*4.2. Quantum Chemical Topology Analyses*

In order to further deepen into the origin of the observed trends in the evolution of the binding energies reported in Table 1, we examined the non-covalent interactions established between both monomers using QCT techniques. For the sake of convenience, the nomenclature shown in Figure 3 will be used to refer to the atoms involved in the intermolecular bonding pattern of these dimers. We first consider the electron redistribution of electron charge due to the formation of the investigated H-bonds. We point out that the formation of an HB is associated with a reorganisation of the electronic density of the moieties involved in this interaction. There is, indeed, a transfer of electron charge from the HB acceptor to the HB donor, with the proton acting as a bridge. For small HB dimers such as (H2O)2 or (HF)2, two of the simplest HBs, such charge displacement makes the HB acceptor a better proton donor. Ditto for the HB donor becoming a better proton acceptor. Notwithstanding, the present work deals with dimers where each molecule acts simultaneously as an HB donor and an HB acceptor and hence, there is no effective charge transfer between the monomers. However, the presence of an HB induces a rearrangement of the electron density that interacts with the *π* clouds in each molecule of the studied systems.

**Figure 3.** Hydrogen bond connectivity pattern involved in the formation of the dimers. A and D stand for the acceptor and donor HB moieties, respectively.

Let us start by examining the major changes undergone upon the dimerisation of the non-aromatic reference systems: formamide and formamidine. The complexation process is accompanied by a significant electron redistribution, as reflected by the change in the QTAIM atomic charges, collected in Table 2. The formation of the non-covalent interactions leads, in both cases, to a noticeable electron enrichment of the D and A atoms (between 0.03 and 0.09 electrons) at the cost of decreasing the electron population of the H atom by ≈0.04–0.09 electrons. On the other hand, the central C atom undergoes a noticeable change in its average electron number of −0.09–0.01 a.u. depending on the nature of the acceptor moiety.

**Table 2.** Change in the QTAIM electron populations of the atoms involved in the HB contacts upon the formation of the dimers for (i) the hydrogen bond Acceptor Contained within the Ring (ACR) and (ii) the hydrogen bond Donor Contained within the Ring (DCR) cases. The labelling of the atoms is shown in Figure 3. All values are reported relative to the monomers which were used as reference. Atomic units are used throughout.


These observations, and with the particular exception of the bridging C atom, are very similar for both NCO and NCN bonding patterns and evidence a conspicuous rise in the polarisation of the system due to the formation of the corresponding dimers. Such an increase in the local polarisation of the terminal atoms enhances the electrostatic interaction in the HB contacts, as reflected by the large classical components of the A···H interaction as reported in Figure S2.

We also considered the change in the number of electrons shared among bonded atoms, as measured by the delocalisation index (DI) (see SI Figure S5 for more details), as gathered in Table 3. The D–H bond order decreases significantly (≈0.16–0.25) upon dimerisation, thus weakening the covalent component to the D–H interaction, as evidenced by the prominent destabilisation of ≈30–40 kcal/mol found for *<sup>V</sup>*D–H xc . A similar, yet more subtle, weakening of the covalent component can also be observed for the C–A bond. On the other hand, the DI(D–C) is increased by 0.07–0.10 electron pairs, going from a single D–C bond to a slightly higher bond order (≈1.1 in the general case). These results point out that hydrogen bonding reinforces the D–C double bond character at the expense of decreasing that of the C–A interaction. We observed a similar effect in our analysis of RAHB in which the DI corresponding to double bonds decrease while that of single bonds have the opposite behaviour after the formation of the RAHB [22].

This last observation is fulfilled for all the systems and suggests that the formation of the dimers may trigger two opposed effects. Because the A–C bond is contained within the ring in ACR dimers, and ΔDI(C–A) < 0 as indicated in Table 3, we would expect that the formation of the H-bond would decrease the number of *π* electrons in the associated ACR cyclic structures as represented in Figure 4. On the contrary, the D–C bond is included in the cyclic structures of DCR dimers, and ΔDI(D–C) > 0 (Table 3), then the number of *π* electrons must increase in the DCR dimers due to the formation of the H-bond. Accordingly, Table S8 indicates that the group energy of the ACR/DCR rings increase/decrease upon

the formation of the corresponding dimers. These changes in electron delocalisation affect the aromaticity and antiaromaticity of the investigated systems as discussed below.

**Table 3.** Change in the electron delocalisation index of the atoms involved in the HB contacts (Figure 3) upon the formation of the dimers with (i) the hydrogen bond acceptor contained within the ring (ACR) and (ii) the hydrogen bond donor contained within the ring (DCR). These changes are computed with respect to the values of the monomers, which were used as references. Atomic units are used throughout.


**Figure 4.** Representation of the major electronic changes induced by the dimerisation process in systems with (i) the hydrogen bond acceptor contained within the ring (ACR) and (ii) the hydrogen bond donor contained within the ring (DCR) displayed in the left and right parts of the Figure, respectively. Red and blue colors indicate QTAIM atoms with electronic charge accumulation and depletion, respectively, due to the formation of the molecular cluster. Ditto for green and orange bonds, employed to highlight an increase or decrease in the DI values. Purple is used to show not clearly established scenarios in this regard.

## *4.3. Perturbation of the Aromaticity of the π Skeleton*

We consider now the interplay between aromaticity and antiaromaticity with the inter-molecular HB contacts of Figure 2. Table 4 gathers the change in the aromaticity indices of the intra-molecular *π* skeleton upon dimerisation, as measured by the MCI and FLU indices (further details about these indices can be found in Section 1 of the SI).


**Table 4.** Change in the MCI and FLU aromaticity indices, along with the change in aromatic/antiaromatic character (Γ), induced by the formation of the dimers in Figure 2. If ΔΓ > 0, there is either (i) an increase of aromaticity or (ii) a reduction of antiaromaticity; vice versa when ΔΓ < 0.

Before discussing in detail the changes in the aromatic character of the spectator groups, it may be enlightening to provide a grasp of the FLU and MCI aromaticity indices. The former measures the electron sharing between neighbouring atoms in a ring as well as its similarities between the constituents of the cyclic structure. Thus, a FLU value of zero corresponds to an "ideal" aromatic system, while positive values evidence a deviation from aromaticity. On the other hand, the MCI index measures the collective electron delocalisation along a collection of M centres. As opposed to the FLU, large MCI values suggest a high aromatic character, whereas any other situation usually results in vanishing MCI indexes. Although these metrics were specifically designed to measure aromaticity, they have been successfully used to study antiaromaticity as well [67]. We used the Hückel rule to assign the aromatic or antiaromatic character of the examined monomers as shown in Figure 5. The aromaticity metrics of the monomers (see Table S1) are in agreement with the aromaticity or antiaromaticity label as determined by the Hückel rule. Indeed, the DCR form of AZH and AZA is more aromatic than their ACR counterparts. Likewise, the ACR tautomer of 2HP and 2AP is more aromatic than the corresponding DCR structures.

**Figure 5.** Aromatic and antiaromatic character of the monomers shown in Figure 2 determined with Hückel rule.

We discuss now how the aromaticity and antiaromaticity of the corresponding monomers change due to the formation of the HB interactions. We mention that apart from the DCR-AZH scaffold, all dimers adopt a nearly fully planar disposition which is key for the delocalisation of *π* electrons and it is optimal for the formation of the AMHB. Except for a slight discrepancy within the results of (i) the FLU on one hand and (ii) MCI and NICS [32] on the other for the change in the aromatic character between the DCR systems AZH and AZA, there is a good agreement between the computed sign of ΔΔ*E* and the changes of aromaticity and antiaromaticity in the examined systems, as shown in Figure 6.

**Figure 6.** Values of ΔΔ*E* and changes in aromaticity and antiaromaticity (ΔΓ) for the AMHB investigated herein.

In short, we observe that the condition ΔΔ*E* < 0, i.e., a more favourable formation of the HB with respect to the reference is related with a reduction in the antiaromatic character of the monomers. Correspondingly, when ΔΔ*E* > 0, i.e., a less favourable HB with respect to the reference is accompanied by a reduction of the aromaticity of the monomers. These observations based on QCT are consistent with those of Wu and coworkers [32]. We note

some flexibility in the interpretation of these results. Namely, the decrease in aromaticity in the HB formation of the AZA and AZH dimers in DCR configuration was interpreted in Reference [32] as an intensification of antiaromaticity (Figure 7). Similarly, the decrease in antiaromaticity of the DCR systems 2HP and 2AP after the generation of the corresponding dimers was interpreted as an HB reinforced by an increase in aromaticity [32]. This observation suggests that aromaticity and antiaromaticity can be put in a similar scale using electron delocalisation tools within the conceptual framework of QCT.

We consider now a further QCT description of the aromatic and antiaromatic moieties considered herein (Figure 5). We observed important differences concerning the atoms directly involved in the H-bond depending on the aromatic or antiaromatic character of the interacting monomers. For the sake of clarity, we will generally refer to the changes of QTAIM and IQA properties upon dimerisation, with respect to the NCN or NCO reference systems.

As expected, the relative change in the atomic charges and delocalisation indices reported in Tables 2 and 3 have a notable impact on the covalent and ionic components to the total IQA interaction energies in the atoms directly involved in the H-bond. Figure 8 collects the change in the classical and exchange-correlation components of *E*int for the atoms entailed directly in the inter-molecular contact, reported relative to the NCO or NCN reference systems. The aromatic scaffolds (see Figure 5) seem to consistently stabilise the *V*C–A xc and *V*D–H xc contributions over the rest of the interactions. The classical term of other interactions shows, on the other hand, trends that are more complicated to interpret. The interplay between these two contribution results in a net stabilisation of the C–A component at the cost of partially disrupting the D–C bond, when measured with respect to their references.

**Figure 7.** Alternative interpretation of the AMHB in the DCR dimers compared to that offered in Figure 6 as increases in aromaticity and antiaromaticity in the (i) 2HP and 2AP on one hand and (ii) the AZA and AZH on the other, respectively.

**Figure 8.** Relative changes in the IQA interaction energies upon dimerisation, all of the values are reported relative to their NCO or NCN reference systems. The tautomeric forms ACR and DCR are indicated by the a and d subscripts, respectively.

Antiaromatic systems have a different behaviour concerning the weakened and strengthened interactions in the inter-molecular region due to the formation of the examined H-bonds. These monomers (Figure 5) generally strengthen both the classical and exchange-correlation components of the H···A contact further than the reference systems. This fortifying of the HB interaction is accompanied by a noticeable destabilisation of the covalent component of the C–A bond, which, as previously discussed, is more strengthened in the aromatic compounds than it is in the reference compounds.
