Angstroms. All structures are neutrally charged. The pair of dots (· ·) indicates the electron density (number of electrons/2), from lone pair electrons, at the atom which would carry the positive charge in the classical cation structure.

Conclusions

The potential-energy surface calculated for C₄H₇⁺ agrees with experiment (Myhre, 1990; Roberts, 1990; Vancik, 1993) and with previous high-level theoretical calculations (Koch, 1988; McKee, 1986; Roberts, 1990; Saunders, 1988) (within the expected uncertainty of 2 kcal/mol). The complete potential-energy surface calculated for C₄H₇^· was calculated and predicts that the VI-to-VIII transition is the only one that will be observed, agreeing with experiment (Beckwith, 1994; Engel, 1997; Newcomb, 1993).