Regium Bonds between Silver(I) Pyrazolates Dinuclear Complexes and Lewis Bases (N 2 , OH 2 , NCH, SH 2 , NH 3 , PH 3 , CO and CNH)

: A theoretical study and Cambridge Structural Database (CSD) search of dinuclear Ag(I) pyrazolates interactions with Lewis bases were carried out and the e ﬀ ect of the substituents and ligands on the structure and on the aromaticity were analyzed. A relationship between the intramolecular Ag–Ag distance and stability was found in the unsubstituted system, which indicates a destabilization at longer distances compensated by ligands upon complexation. It was also observed that the asymmetrical interaction with phosphines as ligands increases the Ag–Ag distance. This increase is dramatically higher when two simultaneous PH 3 ligands are taken into account. The calculated 109 Ag chemical shielding shows variation up to 1200 ppm due to the complexation. Calculations showed that six-membered rings possessed non-aromatic character while pyrazole rings do not change their aromatic character signiﬁcantly upon complexation.


Introduction
Non-covalent interactions (NCIs) are present in complexes formed between two or more Lewis acids and Lewis bases. In fact, those interactions are commonly named by the Lewis acid [1]. On the one hand, a Lewis base (LB) or a LB motif is associated with a region of space where there exists an excess of negative charge (i.e., electron density) in the proximity of an atom or atoms within a molecule. This is predominant in anions and in some neutral molecules such as those that exhibit lone pairs (LP: carbenes, amines, phosphines, N-oxides, etc.), multiple bonds (olefins, acetylenes, benzenes and other aromatic molecules), single bonds (alkanes, dihydrogen, etc.), radicals, metals (rarely), etc. On the other hand, a Lewis acid (LA) is associated with a region of space where there is an excess of positive charge (in other words a deficit of negative charge or electron deficiency) in the proximity of an atom or atoms in a molecule. This can be found in cations, molecules or atoms exhibiting σand π-holes and metals (frequently). The concept of a σ-hole was introduced by Politzer et al. [2][3][4][5] to describe regions of positive potential along the vector of a covalent bond. It was latter extended to other situations, for example, the π-hole [6][7][8][9][10][11] (positive electrostatic potential perpendicular to an atom of a molecular framework) and lone pair hole (similar to σ-hole but along the atom-lone pair direction) [12][13][14]. The maximum value of the molecular electrostatic potential within a given molecular density isosurface surface, V S,max , has also been classified based on the nature of the orbitals (s-, p-and d-orbitals) and it is associated with the corresponding deficiencies (σ s -, σ p -and σ d -holes) [15].
In this article, the dinuclear silver(I) pyrazolates (Scheme 1) have been considered following a previous paper on trinuclear silver(I) pyrazolates [46]. The effect of the ligands on the silver atom and the substituents on the pyrazole rings on the structure, energetic, electronic and magnetic properties have been analyzed.

Cambridge Structural Database Search
The Cambridge Structural Database (CSD) [47] version 5.40 with updates from November 2018, May 2019, and August 2019 was searched for systems with cyclic (Pz-Ag)2 structure. The geometrical characteristics of these systems have been analysed.

Ab Initio Calculations
The geometry of the systems have been fully optimized using the MP2 computational method [48][49][50][51] and the jul-cc-pVDZ basis set [52][53][54] for all the atoms except for the silver atoms, of which the aug-cc-pVDZ-PP effective core potential basis set [55] has been used. Frequency calculations have been carried out at the same computational level to verify that the obtained structures correspond to energetic minima. Dissociation energies have been obtained as the difference between the sum of the energies of the isolated monomers in their optimized geometry and the energy of the complex. These calculations have been carried out with the Gaussian-16 program [56].
The molecular electrostatic potential on the 0.001 au electron density isosurface (MESP) was calculated with the Gaussian16 program, analyzed with the Multiwfn program [57] and represented with the JMol program. [58] The topological analysis of the electron density was carried out by means of the quantum theory of atoms in molecules (QTAIM) method [59][60][61][62] using the AIMAll program [63]. This method identifies the points of space where the gradient of the electron density vanishes (critical points) and based on the number of positive curvatures characterizes them as nuclear attractors (3, -3), bond critical points (3, -1), ring critical points (3, +1) or cage critical points (3, +3). By connecting the bond critical points with the nuclear attractor, following the minimum gradient path, the molecular graph is obtained. Scheme 1. General structure of the complexes under study and their corresponding names.

Cambridge Structural Database Search
The Cambridge Structural Database (CSD) [47] version 5.40 with updates from November 2018, May 2019, and August 2019 was searched for systems with cyclic (Pz-Ag) 2 structure. The geometrical characteristics of these systems have been analysed.

Ab Initio Calculations
The geometry of the systems have been fully optimized using the MP2 computational method [48][49][50][51] and the jul-cc-pVDZ basis set [52][53][54] for all the atoms except for the silver atoms, of which the aug-cc-pVDZ-PP effective core potential basis set [55] has been used. Frequency calculations have been carried out at the same computational level to verify that the obtained structures correspond to energetic minima. Dissociation energies have been obtained as the difference between the sum of the energies of the isolated monomers in their optimized geometry and the energy of the complex. These calculations have been carried out with the Gaussian-16 program [56].
The molecular electrostatic potential on the 0.001 au electron density isosurface (MESP) was calculated with the Gaussian16 program, analyzed with the Multiwfn program [57] and represented with the JMol program [58].
The topological analysis of the electron density was carried out by means of the quantum theory of atoms in molecules (QTAIM) method [59][60][61][62] using the AIMAll program [63]. This method identifies the points of space where the gradient of the electron density vanishes (critical points) and based on the number of positive curvatures characterizes them as nuclear attractors (3, −3), bond critical points (3, −1), ring critical points (3, +1) or cage critical points (3, +3). By connecting the bond critical points with the nuclear attractor, following the minimum gradient path, the molecular graph is obtained. The natural bond orbital (NBO) method [64] was employed using the NBO-3 program to evaluate atomic charges, and to analyze charge-transfer interactions between occupied and unoccupied orbitals.
The TOPMOD program [65] has been used to analyze the areas of electron concentration in terms of the electron localization function (ELF) [66]. For the three-dimensional plots, a convenient ELF value of 0.7 was used [67].
Relativistic-corrected NMR chemical shielding values for the geometries optimized at MP2 level were obtained using the relativistic ZORA spin-orbit Hamiltonian [68,69], the BP86 functional [70][71][72], and the full electron QZ4P basis [73]. In addition, the nuclear-independent chemical shift (NICS) [74] at the centre and 1 Å above the centre of the ring formed by the two Ag atoms and the pyrazole moieties were calculated to study the aromaticity of all systems. In addition, a scan of the NICS values from 0 to 2 Å in steps of 0.25 Å for (PzAg) 2 was calculated [75,76]. These calculations have been performed with the ADF-2017 program [77].

Results and Discussion
In this paper we will report the study of nineteen compounds corresponding to the general formula depicted in Scheme 1 and named by a simple code that allows to easy identification: (R-pzAg) 2 (L) n . Firstly, the substituents (R) on the pyrazole ring are indicated: Pz, 4NO 2 pz, DMepz and 4Clpz for the unsubstituted (H), 4-nitro, 3,5-dimethyl, and 4-chloro derivatives, respectively. Secondly, the ligands (L) interacting with the silver atoms are indicated. Thus, for instance, (4NO 2 pzAg) 2 (PH 3 ) 4 corresponds to the (4-nitropyrazole:Ag) 2 cyclic structure with four phosphines interacting with the silver atoms (two phosphines per each silver atom).

CSD Search
A search in the Cambridge Structural Database has been carried out and the resulting compounds with the structure represented in Scheme 1 were summarized and reported in Table 1 ordered by increasing Ag-Ag intramolecular distance. Two simplified views of these structures can be found in Table S1 of the Supplementary Materials. Structures with dinuclear silver(I) pyrazolates without ligands were not found within the CSD search.
As observed by the crystallographic data, the Ag-Ag distance ranges between 4.305 Å (ZIGSEQ) and 3.392 Å (FINWOR). The shortest distances correspond to complexes with only two ligands present concomitantly with the 3,5-bis-CF 3 substituent on the pyrazole. It is also clear that the larger the number of ligands, the longer the intramolecular Ag-Ag distance. Moreover, it is noteworthy that, in structures with four phosphines ligands, the Ag-Ag distance increases as the substitutes of the pyrazole are H < 4-Cl < 4-NO 2 . These results will be compared against our theoretical calculations in the following sections.
Concerning the conformation of the six-membered rings in the crystal structures (similar to those of 1,4-cyclo-hexadiene, 9,10-dihydroanthracenes, phenothiazines, etc.), very little energy differences between those conformations in the crystals were found, in agreement what was proposed by Rasika Dias [78]. Regarding the tetrahedral configuration of Ag(I) atom, we have used Houser's τ 4 index [79] as recommended by Raptis [80]. This index is defined as: where, α and β correspond to the two largest angles of the six angles around the tetrahedral silver atom. The values of τ 4 range from 1 (perfect tetrahedral geometry, i.e., τ 4 = [360 − (109.5 + 109.5)]/141 = 1) to 0 (perfect square planar geometry). The τ 4 values obtained for three substituents, 0.9 (RATFEX and ZIGSIU) are very similar to those for 4 substituents (ZIGROZ and ZIGSEQ, 0.89 and 0.88 respectively) with both identical Ag atoms. The two lowest values (0.78) correspond to a substituent that differs from PPh 3 (see notes (a) and (b) of Table 1).
Two P ligands that corresponds to four P atoms.
(b) A P ligand in an adamantane type molecule: As observed by the crystallographic data, the Ag-Ag distance ranges between 4.305 Å (ZIGSEQ) and 3.392 Å (FINWOR). The shortest distances correspond to complexes with only two ligands present concomitantly with the 3,5-bis-CF3 substituent on the pyrazole. It is also clear that the larger the number of ligands, the longer the intramolecular Ag-Ag distance. Moreover, it is noteworthy that, in structures with four phosphines ligands, the Ag-Ag distance increases as the substitutes of the pyrazole are H < 4-Cl < 4-NO2. These results will be compared against our theoretical calculations in the following sections.
Concerning the conformation of the six-membered rings in the crystal structures (similar to those of 1,4-cyclo-hexadiene, 9,10-dihydroanthracenes, phenothiazines, etc.), very little energy differences between those conformations in the crystals were found, in agreement what was proposed by Rasika Dias [78]. Regarding the tetrahedral configuration of Ag(I) atom, we have used Houser's τ4 index [79] as recommended by Raptis [80]. This index is defined as: where, α and β correspond to the two largest angles of the six angles around the tetrahedral silver atom. The values of τ4 range from 1 (perfect tetrahedral geometry, i.e., τ4 = [360 − (109.5 + 109.5)]/141 = 1) to 0 (perfect square planar geometry). The τ4 values obtained for three substituents, 0.9 (RATFEX

Electronic Properties of the Isolated (PzAg) 2 System
The isolated (PzAg) 2 system calculated at MP2 level presents a planar structure with D 2h symmetry, where the silver atoms are equidistant to the pyrazole rings with an intramolecular Ag-Ag distance of 2.801 Å (Table S2). This intramolecular distance is slightly larger than that of the Ag 2 cluster (2.514 Å) calculated at the same computational level. In fact, its molecular graph (Figure 1a) shows the presence of an Ag-Ag bond with the corresponding bond critical point. However, other electronic analysis carried out: ELF ( Figure 1b) and NBO, do not indicate the presence of such bonds between the silver atoms due to the absence of the corresponding basin concomitantly with the Wiberg bond index of the Ag-Ag contact in (PzAg) 2 -being 0.01, while that of Ag 2 is 0.92.
being the most positive value in the whole isosurface. Interestingly the LUMO orbital ( Figure 1d) is solely associated to the Ag atoms and located in a region that coincides with the location of the VS,max. Thus, the silver atoms of this molecule should act as a Lewis acid using both the electrostatic and orbital criteria.

The (PzAg)2 Free Complex and the Effect of the Ag-Ag Distance on its Stability
The complex without ligands, (PzAg)2, has in its minimum energy structure a distance between both silver atoms of 2.801 Å (Table S2) which is considerably shorter than any of the Ag-Ag distances found in the CSD. As we have already indicated in the CSD search, this distance is very sensitive to the environment (substituents of the pyrazole and ligands of the silver atoms) and can vary almost 1.0 Å from the shortest to the longest distance (3.392 and 4.305 Å, respectively). In order to explore the energetic penalty due to the elongation of the intramolecular Ag-Ag distance, the (PzAg)2 system has been optimized while keeping fixed the Ag-Ag distance from 2.6 to 4.0 Å in steps of 0.2 Å (Table S2 and Figure 2). The energetic results indicate that lengthening the distance 1 Å, up to 3.8 Å, considerably decreases the stability of the system by 75 kJ·mol -1 which should be compensated by the interactions with the ligands. The molecular electrostatic potential on the 0.001 au electron density isosurface (MESP) of the (PzAg) 2 system (Figure 1c) shows negative values above and below the pyrazole ring while the MESP in the extension of the CH and Ag-Ag bonds are positive. Moreover, the calculated maximum value on the MESP, V S,max , in the vicinity of the silver atoms is +0.03 a.u. (marked with black dots) being the most positive value in the whole isosurface. Interestingly the LUMO orbital ( Figure 1d) is solely associated to the Ag atoms and located in a region that coincides with the location of the V S,max . Thus, the silver atoms of this molecule should act as a Lewis acid using both the electrostatic and orbital criteria.

The (PzAg) 2 Free Complex and the Effect of the Ag-Ag Distance on its Stability
The complex without ligands, (PzAg) 2 , has in its minimum energy structure a distance between both silver atoms of 2.801 Å (Table S2) which is considerably shorter than any of the Ag-Ag distances found in the CSD. As we have already indicated in the CSD search, this distance is very sensitive to the environment (substituents of the pyrazole and ligands of the silver atoms) and can vary almost 1.0 Å from the shortest to the longest distance (3.392 and 4.305 Å, respectively). In order to explore the energetic penalty due to the elongation of the intramolecular Ag-Ag distance, the (PzAg) 2 system has been optimized while keeping fixed the Ag-Ag distance from 2.6 to 4.0 Å in steps of 0.2 Å (Table S2 and Figure 2). The energetic results indicate that lengthening the distance 1 Å, up to 3.8 Å, considerably decreases the stability of the system by 75 kJ·mol −1 which should be compensated by the interactions with the ligands.

Effect of the Ligands and Substituents on the Structure and Dissociation Energy (De)
After studying the free (PzAg) 2 system , the complexes with two ligands simultaneously interacting with the (PzAg) 2 system have been optimized, i.e., each silver atom interacts with a single ligand. The minimum structures obtained show that the interacting atom of the ligand is coplanar with the plane defined by the (PzAg) 2 system. The molecular graphs of two illustrative examples are shown in Figure 3 and the Cartesian coordinates of all of them were summarized in Table S3.

Effect of the Ligands and Substituents on the Structure and Dissociation Energy (De)
After studying the free (PzAg)2 system, the complexes with two ligands simultaneously interacting with the (PzAg)2 system have been optimized, i.e., each silver atom interacts with a single ligand. The minimum structures obtained show that the interacting atom of the ligand is coplanar with the plane defined by the (PzAg)2 system. The molecular graphs of two illustrative examples are shown in Figure 3 and the Cartesian coordinates of all of them were summarized in Table S3. The complexation with ligands produces an elongation of the Ag-Ag distance up to 1.1 Å ( Table 2) in agreement with the large variety of structures and the range of Ag-Ag distances found in the CSD search. For example, the crystal structure RATPAT (L = (PPh3)2) shows an Ag-Ag distance of 3.870 Å while in the calculated (PzAg)2(PH3)2 complex this value is 3.689 Å. It is known that ligands coordinated by O, C or P atoms are strong, while those ligands coordinated by N atoms are weak. In the cases of CNH and HCN ligands are both -donors and -acceptors but the former is coordinated by the C atom while the latter is coordinated by the N atom. This results in larger dissociation energies for the (PzAg)2(CNH)2 complex in comparison with (PzAg)2(NCH)2. Similarly, this happens with CO and NH3 ligands (both -donors), but while CO is a good -acceptor NH3 is not, resulting in smaller values of De. In the case of the PH3 ligand, it is both a -donor and -acceptor

Effect of the Ligands and Substituents on the Structure and Dissociation Energy (De)
After studying the free (PzAg)2 system, the complexes with two ligands simultaneously interacting with the (PzAg)2 system have been optimized, i.e., each silver atom interacts with a single ligand. The minimum structures obtained show that the interacting atom of the ligand is coplanar with the plane defined by the (PzAg)2 system. The molecular graphs of two illustrative examples are shown in Figure 3 and the Cartesian coordinates of all of them were summarized in Table S3. The complexation with ligands produces an elongation of the Ag-Ag distance up to 1.1 Å ( Table 2) in agreement with the large variety of structures and the range of Ag-Ag distances found in the CSD search. For example, the crystal structure RATPAT (L = (PPh3)2) shows an Ag-Ag distance of 3.870 Å while in the calculated (PzAg)2(PH3)2 complex this value is 3.689 Å. It is known that ligands coordinated by O, C or P atoms are strong, while those ligands coordinated by N atoms are weak. In the cases of CNH and HCN ligands are both -donors and -acceptors but the former is coordinated by the C atom while the latter is coordinated by the N atom. This results in larger dissociation energies for the (PzAg)2(CNH)2 complex in comparison with (PzAg)2(NCH)2. Similarly, this happens with CO and NH3 ligands (both -donors), but while CO is a good -acceptor NH3 is not, resulting in smaller values of De. In the case of the PH3 ligand, it is both a -donor and -acceptor The complexation with ligands produces an elongation of the Ag-Ag distance up to 1.1 Å (Table 2) in agreement with the large variety of structures and the range of Ag-Ag distances found in the CSD search. For example, the crystal structure RATPAT (L = (PPh 3 ) 2 ) shows an Ag-Ag distance of 3.870 Å while in the calculated (PzAg) 2 (PH 3 ) 2 complex this value is 3.689 Å. It is known that ligands coordinated by O, C or P atoms are strong, while those ligands coordinated by N atoms are weak. In the cases of CNH and HCN ligands are both -donors and -acceptors but the former is coordinated by the C atom while the latter is coordinated by the N atom. This results in larger dissociation energies for the (PzAg) 2 (CNH) 2 complex in comparison with (PzAg) 2 (NCH) 2 . Similarly, this happens with CO and NH 3 ligands (both -donors), but while CO is a good -acceptor NH 3 is not, resulting in smaller values of De. In the case of the PH 3 ligand, it is both a -donor and -acceptor ligand, which is in agreement with the (PzAg) 2 (PH 3 ) 2 complex, showing the second largest De. Chalcogen ligands, OH 2 (-donor) and SH 2 (-donor) present similar trends. In addition, it is also observed that the distance between both pyrazole rings decreases with the elongation of the Ag-Ag distance.  2 3.689 3.657 2.373 122.9 (PzAg) 2 (CO) 2 3.845 3.519 1.998 90.0 (PzAg) 2 (CNH) 2 3.880 3.534 2.018 136.6 a Measured as the distance between the smallest N-N distance in two pyrazole rings. b Z is the atom of L directly interacting with the silver atoms.
But, does this have any impact on the dissociation energy? Or, in other words, is there any relationship between the Ag-Ag intramolecular distance and the dissociation energy? The dissociation energy values corresponding to the (PzAg) 2 L 2 complexes range between 23 kJ mol −1 for L = N 2 to 137 kJ mol −1 for L = CNH (Table 2). In general, despite observing a trend between the dissociation energy and the elongation of the Ag-Ag distance ( Figure S1), no good fitting has been found (logarithmic fitting, R 2 = 0.73). The only outlier corresponds to the (PzAg) 2 (CO) 2 complex, and when this point is neglected the fitting is more evident (R 2 = 0.90). A better exponential relationship has been found for the distance between the pyrazole rings vs. De (De = 5.55·e −0.114Pz-Pz , R 2 = 0.92) which may suggest that the repulsion between pyrazole groups is partially responsible for the increase of the De energy but somehow compensated by the Ag-Ligand interaction.
The effect of the substituents (pyrazole ring) within the Ag-Ag distance and dissociation energies have been explored by considering four different pyrazole derivatives, R = H, 3,5-di(CH 3 ), 4-Cl and 4-NO 2 . These isolated (R-pzAg) 2 systems have been fully optimized as well as in the presence of two phosphine molecules ligands with the results gathered in Table 3. Table 3. Effect of the 3,5-dimethyl, 4-chloro and 4-nitro substituents (pyrazole ring) within the Ag-Ag distance and on the dissociation energy, De.
So far, only the systems with no ligands, (R-pzAg) 2 , and complexes with two identical ligands, (R-pzAg) 2 L 2, have been studied. In Table 4, the results corresponding to the complexes with 0, 1, 2 and 4 phosphine ligands bonded to (PzAg) 2 and (4NO 2 pzAg) 2 systems are shown. In both series, the intramolecular Ag-Ag distance increases with the number of phosphine ligands bonded. When one PH 3 is bonded the increase in the Ag-Ag distance with respect to the isolated system is 0.286 and 0.311 Å for (PzAg) 2 and (4NO 2 pzAg) 2 systems respectively. However, the change is two times larger when moving from 1 to 2 simultaneous ligands. Finally, the change is again moderate when 4 PH 3 ligands interacting with each Ag are considered. The CDS search also shows an increase of the Ag-Ag distance with the number of ligands. For example in the unsubstituted complexes (R=H) the Ag-Ag distance in RATFET (L = 3) is 3.706 Å and increases to 3.900 Å with four ligands (KIRXIV). This is also observed for substituted complexes with R=4-Cl: ZIGRUF (L = 3) 3.707 Å to ZIGROZ (L = 4) 4.209 Å and for R = 4NO 2 : ZIGSIU (L = 3) 3.827 Å) to ZIGSEQ (L = 4) 4.305 Å. In addition, it is interesting to notice that in complexes with four phosphines, the (R-PzAg) 2 system adopts a chair conformation (Figure 4) vs. the planar one observed with one or two ligands. The experimental Ag-Ag distances are longer than the calculated ones; this could be due to the fact that the ligand found in the crystals are bulkier (for instance PPh 3 vs. PH 3 ). This can be also related with the ratio of σ-donor/π-acceptor capacity in phosphorus ligands. In principle P(CH 3 ) 3 is a better σ-donor than PH 3 , while the latter is a better π-acceptor [81]. However, the σ-donor/π-acceptor ratio indicates that the P(CH 3 ) 3 is a stronger ligand than PH 3 . The same can be expected for PPh 3 and therefore the Ag-Ag distance will be larger for complexes that are PPh 3 -coordinated compared with those for PH 3 ones.  2 3.782 150.8 So far, only the systems with no ligands, (R-pzAg)2, and complexes with two identical ligands, (R-pzAg)2L2, have been studied. In Table 4, the results corresponding to the complexes with 0, 1, 2 and 4 phosphine ligands bonded to (PzAg)2 and (4NO2pzAg)2 systems are shown. In both series, the intramolecular Ag-Ag distance increases with the number of phosphine ligands bonded. When one PH3 is bonded the increase in the Ag-Ag distance with respect to the isolated system is 0.286 and 0.311 Å for (PzAg)2 and (4NO2pzAg)2 systems respectively. However, the change is two times larger when moving from 1 to 2 simultaneous ligands. Finally, the change is again moderate when 4 PH3 ligands interacting with each Ag are considered. The CDS search also shows an increase of the Ag-Ag distance with the number of ligands. For example in the unsubstituted complexes (R=H) the Ag-Ag distance in RATFET (L = 3) is 3.706 Å and increases to 3.900 Å with four ligands (KIRXIV). This is also observed for substituted complexes with R=4-Cl: ZIGRUF (L = 3) 3.707 Å to ZIGROZ (L = 4) 4.209 Å and for R = 4NO2: ZIGSIU (L = 3) 3.827 Å) to ZIGSEQ (L = 4) 4.305 Å. In addition, it is interesting to notice that in complexes with four phosphines, the (R-PzAg)2 system adopts a chair conformation (Figure 4) vs. the planar one observed with one or two ligands. The experimental Ag-Ag distances are longer than the calculated ones; this could be due to the fact that the ligand found in the crystals are bulkier (for instance PPh3 vs. PH3). This can be also related with the ratio of σ-donor/π-acceptor capacity in phosphorus ligands. In principle P(CH3)3 is a better σ-donor than PH3, while the latter is a better π-acceptor. [81] However, the σ-donor/π-acceptor ratio indicates that the P(CH3)3 is a stronger ligand than PH3. The same can be expected for PPh3 and therefore the Ag-Ag distance will be larger for complexes that are PPh3-coordinated compared with those for PH3 ones.

Electron Density
The critical points of the electron density of all systems have been characterized using the quantum theory of atoms in molecules (QTAIM) method. As aforementioned, a bond critical point (BCP) is obtained for the (PzAg) 2 system in between both Ag atoms. A similar feature has been obtained for all the systems with Ag-Ag distances shorter than 3.3 Å. The electron density values, ρ BCP , of this BCP (Table S4) range between 0.032 and 0.014 a.u., with positive values of the Laplacian, ∇ 2 ρ (between 0.098 and 0.041 a.u.) and negative values of the total energy density, H BCP , except for the system with the largest Ag-Ag bond in this set ((PzAg) 2 (NH 3 ) 2 ). Excellent exponential relationships are obtained between the ρ BCP and ∇ 2 ρ BCP and the interatomic distance in agreement with previous reports ( Figure 5) [82,83]. but the differences, i.e., the anti-cooperativity, is rather smaller (4.0 kJ·mol -1 ) for (4NO2pzAg)2(PH3)2 and slightly larger for (4NO2pzAg)2(PH3)4 (57.9 kJ·mol -1 ).

Electron Density
The critical points of the electron density of all systems have been characterized using the quantum theory of atoms in molecules (QTAIM) method. As aforementioned, a bond critical point (BCP) is obtained for the (PzAg)2 system in between both Ag atoms. A similar feature has been obtained for all the systems with Ag-Ag distances shorter than 3.3 Å. The electron density values, ρBCP, of this BCP (Table S4) range between 0.032 and 0.014 a.u., with positive values of the Laplacian, ∇ 2 ρ (between 0.098 and 0.041 a.u.) and negative values of the total energy density, HBCP, except for the system with the largest Ag-Ag bond in this set ((PzAg)2(NH3)2). Excellent exponential relationships are obtained between the ρBCP and ∇ 2 ρBCP and the interatomic distance in agreement with previous reports ( Figure 5) [82,83]. Concering the Ag-L bonds, the corresponding bond critical points between the silver atom and the different ligands have been gathered in Table S5. In all the cases, the interactions exhibit positive values of the ∇ 2 ρBCP and negative values of HBCP, which indicates a partial covalent character of the bond formed [84,85]. The only exception corresponds to the weakest complex, (PzAg)2(N2)2, which shows a small positive HBCP value. The 14 unique Ag-P contacts found in this set show similar relationships between ρBCP and ∇ 2 ρBCP vs. the Ag-P distance to those previously mentioned for Ag-Ag BCPs. Concering the Ag-L bonds, the corresponding bond critical points between the silver atom and the different ligands have been gathered in Table S5. In all the cases, the interactions exhibit positive values of the ∇ 2 ρ BCP and negative values of H BCP , which indicates a partial covalent character of the bond formed [84,85]. The only exception corresponds to the weakest complex, (PzAg) 2 (N 2 ) 2 , which shows a small positive H BCP value. The 14 unique Ag-P contacts found in this set show similar relationships between ρ BCP and ∇ 2 ρ BCP vs. the Ag-P distance to those previously mentioned for Ag-Ag BCPs.

Magnetic Properties and Aromaticity
Among the different nuclei suitable for NMR spectra present in these systems ( 1 H, 13 C and 15 N), 109 Ag is the one with the largest range of chemical shifts. The calculated 109 Ag chemical shielding for all the systems studied in this article are listed in Table 5. It is worth noting that upon complexation 109 Ag can change its chemical shielding by more than 1200 ppm to lower field, from 3765 ppm ((PzAg) 2 ) to 2550.48 ppm ((PzAg) 2 (PH 3 ) 4 ). Furthermore, a good relationship between the chemical shielding and the intramolecular Ag-Ag distance (only for complexes with two ligands) was found ( Figure S3). Unfortunatelly, there are no experimental data on (PzAg) 2 compounds but a recent report on (PzAg) 3 derivatives show that the methodology used here provides δ 109 Ag values within 10 ppm of the experimental ones [46]. In order to explore the potential aromaticity of the six membered ring formed by the nitrogen atoms of the pyrazoles and the two silver atoms, the NICS(0) and NICS(1) have been calculated (Table 5). Despite NICS isotropic values being widely used and well established, there is still a controversy about the reliability of NICS values for assessment of the aromaticity of certain molecules [86,87]. Nevertheless, and following our previous experience, the isotropic values have in several cases been shown to present an accurate description of the aromatic behaviour in poly-aromatic systems [88][89][90].
Despite almost all the systems studied presenting negative NICS(0) values in the six-membered ring, only those with short Ag-Ag distances (2.8−3.0 Å) present very negative values close to the benzene molecule (−8 ppm) [91], and, also, those NICS(0) values decrease in absolute value as the Ag-Ag distance increases. NICS(1) are smaller, in absolute value, than NICS(0), but follow the same trend as the latter. Also, NICS(1) are very small compared with benzene ones (−10.2 ppm) [91] which suggest non-aromatic character. But, those values should be taken carefully, since the two silver atoms are very close and the proximity of the nuclei may affect the NICS(0) measure. To provide a further insight on this, Figure 6 clearly shows that there is a unique dependence between the NICS(0) and NICS(1), the distance between the location where the NICS is measured and the silver atom. The scan of the NICS values for (PzAg) 2 from 0 to 2.0 Å above the centre of the six-membered ring (Table S6) have also been plotted in Figure 6 showing a similar evolution to the NICS(0) and NICS(1) of the rest of the molecules. This indicates that, as aforementioned, this ring is not aromatic but the NICS values obtained are somehow affected by the proximity of the silver atom.
Crystals 2020, 10, x FOR PEER REVIEW 11 of 16 NICS(1) of the rest of the molecules. This indicates that, as aforementioned, this ring is not aromatic but the NICS values obtained are somehow affected by the proximity of the silver atom. In contrast the NICS(0) and NICS(1) values of the pyrazole rings are negative and large in all cases and very close to the ones obtained by Kusakiewicz-Dawid (−13.5 and −11.4 ppm respectively) [92].
On the other hand, the substituents on the pyrazole ring have a greater effect on the NICS values than the different ligands on the (PzAg)2L2 complexes, most likely due to changes in the electron density on the ring current electrons. For instance the NICS(0) in the (Rpz-Ag)2 systems ranges between −10.5 and −13.1 ppm while in the (PzAg)2L2 complexes it is between −12.3 and −13.1 ppm.

Conclusions
A theoretical study and CSD search of the different effects provoked by substituents and ligands upon complexation with dinuclear Ag(I) pyrazolates has been carried out and the structural, energetic, electron density and magnetic features analyzed.
The CSD search shows a great variability of the Ag-Ag distance in the crystal structures. These results have been rationalized based on the number of ligand interactions with the Ag atoms and the substituents of the pyrazole ring.
In the isolated (PzAg)2 system, it was observed that for Ag-Ag the longer the distance, the lesser the stability of the unsubstituted complex with no ligands. This decrease in the stability is somehow compensated by the ligands upon complexation.
Furthermore, complexation with ligands through the Ag atoms increases the intramolecular distance Ag-Ag. In fact, considering the PH3 ligand, the increase of the Ag-Ag distance was found In contrast the NICS(0) and NICS(1) values of the pyrazole rings are negative and large in all cases and very close to the ones obtained by Kusakiewicz-Dawid (−13.5 and −11.4 ppm respectively) [92].
On the other hand, the substituents on the pyrazole ring have a greater effect on the NICS values than the different ligands on the (PzAg) 2 L 2 complexes, most likely due to changes in the electron density on the ring current electrons. For instance the NICS(0) in the (Rpz-Ag) 2 systems ranges between −10.5 and −13.1 ppm while in the (PzAg) 2 L 2 complexes it is between −12.3 and −13.1 ppm.

Conclusions
A theoretical study and CSD search of the different effects provoked by substituents and ligands upon complexation with dinuclear Ag(I) pyrazolates has been carried out and the structural, energetic, electron density and magnetic features analyzed.
The CSD search shows a great variability of the Ag-Ag distance in the crystal structures. These results have been rationalized based on the number of ligand interactions with the Ag atoms and the substituents of the pyrazole ring.
In the isolated (PzAg) 2 system, it was observed that for Ag-Ag the longer the distance, the lesser the stability of the unsubstituted complex with no ligands. This decrease in the stability is somehow compensated by the ligands upon complexation.
Furthermore, complexation with ligands through the Ag atoms increases the intramolecular distance Ag-Ag. In fact, considering the PH 3 ligand, the increase of the Ag-Ag distance was found moderate when going from no ligand to one ligand. However, when two simultaneous PH 3 are interacting, the Ag-Ag distance increases dramatically. Nevertheless, when four PH 3 are considered, the increase is again moderate.
In terms of the QTAIM analysis, it is noteworthy the presence of a BCP between both Ag atoms among all the systems with Ag-Ag distances shorter than 3.3 Å.
Finally, regarding the aromatic/non-aromatic properties, six-membered rings containing the Ag-Ag motif show negative NICS values but those reveal a non-aromatic character mainly affected by the proximity of the Ag nuclei. This was confirmed by the relationship found between the NICS values and the Ag-NICS distance. On the other hand, pyrazole rings maintain their aromatic behaviour with slight changes.