Computational Strategy. Impact of London Dispersion Forces

For a given charge state Z=+2 (native), Z=+1 (one-electron reduced) or Z=0 (two-electron reduced), the stabilization energy (▵E_stab, kcal/mol) of folded (U-shaped) conformers with respect to their corresponding all-trans (extended) conformers is calculated as ▵E_stab=(E_folded−E_all-trans). The energy of each conformer is computed using the large def2-TZVPP basis set³² together with the PBE0 global hybrid density functional³³ after structure optimization at the same level of theory. The D3(BJ) empirical dispersion correction,³⁴ later called D3, is added to the DFT computation to properly account for the effects of weak—medium range—forces such as LD, which decreases as a function of 1/d_LD⁶ (where d_LD is the distance between interacting molecular moieties).

Table 1 summarizes ▵E_stab values calculated without (PBE0) and with the empirical D3 corrections (PBE0-D3), for the superelectrophores distinguished both by their oligomethylene linker and by the connectivity (α-α/γ-γ) of their corresponding cyclomers (from C4γ to C11α; Figure 4). The more negative ▵E_stab value is, the more stable is the folded conformation. The stabilization of the folded native structure (Z=+2) is systematically increased by about 1 to 5 kcal/mol (depending on the linker) by the addition of the dispersion correction. From a structural point of view, this energy stabilization is reflected in a tighter stacking of the pyridinium rings in the transition from PBE0 to PBE0-D3. The same behavior is observed for one- and two-electron reduced species (Z=+1 and 0), respectively. For Z=+2, folding is referred to as a “promoted pre-organization”. It follows that the dispersion forces play a determining role regardless of Z, making folded and extended conformers nearly isoenergetic (within 1 kcal/mol) for dicationic native species. Notably, folding is unnatural for short n-alkanes, which tend to adopt the all-trans conformation,^35-37 hence the role of pyridinium termini as LD-responsive components.

As LD is shown to be a relevant attractive force for the series of superelectrophores, it is worth considering hereafter the correspondingly computed data (PBE0-D3) that include it.

The second interesting general observation is that the stabilization of the folded structure increases with the reduction state, i.e. from Z=+2 to Z=+1 to Z=0, regardless of whether LD is considered. Exceptions to this trend (bold in Table 1) will be explained below. This tendency thus indicates the existence of a stabilizing interaction unrelated to SSI when considering the intermediate ranking of ▵E_stab(Z=+1) involving a single electron. We ascribe this role to the intervention of a supramolecular orbital (supra-MO) and, more specifically, the semi-occupied supra-MO, i.e. SupSOMO of the radical cation (see Figure 5).

Supramolecular Orbitals

The first question to be answered is whether a negative value for ▵E_stab(Z=+2), implying that a folded (cisoid) conformer is more stable than its extended (transoid) counterpart, indicates the existence of a SupLUMO. We show that in the case of the ethylene linker, ▵E_stab(Z=+2) is negative (−0.38 kcal/mol), while supra-MOs do not exist for symmetry reasons (see Figure 5). On the contrary, for derivatives of propylene and butylene having ▵E_stab(Z=+2)>0 as well as for pentylene with ▵E_stab(Z=+2)<0, SupLUMOs are effectively computed, as shown in Figure 5.

Mechanisms of Cyclomerization

Returning to the case study of the ethylene linker (molecule 2), computations show that the cyclobutane derivative (C4γ; Figure 4) is indeed more stable than the extended parent diradical by 19 kcal/mol (Table 1), despite a significantly elongated σ bond (d(Cγ−Cγ)=1.629 Å). Regarding the higher-membered cyclomer with a σ bond at α sites, namely C8α, the compound is computed to be barely stable (if formed) with an ultralong C(α)−C(α) bond of 1.679 Å.³⁸ From the deprivation of supra-MOs, it follows that the formation of C4γ is expected, but following a cyclization mechanism reminiscent of that of the bimolecular σ dimerization, with σ bond formation occurring intramolecularly⁸ as a result of SSI (see also Figure S2 in the SI). Moreover, as far as the folded radical cation is concerned, inspection of Figures 5 and 6 reveals that the SOMO and spin density are localized on one of the two electrophoric subunits, instead of being delocalized over both heterocyclic spin carriers, as is the case for the other prevalent species that are conversely equipped with supra-MOs. This localization is likely to promote reactivity such as (bimolecular) σ dimerization. The above mechanism based on the SSI, referred to as EC² mechanism in electrochemistry,^{39, 40} also implies that we are basically dealing with one-electron processes.

A single one-electron process is involved in the formation of the σ dimer, or two one-electron processes are involved in the formation of the cyclomers C4γ and C8α.

On the other hand, for those superelectrophores with a SupLUMO, a one-step two-electron process of reductive σ bond formation is postulated, as is typical for the Structronic paradigm (Figures 2, 3 and 4).²⁰ This potential inversion (PI) makes the radical cation elusive, which explains why no EPR signal could be detected from it, as noted in the literature.^10-12 It is at the SupSOMO stage that U-shaped (cisoid) conformers are significantly stabilized as reflected by the considerable variation of both ▵E_stab(Z=+1) and distances between C atoms about to be involved in σ bond formation, which approach the critical distance 3.40 Å of van der Waals contact (Table 1). Although the folded radical cations (Z=+1, SupSOMO) and the corresponding folded dicationic conformers (Z=+2, SupLUMO) have geometries that are close to each other except for the distances between the proximal carbon atoms, it is from the radical cation that the ▵E_stab(Z) values begin to diverge, again highlighting the key role of SupSOMO in determining the formation of the γ-based or α-based cyclomers. The main structural relaxation, i.e. the driving force for PI, is the tetrahedralization (hybridization change from sp² to sp³) of two C atoms directly involved in σ bond formation, which occurs upon the second electron transfer. Thus, the increase in energy stabilization ▵E_stab(Z) in the transition from radical cation to two-electron reduced cyclomer (Z=0, SupHOMO) is of greater magnitude than that obtained in the first one-electron reduction, except for the lower-membered cyclomer C7γ from the pentylene derivative, which is predicted to simply not exist (▵E_stab(Z=0)=+5.33 kcal/mol and d(Cγ−Cγ)=3.580 Å). Again, this result is consistent with the experimental work on cyclomer formation from 4,4′-pentylene bipyridinium reduced with Na/Hg (Figure 4).¹³

Regarding the reduction processes, they are governed first by the (Sup)LUMO of different proto-cyclomers and then by the (Sup)SOMO for the second one-electron reduction. Thus, their energies are computed, together with those of the (Sup)HOMOs, as a function of the limiting conformations, namely the fully extended (i.e. all-trans) or the pre-organized for cyclomer formation, i.e. folded (Figure 7 and Table S1 in the SI).

First of all, it should be recalled that extended all-trans conformers are described by regular MOs, whereas the reactivity of folded (pre-organized) conformers is governed by supra-MOs, with the exception of the ethylene derivative (deprived of supra-MOs). Looking at Figure 7, it appears that the energy of the (supra-)MOs depends on the conformation of the electrophores.

For all-trans conformations, an energy decrease of about 0.75 eV (from about −2.6 eV to −3.35 eV) is first observed in going from LUMOs to SOMOs, before a plateau is reached in the transition to HOMOs. The only discrepant behavior is related to the ethylene linker, which shows a decrease in energy when going from LUMO (−2.77 eV) to SOMO (−3.50 eV), followed by a marked destabilization of the HOMO (−2.72 eV). In other words, the diradical of the ethylene derivative (whether singlet or triplet, see Table S1 in the SI) is unstable.

On the contrary, for folded conformers, the energy decreases steadily by 2.1 eV from (Sup)LUMOs (ca. −2.70 eV) to (Sup)HOMOs (ca. −4.80 eV) via (Sup)SOMOs (ca. −3.75 eV). Notably, SupSOMOs are on average 0.25 eV more stable than SOMOs. The ethylene-linked electrophore (red trace in Figure 7) is also distinguishable, but this time only moderately.

Taken together, we observe that the folding pathway of conformers is energetically favored over that of extended diradical species, regardless of the spin multiplicity of the latter (see Table S1 in the SI). This is also true for the ethylene derivative deprived of supra-MOs. These results indicate that the process of prior folding is critical in the formation of cyclomers. Radical cations are shown to play a key role by taking over LD forces. This scenario is not in line with the mechanism proposed so far in the literature, which assumes the formation of diradical species prior to the folding process.^9-13

Electrochemical Correlations

As a final step of our computational analysis, the standard potentials for the first (E₁⁰) and second (E₂⁰) one-electron reduction are calculated on the basis of our mechanistic model of cyclomer formation (Table 2; see also Section S1.3 and Table S2 in the SI). Indeed, these results can be confronted with experimental results based on cyclic voltammetry in order to assess the relevance of the description of the cyclomerization process based on the pre-organization of superelectrophores and supra-MOs.

Except for the peculiar case of 2, folded conformers are systematically easier to reduce than the corresponding all-trans conformers.

Let′s first consider the all-trans conformers that end up as diradical species after two-electron reduction. Again, the ethylene derivative (2) shows a discrepant behavior with a potential gap (▵E⁰=E₁⁰−E₂⁰) of about +1 V, indicating that the second reduction is much more demanding than for other derivatives (▵E⁰ ranges from +0.09 V to +0.22 V; average +0.147 V), while the E₁⁰ values of 3 to 5 are all close to the average value of −1.95 V. This observation suggests that in the case of 2, two one-electron reductions are not independent, indicating communication of redox centers via the alkane chain. Incidentally, STM-break junction experiments performed on the transoid dipyridyl-ethane analog allowed the measurement of a conductance value.⁴¹ In this sense, it is also worth noting that, with the exception of 2, calculated E₂⁰ values assuming triplet states for diradical superelectrophores are quasi-degenerate with those assuming singlet states (Table 2), consistent with the absence of through-bond mediated magnetic coupling between pyridinyl termini as radical carriers. In contrast to the ethylene linker, the other all-trans linkers behave as true “insulators”.

For the folded conformers that give rise to cyclomers according to our mechanistic model, E₁⁰ and E₂⁰ potentials are inverted (▵E⁰<0 V) giving ▵E⁰(3)=−0.22 V and ▵E⁰(4)=−0.17 V or coalescent giving ▵E⁰(5)=+0.03 V. This is in contrast to the cisoid ethylene derivative for which ▵E⁰(2)=+0.23 V. PI and, to a lesser extent, potential coalescence are diagnostic features for the presence of supra-MOs, which are electrochemically translated as single-step two-electron transfer processes.^{30, 31} PI also explains why the radical cation is elusive, as observed experimentally (no EPR signal detected during the reduction process).^10-12 Regarding 2, it is expected that the two-electron reduction will proceed stepwise with two peaks close to each other.

Electrochemistry

The main characterization of the four superelectrophores (2 to 5) together with the reference N-methyl-4-picolinium hexafluoro-phosphate (1) is achieved by performing their cyclic voltammetry under different conditions (mainly concentration C and scan rate v). Typical cyclic voltammograms (CVs) recorded under the same conditions are shown in Figure 8.

The first observation is that all compounds exhibit a single irreversible reduction peak (E_pc(1)~−1.90 V; E_pc(3)~−1.80 V; E_pc(4)~−1.84 V and E_pc(5)~−1.86 V), as expected, with the exception of 2, with ethylene linker, which exhibits two closely spaced cathodic peaks at E_pc1(2)~−1.78 V and E_pc2(2)~−1.89 V.

Exhaustive electrolysis (coulometry) of two representative molecules 2 and 4 at −2.1 V allows to verify that both reduction processes involve two electrons per electrophoric assembly confirming transfer of two electrons for 2 to 5.

Let′s first consider the special case of 2. The experimental results are as follows. First, the variation of the first cathodic peak potential E_pc1(2) as a function of the logarithm of the scan rate (see Figure S3 in the SI) shows that E_pc1 shifts cathodically by 19.5±0.6 mV per decade, which is consistent with a variation of 19.7 mV/n per decade, where n is the number of electrons equal to 1 under the EC² mechanistic scheme assumption corresponding to the intermolecular dimerization process.⁴⁰ Secondly, E_pc1 shifts by 18.8±1.2 mV per decade to positive potentials with increasing concentration of 2 (Figure 9), to be compared with the theoretical value of 19.7 mV per decade under the EC² assumption.⁴⁰ For example, in the case of dimerization of reference molecule 1, E_pc shifts by 21.6±1.3 mV per decade to positive potentials with increasing concentration. Finally, Figure 8 shows that the heights of the peak currents for E_pc(1) and E_pc1(2) are virtually identical, indicating that a single electron is involved. Taken together, these results point to the same fate for the first one-electron transfer, i.e., intermolecular (bimolecular) σ dimerization. This is consistent with the computations, since the spin density mapping of 2 (Z=+1) shows that the first electron is localized on one of the two pyridiniums (see Figure 6), yielding a reactive pyridinyl radical (but see caveat in section S4.2 of the SI). Regarding the fate of the second one-electron process leading to the diradical species: the latter is computed to be much more stable in its folded conformation than in its all-trans conformation (Figure 7 and Table S1 in the SI). Under this assumption of a dominant folding for 2, the two separate and closely spaced standard potentials computed for two one-electron reduction processes (Table 2) agree well with the experiment (Figure 8). According to Table 1, the more stable cyclomer of the two possibilities is by far the γ-linked one (2-C4 in Figure 4).

Concerning the superelectrophore 4, as a representative of the redox-active assemblies 3 to 5, it should be noted that the height of its peak current i_pc(4) for E_pc(4) is ~12 μA, i.e. higher than for E_pc(1) and E_pc1(2), as is expected for a two-electron process. Importantly, and in contrast to molecules 1 and 2, the peak potential E_pc(4) of the chemically irreversible reduction does not shift with increasing concentration, as shown in Figure 9. This finding suggests that we are dealing with an intramolecular process. Considering the electrochemical irreversibility, we deduce that the reduction mechanism consists of two one-electron transfer steps followed by a chemical step, i.e. an EEC process of intramolecular σ bond formation. The question now is whether the electron transfers are independent, as postulated by the SSI assumption (previous work), or not as suggested by the present theoretical study, which evidenced supra-MOs within folded superelectrophores (Figure 5) and which calculated the potential inversion for 3 and 4 (see Table 2). First reduction process in Figure 8 for all molecules 1 to 5 was submitted to semi-integration analysis to provide the semi-integrated current versus potential curves known as neopolarograms, see Figures S4 to S6 of the SI. The log-plot analysis of thus computed neopolarograms for molecules 1 and 2 give a straight line (Figures S4 and S5 in the SI) providing a variation of 56 mV per decade for 1 and 51 mV per decade for 2, which is close to the expected value of 59 mV per decade for one-electron transfer process.

Figure 10 shows the log-plot analysis of neopolarograms for 3 to 5 confirming the potential inversion for these molecules. The neopolarogram of 4 (Figure S6) has a simple shape, indicating that the first and second electrons are strongly correlated, as expected for electrons in the same orbital (supra-MOs). For both molecules 3 and 5, the neopolarograms show a post-reduction adsorption wave, which is clearly visible as bumps in the corresponding CVs (Figure 8). The effects of this adsorption, which have been properly accounted for (see section S4 in the SI), do not change the fact that the two monoelectronic transfers are also strongly correlated. The log-plot analysis in Figure 10 does not yield a straight line with a variation of about 30 mV per decade, which would be consistent with potential inversion of a large magnitude,^{4, 42, 43} as predicted by calculations (about 200 mV; see Table 2). Instead, as in the case of 5 (for which potential coalescence was predicted computationally), traces of a barely discernible S-shape (diagnostic feature for moderate potential inversion) are obtained, delimited by two asymptotes with the same slope approaching 59 mV per decade. This is an experimental finding that not only indicates that two one-electron transfers are fast, but also allows the determination of their formal reduction potentials E₁⁰ and E₂⁰. It turns out that these potentials are indeed experimentally inverted, but only moderately, with values of ▵E⁰=E₁⁰−E₂⁰ of −16 mV for 3, −28 mV for 4, and −26 mV for 5. Further discussion related to the EEC mechanism is provided in the section S4.3 of the SI. In short, the electron transfer steps are electrochemically-reversible with the rate-limiting step being the diffusion of molecules to the electrode. The fact that the process appears irreversible (no counter-wave) is due to chemical irreversibility: the reduced species are converted to the product(s) very quickly, which is consistent with our proposal. In summary, potential inversion has been experimentally confirmed for molecules 3 to 5, but due to the apparent weakness of the phenomena,⁴⁴ the potential difference is only within the range −10 mV to −30 mV.