**3. Results/Discussion**

## *3.1. Discussion of Model Chemistry*

Table 1 shows experimental (*exp*) normal mode frequencies (*<sup>ω</sup>exp*) and theoretical normal mode frequencies (*ωμ*) for the water-hexafluorobenzene dimer (system **1**). Theoretical *ωμ* were computed using Møller–Plesset perturbation theory of second order (MP2) and the *ω*B97X–D functional combined with aug–cc–pVTZ, aug–cc–pVQZ, and def2–TZVPP basis sets. In addition, scaling factors were applied to theoretical frequencies to correct for approximations to the full electronic configuration interaction and the harmonic approximation to the Morse potential [126–132]. In parentheses directly to the right of each theoretical frequency, are % error values calculated with respect to *exp*. It turns out that *ω*B97X–D/aug–cc–pVTZ calculations were in closest agreemen<sup>t</sup> with *exp*. MP2 calculations performed best for the highest frequencies, but were less accurate for low frequencies. The opposite is true of calculations carried out using *ω*B97X–D. The use of the def2-TZVPP basis set was computationally more efficient, but the aug–cc–pVTZ basis set significantly improved accuracy.

Table 2 compares LVM data calculated at the *ω*B97X–D/aug–cc–pVTZ, *ω*B97X–D/aug–cc–pVQZ, and MP2/aug–cc–pVTZ levels of theory for *π*–hole system **1**, where O···C6 denotes the pure *π*–hole interaction between the acceptor O–atom and the geometric center of the C–atoms composing the six–membered ring (acceptor···donor). O···C6F6 is very similar to the interaction just described, except in this case the six F–substituents are included. <sup>H</sup>···C6 denotes the interaction between one acceptor H–atom and the geometric center of the six donor C–atoms; whereas <sup>H</sup>···C6F6 denotes a similar interaction, but with the six F–substituents included (analogous to the O···C6/O···C6F6 comparison). The *π*–hole interactions in the remainder of this work are defined using the first notation (O··· C6) in Table 2: the pure *π*–hole interaction between the central acceptor atom and the geometric center of the donor six–membered ring, not including aryl substituent atoms.

In comparison with the *ω*B97X–D/aug–cc–pVTZ calculations, adding a larger basis set (aug–cc–pVQZ quality) resulted in a modest *r* increase of 0.009 Å and a slight decrease in *ka* of 0.008 mdyn/Å. On the other hand, MP2/aug–cc–pVTZ results gave significantly shorter *r* (by 0.140 Å and 0.149 Å), slightly weaker bond strength (by 0.003 mdyn/Å) compared to *ω*B97X–D/aug–cc–pVTZ, and slightly stronger bond strength (by 0.005 mdyn/Å) compared to *ω*B97X–D/aug–cc–pVQZ. This result is erratic in the case of MP2/aug–cc–pVTZ. The *ω*B97X–D/aug–cc–pVQZ level of theory has large computational cost with small increase of accuracy compared to *ω*B97X–D/aug–cc–pVTZ. Therefore, we have chosen in this study the *ω*B97X–D/aug–cc–pVTZ level of theory as a compromise between accuracy and computational efficiency. Note that for the remainder of this work, the terms '*ka*' and 'bond strength' are used interchangeably. In addition, the term secondary bonding interaction (SBI) refers to any interaction between a single atom of the acceptor molecule and a single atom of the donor molecule which contains a physically meaningful LVM.

## *3.2. Overall Findings and General Trends*

Table 3 summarizes the LVM data of the *π*–hole interactions in **1–14** and two reference NCIs: **R1** (water dimer) and **R2** (Ar··· C6H6). Figure 3 (top) shows molecular geometry of each system, *r* (shown in green), *ka* (blue), *ω<sup>a</sup>* (red), and symmetry point group (black) for **R1**, **R2**, and **1–14**. **R1** and **R2** have been incorporated to provide a frame of reference from well characterized compounds: The H2O dimer represents complex containing a strong HB with non-negligible covalent character, and Ar··· C6H6 represents a weak NCI [133]. Also in Figure 3 (bottom), selected NBO charges are given, where charges in green represent C–atoms, O–atomic charges are red, H–atomic charges are black, N–atomic charges are blue, and F–atomic charges are light blue. Bond length *r* and NBO charge on the acceptor O and N–atoms are plotted with respect to *ka* in Figure 4a and Figure 4b, respectively. Shown as red plot points are interactions where H2O is the acceptor (**1–4**), light blue points represent HCN acceptor systems (**5–7**), in green are NH3 (**8–11**), blue points are the NO – 3 anion– *π*–hole interactions (**12–14**), and black points indicate **R1** and **R2**. This color convention is maintained in the subsequent plots.


**Table 3.** Summary of LVM data: *π*–hole interaction distances *r*, *ka*, *<sup>ω</sup>a*, charge transfer CT, and BSSE counterpoise corrected binding energies BE.

Calculated at *ω*B97X–D/aug–cc–pVTZ level of theory. Units for reported data as follows: *r* in Å, *ka* in mdyn/Å, *ω<sup>a</sup>* in cm<sup>−</sup>1, CT in milli-electron ( *me*), and BE in kcal/mol.

**Figure 3.** Schematics for **R1**, **R2**, and **1–14**, showing: (**top**) molecular geometries, distances *r* given in green font with units of Å, local vibrational mode LVM force constants *ka* (blue font) given in mdyn/Å, corresponding LVM frequencies *ω<sup>a</sup>* (red) given in cm<sup>−</sup>1, point group (shown in black); and (**bottom**) selected NBO charges: C–atomic charges given in green, O–atomic charges in red, N–atomic charges in blue, F–atomic charges in light blue, and H–atomic charges are shown in black. NBO charges are given in A.U.

**Figure 4.** Calculated at the *ω*B97X–D/aug–cc–pVTZ level of theory, (**a**) *r*, and (**b**) NBO charges of the central acceptor atoms O and N; plotted with respect to *ka* of *π*–hole interactions in **1–14**.

There is weak correlation at best between *r* and *ka*, which becomes weaker by presence of **R1** and **R2**. The *π*–hole interaction length in **14** is 1.000 Å longer than the HB in **R1**, ye<sup>t</sup> the former has a *ka* value 0.100 mdyn/Å larger than the latter. The Ar···C6H6 interaction in **R2** is at least 0.200 Å longer than all 14 *π*–hole interactions, but is stronger than **2**, **6**, and **11**. Figure 5a,b show charge transfer (CT) and BE counterpoise corrected for basis set superposition error; both plotted with respect to *ka*. CT was calculated as the transfer of charge between the acceptor lp–donor atom and the aryl ring. Both of these parameters correlate weakly with bond strength in terms of *ka*, but BE and *ka* show the best correlation of any properties considered in this work. Increase in magnitude of BE weakly correlates with increase in bond strength. The HB in **R1** has a *ka* value three times larger than the weakest *π*–hole interactions (**2** and **6**). On the other hand, **14** contains the strongest *π*–hole interaction in this work with a *ka* value 60% larger than *ka* of the HB in **R1**.

**Figure 5.** (**a**) CT (from central *π*–hole acceptor atom O in **1–4**, N in **5–14**, Ar in **R2**, and HB acceptor atom O in **R1** −−→ donor), and (**b**) BE, counterpoise corrected for basis set superposition error; both plotted with respect to *ka* of *π*–hole interactions in **1–14.**

In Figure 6a, the Laplacian of the electron density (∇<sup>2</sup>*ρ*(**<sup>r</sup>***CCP*)), where CCP is a cage critical point encompassing N– or O–atoms from the acceptor and aryl C or N–atoms from the donor, is plotted with respect to *ka*. <sup>∇</sup><sup>2</sup>*ρ*(**<sup>r</sup>***CCP*) tracks regions of local charge concentration/depletion [134]. <sup>∇</sup><sup>2</sup>*ρ*(**<sup>r</sup>***CCP*) increases with increasing strength of the *π*–hole interaction. In other words, increased local concentration of charge at the CCP corresponds to a stronger *π*–hole interaction. Figure 6b shows

correlation between *r* of the HBs and their *ka* values, where increased bond length corresponds to weakening of the HB. Figure 7 shows combined *ka* values of all SBIs/HBs per *π*–hole system, plotted with respect to *ka* of the *π*–hole interaction; where the larger quantity of stronger SBIs/HBs weakly correlate with stronger *π*–hole interactions. This correlation is weak because the HB can strengthen or weaken the *π*–hole, depending on the directionality of HB donation; a topic which is discussed further in Section 3.5.

**Figure 6.** (**a**) The Laplacian of the electron density at the CCP (∇2*ρ*(**<sup>r</sup>***CCP*)), and (**b**) *r* of the HBs; both plotted with respect to *ka* of *π*–hole interactions in **1–14** and **R2**.

**Figure 7.** Combined *ka* values of all SBIs (including HBs) plotted with respect to *ka* of *π*–hole interactions in **1–14** and **R2**.

Table 4 summarizes LVM data for all HBs between donor/acceptor pairs. Figure 8 is a schematic of the atom labelling/numbering convention used in subsequent tables and figures. HBs were not found in systems **2–3, 5–7, 12,** and **14**. In **2**, the aryl C–atom bound to the lone H–substituent has a charge of −0.326 e, but the orientation of the water molecule eliminates the possibility of HB. The other five aryl C–atoms all carry positive charges with values between +0.277 e and +0.361 e. A bonding interaction between positive charges on aryl C–atoms and positive charges on acceptor H–atoms (+0.469 e) is not favored. Acceptor H–atoms in **3** do not form HBs because they are oriented such that they are not in plane with any of the aryl atoms or substituents and their distance from aryl C–atoms is maximized at the given conformation. It was expected that **5–7**, **12**, and **14** would not form HBs for obvious reasons. Interestingly, **13** is the only example of HB between acceptor and aryl-substituents, where the three *π*–hole acceptor nitro O–atoms interact weakly with the three aryl H–atoms.

**Figure 8.** Schematic showing all atom numbers in **1–14**, for use as a reference to Tables 4 and 5.

**Table 4.** LVM analysis: *r*, *ka*, and *<sup>ω</sup>a*, for secondary bonding interactions SBI involving hydrogen atoms in **1, 4, 8–11,** and **13**.


Units for LVM data are given as follows: *r* in Å, *ka* in mdyn/Å, and *ω<sup>a</sup>* in cm<sup>−</sup>1.

Table 5 summarizes LVM data for all SBIs found in **1–14**, excluding HBs; where 13 of the 14 *π*–hole systems contains at as few as six non–HB SBIs (systems **1–3**, **5–11**, and **14**), 12 non-HB SBIs (**12**), and as many as 15 (system **13**) non–HB SBIs of the following type: C···O, C···N, or N···N; where the first atom listed (C/N) is from the *π*–hole donor and the second atom (O/N) is from the acceptor (donor···acceptor). In most cases, there is a LVM between the *π*–hole acceptor and all six atoms of the aryl ring; with **4** being the exception. The remainder of this section is divided into four subsections pertaining to significant factors for modulation of molecular geometry, bond strength, and the intrinsic nature of the *π*–hole interactions: *(3.3) Aryl Substituent Effects, (3.4) Nature of the Aryl Rings, (3.5) Secondary Bonding Interactions and (3.6) Characterization of Normal Vibrational Modes.*


**Table 5.** Summary of LVM data: *r*, *ka*, and *ω<sup>a</sup>* for secondary bonding interactions SBI not including hydrogen atoms, for **1–14**.

Units for computational data are given as follows: *r* in Å, *ka* in mdyn/Å, and *ω<sup>a</sup>* in cm<sup>−</sup>1.

## *3.3. Aryl Substituent Effects*

Systems **1–4** are a good starting point to systematically analyze substituent effects. The donor in **1** is C6F6, the donor in **2** is C6F5H, C6F4H2 in **3**, and C6F3H3 in **4**. One effect is the physical response of acceptor to decreasing the number of aryl F-substituents. Each of the four water molecules in **1–4** is oriented quite differently from one another with respect to the aryl ring. **1** has C2v symmetry, with the acceptor H–atoms pointing opposite the aryl ring. Each atom of the water molecule rests in plane with two aryl C−F groups positioned para to each other. Unexpectedly, the *π*–hole interaction in **1** is not particularly strong (*ka* = 0.090 mdyn/Å) compared to the rest of H2O acceptor group **2–4**, systems **5–14**, and even **R1** and **R2**. The six aryl F–substituents induce a sizable *π*–hole with large positive ESP which therefore should promote stronger *π*–hole interactions, but this effect is countered by a lack of cooperation between atoms of the aryl donor and atoms of the H2O acceptor in forming SBIs [135]. Furthermore, there are two weak C···H donor/acceptor SBIs (*ka* = 0.008 mdyn/Å) in **1** (see Table 4).

Compared to **1**, the acceptor H–atoms in **2** are rotated nearly 90◦ to avoid repulsive forces from the donor H–atom. The aryl C–atom bound to H has a negative charge of −0.326 e; whereas the aryl–C atom para to the lone C−H bond has a charge of +0.317 e. In contrast, all C–atoms in **1** have positive charges (see Figures 3 and **8**) of +0.295 e (C2 through C5) and +0.291 e (C1 and C6). The negative charge on the C–atom in **2** repels the electron rich acceptor O–atom toward the opposite end of the ring, resulting in the *π*–hole interaction distance increasing 0.072 Å compared to **1**. The O–atom is also no longer directly over the *π*–hole, which decreases orbital overlap. Instead, the O–atom is 0.407 Å closer to the C–atom para to C−H. Furthermore, the *π*–hole should migrate closer to the C−F group, and become weaker its ESP becomes more negative. The cumulative effect is that substitution of a single aryl F–atom for H disrupts the molecular symmetry, hinders the reactivity of the *π*–hole, and decreases *ka* of the *π*–hole interaction in **2** by nearly 50% compared to **1**; the *π*–hole interaction in **2** is the weakest of the H2O acceptor systems **1–4**. Although system **2** is an extreme case, where the other aryl rings of **1, 3–4** are significantly more symmetric, there is clear indication that substituent effects involving the aryl ring can significantly weaken/strengthen the *π*–hole interaction and drastically alter the molecular geometry of the system. The strongest *π*–hole interaction among systems **1–4** occurs in **3**, which has C2v symmetry with acceptor H–atoms still oriented away from the aryl ring. The water molecule forms a plane perpendicular to the two FC−−CF bonds of the donor. The water O−H bonds in **3** (*ka* = 8.549 mdyn/Å), are stronger than the O−H bonds in **1, 2**, and **4** (between 8.532 and 8.547 mdyn/Å). This increase in O−H bond strength has a net stabilizing effect on the whole system, which extends to the *π*–hole interaction. The orientations of H2O and the aryl F–substituents also benefit the *π*–hole interaction in system **3**; as any possible repulsive forces between the donor/accepter occur over maximum distances compared to **1, 2,** and **4**, and the position of the *π*–hole is not affected due to the symmetry of the C6F4H2 ring.

**4** has 3 aryl F–substituents and 3 H–substituents, arranged symmetrically in an alternating pattern. Addition of the third H–substituent resulted in inversion of the acceptor H–atoms, which now point toward the aryl ring. H2O is coplanar with para aryl C−H and C−F groups (C2−H12 and C5−F8). The acceptor atom H14 points downward toward C2, which is caused by the charge of −0.382 e on C2. The opposite occurs between acceptor–H15/donor–C5, where positive charges on each atom are repulsive. The H15···C5 distance is 0.388 Å longer than H14···C2 as a result. The *π*–hole interaction in **4** is only slightly weaker than **1** (by 0.004 mdyn/Å), an unexpected result based on substituent effects alone; as the O acceptor in **1** should interact much more strongly with its *π*–hole. However, other factors must be considered. The acceptor H14 in **4** can HB with the negatively charged C2 donor atom, ye<sup>t</sup> is in close enough proximity to bind to C4. A third HB was found in **4**; all three HBs are of the C···H−O type, and are among the strongest HBs in systems **1–14** (see Table 4). This factor is discussed with more detail in Section 3.5. Perhaps the most surprising substituent effect (or lack there of) is an absence of intermolecular interactions involving aryl–substituents and the *π*–hole acceptors. There is only one such SBI; it is in system **13** and is a O···H−C type HB. This interaction is discussed further in Section 3.5.

## *3.4. Nature of the Aryl Rings*

In **5–7**, the influence of SBIs is minimized, and each aryl ring contains four, three, and three N–atoms, respectively. **5** has C2v symmetry while **6** and **7** have C3v symmetry. N–substitution, atomic nature of the aryl ring, and three-fold symmetric F–substitution do not cause significant symmetry related changes in this case. However, it turns out that **5–7** have the weakest *π*–hole interaction strength on average compared to the H2O acceptor (**1–4**), NH3 acceptor (**8–11**), and NO – 3 acceptor (**12–14**) *π*–hole systems. One key difference between systems **5–7** and the systems just mentioned is the orientation and nature of the HCN acceptor in **5–7**, where the N–atom points downward toward the *π*–hole and the H–atom points in the opposite direction. This eliminates the possibility HB donation and decreases the overall possibility of SBIs. System **6** has one less aryl N–atom compared to **5**. The *π*–hole interaction in **5** is 0.107 Å shorter and has a *ka* value nearly two times larger than the *π*–hole interaction in **6**. Incorporating N–atoms into the aryl ring appears to influence strength of the *π*–hole interaction more than F–substitution. The difference between donors of **6** and **7**, is a three fold F–substitution in the latter, which increases strength of the *π*–hole interaction by 0.024 mdyn/Å. Though significant, triple F–substitution is not able to modulate strength of the *π*–hole interaction as much as insertion/removal of aryl donor N–atoms, as is the case for **5** and **6**. Integration of a fourth N–atom to the aryl ring nearly doubles *ka* of the interaction; whereas substituting three C − H groups for three C −F groups achieves an increase in interaction strength by approximately 50%. The NBO picture suggests the N4C2H2 donor of **5** supports a more delocalized electronic density compared with the N3C3H3 donor in **6** and N3C3F3 in **7**. There is a CT of −30.99 milli-electrons ( *me*) from the acceptor N–atom to the aryl ring in **5**, approximately –10 *me* more than in **6** but roughly −15 *me* less than CT in **7**. In addition to being the weakest interactions and not participating in HB, **5–7** also have the three largest CT values among **1–14**. Correlation between CT and *ka* for **1–14** is very weak, but the general trend is that the *π*–hole interactions are stronger when CT gets closer to zero (see Figure 5a).

**8–14** are not ideal for investigating how the nature of the aryl ring influences the *π*–hole interaction, given that the acceptors in these systems are ammonia and the nitrate anion. Each of the three acceptor H–/O–atoms are able to form SBIs, which make it difficult to assess both substituent effects and how addition of N–atoms into the aryl ring can influence the *π*–hole interaction. This is also evident in the case of **1–7**, where the acceptors each have one less atom than the acceptors in **8–14**. On the other hand, this makes **8–14** ideal for studying the effect of SBIs on *π*–hole interactions.

## *3.5. Secondary Bonding Interactions*

Of all systems **1–14**, **4** is the only *π*–hole system completely void of SBIs between a non-hydrogen acceptor and non-hydrogen donor. However, the unusual orientation of H2O in **4** puts H14 in close proximity to the C1− C2− C4 region of the aryl ring (see Figure 1 and Tables 4 and 5); where H14··· C1, H14··· C2, and H14··· C4 lengths are 2.834, 2.680, and 2.834 Å, respectively. H14 interacts with all three aryl C–atoms, and the resultant HBs are among the strongest found in **1–14**. The effect of these HBs is stabilization and increased strength of the *π*–hole interaction by 0.035 mdyn/Å compared to system **2**, where the acceptor O–atom forms SBIs with the donor aryl C–atoms. **2** and **3** do not have any HBs, but all of their aryl C–atoms interact with the acceptor O–atom. **2** has the weakest *π*–hole interaction of **1–14**, which is largely due to the nature of the donor and the arrangemen<sup>t</sup> of the acceptor water molecule, but a contributing effect is that five of the six C··· O interactions are among the weakest for **1–14** (see Table 5). The HBs in system **1**, are nearly the weakest (0.008 mdyn/Å) interactions found in **1–14**. Although these HBs do stabilize and increase strength of the *π*–hole interaction, stronger HBs will promote stronger *π*–hole interactions [136]. For example, system **4** contains the same type of C··· H HB found in system **1**, but the *ka* values of HBs in the former are 5 times larger than in the latter. Also, system **4** has three HBs while system **1** has two. The *π*–hole in **1** should be larger and have more positive ESP compared to **4**, due to the nature of the aryl substituents (six F–atoms in **1**, three F– and three H–atoms in **4**). In addition, the lp is more accessible in **1**, compared to **4** where the H–atoms point toward the aryl ring. Also, O13 is 0.238 Å closer to the *π*–hole in system **1**. Despite all of these

factors, *ka* of the *π*–hole interaction in **4** is within 0.004 mdyn/Å of the *π*–hole interaction in **1**. This a result of the comparatively strong HBs in system **4** providing stability and increasing strength of the *π*–hole interaction.

Although **5–7** do not provide information on the effect HBs have on the *π*–hole interaction, a clear picture emerges in terms of the role other SBIs play. In terms of strength of the *π*–hole interaction, the sequence is: **5** > **7** > **6**. This matches the trend in *ka* of the N···N and C···N donor/acceptor SBIs. For C···N in **5**, *ka* = 0.044 and 0.047 mdyn/Å, for N···N and C···N in **6**, *ka* = 0.024 and 0.025 mdyn/Å, and for N···N/C···N in **7**, *ka* = 0.038 mdyn/Å. This indicates that non-HB SBIs may play a cooperative role, where they help to strengthen *π*–hole interactions. However, HBs seem to have a more significant effect on the *π*–hole interaction comparatively.

**8–11** are the only *π*–hole acceptors where each system participates in HB and SBIs between aryl C or N atoms and the central acceptor N–atom. In terms of *π*–hole interaction strength, the sequence is: **10** > **9** > **8** > **11**. The *π*–hole interaction in **8** should be stronger than **9** based on the nature of the aryl donor, but the ammonia H–atoms in **8** are staggered such that they are centered above the bonds encompassing the ring. H12 is oriented above an N–N bond, and the other two ammonia H–atoms orient above two of the aromatic C–N bonds. This results in **8** having fewer HBs compared to **9–11**. The N···N and C···N donor/acceptor interactions in **8** are slightly shorter and slightly stronger than comparable interactions in **9**. The same type of inter–monomer N···N and C···N interactions in system **10** are the strongest amongs<sup>t</sup> the NH3–acceptor group by at least 0.026 mdyn/Å for N···N and at least 0.055 mdyn/Å for C···N. Strength of the non–HB SBIs trends similarly to the *π*–hole interaction strength order: **10** > **9** ≈ **8** > **11**. This not the case with individual HB strength. However, when HB strength is considered as a sum of each individual HB per *π*–hole system, the collective HB strength matches the trend of *π*–hole interaction strength. HBs are effecting the system compared to the non-HB SBIs. **11** has the weakest *π*–hole interaction, the weakest collective HB strength, the weakest non–HB SBIs, the fewest N–aromatic atoms (zero), and the most F–substituents of **8–11**. Although SBIs are predominant in **8–11**, it turns out that N–aromatic atoms still play a major role in modulating bond strength; with N–aromatic systems having *π*–hole interaction *ka* values increase 100% compared to the species with a C6 ring. Though even less significant than the aforementioned, effects of F–substitution are again apparent when comparing systems **9** and **10**; where the F–substituents result in a 28% increase in *ka* values.

**12–14** are the only anion *π*–hole systems investigated in this work, and as expected, occupy the strong end of *π*–hole interaction spectrum. In **12**, NO – 3 has a staggered conformation with respect to the C6F6 ring which puts the three acceptor O–atoms are at maximal distances from all C and F donor–atoms. Of course, there is no possibility of HBs in **12**, but the negatively charged O–atoms interact with the positively charged aryl C–atoms. The acceptor N–atom also interacts with the aryl C–atoms. This is possible because N has a lone pair and NO – 3 has an excess of delocalized electrons. The *π*–hole interaction in **13** is substantially weaker than the interactions in **12** and **14**. Regardless, **12–14** have the three strongest *π*–hole interactions among **1–14**, while they have the weakest and fewest number of HBs among each acceptor group. All three acceptor O–atoms in **13** HB with the three H–substituents on the aryl ring. These are the only three HBs where the *π*–hole acceptor is also the HB acceptor. In every other case, the directionality of acceptor/donor in the HB is the reverse direction of the *π*–hole interaction. Because the HB donor in **13** is the aryl C−H, electronic density is transferred to the aryl ring and throughout the *π*–system [137]. This transfer will cause an increase in negative ESP at the *π*–hole, which in turn weakens the *π*–hole interaction. When the HB donor/acceptor roles are reversed, electronic density transfers from the aryl ring to the acceptor molecule. The depletion of electronic density from the aryl ring increases the positive ESP at the *π*–hole, and since the acceptor molecule now has more electronic density, it becomes a better electron donor. When the aryl ring is the HB acceptor, charge on the atoms involved will increase in the positive direction and negative charge is leaving the *π*–system. This explains why each HB except for the O···H−C interactions in **13** help increase the strength of the *π*–hole interaction; whereas **13** has a substantially weaker *π*–hole interaction compared to **12** and **14** which do not have any HBs.

## *3.6. Characterization of Normal Modes*

In addition to providing *ka* and *ω<sup>a</sup>* and related local vibrational mode properties [80], the local mode analysis has led to a new way of analyzing vibrational spectra. The characterization of normal modes (CNM) procedure decomposes each normal vibrational mode into local mode contributions for a non-redundant set of LVMs by calculating the overlap between each local mode vector with this normal mode vector [77–79,82]. In this way, the character of each normal mode can be uniquely assessed [68,80,138]. In this work we performed a CNM decomposition for **R2** and **1** comparing in particular the contribution of the local vibrational *π*–hole-interaction mode to the lower frequency normal modes in both complexes. The corresponding decomposition plots are shown in Figures 9 and 10.

**Figure 9.** Decomposition of normal vibrational modes into % LVM contributions for the Ar··· C6H6 dimer **R2**; (**a**) % LVM contributions to normal vibrational modes 1–11, (**b**) % LVM contributions to normal vibrational modes 12–22, and (**c**) % LVM contributions to normal vibrational modes 23–33.

 **Figure 10.** Decomposition of normal vibrational modes into % LVM contributions for the H2O···C6F6 *π*–hole system **1**; (**a**) % LVM contributions to normal vibrational modes 1–13, (**b**) % LVM contributions to normal vibrational modes 14–26, and (**c**) % LVM contributions to normal vibrational modes 27–39.

The set of local modes used for this purpose was chosen to include all inter-monomer local modes. As shown in Figure S1 of the Supplementary Materials there are 9 possible inter-monomer modes, 3 stretching motions (x, y, z direction), 3 rotations (x, y, z direction), and 3 anti-rotations (x, y, z direction). 6 of them are needed to define the set of inter-monomer modes. We generally use the 3 stretching motions labelled Tx, Ty, and Tz an d 3 rotations Rx, Ry, Rz in the following. **R2** is a special case with one monomer being an atom reducing the number of inter-monomer modes to 3 translational modes (labelled as x, y, z).

## 3.6.1. Normal Modes Related to the *π*–Hole Interaction

In Figure 9a–c, normal modes *ωμ* are decomposed into % LVM contributions for complex **R2**, and the corresponding CNM plots for complex **1** are given in Figure 10a–c. Figure 9a shows CNM

for *ωμ* 1–11 (*ωμ*1 through *ωμ*11) into % LVM contributions for **R2**, where 'x,' 'y,' and 'z' in the Figure legends denote translations of the Ar–atom in the x–, y–, and z–directions with respect to the benzene molecule, as described above. In standard orientation, the Ar···*<sup>π</sup>*–hole interaction is in the z–direction. Correspondingly, the z–component of the three inter-monomer Ar···*<sup>π</sup>*–hole LVM parameters represents the direct Ar···*<sup>π</sup>*–hole interaction, i.e., this is the mode which corresponds to the special force constant *ka*. It is shown in yellow color in the CMN plots in Figure 10a–c for quick reference.

For **R2**, *ωμ*1 through *ωμ*3 and *ωμ*8 are all 100% LVM character corresponding to inter-monomer vibrations, where *ωμ*3 at 65.1 cm<sup>−</sup><sup>1</sup> and *ωμ*8 at 669.0 cm<sup>−</sup><sup>1</sup> are dominated by the *π*–hole interaction in the z–direction, representing the stretching and contraction the argon atom with regard to the center of the benzene ring. *ωμ*3 is characterized by the translational motion of the Ar–atom perpendicular to the plane of the benzene ring, and *ωμ*8 represents wagging of the six benzene H–atoms towards and away from the Ar–atom. This mode also perturbs slightly the benzene C–atoms. Movies of the *ωμ*3 and *ωμ*8 vibrational modes are shown in the Supplementary Materials, see Table S1 for description. Collectively, it is all of the vibrations associated with *ωμ*3 and *ωμ*8 which are required to accurately describe the *π*–hole interaction. These findings clearly emphasize that the special force constant *ka* as defined in this work is meaningful.

System **1** consists of 39 *ωμ* and 39 LVMs, including six parameters describing the inter-monomer translations Tx, Ty, and Tz and the inter-monomer rotations Rx, Ry, Rz, introduced above, see also legend in Figure 10a–c. As with CNM for **R2**, the direct *π*–hole interaction occurs in the z–direction; therefore the z–components of the inter-monomer LVMs are of particular interest and and are represented with a light yellow bar for Tz in Figure 10a,b- and with the darker yellow color for Rz.

The z–components of the inter-monomer LVMs contribute to six of the normal modes: *ωμ*1 (22.8 cm<sup>−</sup>1), *ωμ*4 (89.7 cm<sup>−</sup>1), *ωμ*7 (132.6 cm<sup>−</sup>1), *ωμ*10 (215.0 cm<sup>−</sup>1), *ωμ*16 (375.9 cm<sup>−</sup>1), and *ωμ*17 (376.6 cm<sup>−</sup>1). The Rz LVM composes 87% of *ωμ*1, with C−F/C−C LVMs accounting for the remaining 13% (Figure 10a). The Rz component is much less significant for the *π*–hole interaction, but mixing of C–C, C–F, and Rz contributions to *ωμ*1 imply this mode likely relates more to the HBs found in **1**. The motion of *ωμ*1 involves rotation of acceptor H–atoms about the O–atom, parallel to the plane of the donor ring. Description of *ωμ*4 in **1** is comparable to *ωμ*3 of **R2**, which is translation of the acceptor molecule in the z–direction. This normal mode is of 100% Tz character. Therefore, it could be used in experimental spectra as quick identification of the *π*–hole interaction. The motion of the water molecule in *ωμ*4 perturbs the C6F6 slightly; whereas this does not occur in **R2**. The frequency of the former is also larger than the latter by 24.6 cm<sup>−</sup>1.

The *ωμ*7 contains 46 % Tz character combined with small contributions from the C–F, C–C–F, and H–O–H LVMs. This mode is comparable to *ωμ*8 of **R2**: the z-direction wagging of the aryl substituents. As with the previous comparison, the wagging motion of the aryl–F atoms perturbs the acceptor molecule and the aryl C–atoms in **1**; this not the case for **R2**. On the other hand, similarly to **R2**, this mode is also important for a full description of the *π*–hole interaction. Tz accounts for 88% of *ωμ*10 and describes the translation of the aryl C–atoms in the z–direction. This mode strongly perturbs the acceptor H2O molecule, and there is no comparable mode to this in **R2**. The *ωμ*10 is also a main component of the *π*–hole interaction, as the six aryl C–atoms move in phase and therefore translate the *π*–hole directly toward the acceptor. The *ωμ*16 and *ωμ*17 represent z–rotation of the four equivalent aryl C–atoms and z–rotation of the two equivalent aryl C–atoms, respectively. Rz compose 16% of *ωμ*16 and 64% of *ωμ*17. These modes do not strongly effect *π*–hole interactions. However, *ωμ*17 is related to the HBs between acceptor/donor, where the HB acceptor C–atoms rotate in the direction of the water H–atoms. This explains why the contribution from Rz is much larger for *ωμ*17. As was previously mentioned, the HBs with directionality opposite to that of the *π*–hole interaction effectively stabilize and increase strength of the *π*–hole interaction. This relationship is reflected in the CNM analysis. It is evident that the *π*–hole interaction in **1** is stronger than in **R2** based on the CNM because the inter-monomer LVMs Tz and Rz of **1** compose more of *ωμ*, the comparable frequencies are larger tahnin **1**, and the vibrational modes are much more strongly coupled between monomers in **1**. Movies

of the *ωμ*1, *ωμ*4, *ωμ*7, *ωμ*10, and *ωμ*17 vibrational modes are shown in the Supplementary Materials, see Table S1 for description.

## 3.6.2. Normal Modes Not Related to the *π*–Hole Interaction

The C6−C1−C2−C3 dihedral mode is the only LVM contributor to *ωμ*4 through *ωμ*5 and *ωμ*9. Modes 6–7 are composed of mainly angular C−C−C contributions with small components of the various C−C−H contributions. Mode 10 at 845.0 cm<sup>−</sup><sup>1</sup> consists of a nearly even mixture of C−C−C and C−C−H LVM contributions with small (5%) contribution from C6−C1−C2−C3; whereas mode 11 is C−C−H dominant with minor C−C−C, and C−H character. Figure 9b shows decomposition of *ωμ* into % LVM contributions for *ωμ*12 through *ωμ*22 in **R2**. Again, the C6−C1−C2−C3 dihedral is the sole contribution to *ωμ*12 through *ωμ*13 and *ωμ*16. The six C−C LVMs compose *ωμ*14 and are the largest components of *ωμ*18 and *ωμ*22, with minor components being the C−C−H LVMs. C−C−C LVMs compose 80% of *ωμ*15 with C−C−H contributions accounting for the remaining 20%. At 1126.4 cm<sup>−</sup><sup>1</sup> through 1155.9 cm<sup>−</sup>1, *ωμ*19 through *ωμ*21 are C−C−H LVM dominant with small contributions. In Figure 9c, the remaining *ωμ* (23–33) are decomposed into % LVM contributions for **R2**. C−C−H LVMs are the major contributions to *ωμ*23 through *ωμ*25: 100% of *ωμ*23 at 1330.1 cm<sup>−</sup>1, nearly 80% of *ωμ*24 at 1462.7 cm<sup>−</sup>1, and 76% of *ωμ*25 at 1462.8 cm<sup>−</sup>1. C−C LVM contributions steadily increase from *ωμ*24 through *ωμ*27 (1462.7 cm<sup>−</sup><sup>1</sup> through 1596.9 cm<sup>−</sup>1), where % LVM contributions increase from 20 % of the former to nearly 70 % of the latter. The six highest *ωμ* (28–33) span 3045.2 cm<sup>−</sup><sup>1</sup> through 3080.4 cm<sup>−</sup><sup>1</sup> and are composed entirely of C−H LVMs.

*ωμ*7 (132.6 cm<sup>−</sup>1) and *ωμ*11 through *ωμ*15 (263.0 cm<sup>−</sup><sup>1</sup> through 307.9 cm<sup>−</sup>1; see Figure 10a,b) consist mainly of C−F and C−C−F LVM contributions, with minor contributions from the H2O···*<sup>π</sup>*–hole interaction at 132.6 cm<sup>−</sup><sup>1</sup> and C−C/C−C−C LVMs at 263.0 cm<sup>−</sup><sup>1</sup> through 307.9 cm<sup>−</sup>1. *ωμ*8, *ωμ*9, *ωμ*23, and *ωμ*24 are 100% C1−C2−C4−C6 character. *ωμ*14 and *ωμ*15 once again have LVM contributions from the H2O···*<sup>π</sup>*–hole interaction of 23% and 64%, respectively. From 431.8 cm<sup>−</sup><sup>1</sup> to 581.5 cm<sup>−</sup>1, C−C and C−C−C LVMs are the major contributions, with C−F and C−C−F LVMs being minor components. After the C1−C2−C4−C6 modes at 653.5 cm<sup>−</sup><sup>1</sup> through 733.3 cm<sup>−</sup>1, *ωμ*25 is completely C−C−F character and *ωμ*26 is largely C−F character with small C−C−F and C−C contributions. Figure 10c shows *ωμ*27 through *ωμ*39 for **1**, spanning 992.0 cm<sup>−</sup><sup>1</sup> through 3811.0 cm<sup>−</sup>1.The 992.0 cm<sup>−</sup><sup>1</sup> through 1143.1 cm<sup>−</sup><sup>1</sup> region is C−F and C−C−F dominant, while *ωμ*31 at 1240 cm<sup>−</sup>1, *ωμ*36 at 1639.3 cm<sup>−</sup>1, and *ωμ*37 at 1639.4 cm<sup>−</sup><sup>1</sup> are mainly of C−C and C−C−C character. From 1299.8 cm<sup>−</sup><sup>1</sup> through 1512.0 cm<sup>−</sup><sup>1</sup> a mixture of C−F, C−C−F, C−C, and C−C−C LVMs compose the *ωμ*. At 1570.2 cm<sup>−</sup><sup>1</sup> is the H2O bending mode, and the H2O stretching modes are at 3710.3 cm<sup>−</sup><sup>1</sup> (symmetric) and 3811.0 cm<sup>−</sup><sup>1</sup> (asymmetric). Overall, this discussion shows that the CMN feature offered by the local mode analysis provides a powerful tool for the detailed analysis of a vibrational spectrum.
