**1. Introduction**

The initial concept of "pancake bonding" was constructed by Mulliken and Person as to characterize the overall shape and bonding mechanisms of donoracceptor *π* systems [1]. More recently the term "pancake bonding" has primarily been used to describe the formation of stabilizing parallel *π*–*π* interactions between two or more open-shell free radicals, those of which are typically planar and/or consist of light-atoms [2–4]. Such interactions have received a considerable amount of interest as they allow one to synthesize novel radical-based materials, via electron or hole through-space delocalization, that exhibit unique magnetic [5], optical [6], and electronic properties (i.e. conductive polymers, organic conductors) [7].

Generally, free radical species are short lived and exist in low concentration as two radicals will typically react to form a single covalently bonded dimer, or *σ*dimer. However, when radicals are sterically hindered against approaching within a covalent bonding distance, they can exist as a stable, spin-paired, open shell species. Unlike general non-covalent interactions between closed-shell species (i.e. van der Waals), the open-shell radicals have been said to undergo stabilization with each other via through-space *π*-stacking 2e/mc distributed interactions (i.e. pancake bonding). This 2e/mc bonding (i.e. pancake bonding) is a result of overlapping antibonding (*π* <sup>∗</sup> ) singly occupied molecular orbitals (SOMO) of the two monomer radicals with highly delocalized *π*-electrons [8]. It is noted that magnetic experimental analysis has found the spin pairing of pancake bonded dimers to be diamagnetic with an overall spin density of zero (i.e. singlet electronic state) [9]. The overlapping of antibonding (*π* <sup>∗</sup> ) SOMOs is the basis of pancake bonds as this interaction leads to the following distinctive features [4]: i) contact bond distances that are beyond the usual C(sp<sup>3</sup> )–C(sp<sup>3</sup> ) bond length (1.54 Å) but are also much shorter than the bonds of closed shell dimers that are held together by vdW forces (sum of vdW radii = 3.40 Å) (ii) due to direct atom-to-atom overlap, SOMO-SOMO overlapping strongly favors configurations that yield maximum overlap orientations which lower the energy of the two radical SOMOs iii) low lying singlet ( singlet-singlet) and triplet (singlet-triplet) electronic excited states, iv) negative singlet-triplet splitting energies (i.e., ΔE*ST* = E(singlet) – E(triplet)) for stable open shell singlet pancake bonded complexes [10] and v) interaction energies larger than those of vdW interactions. Bond dissociation energies (BDE) of pancake bonded system have been estimated to be smaller than those of a normal covalent system but larger than dimers subject to typical *π*-stacking where this type of *π*-stacking is observed for DNA base pairs [11] (vdW *π* stacking interactions and pancake bonds are different). Several works analyzed the related binding energies (BE), splitted into two contributions, a destabilizing stabilizing vdW part, E*vdW*, and a stabilizing energy, E*SOMO*, associated with the bonding overlap of the singly occupied SOMO [12]. E*SOMO* yields a reasonable description of the SOMO-SOMO overlap contribution to BE and it has been suggested that E*SOMO* can be estimated from the difference between E(singlet) – E\*(triplet), where E\*(triplet) is the triplet energy evaluated for the singlet geometry [12].

BE, E*SOMO* and SOMO-SOMO overlap have been utilized as to further explain the nature of these systems [8, 13]. It was argued that the dimerization of such radicals exhibit covalent bonding character as the spin-pairing of the electrons in the SOMO leads to a filled highest occupied molecular orbital (HOMO) and a corresponding empty antibonding LUMO [14]. In this situation, the interaction occurs at rigid rotational geometries, due to SOMO-SOMO overlapping, which is different from *π*-stacking in which various rotational orientations are possible [15]. On the other hand, dispersion and/or van der Waals interactions have been suggested to play important roles in the overall stabilization of these dimers [14]. Thus, the nature of pancake bonds between 1,2-chalcogen-3,5-diazol radicals and phenalenyl-based radicals remains in debate to the present day.

A CSD database survey based upon 35 cis-cofacial dimers composed of HCNSSN radicals, with C–C contact distances ranging between 2.75 to 3.50 Å, showed that S⋯S contact bond distances ranged from 2.93 to 3.30 Å [8]. These S⋯S contact bond are much shorter than the vdW distance between two sulfur atoms (4.06 Å) [16], in the case of two spherical sulfur atoms the vdW distance has been computed to be 3.60 Å. A CSD database survey based on 12 cis-cofacial 1,2-diselena-3,5 diazolyl dimers, with C⋯C contact distances between 2.80 and 3.50 Å, found the average Se⋯Se contact distance to be 3.26 (s = 0.05) [8]. This average Se⋯Se contact distance is slightly smaller than the vdW distance between spherical Se

atoms (3.32 Å). Previously computed dissociation energies have suggested that dimers of R-CNSeSeN radicals dimers are more binding than dimers of R-CNSSN radicals; relative binding energy values were also observed to be analogous to vdW interactions [8].

1,2-chalcogen-3,5-diazole dimers: Within the past two decades di-chalcogendiazole radicals, such as 1,2-dithia-3,5-diazolyl (i.e. HCNSSN) and 1,2-diselena-3,5 diazolyl (i.e. HCNSeSeN) radicals, and their derivatives have been a subject of many investigations [17]. The rings of HCNSSN and HCNSeSeN are rich in *π*electrons and have *π* <sup>∗</sup> singly occupied molecular orbitals (SOMO). The 1,2-dithia-3,5-diazolyl and 1,2-diselena-3,5-diazolyl radicals have been experimentally observed to result in stable dimerizations in the solid state where, in most cases, the neutral radicals prefer to be oriented with their faces parallel to one another (cis-cofacial) in order to achieve a configuration that supports maximum *π* <sup>∗</sup> -*π* <sup>∗</sup> (SOMO-SOMO) overlapping observed as two electron/eight-center (2e/8c) *π*stacking (i.e. pancake bonding) interactions. A notable feature of HCNSSN and HCNSeSeN dimers are their four long chalcogen-chalcogen bonds (i.e. contacts) ranging between 2.2 and 4.0 Å. HCNSSN and HCNSeSeN dimers have been suggested to stabilize via a combination of *π* and *σ* aromaticity [13].

Phenalenyl-based dimers: In solution, phenalenyl radicals maintain chemical equilibrium via the formation of a *σ*-bonded dimer [18]. Due to the very high symmetry of the radical phenalenyl monomer, a unpaired electron is delocalized across all *α*-positions of the phenalenyl framework excluding the central carbon atom of the monomers [19]. As noted in the work of Kubo [19], the thermodynamic stability of such carbon-centered radical species increases as the delocalization of unpaired electrons across a system increases [19]. Another interesting feature of phenalenyl dimers and their derivatives (i.e. carbon-centered hydrocarbon radicals) is due to the formation of unique two-electron/twelve-center (2e/12c) *π*-stacking interactions between these spin-delocalized hydrocarbon radicals [20] as verified by NMR [21]. The hexagonal arrangement of the SOMO of the phenalenyl radicals enables perfect *π*-*π* overlap in both eclipsed and staggered stacking configurations, the staggered stacking configuration is favored over the eclipsed configuration due to shorter *π*-*π* contacts as a result of less atom-atom repulsion [19]. It is mentioned, that various phenalenyl derivatives, which demonstrate *π*-*π* stacking (i.e. pancake bonding), have been experimentally identified via single crystal X-ray diffraction (XRD) [22]. The formation of *σ*-bonded phenalenyl radical dimer can be inhibited by substituting the carbon atoms of the phenalenyl rings, at the 2,5,8-positions, with *tert*-butyl groups as a *π*-bonded dimer results from the sterically hindered phenalenyl radicals [19]. Moreover, X-ray studies have revealed that the application of sterically hindered substituents (i.e. *tert*-butyl groups) on phenalenyl radicals prevent *σ*-dimerization and results in a *π*-bonded dimer with a face-to-face stacking distance, twice that of the *σ*-bonded dimer, at a length of of 3.2 Å [23]. This *π*-*π* contact (face-to-face) stacking distance is characteristic to pancake bonding as this length is shorter than that of a vdW complex and is beyond the length of a coventional covalent bond. Bond dissociation energy (BDE) for systems containing carbon radicals such as phenalenyl have been estimated to be around 10 kcal/mol [11]. Because *σ*-bonded and *π*-bonded phenalenyl-based dimers are close in energy the existence of the pancake bonded dimer as a fluxional molecule has been reviewed [12].

Although many experimental and computational have been conducted for the dimerizations of 1,2-chalcogen-3,5-diazol and phenalenyl-based radicals, the intrinsic strength of these interactions remains unclear. While popular BDE and its decomposition [24] provides valuable information about the stabilizing forces involved in bond formation (in the case of pancake bond in particular in the

formation of 2e/mc interactions), BDE does not adequately describe the intrinsic strength of a bond [25–27]. Because BDE measures the overall effect of bond breakage it contains the electronic reorganization and geometrical relaxation of the fragments upon dissociation. Therefore, we introduced in this work an intrinsic bond strength measure based on vibrational spectroscopy. Unlike BDE, the local stretching force constant (*k<sup>a</sup>* ), derived from local vibrational modes [25], conserves the geometry and electronic structure of all bonds/interactions. *k<sup>a</sup>* provides a direct description of intrinsic bond strength and has been applied successfully applied to assess the intrinsic bond strengths for a variety of covalent interactions including ultra long C–C bonds, carbon-halogen bonds and non-covalent interactions such as hydrogen, tetrel, pnicogen, chalcogen and halogen bonds; see Ref. [25] and citations therein.

In this study, we applied the local mode analysis [25] complemented with the RING puckering analysis of Cremer and Pople [28] and Bader's quantum theory of atoms in molecules (QTAIM) analysis of the electron density [29] to quantify the strength of the pancake bonds in six spin-paired, open-shell singlet state dimers **1**–**6** (shown in **Figure 1**) and and to learn more about their nature. Species **1**–**3** are 1,2 chalcogen-3,5-diazole dimers which contain sulfur (**1**), selenium (**2**), and tellurium atoms (**3**); it is noted that **3** is a prototypal (i.e. theoretical) species. Species **4**–**6** are phenalenyl-based dimers in which the bulkiness of substituents increase as follows: phenalenyl dimer (**4**) < 2,5,8-trimethylphenalenyl dimer (**5**) < 2,5,8–*tert*butylphenalenyl (**6**). The aromatic character of the dimer species (**4**–**6**) was also explored, in particular the role of the aromaticity for the stabilization of phenalenylbased dimers. In summary, special focus was on: i) to assess the intrinsic bond strengths of the 2e/mc interactions for selected species, ii) to quantify the ring strengths of the selected species, iii) to determine if the pancake bonds of these species are covalent in nature, iv) to elucidate on the effect of substituents on the aromaticity of phenalenyl-based species, v) to determine, for phenalenyl-based dimers, the effect of dimerization on the aromaticity for phenalenyl-based species,

#### **Figure 1.**

*Species investigated in this work. 1) 1,2-dithia-3,5-diazolyl (HCNSSN) dimer 2) 1,2-diselena-3,5-diazolyl (HCNSeSeN) dimer. 3) 1,2-tellura-3,5-diazolyl (HCNTeTeN) dimer 4) phenalenyl dimer. 5) 2,5,8-trimethylphenalenyl dimer. 6) 2,5,8-tri-t-butylphenalenyl dimer. Detected pancake bonds (2e/mc) (i.e. targeted contact bonds and interdimer CC bonds) are denoted in red.*

and vi) to determine what bond property, of the phenalenyl-based species investigated, predominately governs changes in aromaticity.
