The Role of Hydrogen Bonds in Interactions between [PdCl4]2− Dianions in Crystal

[PdCl4]2− dianions are oriented within a crystal in such a way that a Cl of one unit approaches the Pd of another from directly above. Quantum calculations find this interaction to be highly repulsive with a large positive interaction energy. The placement of neutral ligands in their vicinity reduces the repulsion, but the interaction remains highly endothermic. When the ligands acquire a unit positive charge, the electrostatic component and the full interaction energy become quite negative, signalling an exothermic association. Raising the charge on these counterions to +2 has little further stabilizing effect, and in fact reduces the electrostatic attraction. The ability of the counterions to promote the interaction is attributed in part to the H-bonds which they form with both dianions, acting as a sort of glue.


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  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][27][28][29][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][32][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][35][36][37][38][39][40][41][42].
As a ubiquitous and powerful force, the HB contributes heavily to assembling and preserving the architecture of supramolecular synthons [15,[43][44][45][46][47][48][49][50][51][52][53][54][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 the very similar but monocationic + NH3CH2CH2CH2CH2NH2 whose effects can likewise be compared with the much smaller NH4 + and with K + 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.

Computational Methods
Quantum calculations were carried out with the aid of the Gaussian 16, Rev. C.01 set of codes [85]. DFT computations employed the PBE0-D3 functional with the explicit inclusion of dispersion corrections, along with the def2TZVP [86][87][88] basis set. The extrema of the molecular electrostatic potentials (MEP) were measured on the 0.001 au isodensity surface using the MultiWFN program [89,90]. NBO analysis (NBO 7.0 [91]) probed the details of charge transfer and supplied natural atomic charges. Bader's AIM methodology [92] elucidated bond paths and quantitative measures of their strength via the AIMAll suite of programs [93]. The decomposition of interaction energies was carried out at the PBE0-D3/ZORA/TZ2P level of theory through the ADF-EDA procedure according to the Morokuma-Ziegler scheme embedded in ADF software [94][95][96]. The solid-state geometries were accessed through the Cambridge Structural Database (CSD, ver. 5.42 with updates) and supporting CCDC software, Mercury and ConQuest [97,98]. Theoretical computations were based on the NETMOO [84] crystal structure. Heavy atoms were fixed in their crystal coordinates, and the H atom positions optimized. The basis set superposition error (BSSE) was corrected via the standard counterpoise procedure [99].

Results
The geometry of the model system, taken directly from the X-ray coordinates, is exhibited in Figure 2a, which surrounds the PdCl4 2− dimer by four counterion ligands. The reader should be aware that a finite excerpt from a full crystal structure is considered here. There are a multitude of H-bonds connecting these counterions to the dianions. To avoid overcomplication of the figure, only very short ones, with R(H···Cl) less than 2.4 Å, are shown explicitly by the broken blue lines. Figure 2b focuses on the dianion dimer itself,

Computational Methods
Quantum calculations were carried out with the aid of the Gaussian 16, Rev. C.01 set of codes [85]. DFT computations employed the PBE0-D3 functional with the explicit inclusion of dispersion corrections, along with the def2TZVP [86][87][88] basis set. The extrema of the molecular electrostatic potentials (MEP) were measured on the 0.001 au isodensity surface using the MultiWFN program [89,90]. NBO analysis (NBO 7.0 [91]) probed the details of charge transfer and supplied natural atomic charges. Bader's AIM methodology [92] elucidated bond paths and quantitative measures of their strength via the AIMAll suite of programs [93]. The decomposition of interaction energies was carried out at the PBE0-D3/ZORA/TZ2P level of theory through the ADF-EDA procedure according to the Morokuma-Ziegler scheme embedded in ADF software [94][95][96]. The solid-state geometries were accessed through the Cambridge Structural Database (CSD, ver. 5.42 with updates) and supporting CCDC software, Mercury and ConQuest [97,98]. Theoretical computations were based on the NETMOO [84] crystal structure. Heavy atoms were fixed in their crystal coordinates, and the H atom positions optimized. The basis set superposition error (BSSE) was corrected via the standard counterpoise procedure [99].

Results
The geometry of the model system, taken directly from the X-ray coordinates, is exhibited in Figure 2a, which surrounds the PdCl 4 2− dimer by four counterion ligands. The reader should be aware that a finite excerpt from a full crystal structure is considered here. There are a multitude of H-bonds connecting these counterions to the dianions. To avoid overcomplication of the figure, only very short ones, with R(H···Cl) less than 2.4 Å, are shown explicitly by the broken blue lines. Figure 2b focuses on the dianion dimer itself, showing clearly that the Cl of the top unit approaches the Pd of the lower to within 3.217 Å.
showing clearly that the Cl of the top unit approaches the Pd of the lower to within 3.217 Å.

Direct Interactions between PdCl4 Units
The initial calculation focuses on the isolated PdCl4 2− dimer, in the absence of any counterions. The first row of Table 1 documents the strong repulsion between the two naked dianions. The interaction energy of the pair within their X-ray structure is +212 kcal/mol. Most of that can be attributed to a highly repulsive electrostatic component of +218 kcal/mol. Indeed, it would be difficult to generate any degree of attraction for the negatively charged Cl atom when the maximum of the MEP above the Pd atom is −371 kcal/mol, especially when coupled with the VS,min on the Cl of −387 kcal/mol. The next rows of Table 1 indicate the effects of adding four neutral ligands around this dianion pair. For the purposes of examining the interactions of the two principal dianions, two of these ligands were assigned to each PdCl4 unit to compose a [PdCl4] 2− L2 subunit. The MEP was computed for each [PdCl4] 2− L2 subunit, and the interaction energy between them was computed as the energy of the dimerization reaction (1) The Ar atoms were placed at the positions of the proximate N atoms of the NH3-(CH2)4-NH3 2+ counterions within the X-ray structure, as were the N atoms of the NH3 units. The H atoms of the latter were optimized and thus engaged in NH···Cl H-bonds with the anions. The Ar atoms have essentially no effect on the repulsive energy between the two anions, diminishing it by only 3 kcal/mol. The Ar atoms do reduce the negative value of VS,max, lowering its magnitude from −371 to −184 kcal/mol. They also strongly reduce the negative potential on Cl, lowering VS,min from −387 to −202 kcal/mol. However, the electrostatic component of the interaction is little changed, dropping from +218 to +206 kcal/mol. Nor does Ar absorb any of the negative charge of these anions, leaving their total charge at −2.00.
The H-bonds connected with NH3 have a larger impact, albeit still fairly small. VS,max drops a bit more, down to −172 kcal/mol, and VS,min is reduced as well, causing a drop in EES to +173 kcal/mol. The NH3 units absorb a small amount of density, leaving the charges on the PdCl4 dianions at −1.96. Nevertheless, the interaction energy remains high at +182 kcal/mol. Extending the NH3 units to the full NH2(CH2)4NH2 ligands, likewise capable of (a) (b) Distances in Å, angles in degs.

Direct Interactions between PdCl 4 Units
The initial calculation focuses on the isolated PdCl 4 2− dimer, in the absence of any counterions. The first row of Table 1 documents the strong repulsion between the two naked dianions. The interaction energy of the pair within their X-ray structure is +212 kcal/mol. Most of that can be attributed to a highly repulsive electrostatic component of +218 kcal/mol. Indeed, it would be difficult to generate any degree of attraction for the negatively charged Cl atom when the maximum of the MEP above the Pd atom is −371 kcal/mol, especially when coupled with the V S,min on the Cl of −387 kcal/mol. The next rows of Table 1 indicate the effects of adding four neutral ligands around this dianion pair. For the purposes of examining the interactions of the two principal dianions, two of these ligands were assigned to each PdCl 4 unit to compose a [PdCl 4 ] 2− L 2 subunit. The MEP was computed for each [PdCl 4 ] 2− L 2 subunit, and the interaction energy between them was computed as the energy of the dimerization reaction (1) The Ar atoms were placed at the positions of the proximate N atoms of the NH 3 -(CH 2 ) 4 -NH 3 2+ counterions within the X-ray structure, as were the N atoms of the NH 3 units. The H atoms of the latter were optimized and thus engaged in NH···Cl H-bonds with the anions. The Ar atoms have essentially no effect on the repulsive energy between the two anions, diminishing it by only 3 kcal/mol. The Ar atoms do reduce the negative value of V S,max , lowering its magnitude from −371 to −184 kcal/mol. They also strongly reduce the negative potential on Cl, lowering V S,min from −387 to −202 kcal/mol. However, the electrostatic component of the interaction is little changed, dropping from +218 to +206 kcal/mol. Nor does Ar absorb any of the negative charge of these anions, leaving their total charge at −2.00.
The H-bonds connected with NH 3 have a larger impact, albeit still fairly small. V S,max drops a bit more, down to −172 kcal/mol, and V S,min is reduced as well, causing a drop in E ES to +173 kcal/mol. The NH 3 units absorb a small amount of density, leaving the charges on the PdCl 4 dianions at −1.96. Nevertheless, the interaction energy remains high at +182 kcal/mol. Extending the NH 3 units to the full NH 2 (CH 2 ) 4 NH 2 ligands, likewise capable of engaging in NH···Cl H-bonds, has a further stabilizing effect. These longer species absorb a bit more of the anion's charge, reducing it to −1.92, and raises V S,max a small amount, up to −162 kcal/mol and also reducing the magnitude of V S,min . The electrostatic component and interaction energy are accordingly reduced as well, both down below +160 kcal/mol.
In order to distinguish the effects of this longer ligand arising from purely electrostatic considerations, from H-bonding, polarization, dispersion, and so on, these four NH 2 (CH 2 ) 4 NH 2 ligands were each replaced by a series of point charges. There was a oneto-one replacement of each atom of the ligand by such a charge, which was superimposed on the atomic position, and was assigned the natural charge equal to that of the ligand within the complex. The next row of Table 1 shows that this constellation of point charges has a small stabilizing effect on the dianion repulsion, less than that of the true ligand, and only roughly equivalent to the much smaller NH 3 molecule. Of course, as simply a collection of charges, these pseudoligands cannot absorb any charge, so that of each ligand remains at −2.00. So, it is clear that the H-bonds connected with the full ligands, as well as any charge which they can accept from the anions, have a significant effect on stabilizing the anion pair, albeit far too small to make this interaction attractive.
A second iteration of this analysis would involve placing a positive charge on each ligand. The simplest such counterion, and one incapable of engaging in a H-bond, would be a monatomic cation such as K + . As exhibited in the next row of Table 1, the inclusion of four such cations makes the interaction exothermic with negative values of E int . The presence of these cations also strongly reduces the negative values of both V S,max and V S,min , both smaller in magnitude than −80 kcal/mol. These changes are partly responsible for the negative, attractive electrostatic component at −94 kcal/mol.
The E int of −97 kcal/mol is enhanced to −111 kcal/mol if the K + is morphed into the NH 4 + ion of roughly the same size, but with the added capability of forming NH···Cl Hbonds. This mutation has a small reducing effect on the MEP extrema and drops the formal charge on the PdCl 4 unit to −1.80, adding to a slightly more negative E ES , which contributes to the more negative interaction energy. Enlarging the ammoniums to the much longer NH 2 (CH 2 ) 4 NH 3 + was done in such a way that it is the positively charged NH 3 + unit that is placed close to the PdCl 4 species. This counterion enlargement has a slightly deleterious effect, increasing E int from −111 to −101 kcal/mol, as well as dropping the electrostatic attraction energy by 5 kcal/mol. This rise in E int may be due to the lesser concentration of the positive charge in the larger cation. Each of these counterions, whether NH 4 + or NH 2 (CH 2 ) 4 NH 3 + , involves itself in two NH··Cl H-bonds for a total of four such bonds with each PdCl 4 unit. The replacement of the ligand atoms by their corresponding point charges is just slightly less effective, with E int becoming less exothermic by 3 kcal/mol.
It may be noted further from Table 1 that the electrostatic components are all quite attractive when any of these monocations are added, nearly −100 kcal/mol. On the other hand, even with these counterions present, V S,max on Pd remains negative by between 53 and 58 kcal/mol. This contradiction argues against taking a positive V S,max as a condition for an exothermic association. In sum, adding a +1 charge to the surrounding ligands enables them to absorb a bit more of the (PdCl 4 ) 2− dianion's negative charge. Most effective in this regard is the set of four ammonium cations which drop the dianion's charge down to −1.80.
The last few rows of Table 1 refer to the addition of two dications instead of the four monocations. (The smaller number of the former is necessary in order to maintain overall electroneutrality. Equation (1) must be modified to describe each subunit as [PdCl 4 ] 2− L 2+ ). Despite the reduction in their number, the dications prove more effective at promoting a more exothermic association. Whether the small monatomic Ca 2+ or the much larger NH 3 (CH 2 ) 4 NH 3 2+ dications, E int drops below −120 kcal/mol. Even the collection of point charges designed to mimic the long ligand dication is effective in this regard, with E int of -108 kcal/mol. The comparison of the Ca 2+ and the L 2+ systems enables some assessment of H-bonds, which are only possible for the latter. It is intriguing that although the Hbonding ligands leave both V S,max and V S,min more negative, the total electrostatic term is nonetheless more attractive when compared to Ca 2+ .
In fact, upon moving the analysis to the dications, the electrostatic component diverges substantially from the full E int . These two quantities differ by some 60-80 kcal/mol. More specifically, while the transition from monatomic K + monocation to Ca 2+ dication makes E int more negative, and the same sort of change can be seen from L + to L 2+ , it has the opposite effect on E ES , which becomes substantially less negative. This change to a less attractive electrostatic term contrasts with the less negative V S,max quantities associated with the monatomic ions. Despite their smaller number, the Ca 2+ dications are much better at dispersing the negative charge on the central dianions than K + , dropping this charge down to −1.73. There is much less distinction between the longer ligands, where the dications are slightly poorer at absorbing this charge than the monocations.

Secondary Interactions
It must be understood that the interaction energy between the two subunits is not wholly due to the Pd···Cl bond. The electrostatic term arises from interactions between the entire charge distributions of each subunit, which includes not only the central PdCl 4 but also any ligands appended to it. There are also polarization and dispersion energies that involve the entire electron clouds. Added to that are a number of specific noncovalent contacts as well. For example, the NH groups on the small NH 3 and NH 4 + entities on one subunit can engage in H-bonds with the Cl atoms of the PdCl 4 of the other, but also NH··N H-bonds with one another. The same is true of the larger ligands comprised of NH and CH proton donors. Even the monatomic counterions, such as K + and Ca 2+ , are capable of forming specific bonding contacts with the Cl atoms of the opposite subunit.
One can elucidate such interactions via the examination of AIM bond paths. In order to convey some sense of the number of these bonds, the AIM diagram of the system containing four full monocationic ligands is provided in Figure 3. There is a multitude of H-bonds and other noncovalent interactions between these ligands and both PdCl 4 units, and even between one another. The inset to Figure 3 focuses on the dianion pair and one of these ligands for greater clarity. Figure 4 places clearly in evidence the various H-bonds that arise when these ligands are replaced by the smaller NH 4 + , NH 3 , and K + species. The chief markers of the strengths of these various interactions are contained in Table 2. The first column displays the density of the Pd···Cl bond path between the two subunits, which seems to be relatively constant at 0.013 au. This nearly fixed amount is not surprising in view of the fact that the intermolecular Pd···Cl distance was held constant at its X-ray value regardless of the addition of any ligands, and ρ BCP has been shown repeatedly in the literature [12,100,101] to be very sensitive to this interatomic distance. Prior works in the literature [102] have found and used a relationship that ties the energy of a noncovalent bond to 1 2 V, where V represents the potential energy density at the bond critical point. The use of this relationship leads to an estimate of the Pd··Cl bond energy as roughly 3 kcal/mol, as listed in the penultimate column of Table 2. of these ligands for greater clarity. Figure 4 places clearly in evidence the various H-bonds that arise when these ligands are replaced by the smaller NH4 + , NH3, and K + species. The chief markers of the strengths of these various interactions are contained in Table 2. The first column displays the density of the Pd···Cl bond path between the two subunits, which seems to be relatively constant at 0.013 au. This nearly fixed amount is not surprising in view of the fact that the intermolecular Pd···Cl distance was held constant at its X-ray value regardless of the addition of any ligands, and ρBCP has been shown repeatedly in the literature [12,100,101] to be very sensitive to this interatomic distance. Prior works in the literature [102] have found and used a relationship that ties the energy of a noncovalent bond to ½ V, where V represents the potential energy density at the bond critical point. The use of this relationship leads to an estimate of the Pd··Cl bond energy as roughly 3 kcal/mol, as listed in the penultimate column of Table 2.   of these ligands for greater clarity. Figure 4 places clearly in evidence the various H-bonds that arise when these ligands are replaced by the smaller NH4 + , NH3, and K + species. The chief markers of the strengths of these various interactions are contained in Table 2. The first column displays the density of the Pd···Cl bond path between the two subunits, which seems to be relatively constant at 0.013 au. This nearly fixed amount is not surprising in view of the fact that the intermolecular Pd···Cl distance was held constant at its X-ray value regardless of the addition of any ligands, and ρBCP has been shown repeatedly in the literature [12,100,101] to be very sensitive to this interatomic distance. Prior works in the literature [102] have found and used a relationship that ties the energy of a noncovalent bond to ½ V, where V represents the potential energy density at the bond critical point. The use of this relationship leads to an estimate of the Pd··Cl bond energy as roughly 3 kcal/mol, as listed in the penultimate column of Table 2.    The other two columns of Table 2 report the bond critical point density and potential energy density as a sum of all bond paths that stretch between the two subunits, exclusive of Pd···Cl. These values suggest that the Pd···Cl bond is only part of the story, and that in a number of cases, AIM would suggest that the sum of the other noncovalent interactions exceeds this primary component. This energy sum from the last column of Table 2 shows a clear progression from monatomic species, such as Ar and K + , to the small H-bonding NH 3 and NH 4 + , up to the largest ligands containing the connecting butyl chain. Notice also that this auxiliary sum drops off for the dicationic species. Given the magnitude of the collective auxiliary noncovalent bonds within these complexes, it is perhaps not surprising that neither the total interaction energy nor even the full electrostatic term in Table 1 can always be closely related to the magnitudes of the MEP extrema on the Pd and Cl atoms, which concern only one of several noncovalent bonds. The ability of the counterions within the crystal structure to stabilize the entire lattice is mainly due to their effects on the PdCl 4 units. Moreover, these ligands also act as a glue between PdCl 4 units by forming H-bonds with both. This glue is further augmented by H-bonds between the counterions themselves.
As the electrostatic is the leading force stabilizing the current crystal, it is worth comparing the bonding between adjacent, oppositely charged atoms here with that which occurs within a common salt such as NaCl. The atoms of the [PdCl 4 ] 2− dianions that come closest to one another are the Cl of one unit and the Pd of the other. This R(Pd···Cl) distance is 3.217 Å in the system described above, which is considerably shorter than 3.97 Å which corresponds to the sum of Pd and Cl vdW radii [103]. In NaCl, the R(Na···Cl) distance is only 2.8 Å in the crystal, also smaller than their vdW radii sum of 4.3 Å, so in this sense NaCl offers local attractive behavior that is parallel to that in the system under investigation here. If one now extracts a structure similar to the crystallographic arrangement of two [PdCl 4 ] 2− from the NaCl lattice, i.e., a [NaCl 4 ] 3− ···[NaCl 4 ] 3− arrangement on lattice sites, it will become repulsive between the two units, just like in the Pd case. Adding neutral solvents will not cure this, but adding counterions will. So, qualitatively, a simple salt crystal will behave similarly to the Pd dianion system upon increasing the fragments investigated. Of course, the covalency of the PdCl bonds in the dianions along with the presence of the hydrogen bonds make a difference, but only on a quantitative level, which is explored in this work.

Conclusions
The interaction between the two naked PdCl 4 2− dianions is clearly highly repulsive. The introduction of neutral ligands reduces the magnitudes of the negative MEP maxima and minima, which helps lower the electrostatic repulsion energy, but the interaction en-ergy remains quite positive nonetheless, roughly equal to its total electrostatic component. Adding a positive charge to the ligands further reduces the magnitudes of the MEP extrema, although they remain negative. Nevertheless, these cations reverse the sign of the electrostatic and interaction energies, turning the latter exothermic by roughly 100 kcal/mol. Changing from four monocationic ligands to two dications reduces the total electrostatic attractive component but makes the total interaction energy a bit more exothermic. The stabilizing effect of the counterions is only partly due to the dispersal of the negative charges on the PdCl 4 2− units or the reduction of the negative value of the π-hole on Pd. In a more global sense, the addition of the cationic ligands changes the formal charge on the entire subunit from −2 for naked PdCl 4 2− unit to 0 after their introduction. The ability of these counterions to engage in H-bonds with both PdCl 4 units further acts as a glue holding them together. A partial contribution to this structure cohesion is achieved via NH···Cl hydrogen bonds.

Data Availability Statement:
The data presented in this study are available on request from the corresponding authors.