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. 2026 Sep 30;27(19):e70585. doi: 10.1002/cphc.70585

Comparing Homodimer and Homotrimer Noncovalent Complexes Involving Coinage Metals: A Theoretical Perspective

Sergi Burguera 1, Antonio Bauzá 1,✉
PMCID: PMC13626965  PMID: 42816441

Abstract

In this study, the geometry and physical nature of regium···regium contacts have been studied at the RI‐PBE0‐D4/def2‐TZVPD level of theory. In more detail, diatomic M clusters (M = Cu, Ag, and Au) were used in a series of homodimer and homotrimer complexes, where they acted as both electron donor and acceptor counterparts. The results were analyzed using several state of the art quantum chemistry tools, such as the quantum theory of atoms in molecules (QTAIM), the natural nond orbital (NBO), the non‐covalent interaction plot (NCIplot), and the energy decomposition analyses (EDA) methodologies. We believe the results reported herein will be useful for those scientists devoted to the field of physical chemistry, owing to the importance of molecular electronic structure in the formation of weak interactions.

Keywords: σ‐hole chemistry, coinage metals, noncovalent interactions, regium bonding, theoretical study

In this study, the ability of M2 clusters (M = Cu, Ag, and Au) to form noncovalent homodimers/homotrimers has been evaluated at the RI‐PBE0‐D4/def2‐TZVPD level of theory. The results have been complemented with QTAIM, NCIplot, and EDA analyses, which helped to shed light on the physical nature of these unconventional noncovalent contacts.

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1. Introduction

Noncovalent interactions govern the structure and energetics of molecular assemblies across chemistry [1], biology [2], and materials science fields [3], among others. In this context, the energetic stability of supramolecular systems arises from a complex balance of electrostatics, polarization, charge transfer, and dispersion terms, which are closely related to their molecular composition and geometry [4, 5]. While hydrogen bonding [6] and dispersion‐driven interactions, such as π–π stacking [7] or C─H···H─C interactions [8], have been extensively characterized, comparatively less attention has been devoted to noncovalent interactions involving metal atoms, particularly from a quantitative physical chemistry perspective.

In this regard, coinage/noble metals (Cu, Ag, and Au), although frequently encountered in closed‐shell or near‐closed‐shell electronic configurations, can engage in a variety of attractive intermolecular interactions at distances exceeding conventional covalent or coordination bonds. Such interactions typically include Regium Bonds (RgBs, an attractive noncovalent force involving a noble metal atom and an electron‐rich moiety, such as an anion or a lone pair) [9, 10, 11] and Regium–π (Rg–π) contacts [12, 13] (where the electron‐rich partner is usually played by a conjugated π‐system), often coexisting with classical coordination bonds, which complicates their identification and interpretation using purely structural criteria.

From a theoretical perspective, these interactions are supported by an interplay of physical contributions, such as electrostatics, which reflects anisotropies in the electrostatic potential over metal atoms (also known as σ‐holes). In this regard, the presence of both electron‐rich (nucleophilic) and electron‐poor (electrophilic) regions over noble metal atoms has been associated to their catalytic activity in the pioneering studies from Stenlid and collaborators, which served as an introduction to the Regium bond [9, 10]. The presence of σ‐holes in metals is not exclusive of the elements from Group 11, since other NCIs based on this concept have been theoretically and experimentally characterized during the past decade, such as the spodium bond [14, 15], the osme bond [16], the matere bond [17], the wolfium bond [18], the erythronium bond [19] and very recently, the titan Bond [20]. On the other hand, polarization and charge‐transfer effects are enhanced by the high polarizability of coinage metals and the availability of low‐lying virtual orbitals [20]. Moreover, the contribution of dispersion is particularly significant, especially for Ag and Au clusters, where correlation effects contribute substantially to the total interaction energy [21, 22].

In addition to the increasingly recognized RgBs established between coinage metals and conventional Lewis bases, attractive interactions between two Group 11 metal centers have emerged as a fundamental topic in supramolecular and coordination chemistry [23, 24, 25]. These metal···metal/regium···regium contacts, commonly termed aurophilic, argentophilic, and cuprophilic interactions for Au···Au, Ag···Ag, and Cu···Cu pairs, respectively, originate from a combination of dispersion, electrostatic, polarization, charge‐transfer, and relativistic effects [26, 27, 28]. Among them, aurophilic interactions are particularly prominent due to the strong relativistic stabilization and contraction/expansion effects associated with gold orbitals, which significantly enhance the polarizability of Au(I) centers and lead to interaction energies frequently comparable to those observed for medium‐strength hydrogen bonds [29, 30]. Although generally weaker, analogous Ag···Ag and Cu···Cu interactions have also been extensively identified in coordination compounds, supramolecular aggregates, molecular crystals, and metal nanoclusters, where they contribute to conformational stabilization and electronic communication between metal centers [31, 32, 33]. From a conceptual perspective, these metallophilic contacts can also be understood as a particular manifestation of the unique noncovalent chemistry exhibited by coinage metals and are closely connected to the broader framework of regium bonding since both phenomena arise from the anisotropic electron density distribution and remarkable polarizability characteristic of Group 11 elements.

Despite the growing recognition of coinage‐metal noncovalent interactions over the past decade, their quantitative description remains challenging since their interaction energies are often modest, sensitive to geometry, and strongly method‐dependent. To tackle this, we have carried out a systematic theoretical investigation using a common computational framework (RI‐PBE0‐D4/def2‐TZVPD level of theory) to study noncovalent M···M interactions (M = Cu, Ag, and Au) across three representative supramolecular motifs (side‐to‐side, T‐shaped, and parallel geometries; see Figure 1) and two types of complexes (homodimers and homotrimers). Beyond establishing general energetic trends, this comparative framework allows us to determine whether increasing the aggregation state simply strengthens the pre‐existing M···M interaction or can instead induce a qualitative change in its physical nature, thereby revealing the interplay between molecular geometry, aggregation, and the transition from noncovalent interactions toward incipient metallic bonding.

FIGURE 1.

FIGURE 1

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Complexes 1 to 18 used in this study.

More in detail, the electronic nature of the interactions has been examined using the quantum theory of atoms in molecules (QTAIM) method, where NCI contacts involving metal atoms are commonly characterized by the presence of bond critical points exhibiting low electron density values and kinetic/potential energies ratios (−G/V) close to 1, the non covalent interaction plot (NCI) methodology, which provides a unique perspective by visualizing regions of weak interaction in terms of reduced density gradients, allowing attractive and repulsive contributions to be distinguished and spatially correlated with molecular geometry, and the natural bond orbital (NBO) analyses, providing complementary real‐space descriptors of the noncovalent contacts studied herein. Last, energy decomposition analyses (EDA) were carried out to quantify the relative importance of electrostatics, orbital, dispersion, and electron correlation terms in both types of supramolecular complexes studied herein.

2. Computational Methods

Complexes 1– 9 involving metallic homodimers were obtained by first relaxing the metallic cluster and then performing a distance scan between each pair of units. These scans started from either side to side (complexes 1 –3), T‐shape (complexes 4 –6), or parallel geometries (complexes 7 –9), and the intermolecular M···M distance (M = Cu, Ag, and Au) scanned from 2.0 to 4.0 Å, with increments of 0.1 Å. These distances implied atom‐to‐atom contacts except for the T‐shaped complexes, where the scanned distance corresponded to the M···centroidM–M distance. In addition, complexes 10 –18 were obtained by using the homodimer systems as starting points and performing a distance scan with an additional M2 moiety (see Figure S1 for the graphical representation of the total energy (E, in hartrees) versus M···M distance (d M···M, in Å) involving complexes 5 (in blue) and 14 (in orange)). Once a stable adduct was obtained, we let the previously formed dimer to slightly adjust the M···M distance by using distance increments of 0.05 Å.

For both sets of complexes, calculations were performed at the RI‐PBE0 [34]‐D4 [35]/def2‐TZVPD [36, 37] level of theory using the TURBOMOLE 7.7 software [38]. This method and basis set combination is strong enough to study the noncovalent nature of s‐hole‐based interactions, as it has been demonstrated in previous studies [9, 10, 39]. Although the PBE0‐D4/def2‐TZVPD level provides a balanced description of electrostatic, orbital, and dispersion contributions and has been successfully applied to σ‐hole interactions involving coinage metals, the calculated interaction energies should not be regarded as having sub‐kcal/mol accuracy. Residual uncertainties may arise from the approximate exchange‐correlation functional, the treatment of dispersion and relativistic effects, and basis‐set incompleteness, particularly for the heavier Ag and Au systems. Consequently, the interaction energies reported herein are primarily interpreted in terms of relative trends across systems calculated consistently at the same theoretical level rather than as quantitatively exact absolute values. In this regard, single‐point calculations using the MP2, SCS‐MP2, and CCSD(T) methods were carried out (see the results in parenthesis and brackets in Table 1 for some representative cases) to provide a quantitative assessment of the interaction energies reported herein. The results indicated a general agreement between PBE0‐D4 and the rest of the methodologies, providing reliability to the results discussed below.

TABLE 1.

BSSE‐corrected interaction energies (ΔE BSSE, in kcal/mol), intermolecular distances (d, in Å) and values of the density at the bond critical point that characterizes the interaction (ρ, in a.u.) and of the kinetic/potential energy ratio (−G/V) for complexes 1–18 at the RI‐PBE0‐D4/def2‐TZVPD level of theory. ΔE values corresponding to MP2/SCS‐MP2 and CCSD(T) methods are given in parenthesis and brackets, respectively.

Complex ΔE d a ρ x100 −G/V
1 −9.1 (−6.6/−6.2) [−8.8] 2.501 3.20 0.85
2 −7.8 (−9.0/−7.0) [−8.3] 2.802 3.01 0.84
3 −10.2 (−15.8/−12.4) [−11.7] 2.703 4.94 0.82
4 −23.9 (−25.3/−22.7) [−24.3] 2.480 4.03 0.81
5 −19.0 (−25.2/−20.8) [−21.3] 2.813 3.40 0.81
6 −29.7 (−40.3/−33.2) [−31.5] 2.713 5.45 0.80
7 −7.2 (−4.2/−2.2) [−5.5] 2.705 2.64 0.78
8 −4.9 (−4.9/−2.4) [−3.6] 3.300 1.38 0.96
9 −12.2 (−14.1/−8.9) [−8.7] 2.903 3.83 0.85
10 −10.0 2.504 3.28 0.84
11 −8.4 2.803 3.03 0.84
12 −11.2 2.700 4.99 0.82
13 −24.0 2.482 4.03 0.80
14 −19.0 2.813 3.39 0.81
15 −29.7 2.715 7.99 0.74
16 −54.4 (−55.7/−46.4) [−54.7] 2.300 5.18 0.83
17 −5.1 (−10.8/−5.3) [−7.6] 3.302 1.41 0.95
18 −55.3 (−86.7/−64.5) [−61.8] 2.603 6.73 0.77
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a

Distances given as the shortest atom to atom values.

All the interaction energies were obtained using the supermolecule approximation, as the energetic difference between the complex and the optimized monomers, and corrected using the Boys and Bernardi counterpoise technique [40]. No frequency calculations were performed since the final geometries were obtained through the distance scan procedure described above; thus, these corresponded to higher‐order stationary points rather than to true minima. All systems were treated as singlet, and the wavefunction used was restricted; therefore, we have not found spin contamination (stable against spin‐symmetry breaking). In addition, stability analyses of the closed‐shell wavefunctions confirmed that the restricted singlet solutions corresponded to electronically stable closed‐shell states.

On the other hand, the topological analyses (including both QTAIM [41] and NCIplot [42] studies) of the wavefunction corresponding to each complex were performed by means of the Multiwfn software [43] and the results visualized using the Visual Molecular Dynamics (VMD) software [44]. It should be noted that the presence of a bond critical point (bcp) and an associated bond path within the QTAIM framework should not, by itself, be regarded as definitive evidence for the existence or strength of a noncovalent interaction. Particularly for weak interactions, QTAIM descriptors are therefore interpreted herein in conjunction with energetic and complementary electronic‐structure analyses (such as NCIplot), rather than as standalone bonding criteria. Furthermore, NBO [45] calculations (see Supporting Information for discussion) were performed by means of the Gaussian 16 Revision C.01 software package [46], paired with the NBO7.0 software extension [47]. Last, the (EDA) [48, 49] scheme was used to understand the role of electrostatics, exchange‐repulsion, orbital, dispersion and electron correlation contributions in the formation of the noncovalent complexes studied herein at the RI‐PBE0‐D4/def2‐TZVPD level of theory, also using TURBOMOLE 7.7 software.

3. Results and Discussion

3.1. Molecular Electrostatic Potential Analyses

We started by computing the Molecular Electrostatic Potential (MEP) surfaces of the M2 molecules (see Figure 2), and as noted, we found both positive and negative electrostatic potential values, as expected. In more detail, an electropositive region was located on each side of the molecule, along the vector of the M–M coordination bond, denoting the presence of two σ‐holes. The magnitude of the MEP value over these regions decreased (became less positive) from Cu2 (+27.6 kcal/mol) to Ag2 (+23.2 kcal/mol) and then increased (became more positive) from Ag2 (+23.2 kcal/mol) to Au2 (+28.8 kcal/mol), in line with the results obtained in past studies [12, 38]. On the other hand, each of the M2 moieties also showed an electronegative region located around the M─M coordination bond, with similar magnitudes among them spanning from −12.8 to −11.6 kcal/mol. From this analysis, we concluded that certain interaction modes will be more energetically favored than others, such as complexes 4 –6 involving T‐shape geometry, owing to better electrostatic complementarity between both interacting partners. Since the rest of the complexes studied herein involve the interaction between two electropositive (side‐to‐side) or electronegative regions (parallel arrangement), other energy components (e.g., orbital interactions or electron correlation) might likely be crucial for their formation and stabilization. Last, from this analysis, it is expected that those complexes involving Cu2 and Au2 will present more favorable interaction energy values than those involving Ag2, in line with their more positive σ‐hole MEP values and the slightly more negative belt MEP values (in the case of Cu2).

FIGURE 2.

FIGURE 2

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MEP surfaces of Cu2, Ag2 and Au2 molecules. Energy values at concrete points of the surface are given in kcal/mol (0.001 a.u.).

3.2. Energetic Study

In Figure 3 and Table 1, the results obtained from the theoretical calculations at the RI‐PBE0‐D4/def2‐TZVPD level of theory are shown. As can be observed, in the case of M2···M2 homodimer complexes (1 to 9), the proposed interaction modes are well conserved in the final geometries, with complexes 2, 4, and 9 being an example of each of the interacting patterns proposed (see Section 2 for specific details). On the other hand, complexes 11, 13, and 18 represent their analogous systems and also conserved the geometrical pattern, with minimal differences in the intermolecular distance values, except for complex 18 involving Au2 where a noticeable shortening of the Au···Au distance was observed (from 2.903 to 2.603 Å). This behavior was also observed between complexes 7 (2.705 Å) and 16 (2.300 Å) involving Cu2 and will be analyzed in further detail in the NBO and EDA analyses (see below). In the rest of homotrimer systems (10, 12, and 14 –17), the initial interaction pattern was also conserved, with negligible differences in the intermolecular M···M distances.

FIGURE 3.

FIGURE 3

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Final geometries of complexes 2, 4, 9, 11, 13, and 18 at the RI‐PBE0‐D4/def2‐TZVPD level of theory.

Regarding the energetics, the results shown in Table 1 indicate that in all cases, the interaction strength is favorable (negative), ranging from −55.3 kcal/mol (complex 18) to −4.9 kcal/mol (complex 8). In general, the homodimer complexes involving M2 moieties (1 –9) achieved interaction energy values spanning from large (–29.7 kcal/mol, complex 6) to moderately strong ones (–4.9 kcal/mol, complex 8), being those complexes involving Au (3, 6, and 9) the most favorable ones (−10.2, −29.7, and −12.2 kcal/mol, respectively), followed by Cu and Ag complexes, in agreement with the MEP analysis discussed above. Among these three different interacting geometries, the T‐shape geometry (complexes 4 to 6) yielded the stronger RgBs, in line with the electrostatic complementarity shown in the MEP analyses discussed above. In addition, between the side‐to‐side (complexes 1 to 3) and parallel (complexes 7 –9) geometries, the former resulted in a stronger RgB for Cu and Ag, while the opposite was observed in the case of Au complexes 3 (–10.2 kcal/mol) and 9 (–12.2 kcal/mol). This will be further analyzed in the EDA study (see below).

On the other hand, complexes 10 –18 involving the homotrimer systems also achieved favorable interaction energy values in all cases (ranging from −55.3 to −5.1 kcal/mol), meaning that the addition of a third M2 cluster to the homodimer complex is a favorable process from an enthalpic perspective. Interestingly, in most of the complexes, we observed that the formation of the trimer was energetically more favorable than the formation of the dimer. This energy difference varied from 0.2 kcal/mol (between complexes 8 and 17) to 1.0 kcal/mol (between complexes 3 and 12). Furthermore, this difference was much more noticeable in the case of complexes 16 and 18 (47.2 and 43.1 kcal/mol, respectively), in agreement with their shorter M···M distance values compared to those of complexes 7 and 9. Last, in the case of T‐shape complexes 14 and 15, the interaction energy values obtained related to the formation of this second RgB achieved the same value as the formation of the dimer complex (5 and 6).

3.3. QTAIM and NCIplot Analyses

In Figure 4, the combined QTAIM and NCIplot analyses for some representative complexes (2, 4, 9, 11, 13, and 18) are shown. As noted, in the homodimer complexes 2, 4, and 9, one or two bond critical points (bcps) and bond paths connect both interacting units. More in detail, in complexes 2 and 4 involving Ag2 and Cu2, one/two bcps and bond paths connect the M2 molecules, thus denoting the presence of a single and a bifurcated RgB, respectively. In addition, in complex 9, two bcps and bond paths connected both Au2 molecules, thus characterizing a double RgB interaction.

FIGURE 4.

FIGURE 4

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Distribution of intermolecular bcps and bond paths for complexes 2, 4, 9, 11, 13, and 18. The value of the density at the bcps that characterize each supramolecular complex is also indicated in a.u (multiplied by 100). NCIplot color range −0.04 au ≤ (signλ2)ρ ≤ +0.04 au. Isosurface value RDG = 0.5 and ρ cutoff 0.05 a.u.

On the other hand, in the case of the homotrimer complexes 11, 13, and 18, two (in complex 11) and four (in complexes 13 and 18) bcps and bond paths connected the M atoms between them, also following the interacting pattern observed in the homodimer systems. Moreover, it is also worth noting that all complexes exhibited a kinetic/potential energy ratio (–G/V) close to 1, in line with the formation of a noncovalent complex [50]. Last, for all these complexes, the NCIplot analyses were also carried out, revealing the presence of bluish isosurfaces between the interacting M2 clusters, which denoted a moderately strong and attractive interaction, in line with the interaction energy values obtained for the RgB complexes studied herein (see above).

3.4. NBO Study

In Table 2, the donor and acceptor orbitals involved in the formation of complexes 1 to 18 are shown, along with the values of the second‐order perturbation energy (E (2) in kcal/mol). Also, in Figure 5, the graphical representation of the donor and acceptor NBOs involving the major orbital contributions present in complexes 2, 4, 9, 11, 13, and 18 is shown. For consistency across the series, the NBO discussion focuses on the dominant donor–acceptor interaction present in each M···M complex, which provides a straightforward descriptor of the principal charge‐transfer pathway. Consequently, the reported E(2) values should not be interpreted as the total orbital stabilization since additional secondary donor–acceptor contributions may also be present. Hence, the overall orbital contribution is instead evaluated through EDA analysis.

TABLE 2.

Donor and acceptor NBOs in complexes 1–18, including the value of the second order perturbation energy (E (2), in kcal/mol) at the RI‐PBE0‐D4/def2‐TZVPD level of theory. Only those values involving the highest energetic orbital charge transfer are shown. BD, BD*, LP, and Ry* stand for bonding orbital, antibonding orbital, lone pair, and empty Rydberg orbital, respectively.

Complex Donor Acceptor E (2)
1 BD Cu─Cu BD* Cu─Cu 7.72
2 BD Ag─Ag BD* Ag─Ag 8.21
3 BD Au─Au BD* Au─Au 8.39
4 LP Cu BD* Cu─Cu 3.08
5 BD Ag─Ag BD* Ag─Ag 48.91
6 BD Au─Au BD* Au─Au 81.0
7 LP Cu BD* Cu─Cu 1.40
8 BD Ag─Ag Ry* Ag 1.18
9 LP Au BD* Au─Au 3.72
10 BD Cu─Cu BD* Cu─Cu 7.22
11 BD Ag─Ag BD* Ag─Ag 7.45
12 BD Au─Au BD* Au─Au 8.25
13 BD Cu─Cu BD* Cu─Cu 62.41
14 BD Ag─Ag BD* Ag─Ag 71.70
15 BD Au─Au BD* Au─Au 149.59
16 LP Cu BD* Cu─Cu 1.22
17 BD Ag─Ag Ry* Ag 1.42
18 LP Au BD* Au─Au 10.80
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FIGURE 5.

FIGURE 5

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Graphical representation of the donor and acceptor NBOs involving the major orbital contributions present in complexes 2, 4, 9, 11, 13, and 18.

First, for complexes 1 –3 involving two M2 molecules in a side‐to‐side geometry, we found an orbital interaction between a σ M─M bonding orbital (BD M–M) and a σ‐antibonding M orbital (BD* M–M), with a very similar magnitude among the three metals used, spanning from 7.72 to 8.21 kcal/mol. Second, in the case of complexes 4 –6 involving a T‐shaped geometry, we found the same orbital interaction than for complexes 1 –3 except for complex 4, which showed an orbital donation from a Cu lone pair (LP) to a σ antibonding Cu─Cu orbital (BD* Cu–Cu), with a magnitude of 3.08 kcal/mol. For complexes 5 and 6, the magnitude of the orbital interactions obtained was much larger than in previous cases (48.91 and 81.0 kcal/mol, respectively), likely due to the parallel arrangement of the two interacting units.

Third, in the case of complexes 7 –9 involving a parallel geometry, an orbital interaction between a metal LP and a BD* M–M orbital was found in complexes 7 and 9, with magnitudes of 1.40 and 3.72 kcal/mol, respectively. Additionally, complex 8 exhibited an orbital donation from BD Ag─Ag to an empty Rydberg orbital (Ry*) of an Ag atom, with a magnitude of 1.18 kcal/mol. In this regard, the involvement of a BD* orbital belonging to either a Cu/Ag/Au atom in most of the cases agrees well with other σ‐hole‐based NCIs involving elements from the p‐block, such as the halogen [51] or chalcogen bonds [52] and reinforces the participation of metal σ‐holes in the formation of M···M interactions.

On the other hand, in the case of complexes 10 –18 involving the homotrimer systems, we found the same type of main orbital interaction in the case of complexes 10 –12 (BD M–M to BD* M–M) ranging between 7.22 and 8.25 kcal/mol. In addition, in complex 13, we found the donation from a BD Cu─Cu to a BD* Cu─Cu, contrary to the homodimer complex (4) where a Cu LP was involved as a donor and exhibiting a much larger magnitude (62.41 kcal/mol). Also, in line with complexes 5 and 6, homotrimer complexes 14 and 15 exhibited the largest orbital magnitudes of the study (71.70 and 149.59 kcal/mol, respectively). Last, in complexes 16 –18, the same type of donor and acceptor orbitals were found compared to complexes 7 –9 (BD M–M to BD* M–M), showing very similar energetic magnitudes except for complex 18 involving Au2, which achieved a larger orbital interaction compared to complex 9.

3.5. EDA Study

As the final part of our study, we carried out EDA on the RgB complexes studied herein (see Tables 3 and 4). These analyses allowed us to compare the percentage of each attractive energy term for both types of complexes. Interestingly, in complexes 1 –3 (side‐to‐side geometry), we found that the orbital term was the most prominent contributor, followed by electrostatics in complexes 1 and 2 and by electron correlation in complex 3. Additionally, in complexes 1 and 2, the electron correlation term was much more favorable than dispersion, while in the case of complex 3, correlation predominated over electrostatics, with dispersion also playing a minor role.

TABLE 3.

Percentage representation of each attractive energy term (electrostatics (%ele), orbital (%orb), electron correlation (%cor), and dispersion (%disp)) for complexes 1–9 involving Cu2, Ag2, and Au2 at the RI‐PBE0‐D4/def2‐TZVPD level of theory.

Complex %ele %orb %cor %disp
1 31.7 48.5 17.5 2.3
2 25.3 52.9 18.8 3.1
3 13.1 63.0 20.8 3.0
4 62.8 23.6 12.9 0.8
5 46.9 35.1 16.5 1.4
6 38.2 45.3 15.2 1.3
7 55.9 28.1 13.8 2.2
8 33.7 37.0 19.2 10.1
9 38.3 41.5 15.2 5.1
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TABLE 4.

Percentage representation of each attractive energy term (electrostatics (%ele), orbital (%orb), electron correlation (%cor), and dispersion (%disp)) for complexes 9–18 involving Cu2, Ag2, and Au2 at the RI‐PBE0‐D4/def2‐TZVPD level of theory.

Complex %ele %orb %cor %disp
10 30.0 51.0 16.7 2.3
11 23.3 55.3 18.3 3.1
12 13.8 63.3 19.8 3.0
13 59.3 26.7 12.6 1.3
14 43.9 37.9 16.1 2.0
15 35.8 47.3 14.9 2.0
16 53.7 38.2 7.4 0.8
17 35.2 36.4 18.9 9.5
18 33.1 55.8 9.0 2.1
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In complexes 4 –6 (T‐shaped geometry), electrostatics predominated over the rest of energy terms in 4 and 5, while the formation of complex 6 was dominated by orbital interactions. The second most prominent energy terms in complexes 4 and 5 were orbital and correlation contributions, with dispersion again playing a minor role. In contrast, in complex 6, electrostatics also played a prominent role, followed by correlation and dispersion. Moreover, among complexes 7 –9 involving a parallel geometry, we found electrostatics as a predominant term in complex 7, followed by orbital and correlation terms, leading to dispersion making a very minor contribution.

Last, in complexes 8 and 9, electrostatics and orbital terms achieved a similar strength, followed by correlation and dispersion contributions. In general, electrostatics, orbital, and electron correlation terms were more favorable for those complexes involving the T‐shape geometry (4 –6), followed by complexes 7 –9 involving (parallel) and 1 –3 (side to side) geometries. Interestingly, electrostatics resulted in attractive results even for complexes 1 to 3, where two metal σ‐holes were confronted, likely due to the participation of the electronegative M2 belts in the complex formation (see the MEP study above).

In Table 4, the results from the EDA study involving complexes 10 –18 are shown. In general, very similar tendencies were observed compared to the homodimer complexes 1 –9 previously discussed, thus indicating that the formation of a previous RgB complex affects in a minimal way the physical nature of a second RgB. On the other hand, in complexes 16 and 18, a very noticeable contribution of electrostatics and orbital terms was found, leading to the large and negative interaction energy values obtained for these systems (see Table 1), thus exhibiting a noticeable change in the physical nature upon the addition of a third M2 molecule. As has been discussed above, the geometric and energetic features exhibited by these two complexes, together with the enhanced attractive contributions revealed by EDA analyses, indicate a marked increase in metal–metal bonding character induced by the addition of the third M2 unit. Nevertheless, their respective QTAIM − G/V values (0.83 and 0.77) indicate that a significant noncovalent character was retained. Taken together, these results suggest that both systems approach a regime of incipient metal–metal bonding while retaining an important noncovalent component rather than representing conventional metallic bonds.

4. Conclusion

Herein, we have computationally investigated Regium···Regium interactions involving M2 clusters (M = Cu, Ag, and Au) in homodimer and homotrimer complexes adopting side‐to‐side, T‐shaped, and parallel arrangements. The resulting M···M interactions range from moderate to strong, with complexes 16 (−54.4 kcal/mol) and 18 (−55.3 kcal/mol) exhibiting a marked increase in metal–metal bonding character and approaching a regime of incipient Cu─Cu and Au─Au bonding while retaining a significant noncovalent component. Importantly, these results demonstrate that increasing the aggregation state does not necessarily produce a simple additive strengthening of the pre‐existing Rg···Rg interaction but can induce a qualitative change in its physical nature depending on the molecular arrangement. Homotrimer formation was energetically favored over the corresponding dimer in most cases, suggesting possible non‐additive effects associated with aggregation that warrant further investigation. QTAIM and NCIplot analyses supported the presence and spatial distribution of the M···M contacts, while NBO and EDA revealed that orbital interactions and electrostatics constitute the major contributions to their stabilization. We believe the results reported herein will be useful for those scientists devoted to the field of physical chemistry owing to the importance of electronic structure and molecular synergies in the formation of novel supramolecular bonds.

Funding

This study was supported by MICIU/AEI (PID2023‐149492NB‐I00).

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

The file contains: Graphical representation of the total energy (E, in hartrees) vs M···M distance (dM···M, in Å) profile for complexes 5 (in blue) and 14 (in orange). Cartesian coordinates of complexes 1 to 18.

CPHC-27-e70585-s001.pdf (357.6KB, pdf)

Acknowledgments

A.B. thanks the MICIU/AEI for financial support (PID2023‐149492NB‐I00) and the “Centre de Tecnologies de la Informació” (CTI) UIB for computational support.

Data Availability Statement

The data that support the findings of this study are available in the Supporting Information of this article.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

The file contains: Graphical representation of the total energy (E, in hartrees) vs M···M distance (dM···M, in Å) profile for complexes 5 (in blue) and 14 (in orange). Cartesian coordinates of complexes 1 to 18.

CPHC-27-e70585-s001.pdf (357.6KB, pdf)

Data Availability Statement

The data that support the findings of this study are available in the Supporting Information of this article.

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