**1. Introduction**

More than a century of study of the hydrogen bond (HB) [1,2] has yielded a myriad of facts, ideas, and principles concerning this crucial linker in the microscopic world. It has been learned [3–25] that Coulombic forces are a critical factor, wherein the polarization of the R-H covalent bond induces a partial positive charge on the H which attracts an approaching nucleophile. This basic attraction is supplemented by a charge transfer from the lone electron pair of the nucleophile to the antibonding σ\*(R-H) orbital of the acid. Other contributing factors arise from mutual polarization of the two subunits and London dispersion. The HBs that result from this confluence of phenomena are both ubiquitous and of enormous importance [7,12,14,26–30], essential for life, occurring within such biomolecules as proteins or nucleic acids, enzymatic reaction pathways, catalytic intermediates, and of course in water.

Of more recent interest are a number of other closely related noncovalent interactions, where the bridging H of the HB is replaced by any of a long list of atoms that lie on the right of the periodic table. These bonds are typically classified by the family of the bridging atom, e.g., halogen, chalcogen, tetrel, pnicogen, and triel bonds. However, they share with the HB many of the same contributing factors [31–33]. The bridging atom acquires a positive region, differing from the H only in that this region is more localized, which can similarly attract a nucleophile. Moreover, like the HB, these other noncovalent bonds are likewise stabilized by charge transfer, polarization, and dispersion [34–42].

As a ubiquitous and powerful force, the HB contributes heavily to assembling and preserving the architecture of supramolecular synthons [15,43–55]. Of the sorts of assemblies to which H-bonds contribute, among the most intriguing are those that contain "like–like charge" interactions where ions of like charge lie adjacent to one another. These

**Citation:** Wysoki ´nski, R.; Zierkiewicz, W.; Michalczyk, M.; Maris, T.; Scheiner, S. The Role of Hydrogen Bonds in Interactions between [PdCl4] <sup>2</sup><sup>−</sup> Dianions in Crystal. *Molecules* **2022**, *27*, 2144. https://doi.org/10.3390/molecules 27072144

Academic Editor: Miroslaw Jablonski

Received: 25 February 2022 Accepted: 22 March 2022 Published: 26 March 2022

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anion···anion [56–72] and cation···cation [72–77] interactions are counterintuitive and have generated recent and extensive scrutiny [5,77–81], being called among other names an "anti-electrostatic" hydrogen bond (AEHB) [78,82]. In one picture, the Coulombic repulsion is overcome by resonance-type covalency represented by n→π\*/n→σ\* charge transfer [82]. Another view claims that the nominal point charge-point charge repulsion is oversimplified [83], and the full electrostatic term is more complicated, arising from the charge distribution over the entire subunit, as well as charge penetration effects. Another factor helping to overcome the implicit repulsion is the cooperativity of hydrogen bonding not only in simple dimers, but also within larger clusters [73], with a supporting role played by dispersion. Such cooperativity has been confirmed from both spectroscopic (IR, NMR) and computational perspectives [73].

Our own group has extended the understanding of this question to anion–anion interactions that involve π-holes [59–61,66–71]. These systems were stabilized by an assortment of noncovalent bonds, including pnicogen, triel, spodium, noble gas, and alkali earth bonds. The calculations showed that these complexes were metastable in the gas phase, wherein the dissociation was impeded by an energy barrier, but were fully stable in solution, despite their like charges. The innate attractive forces in systems such as tetrachloridopalladate(II) or trichloridomercurate(II) units) [66,68] were demonstrated by AIM, NBO, and NCI analyses, supported by experimental data. The results showed how the presence of counterions could stabilize these anion–anion interactions, in large part through the attenuation of the charges residing on the interacting anions.

One very recent study in particular [66] explored the interaction between a pair of [PdCl4] <sup>2</sup><sup>−</sup> dianions. The double charge on each makes for a particularly repulsive naked Coulombic repulsion. Indeed, in the absence of any surrounding environment, these two dianions strongly repel one another. However, the inclusion of a few of the surrounding counterions, along with the H-bonds which they form to these dianions, enables the entire system to be held together as seen in the crystal. In this case, the two dianions are held together in part by a charge transfer from the Cl lone pair of one unit to the vacant Pd π-orbitals lying above the plane of the other.

The present work is designed to explore the precise mechanism whereby this nominally highly dianion–dianion repulsion can be overcome by such external species. What is the relative importance of the charges on these surrounding molecules as compared to the H-bonds which they form with the anions? Is this a purely electrostatic phenomenon, or are there strong elements of polarization and charge dispersal which are important? Are there any specific stabilizing interactions between the pair of [PdCl4] <sup>2</sup><sup>−</sup> units which can act to hold them together if the overall Coulombic repulsion can be overcome, and how might these noncovalent bonds be affected by the surrounding molecules?

The analysis is designed to focus on a specific system whose crystal structure has been determined as an example. Figure 1 displays the relevant portion of the NETMOO [84] system, which shows some of the most important interactions. One can see the contact between the Cl of the upper unit and the Pd of that below. Quantum calculations attributed this arrangement to a π-hole bond wherein Cl lone pairs of one unit transfer charge to vacant orbitals above the Pd center of its neighbor [66]. It is also apparent that the NH groups of the counterion can engage in NH···Cl H-bonds with either of the dianions. As a starting point, the two [PdCl4] <sup>2</sup><sup>−</sup> anions are placed in the positions which they occupy in the crystal. Then, various models, of various size and complexity, of the counterions are added to the system in stages, monitoring the strength and nature of the interactions. The size of the counterion is examined by the comparison of the full +NH3CH2CH2CH2CH2NH3 + species which occurs in the crystal with shorter versions such as Ca2+. Not only is the latter much smaller, but it is unable to engage in H-bonds. Other model ligands were considered of charge +1 and 0 so as to monitor the effect of the overall ligand charge. For example, removing a proton from +NH3CH2CH2CH2CH2NH3 <sup>+</sup> yields the very similar but monocationic +NH3CH2CH2CH2CH2NH2 whose effects can likewise be compared with the much smaller NH4 <sup>+</sup> and with K<sup>+</sup> as a non-H-bonding cation. The models can be

extended to those with no charge at all, such as NH2CH2CH2CH2CH2NH2, NH3, and Ar. Lastly, one can isolate the effects of a purely electrostatic treatment by replacing any of these species with a series of point charges, incapable of accepting any charge from any of the participating units, or engaging in any noncovalent bonding of any sort.

**Figure 1.** View of the crystal structure of the studied system (CSD REFCODE: NETMOO [84]). (**a**): View of the PdCl4 <sup>2</sup><sup>−</sup> anions (**b**) View of one unit-cell content showing the layered character of the structure. In both views, the shortest hydrogen bonds are shown as dashed lines.
