The nature of resonance-assisted hydrogen bonds: a quantum chemical topology perspective

Resonance Assisted Hydrogen Bonds (RAHB) are particularly strong H-Bonds (HB) which are relevant in several fields of chemistry. The traditional explanation for the occurrence of these HBs is built on mesomeric structures evocative of electron delocalisation in the system. Nonetheless, there are several theoretical studies which have found no evidence of such electron delocalisation. We considered the origin of RAHBs by employing Quantum Chemical Topology tools, more specifically, the Quantum Theory of Atoms in Molecules (QTAIM) and the Interacting Quantum Atoms energy partition. Our results indicate that the π-conjugated bonds allow for a larger adjustment of electron density throughout the H-bonded system as compared with non-conjugated carbonyl molecules. This rearrangement of charge distribution is a response to the electric field due to the H atom involved in the hydrogen bonding of the considered compounds. As opposed to the usual description of RAHB interactions, these HBs lead to a larger electron localisation in the system, and concomitantly to larger QTAIM charges which in turn lead to stronger electrostatic, polarization and charge transfer components of the interaction. Overall, the results presented here offer a new perspective into the cause of strengthening of these important interactions.

developed by Alkorta and coworkers [30,31] showed that neither the proton chemical shifts nor the coupling constants point out to significant contributions from π-resonance to these interactions. On the contrary, it is put forward that the reason behind the RAHB strengthening lies in the σ−skeleton of the system [32,33] . Grabowski and coworkers used QTAIM analyses [34] to examine the nature of the stabilization in RAHBs. While they initially argued that the dissociation energy of intramolecular RAHBs depends mainly on the π−electron delocalisation [35] , they suggested in subsequent studies that the enhancement of electron delocalisation and the equalization in length of the conjugated bonds do not correspond to a strengthening of the HB [36] . In addition, ELF analyses have shown no difference between RAHBs and other intramolecular HBs found in molecules with saturated chains [37] . As a final example, valence bond theory has also been used to study dimers of carboxylic acids, amides and malonaldehyde along with its substituted derivatives as model structures for intermolecular and intramolecular RAHBs [38][39][40] . Whereas most of the aforementioned investigations suggest that the resonance stabilization energies are negligible as compared with the total interaction energies and that Hbond covalency is not significatively increased in RAHB interactions [38,39] other studies propose that the delocalisation energy term is the main source of stabilisation in these HBs [40] .
With this background, the aim of this study is to contribute to the elucidation of the nature of RAHBs. For this purpose, we have considered a set of representative H-bonded molecules and clusters as shown in Figure 2: (i ) systems with intramolecular RAHBs (Figures 2(a)-2(b)), along with their non-conjugated counterparts (Figures 2(c)-2(d)), and (ii ) the dimers of formic acid and formamide whose interactions have also been associated with intermolecular RAHB formation (Figures 2(e)-2(f)). We have used a battery of quantum chemical topology (QCT) tools, more specifically the Quantum Theory of Atoms in Molecules (QTAIM) [34] and the Interacting Quantum Atoms (IQA) energy partition, [41,42] which together provide a good description of both long and short range interactions, a critical issue in this research. The QTAIM and IQA wavefunction analyses (i ) are based on orbital invariant quantities, namely, the reduced first order density matrix ̺ 1 (r 1 , r ′ 1 ) along with the pair density ̺ 2 (r 1 , r 2 ) and (ii ) provide a division of the electronic energy in components endowed with a well-defined physical meaning. In this sense, they offer answers which do not depend on the specific method used to perform the underlying calculations. Their conclusions are thus free of method biases. Most importantly, IQA and QTAIM have already been used successfully to analyze the nature of the hydrogen bond [5] and to study H-bond cooperative [43] and anticooperative [44] effects in water clusters. In this sense, they open a unique reference-free window to understand the reasons chemical bonds such as covalent and intermolecular interactions can be characterized by means of QTAIM on the same rigorous basis. With this intention in mind, we determined the charges associated to each QTAIM atom, and carried out the integration of the Fermi and Coulomb holes [46] over one or two different basins to determine the localisation (LI) and delocalisation (DI) indices [34] used to analyse the change in the number of delocalised electrons across the system due to the formation of every considered HB.
In addition, we used the QTAIM division as a starting point to perform the partition of the Born-Oppenheimer electronic energy, E, in accordance with the IQA approach [41,42] in which E A net represents the net energy of basin A and E AB int denotes the interaction energy between atoms A and B. In addition, E A net and E AB int can be further splitted in wherein T A is the kinetic energy of atom A whilst V XY νµ indicates the potential energy due to the interaction of ν in atom X with µ in atom Y, with the labels ν and µ representing either electrons or nuclei. The detailed expressions of every term in the RHS of equations (2) and (3) as functionals of the first order reduced density matrix, ̺ 1 (r 1 , r ′ 1 ), or the pair density, ̺ 2 (r 1 , r 2 ) are given in references [41] and [42].
The interaction between two atoms can be further characterized by considering the Coulombic (J) and exchange-correlation (xc) components of the pair density Since the Hartree-Fock (HF) method considers only Fermi correlation, V AB xc becomes V AB x when we consider density functions derived from this approximation.

Computational details
The geometry of every system was optimized with the MP2 [47] approximation in its efficient RIJCOSX variant [48] along with the aug-cc-pVTZ basis set [49] , as implemented in the Orca program [50] . It has been reported that O−H···O hydrogen-bonded systems are properly described by second order Møller-Plesset perturbation theory together with augmented Dunning basis sets [51] . In order to analyse the changes in a system when an HB is formed or dissociated, we We performed afterwards the IQA and QTAIM wavefunction analyses based on HF density functions. The HF wavefunctions were computed with the Gamess-us [52] program. The IQA partition based on ̺ HF 1 (r 1 , r ′ 1 ) and ̺ HF 2 (r 1 , r 2 ) have been successfully used recently to study the H-bond cooperative effects within small water clusters [43] . Besides, the RAHB formation energies examined in this work are considerably larger in magnitude than those associated to these H 2 O systems [43,53] and hence easier to study from an energetic perspective. Finally, the behaviour of the HF and MP2 electron densities, as reflected in the changes of delocalisation indices upon the formation of the HB, is very similar. The corresponding data are summarized in Table S1 in the Supporting Information.
The IQA partition energy and the QTAIM analyses were performed with the Aimall [54] package. Finally, we used the programs Avogadro [55,56] and Gnuplot [57] for molecular and data visualization respectively.

Results and discussion
A deeply rooted chemical conception establishes that resonance is a stabilizing effect. However, uncovering the physical nature of that purported stabilization is far from trivial, as the severe problems encountered when trying to define resonance (e.g. aromatic) stabilization energies have taught us over the years. [58] As we show below, uncoupling the several effects that charge delocalisation channels open up, may lead to several unexpected results. Fortunately, these effects are easy to isolate using a reference-free real space energy decomposition method like IQA.
As established in previous studies on H-bonds [13,43,44,59,60] , the formation of an RAHB leads   Table S1 in the supporting information Since ̺(r) may be obtained by tracing out one electron coordinate from ̺ 2 (r 1 , r 2 ), it is not surprising that the changes in the latter scalar field will also be evidenced through the analysis of the charge distribution. Table 1 shows the variation in the most relevant QTAIM atomic charges, ∆Q, after H-bond formation. We can clearly sort the systems into conjugated/nonconjugated ones, and within each of these classes into O−H/N−H containing molecules. We note that (i ) that formation of the H-bond induces a charge redistribution that increases the overall total atomic charge (in absolute value), this effect being considerably larger in conjugated systems; (ii ) that the changes in atomic charges are slightly larger than those we have found in small water clusters with σ H-bond cooperative effects. [43] In chemically appealing terms, the π channel allows for a more efficient charge density flow.
We can now examine the origin of the one-electron and pair density reorganization. The IQA approach has been used to show that the charge distribution changes in H-bonded dimers containing only σ-bonds results mostly from the self-consistent classical polarization induced by the interacting monomers on each other. [5] To examine a similar possibility herein, we simulated and the cancelling negative one in close proximity § to the enolic hydrogen atom of (a).
As the central panels of Figure 4 show, the model reproduce very well the DIs of the RAHbonded compound. Once again, the change in the electric field experienced by the system upon H-bond formation is the basic factor determining charge reorganization. Furthermore, these changes explain the pair density variations. This is an important new insight that admits the following interpretation: π channels are more efficient roads to charge flow than σ ones, as expected, but aromatic-like resonance is not needed to understand bond-order equalization and charge transfer in these systems. They come out as the response of the mobile π electrons to the new electric field.
Thus, the role of resonance itself in the stabilization of RAHB systems should be questioned.
Further evidence on this issue is easily found. A remarkably simple argument comes from the total number of delocalised electrons in the system: if resonance is a basic energetic factor in RAHB, the formation of these H-bonds should result in an increase of this quantity. As we show now, the condition that the delocalisation indices are more uniformly distributed among the system does not imply that their total sum is incremented. By considering the division of electrons using LIs and DIs, we note that the total number of delocalised ( 1 2 A =B δ(A, B)) electrons decreases (Table 2) after the formation of the H-bond. Such a reduction is of course accompanied by a concomitant increase in the total number of localised electrons, and becomes a powerful argument against the traditional role assigned to resonance in RAHB, in consistency with previous experimental and theoretical studies which found no indication of an increased electron delocalisation in this type of H-bonds [29][30][31][32][33][36][37][38][39][40] .
Since, as stated above, the interacting quantum atom partition provides a unique combination of electron and energy descriptors, we may now explore the energy features of the formation of the HB associated to the previously discussed changes in ̺ 1 (r) and ̺ 2 (r 1 , r 2 ). Following the previous line of reasoning, we first examine the energetics of resonance and bond order equalization, which are indicated in terms of the covalent, i.e. exchange, IQA contributions. Table  Table 2: Changes in the sum of (i ) delocalisation indices (δ AB ), (ii ) the intra-atomic (V A x ) and (iii ) interatomic (V AB x ) exchange components of the electronic energy upon the formation of the corresponding hydrogen bonds. Energy values in kcal/mol.  Table 2 are considerably larger in the conjugated systems, giving further support to the notion that resonance is not the direct cause of the strengthening of RAHBs.
Besides the energy partition into intra and interatomic contributions just presented, it is also relevant to study the local IQA contributions within the six-membered pseudo-rings in  Table 3. In order to consider suitable references for discussion purposes, we also present results for the (H 2 O) 2 and H 2 O···H 3 N clusters, with no σ or π cooperative effects whatsoever. We point out that classical interaction energies V AB cl include all the electrostatic, polarization and charge transfer contributions to the interaction that are usually considered in perturbation-like approaches, as suggested in reference [43] .
Because all the interactions occur between considerably charged atoms, the classical component dominates over the exchange-correlation in every entry of Table 3 as expected, representing more than 90% of E AB int in the O···H and O···X cases. It also represents more than 65% in all instances of the O−H bonds, comprising a smaller percentage for the N−H bonds in molecules  Table 3: IQA interaction energies along with its classical and exchange parts, for the O···H, O···X, and X−H bonds, directly involved in the intramolecular non-covalent interactions under examination. X represents O in compounds (a), (b) and N in systems (c), (d). The data for the water dimer are also presented for comparison.
The data are reported in kcal/mol. spondence with previous studies. [5,43]  V AB x shown in Table 2 means that the strengthening of the H-bond by means of electron sharing is accompanied by the weakening of other covalent interactions in the system. IQA provides no indication that the O···H−X system is stabilized by means of π-resonance effects. [13,14] To gain further insight, we now consider the role played by the rest of the atoms appart from the O···H−X moiety in the formation of the RAHBs under examination. Indeed, the IQA components involving the atoms that do not participate directly in the HB have to be accounted into the energetics of these interactions as well. [43]  interactions with the repulsive O···O and H···H contacts as explained in the discussion of Figure   10 in reference [43] . This is a relatively common result: the strong electrostatic interactions tend to cancel out within neutral systems. These analyses are obscured in the case of intramolecular H-bonds, for there is no natural partition of the atoms of the molecule into interacting fragments. Nonetheless, we can use the dimers of formic acid and formamide, adducts (e) and (f), to One of the capabilities of IQA and the QTAIM wavefunction analyses is to offer views at different levels of granularity, hence it is also interesting to discuss on some of the specific interatomic interactions, particularly those along the pseudo-rings in Figures 2 (a)-(d). As The aforementioned preponderance of the IQA classical component is in agreement with a previous analysis of resonance assisted hydrogen bonded adenine-thymine pairs and other analogue compounds [61] where no changes in relative stability were found when the aromatic rings were removed. A corresponding EDA analysis [62] shows that the stronger HB in the unsaturated systems is originated by larger electrostatic interactions. Separately, Mo and coworkers, using real space techniques, arrived to similar conclusions [63] . Altogether, our results indicate that a RAHB involves a considerable rearrangement of both the one-and two-particle densities facilitated by the π-conjugated system, that leads to an overall electron localisation and a large separation of charges. It is through the latter, that would be interpreted as a combination of charge transfer and polarization effects, that the RAHB systems acquire their particular stability.
The alternating pattern of V A−−− −B x in the diagrams of Figure 5 is reminiscent of the equalization of DIs previously discussed as a distinctive feature of the stabilization mediated by conjugation atomic counterpart. The IQA and QTAIM analyses indicate that the π channels of the analyzed conjugated carbonlys allow for a greater reorganization of the one and two-electron distributions which lead to important contributions from electrostatics, polarization and charge transfer in the establishment of an RAHB. On the whole, we anticipate that the new interpretation of the resonance assisted hydrogen bonds presented herein, will prove valuable in the understanding of these important interactions in different fields of physical chemistry.

Acknowledgements
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