**1. Introduction**

The term ' *π*–hole interaction' was coined by Murray and Politzer [1–4], and is described as a noncovalent interaction (NCI) between a region of positive electrostatic potential (ESP) located on a *π*–bond (i.e., a ' *π*–hole') [5], and a lone–pair (lp) donor [6–8], anion [9,10], or other electron rich species [11,12]; where the *π*–hole is perpendicular to the molecular framework and electrons from the *π*–hole acceptor interact with an empty *π*<sup>∗</sup> orbital of the donor. Some classic examples of *π*–hole interactions involving aryl groups include the benzene/hexafluorobenzene–water complexes, where an oxygen–lp interacts favorably with the center of the aromatic ring [13–20]. This special type of interaction has been identified in several important and highly relevant areas of modern chemical research, including drug targets [21,22], biological systems [23,24], and molecular crystals/solid state chemistry [25–30]. Interestingly, noble gases have recently been found capable of forming both *σ*– and *π*–hole interactions [31–33]. Ideal *π*–hole donors should contain heavier and more polarizable atoms, as these properties improve accessibility, size, and positive ESP of a *π*–hole [34–37]. Electron withdrawing *π*–hole acceptors can also increase the positive ESP of the *π*–hole [38,39]. The main interaction energy terms describing *π*–hole interactions are: ion induced polarization and a permanent quadrupole moment ( *Qzz*) from the electrostatic forces [40–42]. Though there have been several recent theoretical and experimental studies on *π*–hole interactions [43–56], often the strength of these interactions is discussed in terms of bond lengths (*r*) or binding energies (BE)/dissociation energies (DE). However, these properties are not necessarily qualified as bond strength descriptors. There is an ample number of examples in which the shorter bond is not the stronger bond [57–59]. It is often assumed that BE or DE provide a measure of the intrinsic bond strength of the NCI in question. However this might not even be true in a qualitative sense, as BE and DE are cumulative properties; i.e., they are the sum of all interactions between the monomers, including long–range electrostatic interactions which may even involve the more remote atoms of the monomer [60]. Therefore, it is difficult to single out a specific interaction between atoms or groups of monomers; even computationally this can only be done in a qualitative way via an energy decomposition scheme, which leads to model dependent results [61–65]. In this situation, vibrational spectroscopy provides an excellent alternative for the description of the interactions between the monomers of a complex, and offers a platform for deriving a spectroscopic measure of complex stability. However, as has been frequently pointed out [60,66–69], any description of bond strength based on vibrational modes has to consider that normal vibrational modes are generally delocalized due to the coupling of the motions of the atoms within a molecule or complex [70–74]. Therefore, only decoupled local vibrational modes can serve as bond strength measurements, as was realized in the Local Vibrational Mode (LVM) theory originally formulated by Konkoli and Cremer [75–82]. Local mode stretching force constants (*ka*) are directly related to the intrinsic strength of a bond, and therefore provide a unique measure of bond strength based on vibrational spectroscopy [83]. The local mode procedure was inspired by the isotopic substitution of McKean [84]. McKean found that if an XH fragment in a molecule is replaced by XD, a local X–D stretching mode may be detected in the IR spectrum, and therefore the force constant of the X–H or X–D stretching may be measured. This technology has been used to measure the force constants of many X–H bonds, but it cannot be extended to other systems due to the weak isotope effect. However, theoretical calculations are not limited to natural isotopes, allowing for isotopes of any mass to be "invented." The local mode procedure treats all the atoms which are not involved in a particular local mode as massless particles, so that they can effortless follow the local motion. For each local mode associated with an internal coordinate such as a bond length, bond angle, dihedral angle or puckering coordinate a unique local mode force constant, associated local mode mass and frequency can be obtained. So far, the LVM analysis has been successfully applied to characterize covalent bonds [59,66,83,85–88] and weak chemical interactions such as halogen [89–92], chalcogen [58,93,94], pnicogen [95–97], and tetrel interactions [98]; as well as hydrogen bonding (HB) [67,69,99–102]. For a comprehensive review the reader is referred to Ref. [80].

In this work, LVM theory is utilized to obtain a more accurate measurement of strength and the intrinsic nature of interactions between various aryl systems as *π*–hole donors and a number of small electron rich *π*–hole acceptors; where the *π*–hole either interacts with lp–electrons from a charge neutral acceptor, or an anionic acceptor species. A special inter–monomer LVM stretching force constant is utilized, which directly assesses the strength of the *<sup>π</sup>*–hole··· *π*–hole acceptor interaction. Based on this special inter–monomer *ka* measure, recently and for the first time, the strength of metal–ring interactions in a series of actinide sandwich compounds was quantified [103], and a nonclassical HB involving a BH··· *π* interaction was identified [104,105]. Burianova et al. concurrently verified this type of nonclassical HB involving a BH··· *π* interaction both experimentally and theoretically while performing a mechanistic study involving the nucleophilic addition of hydrazines, hydrazides, and hydrazones to C −−−N groups of boron–based clusters [106].

The current work investigates the interactions of *π*–hole acceptors H2O, HCN, NH3, and NO – 3 , with the following aromatic *π*–hole donors: C6F6, C6F5H, C6F4H2, C6F3H3, N3C3H3, N3C3F3, and N4C2H2 (see Figure 1). Original theoretical works of similar nature date back to 1997, when Alkorta et al. investigated the effects of F–substitution on reactivity of the aromatic rings in systems where small electron-donating molecules interact with the *π*–clouds of benzene and hexafluorobenzene [107]. An extension of this work was reported in 2002, which included a larger array of aromatics and benzene derivatives and several negatively charged electron donors [108]. Simultaneously, a similar phenomenon was reported involving 1,3,5–triazine derivatives interacting with F– , Cl – , and azide (N3) [109], and a computation study was combined with crystallographic evidence to confirm such interactions can favorably occur [110].

**Figure 1.** Schematic of the two references systems, **R1** and **R2**, and *π*–hole systems **1–14** studied in this work showing molecular geometries of each system; calculated at the *ω*B97X–D/aug–cc–pVTZ level of theory.

## **2. Computational Methods**

DFT was utilized to optimize molecular geometries, calculate stationary point normal mode vibrational frequencies (*ωμ*), LVM frequencies (*ω<sup>a</sup>*), *ka* [75,78,79], and Natural Bond Orbital (NBO) charges. Calculations were carried out at the *ω*B97X–D/aug–cc–pVTZ level of theory with tight convergence criteria and superfine integration grid [111–116]. All stationary points were confirmed to be minima by absence of imaginary *ωμ*. Calculated and experimental vibrational frequencies of the H2O···C6F6 [117] system were used to gauge the accuracy of several model chemistries (see Tables 1 and 2). Theoretical vibrational spectroscopy was utilized to quantify the intrinsic strength of *π*–hole interactions in this work. Normal vibrational modes do not give direct measurements of bond strength because of electronic and mass coupling. This results in delocalization of the normal modes in most cases. The electronic coupling is eliminated by solving the Wilson equation of spectroscopy [118] and transforming to normal coordinates. Konkoli and Cremer found that mass coupling can be removed by solving a mass–decoupled equivalent of the Wilson equation, which leads to LVMs. LVMs are associated with internal coordinates: bond length, bond angle, or dihedral angle [76], and lead to a direct relationship between the intrinsic strength of a bond and its *ka* value [83]. For the first time, this theory is applied to *π*–hole interactions. LVM analysis was computed with the program COLOGNE2018 [119]. NBO populations were calculated using NBO6 [120–122]. Calculations of *ρ*(**<sup>r</sup>***CCP*) and <sup>∇</sup><sup>2</sup>*ρ*(**<sup>r</sup>***CCP*) were performed with the AIMAll program [123,124]. All DFT calculations were made with GAUSSIAN16 [125].

**Table 1.** Comparison of experimental *exp* normal mode vibrational frequencies *<sup>ω</sup>exp*, with theoretical normal mode vibrational frequencies *ωμ* for **1** computed at the *ω*B97X–D/aug–cc–pVTZ, *ω*B97X–D/aug–cc–pVQZ, *ω*B97X–D/def2–TZVPP, MP2/aug–cc–pVTZ , and MP2/def2–TZVPP levels of theory.


*<sup>ω</sup>exp* and *ωμ* are reported in cm<sup>−</sup><sup>1</sup> and errors are given as % with respect to *exp* in parentheses next to each *ωμ*. Scaling factors are as follows: 0.957 (*ω*B97X–D/aug–cc–pVTZ), 0.957 (*ω*B97X–D/aug–cc–pVQZ), 0.955 (*ω*B97X–D/def2-TZVPP), 0.953 (MP2/aug–cc–pVTZ), and 0.952 (MP2/def2-TZVPP) [126–132].

**Table 2.** Comparison of local vibrational mode LVM data for *π*–hole system **1**, where O···C6 (acceptor···donor) represents the pure *π*–hole interaction between the acceptor O–atom and the geometric center of the C–atoms comprising the six–membered ring, O···C6F6 denotes similar as above but includes the six F–substituents of the *π*–hole donor, <sup>H</sup>···C6 denotes one acceptor H–atom interacting with the geometric center of the six donor C–atoms, and <sup>H</sup>···C6F6 represents the aforementioned interaction with inclusion of the aryl F–substituents.


bond lengths *r* are given in Å, LVM force constants *ka* in mdyn/Å, and units for LVM frequencies *ω<sup>a</sup>* are cm −1 .

Figure 2 illustrates how the special force constant *ka* is defined for the special case of the *π*–hole interaction involving a six–membered ring as *π*–hole donor. *ka* is defined via the direct interaction between the central O– or N–atom of the *π*–hole acceptor (position X1 in Figure 2) and the geometric center of the six atoms composing the aryl ring of the *π*–hole donor (X2 in Figure 2). A key feature

of the LVM methodology is that the *π*–hole need not be at the X2 geometric center of the ring. If this is the case, and the acceptor atom at X1 is collinear with X2 and the *π*–hole, the value of *ka* will not change because the local modes of X1···X2 and X1···*<sup>π</sup>*–hole are normalized in the LVM theory formalism. In systems **R2**, **1–4** and **11–12**, the ring atoms are all carbon; whereas in systems **6–7**, **9–10**, and **13–14**, three N–atoms and three C–atoms are incorporated into the ring structure. In systems **5** and **8**, the six–membered rings are composed of four N–atoms and two C–atoms.

**Figure 2.** Schematic of how the special LVM force constant *ka* is defined for the *π*–hole interaction involving a six–membered aromatic ring as *π*–hole donor, where X1 is the location of the central atom of the acceptor molecule interacting directly with the *π*–hole located at X2; shown is complex **2**.
