Interaction between Trinuclear Regium Complexes of Pyrazolate and Anions, a Computational Study

The geometry, energy and electron density properties of the 1:1, 1:2 and 1:3 complexes between cyclic (Py-M)3 (M = Au, Ag and Cu) and halide ions (F−, Cl− and Br−) were studied using Møller Plesset (MP2) computational methods. Three different configurations were explored. In two of them, the anions interact with the metal atoms in planar and apical dispositions, while in the last configuration, the anions interact with the CH(4) group of the pyrazole. The energetic results for the 1:2 and 1:3 complexes are a combination of the specific strength of the interaction plus a repulsive component due to the charge:charge coulombic term. However, stable minima structures with dissociation barriers for the anions indicate that those complexes are stable and (Py-M)3 can hold up to three anions simultaneously. A search in the CSD confirmed the presence of (Pyrazole-Cu)3 systems with two anions interacting in apical disposition.

More recently, the possibility to find σ-hole regions within atoms of column 11 of the periodic table (the coinage metals Cu, Ag and Au) in organometallic molecules or small nanoclusters was described [22][23][24][25]. However, only a limited number of complexes between derivatives of coinage atoms acting as Lewis acids and electron donors were explored in the literature [26]. This interaction was named the regium bond or metal-coinage bond [27]. However, it is worth noting that gold is an inert metal in bulk but in small nanostructures or in organometallic clusters it becomes a powerful catalyst [28,29].
Focusing on complexes with gold derivatives, the structures of the OC· · · AuX (with X = F, Cl and Br) [30], H 2 S· · · AuI [31] and H 2 · · · AuCl [32] complexes were determined by microwave spectroscopy. Additionally, the interaction of iodoperfluorobenzene derivatives with gold nanoparticles was investigated by Obenchain et al. using different spectroscopic techniques [33]. Besides, the NMR On a different topic, interaction between charged systems, i.e., anion-anion and cation-cation are very interesting, and were found both within the gas phase [69][70][71][72][73][74][75][76][77][78][79] and crystal structures [80,81]. This rare type of interaction, which, in principle should be repulsive, should lead to unstable complexes, i.e., they will revert into the separated ions. However, in the last decade, it was demonstrated that complexes exhibiting positive interaction energies are indeed minima and therefore stable structures [82,83]. Some examples can be found in the literature in which two charged conducting spheres can attract each other when they are in a close range distance [84]. The F···F interaction in negatively charged dimers was studied at MP2 level and using symmetry-adapted perturbation theory [85,86].
In the present work we will focus our efforts in the study of trinuclear regium pyrazolate systems interacting with one, two and three anions simultaneously to infer whether regium bonds or hydrogen bonds can be established and stabilize structures, which in principle should not be stable. We will investigate the interaction of (Pz-M) 3

Isolated (Pz-M)3 Monomers
The isolated (Pz-M)3 compounds (M = Cu, Ag and Au) were optimized at the MP2/aug'-cc-pVDZ computational level. All three of them showed a D3h symmetry with each metal atom (M) located equidistantly between the two nitrogen atoms of the adjacent pyrazole rings. In On a different topic, interaction between charged systems, i.e., anion-anion and cation-cation are very interesting, and were found both within the gas phase [69][70][71][72][73][74][75][76][77][78][79] and crystal structures [80,81]. This rare type of interaction, which, in principle should be repulsive, should lead to unstable complexes, i.e., they will revert into the separated ions. However, in the last decade, it was demonstrated that complexes exhibiting positive interaction energies are indeed minima and therefore stable structures [82,83]. Some examples can be found in the literature in which two charged conducting spheres can attract each other when they are in a close range distance [84]. The F· · · F interaction in negatively charged dimers was studied at MP2 level and using symmetry-adapted perturbation theory [85,86].
In the present work we will focus our efforts in the study of trinuclear regium pyrazolate systems interacting with one, two and three anions simultaneously to infer whether regium bonds or hydrogen bonds can be established and stabilize structures, which in principle should not be stable. We will investigate the interaction of (Pz-M) 3  On a different topic, interaction between charged systems, i.e., anion-anion and cation-cation are very interesting, and were found both within the gas phase [69][70][71][72][73][74][75][76][77][78][79] and crystal structures [80,81]. This rare type of interaction, which, in principle should be repulsive, should lead to unstable complexes, i.e., they will revert into the separated ions. However, in the last decade, it was demonstrated that complexes exhibiting positive interaction energies are indeed minima and therefore stable structures [82,83]. Some examples can be found in the literature in which two charged conducting spheres can attract each other when they are in a close range distance [84]. The F···F interaction in negatively charged dimers was studied at MP2 level and using symmetry-adapted perturbation theory [85,86].
In the present work we will focus our efforts in the study of trinuclear regium pyrazolate systems interacting with one, two and three anions simultaneously to infer whether regium bonds or hydrogen bonds can be established and stabilize structures, which in principle should not be stable. We will investigate the interaction of (Pz-M)3, M = Cu, Ag and Au, with three anions (F − , Cl − and Br − ) limiting to the lowest oxidation states (I) of the three coinage metals ( Figure 2).

Isolated (Pz-M)3 Monomers
The isolated (Pz-M)3 compounds (M = Cu, Ag and Au) were optimized at the MP2/aug'-cc-pVDZ computational level. All three of them showed a D3h symmetry with each metal atom (M) located equidistantly between the two nitrogen atoms of the adjacent pyrazole rings. In

Isolated (Pz-M) 3 Monomers
The isolated (Pz-M) 3 compounds (M = Cu, Ag and Au) were optimized at the MP2/aug'-cc-pVDZ computational level. All three of them showed a D 3h symmetry with each metal atom (M) located equidistantly between the two nitrogen atoms of the adjacent pyrazole rings. In fact, the calculated M-N distances are 1.817, 2.039 and 1.970, Å for the copper, silver and gold derivatives, respectively.
To evaluate the areas of possible electrophilic attack, the molecular electrostatic potential (MESP) was calculated and plotted on the 0.001 au electron density isosurface in Figure 3. MESP showed negative (red) regions above and below of the pyrazole rings while the positive (blue) regions were associated to the hydrogen atoms in the periphery. The values were associated with four stationary points one minimum (over the pyrazole ring) and three maxima: two corresponding to the C-H bonds and one over the center of the system (C 3 axis), for each of the three systems were also indicated in Figure 3 by their corresponding value. It is interesting to notice the dependence of the MESP sign along the C 3 axis with the metal considered. While for the Au derivative the maximum shows a negative value (−22 kJ/mol), it becomes positive for silver (+28 kJ/mol) and very small and negative for copper (−5 kJ/mol) derivatives. Regarding the maxima associated to the CH groups of the pyrazole, in all cases the CH(4) exhibits is less positive MESP values than the CH(3) one, being the values of the former between 70 and 77 kJ/mol and those for the latter between 97 and 80 kJ/mol. To evaluate the areas of possible electrophilic attack, the molecular electrostatic potential (MESP) was calculated and plotted on the 0.001 au electron density isosurface in Figure 3. MESP showed negative (red) regions above and below of the pyrazole rings while the positive (blue) regions were associated to the hydrogen atoms in the periphery. The values were associated with four stationary points one minimum (over the pyrazole ring) and three maxima: two corresponding to the C-H bonds and one over the center of the system (C3 axis), for each of the three systems were also indicated in Figure 3 by their corresponding value. It is interesting to notice the dependence of the MESP sign along the C3 axis with the metal considered. While for the Au derivative the maximum shows a negative value (−22 kJ/mol), it becomes positive for silver (+28 kJ/mol) and very small and negative for copper (−5 kJ/mol) derivatives. Regarding the maxima associated to the CH groups of the pyrazole, in all cases the CH(4) exhibits is less positive MESP values than the CH(3) one, being the values of the former between 70 and 77 kJ/mol and those for the latter between 97 and 80 kJ/mol.

1:1 Complexes
We began by studying the complexes established between a (Pz-M)3 unit and a single halide anion. All the molecular graphs have been included in Tables S1-S3. Three different energetic minima were found for each coinage metal derivative ( Figure 4): (a) 1:1 apical, where the anion is located along the C3 symmetry axes and interacting simultaneously with the three metal atoms, (b) 1:1 planar, where the anion is within the molecular plane and simultaneously interacting with a metal atom and two CH(3) groups and (c) 1:1 CH(4), in which the anion interacts directly with a single hydrogen atom from the CH(4) group.

1:1 Complexes
We began by studying the complexes established between a (Pz-M) 3 unit and a single halide anion. All the molecular graphs have been included in Tables S1-S3. Three different energetic minima were found for each coinage metal derivative ( The binding energies, Eb, obtained as the difference of the energy of the complex minus the energy of the isolated monomers, for the 1:1 complexes were gathered in Table 1 and plotted in Figure 5. All the binding energies are shown to be large and negative as expected for the interaction between an anion with a neutral molecule, ranging between −204 and −35 kJ mol −1 . The most stable complex of each configuration highly depends on the metal atom and on the anion considered. For instance, 1:1 apical complexes were found to be the most stable for silver and copper derivatives, while for gold derivatives 1:1 planar complexes are the most stable. The only exception is the copper complex with Fwhere the planar complex is even more stable than the apical one. In all cases, the complexes showing only the interaction with the CH(4) group are the least stables ones. Considering the same configuration, 1:1 apical and planar complexes, the most stable complex for a given anion corresponds to the silver derivative, followed by the copper and the gold derivative. The only exception corresponds to the 1:1 planar complexes with Fwhere the copper complex is more stable than the silver one and the trend is Au < Ag < Cu. Similar trends, involving the metal atom, were already described in the literature for M···O interactions [87] and ethylene coinage metal complexes [88]. For all of the different 1:1 CH(4) complexes, in order of the most to least stable derivative we found gold, followed by copper and lastly silver. The binding energies, E b , obtained as the difference of the energy of the complex minus the energy of the isolated monomers, for the 1:1 complexes were gathered in Table 1 and plotted in Figure 5. All the binding energies are shown to be large and negative as expected for the interaction between an anion with a neutral molecule, ranging between −204 and −35 kJ mol −1 . The most stable complex of each configuration highly depends on the metal atom and on the anion considered. For instance, 1:1 apical complexes were found to be the most stable for silver and copper derivatives, while for gold derivatives 1:1 planar complexes are the most stable. The only exception is the copper complex with Fwhere the planar complex is even more stable than the apical one. In all cases, the complexes showing only the interaction with the CH(4) group are the least stables ones.  Concerning the geometrical parameters, the M···X -distances within the 1:1 apical and planar configurations were influenced by the size of the metal (Au >Ag > Cu) and the anion (F < Cl < Br) considered (Table 2). Focusing on configurations for the same metal and anion, in all the cases the M-X distance in the planar configuration was about 0.12 Å shorter than the apical one. This can be due to the fact that in the apical configuration, the anion was simultaneously interacting with three metals and thus the interaction was weakening (as observed for the binding energies) and therefore the M···X − distance became longer. In the case of planar configuration, the anion was interacting with only one metal, plus there was a geometrical constraint due to the interactions with the CH(3) Considering the same configuration, 1:1 apical and planar complexes, the most stable complex for a given anion corresponds to the silver derivative, followed by the copper and the gold derivative.
The only exception corresponds to the 1:1 planar complexes with F − where the copper complex is more stable than the silver one and the trend is Au < Ag < Cu. Similar trends, involving the metal atom, were already described in the literature for M· · · O interactions [87] and ethylene coinage metal complexes [88]. For all of the different 1:1 CH(4) complexes, in order of the most to least stable derivative we found gold, followed by copper and lastly silver.
Concerning the geometrical parameters, the M· · · X − distances within the 1:1 apical and planar configurations were influenced by the size of the metal (Au >Ag > Cu) and the anion (F < Cl < Br) considered (Table 2). Focusing on configurations for the same metal and anion, in all the cases the M-X distance in the planar configuration was about 0.12 Å shorter than the apical one. This can be due to the fact that in the apical configuration, the anion was simultaneously interacting with three metals and thus the interaction was weakening (as observed for the binding energies) and therefore the M· · · X − distance became longer. In the case of planar configuration, the anion was interacting with only one metal, plus there was a geometrical constraint due to the interactions with the CH(3) groups. Regarding the CH(4) configurations, the H· · · X − distances found were shorter than those found for the CH(3)· · · X − ones. The dependence on H(3)· · · X distances for a given anion change with the metal atom shows the following trend: Ag > Au > Cu, while for the H(4)· · · X − it is Ag > Cu > Au. Table 2. Geometrical intermolecular M· · · X − and H· · · X − distances (Å) in the 1:1 complexes. No clear relationships between the binding energies and intermolecular distances were found for any of the complexes, except for the CH(4) complexes. This can highlight the complexity of the M· · · X − interaction. The lack of correlation between those quantities can be associated with the electronic repulsion between the anion and the rest of the atoms in the (Pz-M) 3 system. Additionally, the number of simultaneous interactions acting in 1:1 apical and planar configurations can play a role deviating the correlation between E b and M· · · X − distances.
Finally, the QTAIM analysis of the electron density ( Figure 4 and Tables S1-S3) indicates the presence of three symmetrical bond paths in the apical configuration connecting the X − anion and the three metal atoms. In the planar configuration, another three bond paths were found too, but in this case only one of them connects the anion with the metal atom while the other two corresponded to CH(3)· · · X − interactions.
Regarding, CH(4) configuration only one bond path was found between the anion and the CH(4) group. The electron density values of the intermolecular bond critical points (BCP ; Table S4) present positive values of the Laplacian and negative values of the total energy density for the anion-metal bonds in the apical and planar configurations as an indication of the partial covalent nature of the interaction [89,90].

1:2 Complexes
After analyzing the 1:1 complexes and their binding energies, the arising question was: can those negatively charged 1:1 complexes interact with another anion and produce stable structures? To address this question, three different 1:2 configurations were considered using the 1:1 complexes configurations as parental structures, i.e., 1:2 apical, 1:2 planar and 1:2 CH(4). There were a large number of possible combinations for the second anion to interact with the 1:1 complex, but for the sake of simplicity only two similar interactions will be explored simultaneously. In Figure 6 the structures corresponding to the 1:2 (Pz-M) 3 : Br 2 complex were depicted as an illustrative example. In all the cases, the 1:2 apical complexes show D 3h symmetry while the planar and CH(4) ones present C 2v symmetry.
To address this question, three different 1:2 configurations were considered using the 1:1 complexes configurations as parental structures, i.e., 1:2 apical, 1:2 planar and 1:2 CH(4). There were a large number of possible combinations for the second anion to interact with the 1:1 complex, but for the sake of simplicity only two similar interactions will be explored simultaneously. In Figure 6 the structures corresponding to the 1:2 (Pz-M)3: Br2 complex were depicted as an illustrative example. In all the cases, the 1:2 apical complexes show D3h symmetry while the planar and CH(4) ones present C2v symmetry. The binding energies for the 1:2 complexes (Table 3) varied from configuration to configuration ranging from +140 to −82 kJ mol −1 . In the case of complexes with the largest halogen anions, Br − and Cl − , the binding energies are found to be positive indicating a repulsive force within the complex. Only one exception was found for those anions, the 1:2 planar complex with silver were the binding energies were negative (−24 and −26 kJ mol −1 , respectively). In contrast, the complexes with F − present negative binding energies save for the 1:2 apical complexes with gold, which was positive (+113 kJ mol −1 ). As occurred for the 1:1 complexes, for each anion and configuration, the most stable complex corresponded to the silver one, except for the 1:2 planar and CH(4) configuration with F − where the 1:2 planar with copper and gold were a more stable complex.
However, the existence of a stable minimum with positive values of the binding energy resembles to those minima in complexes between molecules with the same charge (anion-anion or cation-cation). When an energy scan corresponding to the separation of one of the anions from the rest of the system (X−···Pz-M-X−) in the apical complexes is done, it is observed a maximum in the potential energy surface that prevents the spontaneous dissociation of the anion ( Figure S1 and Table S5). This fact has been previously described for anion-anion and cation-cation complexes in the literature [69,82,83]. The largest barriers were found for the F− anion, reaching up to 96 kJ mol−1 for the 1:2 (Pz-Cu)3:F2. In contrast, the dissociation barriers for the 1:2 (Pz-Au)3:Cl2. and 1:2 (Pz-Au)3:Br2 were found to be small, 9 and 7 kJ mol−1, respectively. The energetic differences within the 1:2 complexes were clearly influenced by the electronic repulsion between the two X-anions, which was essentially related to their X···X interatomic distance (Table S6). The shortest interatomic distances in the anions were found in the 1:2 apical complexes (between 2.7 and 5.5 Å), followed by the 1:2 planar complexes (between 7.4 and 12.2 Å) being the longest distances in the 1:2 CH(4) ones (between 13.2 and 16.0 Å), this means that the repulsive interaction between anions within the 1:2 apical configuration will be larger than those in the planar and CH(4) configuration. This is clearly observed in the binding energies for apical and planar configurations. However, when CH(4) complexes were taken into account, they did not follow the trend. This can be explained in terms of the type of interaction involved, since the binding The binding energies for the 1:2 complexes (Table 3) varied from configuration to configuration ranging from +140 to −82 kJ mol −1 . In the case of complexes with the largest halogen anions, Br − and Cl − , the binding energies are found to be positive indicating a repulsive force within the complex. Only one exception was found for those anions, the 1:2 planar complex with silver were the binding energies were negative (−24 and −26 kJ mol −1 , respectively). In contrast, the complexes with F − present negative binding energies save for the 1:2 apical complexes with gold, which was positive (+113 kJ mol −1 ). As occurred for the 1:1 complexes, for each anion and configuration, the most stable complex corresponded to the silver one, except for the 1:2 planar and CH(4) configuration with F − where the 1:2 planar with copper and gold were a more stable complex. However, the existence of a stable minimum with positive values of the binding energy resembles to those minima in complexes between molecules with the same charge (anion-anion or cation-cation). When an energy scan corresponding to the separation of one of the anions from the rest of the system (X − · · · Pz-M-X − ) in the apical complexes is done, it is observed a maximum in the potential energy surface that prevents the spontaneous dissociation of the anion ( Figure S1 and Table S5). This fact has been previously described for anion-anion and cation-cation complexes in the literature [69,82,83]. The largest barriers were found for the F − anion, reaching up to 96 kJ mol−1 for the 1:2 (Pz-Cu) 3 :F 2 . In contrast, the dissociation barriers for the 1:2 (Pz-Au) 3 :Cl 2 . and 1:2 (Pz-Au) 3 :Br 2 were found to be small, 9 and 7 kJ mol−1, respectively.
The energetic differences within the 1:2 complexes were clearly influenced by the electronic repulsion between the two X − anions, which was essentially related to their X· · · X interatomic distance ( Table S6). The shortest interatomic distances in the anions were found in the 1:2 apical complexes (between 2.7 and 5.5 Å), followed by the 1:2 planar complexes (between 7.4 and 12.2 Å) being the longest distances in the 1:2 CH(4) ones (between 13.2 and 16.0 Å), this means that the repulsive interaction between anions within the 1:2 apical configuration will be larger than those in the planar and CH (4) configuration. This is clearly observed in the binding energies for apical and planar configurations. However, when CH(4) complexes were taken into account, they did not follow the trend. This can be explained in terms of the type of interaction involved, since the binding energy depended both on the type/strength of the interaction involved plus the electronic repulsion between the anions. It is clear that while in the CH(4) the repulsion between the anions would be very small, the interaction C-H· · · X was also very weak.
One way to analyze the repulsion, or in other words, to narrow down the binding energy contribution of the anion-anion repulsion is to correct the binding energy by subtracting the charge-charge repulsion using the location of the anions with a charge of -1e [82]. As observed for the value in Table S7, corrected binding energies were in all the cases negative, which was aligned with the existence of those minima. However, in planar and CH(4) configurations, these values were smaller, in the absolute value, than twice of those for the 1:1 complexes, as expected for two identical charge transfer interactions over a neutral molecule. In contrast, in most of the 1:2 apical complexes the corrected energy was larger (in absolute value) than twice the 1:1 complexes binding energies, highlighting the importance of the charge transfer and polarization in these complexes.
Regarding the interatomic M· · · X and H· · · X distances (Table 4) in the 1:2 complexes, those were found to be slightly longer than the ones found for 1:1 complexes. This was consistent with the fact that the anions tend to separate from each other (due to Coulombic repulsion) while maintaining the same interaction patterns shown in the 1:1 complexes. The only exception corresponded to the planar 1:2(Pz-Au) 3 :F 2 where each anion exclusively interacted with one of the H(3) atoms as shown in its molecular graph (Table S1). Table 4. Intermolecular M· · · X − and H· · · X − distances (Å) in the 1:2 complexes. Regarding the QTAIM analysis, the electron density properties (Table S8) show similar bond paths between the anions and the (Pz-M) 3 systems in 1:2 complexes than those found for 1:1 complexes, with the aforementioned exception of the planar 1:2(Pz-Au) 3 :F 2 complex. Another interesting feature of the QTAIM analysis in 1:2(Pz-Au) 3 :F 2 complex was the presence of a bond path linking the two F − anions.
As it is clear from the discussed result, the (Pz-Au) 3 was capable of forming stable complexes with two anions simultaneously, but what is the limit of those complexes? Can 1:2(Pz-M) 3 :X 2 form stable complexes with a third anion?

1:3 Complexes
Following the same premises as that of 1:2 complexes, 1:3 complexes for the planar and CH(4) configuration were explored. Energetic minima structures were found for all the 1:3 CH(4) complexes and also for four of the nine 1:3 planar complexes. However, for the rest of the possible complexes, the anions dissociated spontaneously from the (Pz-M) 3 molecule. In those stable cases, all the systems present D 3h symmetry. Two illustrative examples are shown in Figure 7.
As it is clear from the discussed result, the (Pz-Au)3 was capable of forming stable complexes with two anions simultaneously, but what is the limit of those complexes? Can 1:2(Pz-M)3:X2 form stable complexes with a third anion?

1:3 Complexes
Following the same premises as that of 1:2 complexes, 1:3 complexes for the planar and CH(4) configuration were explored. Energetic minima structures were found for all the 1:3 CH(4) complexes and also for four of the nine 1:3 planar complexes. However, for the rest of the possible complexes, the anions dissociated spontaneously from the (Pz-M)3 molecule. In those stable cases, all the systems present D3h symmetry. Two illustrative examples are shown in Figure 7. The binding energies for the 1:3 complexes are gathered in Table 5. As occurred for 1:2 complexes, 1:3 complexes exhibit positive binding energies ranging between 160 and 260 kJ mol −1 , much larger than those for the 1:2 complexes, although the structures found are still minima in the potential energy surface. Binding energies suggest that CH(4) configurations were more stable than the corresponding planar ones, while the other way around happened for the 1:2 complexes. In fact, in the latter, the shorter X − ···X − distances (Table S9) in the planar configuration (between 7.5 and 9.4 Å) penalize the binding energy versus those in CH(4) configuration where the anions were further away (between 13.2 and 15.6 Å). As for the 1:2 cases, removing the X − ···X − electrostatic repulsion term (Table S10) provided with a better picture of the binding energies. Corrected Eb values were negative but smaller than three times the binding energies found for the 1:1 complexes indicating a certain degree of anti-cooperativity in the 1:3 complexes.  The binding energies for the 1:3 complexes are gathered in Table 5. As occurred for 1:2 complexes, 1:3 complexes exhibit positive binding energies ranging between 160 and 260 kJ mol −1 , much larger than those for the 1:2 complexes, although the structures found are still minima in the potential energy surface. Binding energies suggest that CH(4) configurations were more stable than the corresponding planar ones, while the other way around happened for the 1:2 complexes. In fact, in the latter, the shorter X − · · · X − distances (Table S9) in the planar configuration (between 7.5 and 9.4 Å) penalize the binding energy versus those in CH(4) configuration where the anions were further away (between 13.2 and 15.6 Å). As for the 1:2 cases, removing the X − · · · X − electrostatic repulsion term (Table S10) provided with a better picture of the binding energies. Corrected E b values were negative but smaller than three times the binding energies found for the 1:1 complexes indicating a certain degree of anti-cooperativity in the 1:3 complexes. Regarding the intermolecular M· · · X and H· · · X distances, selected geometrical parameters of the 1:3 minima are listed in Table 6. The comparison of these parameters with the corresponding in the 1:1 and 1:2 complexes ( Table 1; Table 3), indicates that within the (Pz-Ag) 3 X − planar configuration there was a lengthening in the M· · · X − distance from 1:3 vs. the 1:2 (∆R M· · · X = 0.084, 0.291 and 0.389 Å for F − , Cl − and Br − respectively), which was larger than the difference between 1:2 and 1:1(Pz-Ag) 3 X − complexes (0.071, 0.111 and 0.127 Å). This is an indication of a larger anti-cooperativity effect in the former than in the later complexes. Additionally, these effects were more acute for heavier halogens than for lighter ones. In case of CH(4) configuration, a similar pattern was found but the increment was milder across the halogen series. Curiously, the metal M also has influence on the H· · · X − along the 1:1, 1:2 and 1:3 series for the CH(4) configuration, with a much larger increase for the (Pz-Cu) 3 X − > (Pz-Ag) 3 X − > (Pz-Au) 3 X − . The molecular graph of the 1:3 planar complexes show three BCP between each anion and the (Pz-M) 3 system, two of them with the CH(3) and one with the metal. In the case of the CH complexes a single bond path per anion was found. The BCPs (Table S11) exhibited similar electron density characteristics to those found for the 1:1 and 1:2 complexes.
A general analysis of all the BCPs extracted from this article shows excellent correlations between the electron density and the interatomic distance for each pair of atoms involved in the interactions ( Figure S2). These results were in agreement with previous reports that have shown similar relationships [91][92][93].
The discrepancies between the interaction energies found for the 1:2 complexes in which planar complexes exhibited more negative interaction energies than apical ones could be due to the presence of counterions in the crystal structures that compensate the charge of the systems and the the repulsion of the anions that are absent in the gas phase calculations and due to crystal packing constraints.
The metal-halogen intermolecular distances were gathered in Table S12. As observed, those distances ranged between 2.38 and 2.61 (Cu-F), 2.34 and 3.06 (Cu-Cl) and 2.50 and 3.06 Å (Cu-Br), being the average distances 2.51, 2.61 and 2.72 Å for the Cu-F, Cu-Cl and Cu-Br interactions, respectively. In the case of Cu-F, the computational distance (2.11 Å) was shorter than the crystal one, while for the Cu-Cl and Cu-Br the computational distances were in fair agreement with the experimental ones. one, while for the Cu-Cl and Cu-Br the computational distances were in fair agreement with the experimental ones.

Materials and Methods
The geometries of systems were fully optimized at the MP2 computational level [103] with a combination of the aug'-cc-pVDZ and the aug-cc-pVDZ-PP basis sets [104,105]. The aug'-cc-pVDZ basis set is built using aug-cc-pVDZ for C, N, F and Cl atoms and cc-pVDZ for the H atoms. For the heavy (coinage) atoms (Cu, Ag and Au) the effective core potential basis set, aug-cc-pVDZ-PP was used. Frequency calculations at the same computational level were carried out to confirm that the structures obtained correspond to energetic minima. These calculations were carried out with the Gaussian-16 program [106]. The electronic energy and geometry of all systems were gathered in Tables S1-S3.
The binding energy was calculated as the difference of the electronic energy of the complexes minus the sum of the energies of the isolated monomers in their minimum energy. Positive and negative values of the binding energies correspond to unfavorable (repulsive) and favorable (attractive) interactions.
The topological characteristics of the electron density were studied within the quantum theory of atoms in molecules (QTAIM) [107,108] framework with the AIMAll program [109]. The molecular electrostatic potential (MESP) of the isolated monomers was represented with the Jmol program [110] and analyzed on the 0.001 au electron density isosurface with the Multiwfn program [111].
A search in the Cambridge Structural Database (CSD) [112] (Version 5.41 with updates of March, May and August 2020) was carried in order to find crystal structures of (Py-M)3 structures with halides. It should be noted that complementary studies focused on the analysis of the crystal structures that show the interaction between (Py-M)3 molecules and the hydroxyl anion are available in the literature [68].

Conclusions
The interactions between trinuclear regium complexes of pyrazolate with anions (F, Cl and Br) were studied by means of MP2 theory.
It was found that pyrazolate complexes could establish three type of modes of interactions, apical, planar and CH(4) involving different type of interactions and the strength of each interactions was highly dependent on the type of metal and anion considered.

Materials and Methods
The geometries of systems were fully optimized at the MP2 computational level [103] with a combination of the aug'-cc-pVDZ and the aug-cc-pVDZ-PP basis sets [104,105]. The aug'-cc-pVDZ basis set is built using aug-cc-pVDZ for C, N, F and Cl atoms and cc-pVDZ for the H atoms. For the heavy (coinage) atoms (Cu, Ag and Au) the effective core potential basis set, aug-cc-pVDZ-PP was used. Frequency calculations at the same computational level were carried out to confirm that the structures obtained correspond to energetic minima. These calculations were carried out with the Gaussian-16 program [106]. The electronic energy and geometry of all systems were gathered in Tables S1-S3.
The binding energy was calculated as the difference of the electronic energy of the complexes minus the sum of the energies of the isolated monomers in their minimum energy. Positive and negative values of the binding energies correspond to unfavorable (repulsive) and favorable (attractive) interactions.
The topological characteristics of the electron density were studied within the quantum theory of atoms in molecules (QTAIM) [107,108] framework with the AIMAll program [109]. The molecular electrostatic potential (MESP) of the isolated monomers was represented with the Jmol program [110] and analyzed on the 0.001 au electron density isosurface with the Multiwfn program [111].
A search in the Cambridge Structural Database (CSD) [112] (Version 5.41 with updates of March, May and August 2020) was carried in order to find crystal structures of (Py-M) 3 structures with halides. It should be noted that complementary studies focused on the analysis of the crystal structures that show the interaction between (Py-M) 3 molecules and the hydroxyl anion are available in the literature [68].

Conclusions
The interactions between trinuclear regium complexes of pyrazolate with anions (F, Cl and Br) were studied by means of MP2 theory.
It was found that pyrazolate complexes could establish three type of modes of interactions, apical, planar and CH(4) involving different type of interactions and the strength of each interactions was highly dependent on the type of metal and anion considered.
Considering the 1:1 complexes, (Pz-Ag) 3 F − stood out as the strongest ones particularly within the planar configuration.
The most interesting question, which lay beneath the study, was: can those negatively charged (PzM) 3 X − complexes interact and form stable complexes with another anion? Energetically speaking, stable 1:2 complexes were found but the binding energy was shown to be positive, which indicates a repulsive interaction. However, once the repulsion between anions was subtracted, the resulting corrected binding energies were negative. Potential energy surfaces corresponding to the removal of one of the anions indicated the existence of a barrier that prevented the anion dissociation.
Going one step forward, and trying to see what is the maximum number of anions that a pyrazolate complex can hold, the 1:3 complexes were also explored, finding that (Pz-Ag) 3 X (F, Cl and Br) were stable with large positive binding energies for both planar and CH(4) configurations.
A search in the CSD shows the presence of thirteen crystal structures of (Py-Cu) 3 systems with two anions interacting in apical disposition.
This study involving anion· · · anion interactions will be very useful to analyze future interaction with transition metals and can bring more insight on these types of interactions, particularly in the crystal structure domain.