What is the nature of the long bond in the TCNE22− π-dimer?

The TCNE22− π-dimer has many of the characteristics of a chemical bond despite its remarkably long (2.9 Å) C–C bondlength, as it exists in crystalline solids.

Using computational methods, we examine the nature of this long bond by obtaining potential energy curves that are, for the first time, in satisfactory agreement with recent experiments.

We find that the unusually long C–C bond observed in π-TCNE22− dimers is in fact an outcome of significant dispersion attractions between the two cofacial monomers, as well as their (partial) intradimerπ-bonding interactions.

Stable organic free radicals under appropriate conditions can dimerize spontaneously to form chemical bonds.1

The tetracyanoethylene anion radical (TCNE˙) is known experimentally to form three types of dimeric products (TCNE22−) in the solid state (Chart 1).2,3

In a σ-bonded dimer, the intradimer C–C distance is 1.61 Å (1), and in the two π-bonded dimers, they are 2.90 (2) and 3.47 Å (3), respectively.

The C–C distance in structure 1 is a long C–C single bond, while the C–C distance in structure 3 is close to van der Waals range (though slightly shorter).

Structure 2, with its intriguing C–C separation that is too short for a van der Waals complex but astonishingly long for a C–C bond, may represent an emerging class of organic molecules that possess unusually long C–C bonding interactions.4

It is the focus of this paper.

In simplest terms, the π-bond in 2 should derive from in-phase bonding interactions between the two half-occupied antibonding π* orbitals from each monomer unit.

Two electrons, one from each of the two open shell anionic monomers, then occupy this overall bonding orbital, which will be the highest occupied molecular orbital (HOMO) of the dimer.

Since these two electrons are distributed over four carbon atoms, it has been suggested to be essentially a four-center two-electron bond.2

Due to the very long bond distance, however, the energy gap between the HOMO and the lowest unoccupied MO (the LUMO, which will be antibonding between the monomers) will surely be small.

The appearance of an allowed absorption in the near-IR region (549–667 nm for different cations) upon dimerization was then assigned to this HOMO to LUMO electronic transition.2

In more detail, the long bond in 2 (TCNE22−) is the net result of a complex interplay between many factors.

In addition to the π-bonding interaction described above, Coulomb and exchange repulsion between filled orbitals on the monomers is also surely important.

Based on electronic structure calculations, it has been suggested that π-TCNE22− dimers are bound only when the electrostatic repulsions due to the overall negative charge are compensated by counter-cations.2,3

In other words, without this cation-mediated attraction, the isolated TCNE22− dimers in the gas phase are unstable with respect to dissociation.

Dispersion interactions, however, were not identified to play a (major) role in forming these exceptionally long bonds,3 although they were considered by some authors later.7

Recently, Lü and co-workers experimentally demonstrated that the inter-monomer separation is remarkably invariant (2.90 ± 0.05 Å) to the use of different counter-cations whose electrostatic forces on TCNE˙ can differ by nearly an order of magnitude.5

This is an indirect indication that the long bonding interaction in π-TCNE22− is an intrinsic property of the dimer dianion as long as the excess Coulomb repulsion is eliminated by some counter-cations.

They also have determined the enthalpy (ΔH) and entropy (ΔS) changes associated with reversible dimer formation in solution to be −8.8 kcal mol−1 (−36.8 kJ mol−1) and −41 cal mol−1 K−1 (−171.5 J mol−1 K−1) respectively, from UV/vis measurements.6

In free energy terms, therefore, the stability of a π-dimer arises from a sufficiently exothermic enthalpy of formation to overcome the reduction in entropy.

Perhaps due to the unusual nature of the interaction, electronic structure calculations to date have not yielded binding energies and C–C distances in good agreement with experiment.2,3,7

This is firstly to do with differences between the models studied and experiment, because typically no explicit counter-ions are present in the calculated potential energy curves (although estimates of their stabilizing effect have been made).2,3,7

Additionally the deviations can also be associated with limitations in the computational methods.

In this work, we address both of these factors, by systematically improving the theoretical method, and by obtaining potential curves using fully optimized model complexes with and without counter-ions.

Satisfactory agreement is obtained between calculated observables at the highest level of theory and experiment.

More importantly, we also find that the unusually long C–C bond observed in π-TCNE22− dimers is in fact an outcome of significant dispersion attractions between the two cofacial monomers, as well as their π-bonding interactions.

In order to realistically compensate electrostatic repulsion between the two TCNE˙ anionic monomers, we have used potassium ions, K+.

There are two possibilities to locate these counter-cations in theoretical models.

First, one can put two K+ ions midway between the two parallel TCNE˙ planes (Fig. 1a).

In this configuration, the monomer (in particular the nitrile groups) will bend substantially toward the center of the dimer at longer intradimer distances, due to electrostatic attractions to the cations.

In an X-ray structure, however, nitriles bend away from the center of the dimer by about 5°.2–6

An alternative, which is our choice of model, is to cap the dianion dimer, 2, with up and down K+ ions, as in Fig. 1b.

With this configuration, monomer structures in a dimer stay roughly as observed in X-ray structures at all bond separations, and therefore the nature of internal bonding in a dimer can be investigated without causing artificial distortion of the monomers at all distances.

Geometries were then fully optimized apart from one constraint which is the C–C inter-monomer distance (R) corresponding to the reaction coordinate for dimer formation or scission.

Symmetry of the dimer was reflected by retaining D2h point group symmetry in all calculations.

A bond formation (or cleavage) reaction, such as the π-dimerization of TCNE˙, necessarily involves a multi-reference description if restricted orbitals are used.

A dimer beyond a certain value of R will become increasingly diradicaloid in nature, and eventually becomes a pure “diradical” (i.e., two radical monoanions, 2TCNE˙) at large enough separations.8

For this reason, we have used a perfect pairing (PP) wave function with one correlated pair of orbitals, which is denoted by PP(1).9

PP(1) is an exact wave function for H2 at all bond separations for a minimal basis, and is expected to capture the main qualitative features involved in a single bond formation such as π-dimerization of TCNE˙.

It does, however, neglect the remaining dynamical correlation effects that correspond to excitations into higher-than-valence virtual orbitals.

Dynamical correlation, which amongst other things is the main quantum mechanical origin of dispersion interactions, turns out to be critical in describing the TCNE22− dimerization as we discuss below.

We include it by second-order perturbative corrections to the PP(1) wavefunction, at PP(1) optimized geometries.

This level of theory is denoted as MRMP2//PP(1).10

The standard 6-31G(d) basis11 was used for all calculations, which were performed using the GAMESS program.12

The 6-31+G(d) basis was also employed for important minima to consider relatively diffuse electron density of anionic species.

In Fig. 2, the potential energy curve for π-dimerization of K+TCNE˙ along the varying inter-monomer distance, R, is shown at the PP(1)/6-31G(d) level.

It is apparent that the interaction between the two K+TCNE˙ monomers is overall dissociative at the PP(1) level.

This result is perhaps unexpected because, in previous literature on this interaction,2,3,7 the instability of a dimer arising from two TCNE˙ units was ascribed to anion-anion Coulomb repulsions, but in our theoretical model these radical anions are now each neutralized by K+ ions (i.e., K+TCNE˙ as in experiments) and yet they are still purely repulsive.

It indicates that the interaction between two K+TCNE˙ units cannot just be regarded as coupling between 2 “giant hydrogen-like radicals” since no bond is found when just two-electron bonding and antibonding interactions between the (neutralized) monomers are permitted.

The answer to the origin of the net binding is found when dispersion interactions are considered.

Dispersion attractions are long-range (and non-local) correlation effects that are associated with instantaneous correlated fluctuations of two electrons, one on each monomer.13

Second-order perturbation theory is the simplest correction beyond the PP(1) level (or indeed beyond the mean field description) that includes dispersion forces, through treating dynamical correlation.

The MRMP2 curve in Fig. 2 shows that the dimer is now stable relative to the reactants (two radical monomers) by −17.3 kcal mol−1 (−72.4 kJ mol−1), with a minimum at Req = 2.6 Å (260 pm).

The message that this result delivers is clear: Dispersion effects, as well as other attractive and repulsive forces, must be considered to correctly describe the π-dimerization of TCNE˙.14–16

In these kinds of π-dimers where bondlengths are unusually long and thus the bond-strength is weak (at least compared to conventional chemical bonds), the binding energy is generally overestimated due to basis set superposition errors (BSSE) associated with finite basis sets.

BSSE is the “borrowing” of basis functions from the second monomer to improve the orbitals of the first monomer relative to that monomer in isolation, an effect which in incomplete basis sets can occur with any chemical interaction between the fragments.

We apply the standard counterpoise (CP) correction to obtain improved results.

At the CP corrected MRMP2 level (denoted as CP-MRMP2 in Fig. 2), the binding energy of the π-dimer is refined to be −11.2 kcal mol−1 (−46.9 kJ mol−1) with Req = 2.7 Å (270 pm).

CP-MRMP2/6-31+G(d) single point energy calculations (with diffuse functions on carbon and nitrogen atoms) at the same geometries as above yield a similar binding energy of −12.5 kcal mol−1 (−52.3 kJ mol−1), which is our best estimate.17

These results (i.e., binding energy and equilibrium geometry) are in satisfactory agreement with recent experiments in solution and in the solid state, in which ΔH = −8.8 kcal mol−1 (−36.8 kJ mol−1) and Req = 2.9 Å (290 pm).5

The fact that Req at CP-MRMP2 occurs at a slightly longer distance than for MRMP2 reflects the fact that without CP correction, the bond length is spuriously shortened due to BSSE.

Although not shown in the figure, the CP corrected PP(1) curve also lies above PP(1), making the dimer even more unbound at that level of theory.

Next, we considered the dimer formation of dianionic forms of TCNE22− in the absence of cations, employing the same theoretical methods used above, namely, PP(1), MRMP2, and CP-MRMP2.

It is clear from Fig. 3 that, at all levels, a dianionic dimer is unstable relative to the two separate monoanions due to the overwhelming anion-anion repulsion, although, at MRMP2 before the BSSE correction, there is a shallow and thick metastable minimum (with a 4.5 kcal mol−1 barrier) appearing at 2.8 Å.

This metastable minimum is found to be spurious confirming that the dimer is bound only in the presence of counter-cations since it almost disappears (with a 0.6 kcal mol−1 barrier) when BSSE is corrected at CP-MRMP2.

Nontheless, the MRMP2 curve still shows that the dispersion interactions stabilized the dimer significantly.

We also note that with the 6-31+G(d) basis, an isolated dianion dimer at R = 2.8 Å is still unstable with respect to the separated monoanions by about 34 kcal mol−1 at the MRMP2 level.

Although crucial as shown above, dispersion forces are only one of the driving forces that allow dimer formation with this unusual 2.9 Å bondlength.

As plots of the calculated PP(1) HOMO and LUMO show in Fig. 4, there indeed is strong bonding interactions in the HOMO between the two TCNE monomers.

These four-center two-electron π-bonding interactions also help to allow the TCNE monomers to come well inside the usual van der Waals envelope, by partly compensating repulsive Coulomb and exchange interactions between filled orbitals that are responsible for normal van der Waals separations.

More quantitatively, the tradeoff between bonding and antibonding interactions can be seen in Table 1, from the tabulated π* LUMO (see Fig. 4) occupation numbers as a function of dimer distance.

At the optimal separation (Req = 2.7 Å), the LUMO occupation number is 0.24 electrons.

This means that the dimer has about 76% π-bonding character and 24% diradical character, which suggests that its bond order at this distance is 0..7618

Despite substantial diradical character, therefore, one can say overall there exists a “chemical bond”, or a partial π-bond more precisely.

However, the fact that the dimer is unstable at Req = 2.7 Å at the PP(1) level indicates clearly that this π-bonding interaction (76%) is not enough by itself to overcome the other effects and to yield a stable dimer.

In fact, it is very rare to find a stable molecule with 24% diradical character, because it would usually like to rearrange to increase its bond character.

Due to the cofacial parallel geometry of the π-dimer, dispersion attractions between the monomers are large enough to stabilize this unusually diradicaloid bond.

The balance between two-electron bonding and dispersion attractions (which favor shorter distances), against Coulomb and exchange repulsions of other filled orbitals (which favor longer distances), results in the observed bondlength.

Hence the electronic structure of the exceptionally long C–C bond found in TCNE22− can be best described as a dispersion-assisted partial four-center two-electron π-bond.

This is supported by the fact that comparable calculations on the neutral TCNE dimer (i.e., including dispersions but not covalent interactions) show much weaker binding.19

Recent experiments on various organic π-radicals showed that, regardless of whether they are negatively charged as in TCNE˙, positively charged (OMB+˙), or neutral (PHEN˙), these radicals all spontaneously self-associate to form π-dimers with Req ≈ 3 Å.5

They have similar exothermic enthalpies of formation ranging between −6 and −10 kcal mol−1, strong allowed optical transitions in the near-IR region due to HOMO to relatively low-lying LUMO excitations, and persist in solution as well as in the solid state.

These common characteristics suggest that these supramolecular π-dimers may have similar electronic structures as well.

We are currently performing detailed investigations on how general the combined effects of partial π-bonding and dispersion attractions are in supramolecular π-dimerizations of organic π radicals.