Theoretical Analysis of Intermolecular Interactions in Cationic π-Stacked Dimer Models of Antiaromatic Molecules

Abstract

We have theoretically examined the intermolecular interactions in the cationic states of π-stacked dimers of 4nπ antiaromatic molecules. The ground state of face-to-face π-dimer models, consisting of cyclobutadienes (CBDs), was analyzed as a function of the stacking distance (d) for their monocationic and dicationic states using multi-reference second-order perturbation theory. Multi-configurational wavefunction analysis in a diabatic representation was employed to understand the electronic structures of the dimer models in terms of the monomer electron configurations. It is found that the monocationic dimer exhibits a local minimum at about d = 2.4 Å in the ground state, where each monomer is represented by a superposition between neutral triplet and cationic doublet electron configurations. Crossing of the ground and excited states occurs through changing d, which is due to the small energy gap between the highest occupied and lowest unoccupied molecular orbitals of antiaromatic molecules.

1. Introduction

Understanding the nature of intermolecular interactions between (anti)aromatic molecular systems has garnered attention for advancing fundamental chemistry and biology, as well as for creating functional molecular materials that exhibit molecular recognition capabilities [1,2,3]. In the field of controlling stimuli-responsive behaviors, tuning the strength and manner of interactions between atomic or molecular entities that change with electron oxidations or reductions is particularly intriguing [4,5,6,7,8,9,10,11,12,13].

It is known that two closed-shell neutral helium atoms do not prefer to form a covalently bonded diatomic molecule, whereas its monocationic state, He₂⁺, can exhibit a bound state with a finite interatomic distance [14,15,16,17]. From a molecular orbital (MO) perspective, this situation can be understood by evaluating the formal bond-order (BO), which is defined as (n_b − n_ab)/2, where n_b and n_ab are the occupation numbers of bonding and antibonding MOs, respectively, constructed by the linear combinations of the 1s atomic orbitals (AOs). The formal BO is 0 for He₂ since n_b = n_ab = 2, whereas the value is 0.5 for He₂⁺ where n_b = 2 and n_ab = 1 in the ground electron configuration. In the dicationic He₂²⁺, electrostatic repulsion between the He⁺ ions becomes considerable although the formal BO is 1 (n_b = 2 and n_ab = 0).

On the other hand, valence-bond (VB) theory and related methods have provided an intuitive understanding of chemical bonds [18,19,20,21,22]. Pauling investigated the potential energy curves of He₂^m (m = +1 and +2), based on the resonance theory [15]. A local minimum of the potential energy curve was found not only in He⁺ but also in He₂²⁺, although the potential curve around the minimum is shallow and is overall repulsive in the latter case. Pauling also demonstrated how the resonance of several valence electron configurations contributes to energy stabilization at the local minimum. In the VB theory, although the BO may not be defined as straightforwardly as the MO theory, it still offers insight into the electron configuration of each structural element (namely, an atomic/molecular unit) that is sometimes difficult to understand with canonical MOs delocalized over the entire system. A comprehensive understanding of chemical bonding can be achieved by translating between the MO- and VB-based interpretations of the wavefunction for the system of interest.

In a series of theoretical studies [23,24,25], we have investigated the mechanism of decreasing antiaromatic characters in closely stacked face-to-face dimers of 4nπ molecules [26,27], a phenomenon usually referred to as stacked-ring aromaticity. In addition to the explanations by several pioneering theoretical and experimental studies [26,27,28,29,30,31,32,33,34,35,36,37,38], we proposed an interpretation using a VB-based multi-configurational wavefunction model, that is, stacked-ring aromatic systems can be viewed as assemblies of T₁-like Baird-aromatic units interacting with each other [23]. Specifically, an electron promotion from the HOMO to the LUMO, which results in a high-spin T₁ configuration, is induced in each monomer by intermolecular interactions. A structural change from the distorted lower symmetry to the higher symmetry of the 4nπ-electron system is also triggered, enabling the HOMO and LUMO of the monomer to become degenerate and mix. Consequently, the 2 × 2 unpaired electrons of T₁-like monomers contribute to form multi-center covalent-like double bonds delocalized over several atomic sites, although their energy gain is not expected to be the main factor in dimer formation. Perhaps these interpretations are similar to “promotion and hybridization,” which explain covalent bond formation in CH₄ [excitation from the ³P (1s²2s²2p²) to the ⁵S (1s²2s¹2p³) states and subsequent s-p atomic orbital mixing]. Recently, we found that a similar change in electronic structure in a 4nπ molecule can also be caused by molecule–surface interactions [39].

From the MO perspective, there is a stacking distance (d_c) at which the order of energy levels of the intermolecular bonding and antibonding MOs reverses [27,28,32,40]. As a result, the formal BOs of the neutral and cationic states can differ between the regions d > d_c and d < d_c (see Figure 1). For example, the formal BOs of the neutral, monocationic, and dicationic states are 0, 0.5, and 1 at d > d_c, which is similar to the helium dimer cases, whereas they are 2, 1.5, and 1 at d < d_c, with multi-center bonding characters. Such variations in the BO of the dimers of antiaromatic molecules can be used to control molecular aggregation and physicochemical properties [41].

Developing a VB-based model of electronic structures and intermolecular interactions in the cationic states of dimers of antiaromatic molecules is expected to improve the understanding of how aromaticity of the system changes through redox reactions. Therefore, we have attempted to apply VB-based multi-configurational wavefunction models to analyze electronic structures and intermolecular interactions in face-to-face dimer models composed of cyclobutadienes (CBDs). We compared our results with those of the helium dimer cations developed by Pauling and discussed their similarities and differences, because the small HOMO-LUMO gaps of antiaromatic molecules cause mixing of several excitation configurations into the ground state that cannot be observed in the helium cases.

2. Computational Details

2.1. Calculated Models

In this study, we have examined the electronic structures of π-stack dimer models consisting of CBDs. CBD has often been used as a model molecule to extract essential information about antiaromatic and stacked-ring aromatic characters [26,29,30,33,34,36]. First, we optimized the geometry of CBD monomer 1 in the neutral singlet ground state at the strongly contracted n-electron valence state second-order perturbation theory [42,43,44], SC-NEVPT2-CASSCF(4e, 4o)/cc-pVTZ level, under the constraints of D_2h and D_4h symmetries, which is the same level as that used in the previous study [23]. The single point calculation results of 1 in the neutral singlet, neutral triplet, and cationic doublet states of the D_2h and D_4h structures are listed in Table S1. Then, we constructed face-to-face π-stack dimer models with various stacking distances d, consisting of CBDs with D_2h or D_4h symmetry. We call these models 1₂(D_2h) and 1₂(D_4h), respectively (see Figure 2). In this paper, we did not account for structural relaxation effects caused by intermolecular interactions or changes in charge state, meaning each monomer geometry was fixed, and no further geometry optimization was performed to focus only on the electronic structure changes. This is because we aim to focus only on the modulation of π-electronic structures of each monomer in the interacting π-dimers.

2.2. Calculation and Analysis Methods

Based on the geometries of the π-stack dimer models with various d, we conducted single-point calculations in the cationic states, 1₂(D_2h)^m and 1₂(D_4h)^m (m = +1, +2). The results of the neutral dimer can be found in our previous paper [23]. We performed the state-averaged (SA-)CASSCF calculation, considering the full-π-valence [(8 − m)e, 8o] active space. The ten lowest doublet (m = +1) and singlet (m = +2) states were included in the state-averaging process. Using the reference SA10-CASSCF wavefunctions, single-point calculations at the quasi-degenerate (QD-)SC-NEVPT2 [45] level were conducted to determine the potential energy surface (PES) along the stacking direction for each system. The cc-pVTZ basis set was used throughout the calculation. All the multi-reference calculations were carried out using the Orca 5.0.4 program package [46].

To understand the d-dependent electronic structures of the π-stack dimer models, we performed a diabatization of the multi-configurational wavefunctions that we previously developed for analyzing the neutral systems [23]. The active (pseudo-)canonical molecular orbitals (MOs) obtained from the SA10-CASSCF calculation were rotated to localize the active π-orbitals on each monomer (see Figure 3a). Details of the orbital rotation procedure are available in the Supporting Information and in the previous paper. Then, the QD-NEVPT2 calculation was carried out using the CASCI wavefunctions, constructed from the localized active orbitals, as the reference wavefunctions. This transformation enables us to analyze wavefunctions based on the electron configurations of monomers. It was reported that the S₀-T₁ gap of the neutral CBD monomer is somewhat underestimated by the NEVPT2 [47]. In the previous paper, we analyzed how this underestimation influences the contributions of T₁-like monomer configurations [23].

Figure 3b shows examples of electron configurations in the monocationic doublet dimer models, 1₂(D_2h)⁺ and 1₂(D_4h)⁺. Although we have considered the full-π-valence CAS(7e, 8o) active space during multi-reference calculations, here we highlight the key electron configurations arising from the CAS(3e, 4o) subspace, which includes the HOMO and LUMO of each monomer. Here, g and d denote monomer configurations where the HOMO or LUMO is doubly occupied, i.e., the ground and doubly excited configurations, respectively. t refers to the single excitation from the HOMO to the LUMO with the triplet multiplicity. c and c* (a and a*) represent the monocationic (monoanionic) states of the monomer, where the HOMO or LUMO is singly (doubly) occupied, respectively. c² indicates the dicationic configuration of the monomer.

The electron configurations of the dimer (diabatic states), such as gc, shown in Figure 3b, are constructed as pairs of monomer electron configurations, considering the total spin symmetry. For example, gc (gc*) represents the diabatic state characterized by a pair of monomer configurations, g and c (c*), where one of the monomers takes the neutral ground configuration and the other takes the monocationic configuration. dc (dc*) and tc (tc*) represent the neutral–monocation pair states, where one of the monomers has the doubly excited singlet and triplet configurations, respectively. ac² and a*c² represent the dimer configurations composed of monoanionic and dicationic monomer pairs. Each electron configuration of the dimer has a corresponding counterpart, such as gc and cg. Then, the weight of each diabatic electron configuration in the adiabatic state wavefunctions was evaluated.

3. Results and Discussion

3.1. Monocationic π-Dimer Models

Figure 4 shows the calculated results of the state energies and wavefunctions of the monocationic dimer models, 1₂(D_2h)⁺ and 1₂(D_4h)⁺. Note that, regarding the total contribution of the linear combination (resonance) between gc and cg, it is represented as “gc” in Figure 4b for simplicity (the same applies to other configurations). First, we examine the results of 1₂(D_2h)⁺ (the left panel of Figure 4). At a sufficiently large d, it is natural to assume that the gc and cg are the lowest-energy configurations among all. Indeed, the lowest ¹B_2g state, as described by this contribution at any value of d, is the ground state of 1₂(D_2h)⁺ for d > 2.7 Å, showing a local minimum at d ~ 3.0 Å. The energy stabilization of the ²B_2g state results from the resonance between gc and cg (gc + cg) which is characterized by the transfer integral between the HOMOs of monomers, t_H-H. This situation is similar to the He₂⁺ case, where the bound state results from the resonance between two configurations corresponding to gc and cg, associated with the energy splitting between the (gc + cg) and (gc − cg) states [15]. A similar energy splitting is observed in the present case between the ²B_2g and ²B_3u states (see Figure S3).

The ²B_2u state is an excited state with a large d, mainly described by the tc, where one monomer is in the triplet state and the other is in the cationic ground state. The weight of gc* in the ²B_2u state increases as d decreases, associated with a rapid energy stabilization. This energy stabilization would result from the resonance between the tc and c*g (ct and gc*), which is also characterized by t_H-H (see Figure 3b). For d < 3.4 Å, where the electron clouds of monomers overlap, the weight of ac² started to increase slightly. Then, the ¹B_2u state exhibits a local minimum at d ~ 2.4 Å.

In the case of 1₂(D_4h)⁺ (the right panel of Figure 4), the doubly degenerate ²B_2u and ²B_3u states are the ground state, slightly lower than the doubly degenerate ²B_2g and ²B_3g states (we treated the system as if it had D_2h symmetry during the calculation to simplify comparison with 1₂(D_2h)⁺). Then, the ground state takes a local minimum at d ~2.4 Å. The minimum point of the ²B_2u/²B_3u states in 1₂(D_4h)⁺ is similar to that of the ²B_2u state in 1₂(D_2h)⁺. Because the HOMO and LUMO of each monomer are degenerated in the D_4h monomer, the ground state of the neutral monomer is described by the linear combination of the configurations g and d with equal weighting, and the energies of the dimer configurations, gc, gc*, dc, and dc*, are identical. Therefore, these configurations describe the ¹B_2g/¹B_3g states. The tc and tc* configurations mix in the ²B_2g/²B_3g states to some extent since the energies of tc and tc* are slightly higher than those of gc, gc*, dc, and dc*. In contrast, the tc and tc* are dominant in the ²B_2u/²B_3u states. The gc* and dc can mix in the ²B_2u/²B_3u states because the energies of these configurations are close to those of tc and tc*. In addition, the gc* (dc) can be generated from the ct (c*t) by one-electron transfer (see Figure 3b). A rapid energy stabilization of the ²B_2u/²B_3u states results from the resonance between tc and c*g (ct and gc*), characterized by t_H-H, and that between tc* and cd (c*t and dc), characterized by t_L-L (transfer integral between the LUMOs of monomers). Although the number of electron configurations contributing to the ²B_2u/²B_3u states increases compared to 1₂(D_2h)⁺, the degree of energy stabilization remains similar: The total energy difference between d = 5.0 Å and d = 2.4 Å (minimum) is −48.92 kcal/mol and −49.72 kcal/mol for 1₂(D_2h)⁺ and 1₂(D_4h)⁺, respectively.

The local minimum of the ²B_2u/²B_3u states in 1₂(D_4h)⁺ is lower than that of the ²B_2u state in 1₂(D_2h)⁺, indicating that the D_4h structure is energetically favored. From these results, in the ground state of the monocationic dimer at sufficiently small d, each monomer is expected to have both neutral triplet and cationic doublet characters as the superposition.

3.2. Dicationic π-Dimer Models

Figure 5 shows the calculated results for the dicationic dimer models, 1₂(D_2h)²⁺ and 1₂(D_4h)²⁺. In both cases, the key electron configuration is cc, where each monomer is monocationic (see Figure 5b), and Coulomb repulsion between the cationic monomers is assumed to describe the PESs of these models. Figure 5c shows the ground state energy of the dimer relative to that at d = 100 Å, i.e., ΔE(d) = E(d) − E(d = 100 Å). Note that the reversed order of ΔE(d) between 1₂(D_2h)²⁺ and 1₂(D_4h)²⁺ is due to the difference in the monomer energy (see Table S1). We also plot the potential energy (V) between point charges with q = +e (e: elementary charge). From the comparison, the PESs of 1₂(D_2h)²⁺ and 1₂(D_4h)²⁺ are modeled well by Coulomb repulsion between the positive charges at large d. However, there is a shallow minimum around d = 2.7 Å. From the wavefunction analysis results (Figure 5b), the contribution of charge-transfer (CT) configurations, that is, neutral–dication pairs, increases at d ≤ 3.4 Å, where the overlap of electron clouds enables electron transfer between the monomers. The electronic coupling between the cc and CT forms is characterized by t_H-H (and t_L-L as well in the case of 1₂(D_4h)²⁺), and the resonance between these configurations causes the energy stabilization at small d. A similar result was obtained for the He₂²⁺ case by Pauling [15]. However, the PES around the minimum is shallow and is essentially repulsive overall. Therefore, a stable dimer is expected to be difficult to form in the dicationic state.

4. Conclusions

Using the high-precision QD-NEVPT2 calculation combined with a VB-based multi-configurational wavefunction expression, we have theoretically analyzed the electronic structures of the cationic states of face-to-face π-stacked dimer models consisting of CBDs as a function of d. In the monocationic state, two types of states with different characters appear in the ground state depending on the stacking distance d. This situation is similar to the neutral state case demonstrated in our previous study. Within the scope of the considered model, the lowest-energy state is provided by a local minimum around approximately 2.4 Å in the D_4h dimer, where the ground state is mainly expressed by the tc and ct configurations. That is, each monomer is represented by a superposition between the neutral triplet and the cationic doublet states.

The ground state of the dicationic state is dominated by the cc configuration at d > 3.4 Å, where each monomer is in the monocationic state, and intermolecular Coulombic repulsion causes a repulsive trend of the PES. On the other hand, in the region where the monomers stack closely together, electron transfer between the monomers becomes possible, leading to an increase in the CT configuration, which stabilizes the energy of the dimer to some extent. The PES near the local minimum is shallow, yet it still exhibits an essentially repulsive PES. While the results for the CBD dimer models have several similarities with those for the helium dimer cases, the state crossing in the monocationic state, attributed to the CBD’s narrow HOMO-LUMO gap, which is comparable to the magnitudes of t_H-H and t_L-L around d_c, is considered unique to stacked systems of antiaromatic molecules.

The insights into electronic configurations obtained in this study are crucial for selecting the appropriate level of approximation required for structural optimization of cationic dimers and magnetic field response calculations. Moving forward, analyzing not only the cationic state but also the photo-accessible electronic excited states is considered a key theme for future work, aiming to pioneer functional switching based on the stacked system of antiaromatic molecules.