*3.2. 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 PdCl4 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 NH3 and NH4 <sup>+</sup> entities on one subunit can engage in H-bonds with the Cl atoms of the PdCl4 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 Ca2+, 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 PdCl4 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 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 <sup>1</sup> <sup>2</sup> 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.

**Figure 3.** The AIM diagram of [NH3-(CH2)4-NH2]4[PdCl4]2. Numbers refer to bond path critical point in au.

**Figure 4.** The AIM diagram of dianion surrounded by (**a**) 4 NH4 +, (**b**) 4 NH3, and (**c**)4K+.

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 NH3 and NH4 +, up to the largest ligands containing the connecting butyl chain. Notice also that this auxiliary sum drops off for the dicationic species.


**Table 2.** AIM properties of bond critical points between subunits in complexes.

<sup>a</sup> between [PdCl4] <sup>2</sup>−Ln units, *n* = 2 for neutral and monocationic ligands, 1 for dications. <sup>b</sup> PC refers to the constellation of point charges that approximate L0, L+, or L2+, respectively.

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 PdCl4 units. Moreover, these ligands also act as a glue between PdCl4 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 [PdCl4] <sup>2</sup><sup>−</sup> 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 [PdCl4] <sup>2</sup><sup>−</sup> from the NaCl lattice, i.e., a [NaCl4] <sup>3</sup>−···[NaCl4] <sup>3</sup><sup>−</sup> 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.

### **4. Conclusions**

The interaction between the two naked PdCl4 <sup>2</sup><sup>−</sup> 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 PdCl4 <sup>2</sup><sup>−</sup> 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 PdCl4 <sup>2</sup><sup>−</sup> unit to 0 after their introduction. The ability of these counterions to engage in H-bonds with both PdCl4 units further acts as a glue holding them together. A partial contribution to this structure cohesion is achieved via NH···Cl hydrogen bonds.

**Author Contributions:** Conceptualization, M.M, W.Z. and R.W.; data curation, R.W. and M.M.; supervision, S.S.; visualization, R.W., S.S., T.M. and M.M.; writing—original draft, R.W. and M.M.; writing—review & editing, S.S., R.W., W.Z., T.M. and M.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Polish Ministry of Science and Higher Education for the Faculty of Chemistry of Wroclaw University of Science and Technology under Grant No. 8211104160/K19W03D10 and by the US National Science Foundation under Grant No. 1954310.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

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

**Acknowledgments:** A generous allotment of computer time from the Wroclaw Supercomputer and Networking Center is acknowledged.

**Conflicts of Interest:** There are no conflict to declare.

**Sample Availability:** Samples of the compounds are not available from the authors.
