The Metallophilicity Phenomenon: The Interplay of Steric, Orbital, Dispersion, and Electronic Correlation Effects in Closed-Shell Metal–Metal Contacts

Abstract

Metallophilic interactions between closed-shell metal centers play a decisive role in the structure, stability, and functional properties of multimetallic systems, yet their physical origin remains controversial. Herein, we present a systematic theoretical study of metal–metal interactions in homo- and heterobimetallic complexes, including [MCl­(PH₃)]₂ (M = Au, Ag, Cu), [HgCl₂]­[MCl­(PH₃)] (M = Au, Ag), and [PdCl₂(PH₃)₂]­[MCl­(PH₃)] (M = Au, Ag). Correlated wave function methods (MP2 and SCS-MP2), together with a broad set of dispersion with density functional approaches, were employed to evaluate equilibrium geometries, interaction energies, and the reliability of commonly used computational models. Energy Decomposition Analysis shows that metallophilicity originates from the compensation of a repulsive steric term (electrostatic and Pauli repulsion) by stabilizing correlation/dispersion, electrostatic, and orbital contributions. Coin metal-containing complexes display enhanced orbital stabilization and weak covalent character, while Pd- and Hg-containing systems depend more strongly on correlation-driven attraction. Although dispersion with DFT methods reproduces the main qualitative trends, significant functional dependence is observed in the balance of Pauli and orbital terms. These results reconcile previous dispersion- and covalence-based interpretations within a unified framework and provide a physically transparent basis for understanding metallophilic interactions.

Introduction

Since the late 1990s, chemistry has increasingly embraced the study of complexity at the molecular scale, motivated by the recognition that collective behavior can emerge from relatively simple interacting units. − This perspective, now broadly framed within systems chemistry, emphasizes that the properties of compounds are not solely determined by its individual molecules, but rather by the organization and interplay of the interactions among them. , At the most fundamental level, noncovalent interactionsarising from electrostatic, inductive, Pauli-exchange repulsion, and correlation electronic and dispersion contributionsgovern self-assembly processes and the formation of extended molecular architectures. , Within this context, heavy metal complexes occupy a particularly intriguing position. Closed-shell d¹⁰, d⁸, and s² metal centers such as Au­(I), Ag­(I), Cu­(I), Pt­(II), Pd­(II), Hg­(II), and Tl­(I) can form stable supramolecular assemblies and inorganic complexes that display unconventional geometries, rich photophysical behavior, and remarkable structural diversity. − The versatility of coordination modes accessible to these metals has enabled a wide range of topologies, many of which are stabilized by weak but highly directional metal–metal contacts. −

These interactions, termed metallophilic interactions, and aurophilic interactions in the specific case of gold, are counterintuitive attractions between formally closed-shell metal centers. − The definition of the metallophilic is the tendency of closed-shell metal centers to form attractive metal–metal contacts at distances shorter than the sum of their van der Waals radii. For Au­(I) systems, their strength has been experimentally estimated to be comparable to that of moderate hydrogen bonds, comparable to Ag­(I), and weaker for Cu­(I). − Metallophilic interactions have been extensively characterized through solid-state X-ray diffraction − and a variety of spectroscopic techniques, including NMR and optical absorption and emission measurements. − The variations in metal–metal distances have been shown to directly modulate luminescent properties in both solution and solid-state environments, underscoring the functional relevance of these weak interactions. Despite decades of experimental and theoretical investigation, the physical origin of metallophilic interactions remains a subject of active debate. Early theoretical interpretations emphasized the role of relativistic effects and London dispersion as the primary sources of attraction, particularly for heavy elements such as gold and platinum. The metallophilic attraction has been understood as the contribution of two terms to the equilibrium distance: dispersion and ionic. − Subsequent studies proposed that orbital interactions arising from (n – 1)­d and ns/np hybridization may also contribute to stabilization. − More recently, Energy Decomposition Analyses (EDA) have highlighted the delicate balance between attractive electrostatic and orbital terms and strong short-range Pauli repulsion, suggesting that metallophilicity cannot be attributed to a single physical mechanism. − In this framework, electronic correlation plays a decisive role, as electronic correlation and dispersion are fundamentally a correlation-driven effect, although not all electronic correlation contributions can be strictly classified as dispersion.

From a computational standpoint, describing metallophilic interactions presents a significant methodological challenge. Post-Hartree–Fock methods such as MP2, Spin-Component-Scaled (SCS)-MP2, MPn, and CCSD­(T) are generally capable of capturing dispersion and electronic correlation effects with high accuracy, but their computational cost limits their applicability to relatively small systems. , Consequently, density functional theory (DFT) has become the method of choice for larger complexes. However, standard DFT approaches are affected by self-interaction and delocalization errors (DE), − which tend to artificially enhance covalency and overestimate the strength of weak interactions, particularly in closed-shell metal–metal systems. While hybrid and range-separated functionals can partially mitigate these errors, substantial inconsistencies persist across different functionals, especially when dispersion corrections are included.

Recent studies have revisited the nature of closed-shell metallophilic interactions, challenging the traditional view that these interactions are governed solely by relativistic effects and London or dispersion. , Detailed molecular orbital and Energy Decomposition Analysis (EDA) demonstrate that, in addition to dispersion, orbital interactions and Pauli repulsion critically determine both the strength and geometry of metal–metal contacts. For simple M⁺···M⁺ dimers (M = Cu, Ag, Au), attractive orbital interactions can outweigh Pauli repulsion, leading to net stabilizing metal–metal bonding contributions. In contrast, in perpendicular molecular dimers such as [H₃P–M–X]₂, increased orbital separation enhances Pauli repulsion, resulting in dominant four-electron repulsive interactions. Also, complementary experimental and theoretical investigations on hetero- and homobimetallic d¹⁰–d⁸ and d¹⁰–d¹⁰ systems in [HgCl₂]­[MCl­(PH₃)] and [PdCl₂(PH₃)₂]­[MCl­(PH₃)] (M = Ag, Au) further reveal that inner d electrons actively participate in metallophilic bonding, exhibiting tunable Lewis’s acidity and basicity. These findings establish a covalent component as an essential and directional factor in closed-shell metallophilicity, providing a unified framework that reconciles dispersion, repulsion, and orbital effects in multimetal cluster assembly. −

Given these unresolved issues, a systematic and comparative assessment of metallophilic interactions using multiple theoretical frameworks is still urgently needed. Boundary cases, where metal–metal distances approach covalent regimes while interaction energies remain within the range of classical metallophilicity, provide an ideal testing ground to disentangle the relative roles of electrostatics, Pauli repulsion, orbital interactions, and electronic correlation or dispersion. Beyond their fundamental importance, a quantitative understanding of metallophilic interactions is increasingly relevant for the rational design of functional materials containing closed-shell metal centers, including coordination polymers, supramolecular assemblies, luminescent materials, and metal–organic frameworks. − In addition, systematic data sets of interaction energies, equilibrium distances, and energy-decomposition components provide valuable benchmarks for the development of next generation computational models, including data-driven and machine-learning approaches to predict metal–metal interactions in complex chemical environments. Consequently, establishing reliable structure and energy relationships and identifying suitable computational methodologies remain important objectives in the study of metallophilic systems. −

In this work, we present a comprehensive theoretical investigation of metal–metal interactions in a series of model systems, including perpendicular dimers of the type [MCl­(PH₃)]₂ (M = Au, Ag, Cu), [HgCl₂]­[MCl­(PH₃)] (M = Au, Ag), and [PdCl₂(PH₃)₂]­[MCl­(PH₃)] (M = Au, Ag). By combining post-Hartree–Fock methods (MP2 and SCS-MP2) with a broad selection of DFT functionals and dispersion corrections, and by employing detailed Energy Decomposition Analysis, Wiberg bond indices, noncovalent interaction indexes, and topological interaction descriptors; we aim to clarify the fundamental nature of metallophilic bonding. Our results provide new insight into the balance of forces governing metal–metal attraction and offer a critical evaluation of the reliability of commonly used theoretical approaches for describing these elusive interactions.

Models and Computational Methods

The models used in this study are represented in Figure : perpendicular dimers [MCl­(PH₃)]₂ (M = Au, Ag, Cu); [HgCl₂]­[MCl­(PH₃)] (M = Au, Ag); [PdCl₂(PH₃)₂]­[MCl­(PH₃)] (M = Au, Ag). All models have C₁ symmetry. The calculations were performed at the post-Hartree–Fock (MP2 and SCS-MP2) and Density Functional Theory (DFT) with Grimme dispersion D3­(BJ) (B3LYP, PBE, TPSS, LC-wPBE, SCAN, and M06L) methods. Also, we have included double-hybrid functionals of the type PBEP86-D4 and BLYP-D4. The metal–metal interaction energy (ΔE _int) and geometric equilibrium distances (R _e) of the complexes were obtained using the counterpoise correction for the basis-set superposition error (BSSE) , from the monomeric unit. A practical justification for employing SCS-MP2 rather than CCSD­(T) in the present study is based on the balance between accuracy, computational feasibility, and methodological consistency within the energy decomposition analysis (EDA) framework. CCSD­(T) is widely regarded as the gold standard for weak intermolecular interactions, particularly when multiple systems, equilibrium scans, and comparisons across numerous density functionals are required. In contrast, SCS-MP2 offers a substantially lower computational cost while correcting the well-known tendency of conventional MP2 to overestimate dispersion-driven interactions. For noncovalent contacts, including metallophilic systems, SCS-MP2 has repeatedly shown improved interaction energies and equilibrium distances relative to standard MP2, often approaching coupled-cluster quality at a fraction of the cost. ,

1.

(a) Models [MCl­(PH₃)]₂ (M = Au, Ag, Cu), (b) [HgCl₂]­[MCl­(PH₃)] (M = Au, Ag), (c) [PdCl₂(PH₃)₂]­[MCl­(PH₃)] (M = Au, Ag).

The calculations were carried out using the Gaussian16 and ADF (Amsterdam Density Functional) program packages. The Gaussian program was used to obtain the equilibrium geometries and electronic properties. The 19 valence-electrons (VE) of Au quasi-relativistic (QR) average spin–orbit coupling of Stuttgart/Cologne pseudopotential (PP) were employed. We used 3f2g-type polarization and diffuse functions on metal. Also, the H, P, and Cl atoms were treated through PPs, using double-ζ basis sets with the addition of two d-type polarization functions. The Natural Bond Orbital (NBO) analysis Wiberg Bond Indices (WBIs) , were performed using the Gaussian16 program. All the indices (WBIs and NBO) and topological analyses here used are built from the density matrices at the MP2 and DFT levels. Quantum theory of atoms in molecules (QTAIM), noncovalent interaction index (NCI), Independent Gradient Model based on Hirshfeld partition (IGMH), and electron localization function (ELF) analysis were performed using Multiwfn 2026.1.12 code. For each model, the electron density employed in the topological analyses was obtained directly from the corresponding electronic structure calculation, ensuring consistency between the wave function and the derived real space descriptors.

To understand the origin of the metal–metal interaction in the models, the energy decomposition analysis (EDA) , was carried out at the HF, MP2, SCS-MP2, and DFT levels using the ADF program version 2022.1. The basis sets used are TZ2P with All Electrons and Hamiltonian ZORA quasi-relativistic without spin–orbit coupling. We have used the geometries optimized from Gaussian program. Hence, the interaction energy is expressed as

$$\mathrm{\Delta }{E}_{\mathrm{int}}=\mathrm{\Delta }{E}_{\mathrm{elst}}+\mathrm{\Delta }{E}_{\mathrm{Pauli}}+\mathrm{\Delta }{E}_{\mathrm{orb}}+\mathrm{\Delta }{E}_{\mathrm{corr}/\mathrm{disp}}$$ 1

where ΔE _elst, ΔE _Pauli, ΔE _orb, and ΔE _corr/disp terms account for the electrostatic, Pauli repulsion, orbital, and correlation electronic (dispersion by DFT) contributions to the interaction energy. In particular, ΔE _elst is the electrostatic interaction energy between fragments. The Pauli repulsion energy (ΔE _Pauli) is the sum of the Hartree–Fock-type exchange term (ΔE _ex) and the repulsion term (ΔE _rep) arising from the orthogonalization of orbitals. It is customary to combine ΔE _elst and ΔE _Pauli under the heading of steric interaction energy as ΔE _steric = ΔE _elst + ΔE _Pauli. The term ΔE _orb is due to orbital interaction contributions and polarization. Finally, the correlation/dispersion term, which is defined as the correlation electronic or dispersion term in the original EDA, arises from the calculations with post-Hartree–Fock methods, for example, second order Møller–Plesset perturbation theory (MP2 and SCS-MP2) methods. This term is defined as the difference between post-Hartree–Fock and Hartree–Fock interaction energies, showing the contribution of the dynamic correlation to the total interaction energy. For the DFT calculations, the last term corresponds to dispersion via D3­(BJ) approximation. In this context, SCS-MP2 represents the highest correlated wave function level that can be consistently integrated with the selected EDA formalism.

Results and Discussion

Metallophilic Interaction

To compare and systemize the analysis of the intermolecular interaction in dimers, we summarized the most relevant results in Tables –,. The geometry and coordinates of the models are provided in the Supporting Information (Table S1). We have included models calculated from previous works in Table . , The purpose is to quantify the effects of electron correlation. On the other hand, the geometries from the MP2 models are also used to obtain the energies at the Hartree–Fock level because with the latter method an equilibrium distance is never converged during optimization.

1. Metal–Metal Distances (pm), and EDA Analysis of the Interaction Energies Calculated with the ADF Program (in kcal/mol) with Basis Set TZ2P/ZORA for Models [AuCl­(PH₃)]₂ (1), [AgCl­(PH₃)]₂ (2), and [CuCl­(PH₃)]₂ (3), at Each Level of Theory .

Models	Method	M–M	ΔE elst (%)	ΔE Pauli	ΔE steric	ΔE orb (%)	ΔE corr/disp (%)	ΔE int
[AuCl(PH3)]2 (1)	HF	291.1	–27.59 (78)	+43.67	+16.08	–7.84 (22)		+8.24
	MP2	291.1	–27.59 (50)	+43.67	+16.08	–7.84 (14)	–19.97 (36)	–11.73
	SCS-MP2	299.7	–21.46 (50)	+34.21	+12.75	–6.70 (15)	–15.39 (35)	–9.34
	PBE-D3(BJ)	313.4	–19.46 (58)	+25.56	+6.28	–8.97 (27)	–4.85 (15)	–7.53
	TPSS-D3(BJ)	307.4	–21.59 (56)	+29.88	+8.29	–10.10 (26)	–7.12 (18)	–8.93
	B3LYP-D3(BJ)	318.5	–16.09 (50)	+23.11	+7.02	–6.80 (21)	–9.08 (28)	–8.66
	LC-wPBE-D3(BJ)	311.8	–17.28 (59)	+22.59	+5.30	–7.26 (24)	–4.89 (17)	–6.85
	M06L	324.8	–11.83 (66)	+11.07	–0.76	–6.02 (34)		–6.78
	SCAN-D3(BJ)	307.4	–22.23 (51)	+30.39	+8.16	–14.64 (33)	–7.12 (16)	–13.60
	PBEP86-D4	313.4	–15.55 (52)	+23.06	+7.51	–5.87 (20)	–8.24 (28)	–6.59
	BLYP-D4	318.5	–13.49 (49)	+20.04	+6.55	–5.20 (19)	–9.01 (33)	–7.66
[AgCl(PH3)]2 (2)	HF	285.8	–15.31 (72)	+26.41	+11.10	–5.82 (28)		+5.27
	MP2	285.8	–15.31 (38)	+26.41	+11.10	–5.82 (14)	–19.56 (48)	–14.29
	SCS-MP2	294.2	–11.96 (38)	+20.86	+8.90	–5.04 (16)	–14.40 (46)	–10.54
	PBE-D3(BJ)	294.2	–20.16 (57)	+25.92	+5.76	–10.14 (29)	–4.81 (14)	–9.20
	TPSS-D3(BJ)	288.2	–22.21 (55)	+30.04	+7.83	–11.36 (28)	–7.01 (17)	–10.55
	B3LYP-D3(BJ)	300.0	–15.95 (49)	+22.68	+6.73	–7.44 (23)	–8.97 (28)	–9.68
	LC-wPBE-D3(BJ)	293.9	–15.95 (56)	+21.37	+5.42	–7.86 (27)	–4.82 (17)	–7.90
	M06L	298.5	–15.03 (67)	+13.24	–1.79	–7.49 (33)		–9.28
	SCAN-D3(BJ)	288.2	–22.00 (64)	+22.88	+0.88	–10.51 (30)	–1.25 (6)	–10.88
	PBEP86-D4	294.2	–13.95 (48)	+21.36	+8.41	–5.98 (21)	–9.13 (32)	–7.71
	BLYP-D4	300.0	–11.88 (44)	+18.21	+6.33	–5.30 (20)	–9.70 (36)	–8.67
[CuCl(PH3)]2 (3)	HF	275.3	–12.95 (68)	+22.98	+10.03	–6.13 (32)		+3.90
	MP2	275.3	–12.95 (38)	+22.98	+10.03	–6.13 (18)	–14.86 (44)	–10.96
	SCS-MP2	289.9	–8.61 (36)	+15.60	+10.03	–4.84 (20)	–10.17 (44)	–8.02
	PBE-D3(BJ)	269.3	–26.60 (58)	+35.07	+8.47	–14.00 (31)	–5.19 (11)	–10.72
	TPSS-D3(BJ)	261.7	–30.60 (56)	+41.65	+11.05	–16.24 (30)	–7.49 (14)	–12.68
	B3LYP-D3(BJ)	279.9	–18.50 (50)	+26.91	+8.41	–9.12 (24)	–9.48 (26)	–10.20
	LC-wPBE-D3(BJ)	270.8	–22.46 (58)	+28.77	+6.31	–10.55 (27)	–5.16 (14)	–9.41
	M06L	270.2	–11.83 (66)	+11.07	–0.76	–6.02 (34)		–6.78
	SCAN-D3(BJ)	261.7	–30.09 (64)	+34.26	+4.17	–15.12 (32)	–1.28 (4)	–12.23
	PBEP86-D4	269.3	–18.06 (51)	+27.99	+9.93	–7.95 (23)	–9.40 (27)	–7.41
	BLYP-D4	279.9	–13.60 (46)	+21.20	+7.60	–6.47 (22)	–9.61 (32)	–8.49

^aMetal–metal distance is pm.

^bThe percentages are calculated with respect to the sum of the stabilizing contributions of ΔE _elst + ΔE _orb + ΔE _corr/disp.

^cHF calculated using the MP2 equilibrium distance. PBEP86-D4 and BLYP-D4 calculated using the PBE-D3­(BJ) and B3LYP-D3­(BJ), respectively.

2. EDA Used the ADF Program (in kcal/mol) and Basis Set TZ2P/ZORA [HgCl₂]­[AuCl­(PH₃)] (4) and [HgCl₂]­[AgCl­(PH₃)]₂ (5) at the Several Levels .

Models	Method	Hg–M	ΔE elst (%)	ΔE Pauli	ΔE steric	ΔE orb (%)	ΔE corr/disp (%)	ΔE int
[HgCl2][AuCl(PH3)] (4)	HF	290.5	–18.40 (67)	+34.0	+15.60	–9.02 (33)		+6.58
	MP2	290.5	–18.40 (41)	+34.0	+15.60	–9.02 (20)	–17.25 (39)	–10.66
	SCS-MP2	298.9	–14.18 (42)	+26.59	+12.41	–7.53 (22)	–12.33 (36)	–7.45
	PBE-D3(BJ)	308.2	–16.07 (52)	+23.16	+7.09	–10.46 (34)	–4.07 (14)	–7.44
	TPSS-D3(BJ)	303.2	–17.42 (50)	+26.57	+9.15	–11.57 (33)	–5.96 (17)	–8.38
	B3LYP-D3(BJ)	312.8	–13.35 (45)	+21.17	+7.82	–8.21 (28)	–7.81(27)	–8.21
	LC-wPBE-D3(BJ)	305.5	–14.49 (52)	+21.12	+6.63	–9.04 (33)	–4.12 (15)	–6.53
	M06L	314.6	–11.25 (59)	+12.00	+0.75	–7.93 (41)		–7.18
	SCAN-D3(BJ)	303.2	–18.13 (60)	+21.17	+3.04	–11.11 (37)	–1.00 (3)	–9.07
	PBEP86-D4	308.2	–12.04 (47)	+20.35	+8.31	–7.12 (28)	–6.45 (25)	–5.26
	BLYP-D4	312.8	–10.57 (44)	+17.87	+7.30	–6.38 (26)	–7.19 (30)	–6.28
[HgCl2][AgCl(PH3)] (5)	HF	289.6	–11.06 (61)	+24.81	+13.75	–7.17 (39)		+6.58
	MP2	289.6	–11.06 (31)	+24.81	+13.75	–7.17 (20)	–17.82 (49)	–11.24
	SCS-MP2	298.9	–8.06 (30)	+18.97	+10.91	–5.93 (22)	–12.78 (48)	–7.79
	PBE-D3(BJ)	300.0	–15.20 (51)	+22.74	+7.54	–10.86 (36)	–4.02 (13)	–7.34
	TPSS-D3(BJ)	295.8	–15.92 (47)	+25.42	+9.50	–11.85 (35)	–5.85 (17)	–8.20
	B3LYP-D3(BJ)	306.3	–11.54 (41)	+19.51	+7.97	–8.04 (29)	–7.63 (27)	–7.70
	LC-wPBE-D3(BJ)	300.0	–11.95 (48)	+18.75	+6.79	–8.56 (35)	–4.02 (16)	–5.79
	M06L	304.5	–10.81 (57)	+11.79	+0.98	–8.33 (43)		–7.34
	SCAN-D3(BJ)	295.8	–16.01 (56)	+19.06	+3.05	–11.20 (40)	–0.99 (4)	–9.14
	PBEP86-D4	300.0	–9.66 (41)	+18.79	+9.13	–6.81 (29)	–7.02 (30)	–4.70
	BLYP-D4	306.3	–7.86 (39)	+15.57	+7.71	–5.99 (30)	–6.40 (31)	–4.68

^aMetal–metal distance is pm.

^bThe percentages are calculated with respect to the sum of the stabilizing contributions of ΔE _elst + ΔE _orb + ΔE _corr/disp.

^cHF calculated using the MP2 equilibrium distance. PBEP86-D4 and BLYP-D4 calculated using the PBE-D3­(BJ) and B3LYP-D3­(BJ), respectively.

3. EDA Used the ADF Program (in kcal/mol) and Basis Set TZ2P/ZORA [PdCl₂(PH₃)₂]­[AuCl­(PH₃)] (6) and [PdCl₂(PH₃)₂]­[AgCl­(PH₃)] (7) at the Several Levels .

Models	Method	Pd–M	ΔE elst (%)	ΔE Pauli	ΔE steric	ΔE orb (%)	ΔE corr/disp (%)	ΔE int
[PdCl2(PH3)2][AuCl(PH3)] (6)	HF	351.1	–8.04 (64)	+18.63	+10.59	–4.55 (36)		+6.04
	MP2	351.1	–8.04 (27)	+18.63	+10.59	–4.55 (15)	–17.14 (58)	–11.09
	SCS-MP2	361.2	–5.87 (27)	+13.78	+7.91	–3.70 (17)	–12.47 (56)	–8.26
	PBE-D3(BJ)	365.7	–7.71 (42)	+11.03	+3.32	–5.25 (28)	–5.80 (31)	–7.74
	TPSS-D3(BJ)	360.3	–8.57 (33)	+13.95	+5.38	–8.87 (34)	–8.62 (33)	–12.10
	B3LYP-D3(BJ)	362.3	–8.09 (34)	+14.15	+6.06	–4.76 (20)	–10.81 (46)	–9.50
	LC-wPBE-D3(BJ)	362.9	–7.00 (40)	+10.46	+3.46	–4.41 (26)	–5.92 (34)	–6.86
	M06L	366.8	–5.79 (54)	+3.67	–2.12	–4.85 (46)		–6.97
	SCAN-D3(BJ)	360.3	–8.45 (53)	+8.54	+0.09	–5.74 (36)	–1.78 (11)	–7.43
	PBEP86-D4	365.7	–5.76 (33)	+11.01	+5.25	–3.67 (21)	–7.90 (46)	–6.32
	BLYP-D4	362.3	–6.44 (33)	+12.48	+6.04	–3.85 (19)	–9.43 (48)	–7.24
[PdCl2(PH3)2][AgCl(PH3)] (7)	HF	357.8	–4.06 (51)	+13.34	+9.28	–3.89 (49)		+5.39
	MP2	357.8	–4.06 (18)	+13.34	+9.28	–3.89 (17)	–15.00 (65)	–9.61
	SCS-MP2	365.5	–2.86 (16)	+10.28	+7.42	–3.32 (20)	–11.16 (64)	–7.06
	PBE-D3(BJ)	365.7	–6.10 (37)	+9.48	+3.38	–5.10 (31)	–5.52 (34)	–7.24
	TPSS-D3(BJ)	360.9	–6.34 (32)	+11.76	+5.42	–5.46 (27)	–8.15 (41)	–8.18
	B3LYP-D3(BJ)	362.4	–5.59 (28)	+11.22	+5.63	–4.30 (22)	–9.91 (50)	–8.58
	LC-wPBE-D3(BJ)	365.7	–4.80 (33)	+8.24	+3.44	–4.07 (28)	–5.52 (38)	–6.14
	M06L	363.7	–4.67 (48)	+3.08	–1.59	–5.05 (52)		–6.65
	SCAN-D3(BJ)	360.9	–6.34 (47)	+6.66	+0.32	–5.50 (40)	–1.72 (13)	–6.89
	PBEP86-D4	365.7	–3.79 (25)	+9.55	+5.76	–3.61 (24)	–7.66 (51)	–5.52
	BLYP-D4	362.4	–3.82 (24)	+9.77	+5.95	–3.53 (22)	–8.70 (54)	–6.29

^aMetal–metal distance is pm.

^bThe percentages are calculated with respect to the sum of the stabilizing contributions of ΔE _elst + ΔE _orb + ΔE _corr/disp.

^cHF calculated using the MP2 equilibrium distance. PBEP86-D4 and BLYP-D4 calculated using the PBE-D3­(BJ) and B3LYP-D3­(BJ), respectively.

4. EDA Used Ref , with the ADF Program (in kcal/mol) and Basis Set TZ2P/ZORA by the Models: [MCl­(PH₃)]₂ (M = Au, Ag, Cu), and [HgCl₂]­[AgCl­(PH₃)]₂ (5) at the Several Levels .

Models	Method	M–M	ΔE elst (%)	ΔE Pauli	ΔE steric	ΔE orb (%)	ΔE disp (%)	ΔE int
[AuCl(PH3)]2	B3LYP-D3(BJ)	315	–19.4 (50)	+29.0	+9.6	–8.6 (22)	–10.8 (28)	–9.9
[AgCl(PH3)]2	B3LYP-D3(BJ)	297	–19.4 (49)	+28.4	+9.0	–9.6 (24)	–10.8 (27)	–11.4
[CuCl(PH3)]2	B3LYP-D3(BJ)	270	–28.9 (50)	+43.1	+14.2	–14.9 (24)	–13.1 (26)	–13.8
[HgCl2][AuCl(PH3)]	LC-wPBE-D3(BJ)	316	–14.45	+20.98	+6.53	–8.89	–4.41	–6.76
[HgCl2][AgCl(PH3)]	LC-wPBE-D3(BJ)	312	–11.87	+17.84	+5.97	–8.20	–4.11	–6.34
[PdCl2(PH3)2][AuCl(PH3)]	LC-wPBE-D3(BJ)	349	–11.56	+16.00	+4.44	–6.99	–6.39	–9.33
[PdCl2(PH3)2][AgCl(PH3)]	LC-wPBE-D3(BJ)	350	–9.04	+13.72	+4.68	–6.98	–6.62	–8.44

^aMetal–metal distance is in pm.

^bThe percentages are calculated with respect to the sum of the stabilizing contributions of ΔE _elst + ΔE _orb + ΔE _corr/disp.

Table presents a comprehensive EDA of the metallophilic interactions in the [MCl­(PH₃)]₂ dimers (M = Au, Ag, Cu), enabling direct comparison among Hartree–Fock, post-Hartree–Fock, and a broad range of dispersion DFT approaches. In all models, the EDA clearly shows that the metal–metal interaction results from a delicate balance between attractive electrostatic and orbital terms and a substantial Pauli repulsion, with electrostatic interactions representing a major fraction of the attractive stabilization; whereas dynamic electronic correlation plays the decisive role in transforming an otherwise repulsive interaction into a bound metallophilic contact. At the Hartree–Fock level, all three dimers are predicted to be weakly repulsive, with positive ΔE _int values despite significant electrostatic (ΔE _elst from −13 to −28 kcal/mol) and orbital contributions (ΔE _orb from −6 to −8 kcal/mol). This demonstrates that static mean-field descriptions are insufficient to account for metallophilicity, as this behavior arises from the absence of dynamic correlations. Since metallophilic interactions are known to depend critically on dynamic electron correlation, the repulsive interaction energies predicted at the Hartree–Fock level are not unexpected. The inclusion of correlation through MP2 and SCS-MP2 provides the additional stabilization required to overcome the positive steric contribution and generate bound metal–metal contacts.

These results confirm that the electronic correlation term is essential for the emergence of net metallophilic attraction. Upon inclusion of dynamic correlation at the MP2 and SCS-MP2 levels, the interaction energies become strongly negative, reaching −9.3 kcal/mol for Au–Au, −10.5 kcal/mol for Ag–Ag, and −8.0 kcal/mol for Cu–Cu at the SCS-MP2 level. These results strongly evidence that electronic correlation is indispensable for stabilizing closed-shell metal–metal contacts. This trend is consistent with the increasing polarizability and relativistic effects along the coinage metal series and agrees well with earlier high-level wave function studies. − ,−

A key result from Table is that, although electronic correlation provides the contribution required to achieve net binding, electrostatic interactions constitute the largest individual attractive component in many of the systems studied, accounting for approximately 35–65% of the total attractive stabilization, depending on the metal and theoretical method. Orbital interactions remain significant, contributing typically between 15 and 35%. The electrostatic contribution decreases systematically along the series Au–Au > Ag–Ag > Cu–Cu and exhibits the strongest dependence on metal identity. By contrast, the correlation/dispersion contribution varies to a lesser extent, while the orbital contribution remains comparatively constant. These results indicate that changes in the electrostatic component are primarily responsible for the differences observed among the coinage-metal dimers. The larger electrostatic contribution observed for the Au–Au dimer reflects the increased polarizability of gold and the influence of relativistic effects on its valence-electron distribution. These factors favor stronger electrostatic stabilization between interacting fragments and contribute to the enhanced metallophilic character commonly observed in Au­(I) complexes compared with their Ag­(I) and Cu­(I) counterparts. The greatest Pauli repulsion is observed in gold systems, with the largest effect at the atomic level. The main difference between Au–Au and Ag–Ag is the decrease in the latter in the electrostatic contribution and Pauli repulsion (steric). The electronic correlation/dispersion terms are of similar magnitudes, and the orbital component is slightly lower for Ag. Although Au–Au interactions benefit from larger electrostatic and orbital stabilization, they are also associated with substantially greater Pauli repulsion. Consequently, the reduced steric penalty in the Ag–Ag dimer leads to a slightly more favorable balance between attractive and repulsive contributions, yielding marginally stronger net interaction energy at the SCS-MP2 level.

Notably, Au–Au interactions display a relatively larger orbital contribution compared to Ag–Ag and Cu–Cu, indicating enhanced metal–metal covalency driven by relativistic orbital contraction and improved d–s mixing. This finding supports modern interpretations that metallophilicity cannot be described as a purely dispersive phenomenon. The DFT results reveal a pronounced functional dependence, particularly in the relative balance between steric (ΔE _elst + ΔE _Pauli) and stabilizing terms. Generalized gradient and meta-GGA functionals corrected with D3­(BJ) or D4 dispersion reproduce the qualitative trends observed at the MP2 and SCS-MP2 levels but often underestimate Pauli repulsion and overemphasize orbital contributions, especially for Au systems. This behavior is particularly evident in SCAN-D3­(BJ), which systematically predicts shorter metal–metal distances and greater stabilization than wave function methods. M06L also yields strong stabilization energies, although its effect on the equilibrium metal–metal distance is less systematic across the coinage-metal series. These trends are consistent with the known tendency of some density functionals to redistribute interaction energy toward orbital stabilization due to delocalization error. Moreover, the present data extends previous analyses by demonstrating that Pauli repulsion remains a dominant opposing force even at equilibrium distances, particularly for lighter metals. For Cu–Cu interactions, the large Pauli term significantly offsets electrostatic stabilization, making dispersion and correlation indispensable for binding. This is because Cu-based metallophilic interactions are generally weaker and more sensitive to the ligand environment than those of Ag and Au. In the Supporting Information, Figure S1 shows the EDA for each level of theory in terms of their steric (ΔE _steric), orbital (ΔE _orb), and correlation/dispersion (ΔE _corr/disp) energy contributions for the models 1–3.

A direct comparison with previous benchmark EDA studies summarized in Table highlights a remarkable consistency in the qualitative decomposition pattern. Earlier works reported that, for [AuCl­(PH₃)]₂ and [AgCl­(PH₃)]₂, electrostatics accounts for approximately 45–50% of the total stabilization, orbital interactions for ∼20–25%, and dispersion for ∼25–30%. The present results reproduce these ratios within a narrow margin, despite differences in functional choice and computational setup. This agreement reinforces the robustness of the EDA framework and confirms that the relative importance of the different physical contributions is not an artifact of a specific methodology. Moreover, the results in Table , when interpreted alongside prior studies in Table , provide compelling evidence that metallophilic interactions arise from a multifaceted interplay of electrostatics, orbital interactions, Pauli repulsion, and correlation/dispersion effects. Dispersion is necessary but not sufficient to explain the magnitude, directionality, and metal dependence of these interactions. Instead, metallophilicity should be viewed as an emergent phenomenon rooted in subtle covalent contributions modulated by relativistic effects and balanced by strong steric repulsion. This unified picture reconciles earlier conflicting interpretations and provides a solid theoretical foundation for understanding metal–metal interactions in more complex multimetallic architectures.

The heterobimetallic models [HgCl₂]­[MCl­(PH₃)] (M = Au, Ag) and [PdCl₂(PH₃)₂]­[MCl­(PH₃)] (M = Au, Ag), provide a valuable extension to the homobimetallic analysis by introducing electronically asymmetric d¹⁰–d¹⁰ and d¹⁰–d⁸ metal–metal contacts. These models allow direct assessment of how metallophilic interactions evolve when symmetry is broken and when metals with different Lewis’s acidity, polarizability, and orbital availability are brought into proximity. The metal–metal distances, interaction energies, and the EDA analysis for the heterobimetallic models are listed in Tables and . For the Hg–M complexes (Table ), Hartree–Fock calculations again predict weakly repulsive or marginally stabilized interactions, with ΔE _int values remaining positive despite sizable electrostatic (from −11 to −18 kcal/mol) and orbital (from −7 to −9 kcal/mol) contributions. As observed for the homobimetallic dimers, this behavior underscores the inability of mean-field approaches to capture the stabilizing mechanisms operative in closed-shell metal–metal interactions. Inclusion of electronic correlation at the MP2 and SCS-MP2 levels leads to substantial stabilization with ΔE _int values of −10.7 and −7.4 kcal/mol for Hg–Au, and −11.2 and −7.8 kcal/mol for Hg–Ag, respectively. As expected, SCS-MP2 reduces the magnitude of the MP2 stabilization, reflecting a more balanced treatment of same-spin and opposite-spin correlation, while preserving the qualitative energetic trends and confirming that correlation remains essential for binding.

The EDA shows that electrostatic interactions remain a major stabilizing component, accounting for roughly 40–50% of the total attractive contributions, while orbital interactions contribute approximately 20–25%. Importantly, these ratios closely mirror those reported in earlier benchmark studies summarized in Table , in which similar Hg–M systems analyzed at the LC-ωPBE-D3 (BJ) level exhibited comparable balances among electrostatics, orbital interactions, and dispersion. This agreement reinforces the conclusion that the relative weight of the stabilizing terms is intrinsic to the nature of the interacting metals rather than an artifact of the computational method. Dispersion DFT methods reproduce the qualitative trends observed at the post-Hartree–Fock level but again display method-dependent variations in the magnitude of Pauli repulsion and orbital stabilization. Meta-GGA and hybrid functionals tend to reduce the steric penalty compared with MP2 and SCS-MP2, resulting in slightly more negative interaction energies at longer Hg–M equilibrium distances. Nevertheless, the overall decomposition pattern remains consistent: strong Pauli repulsion is partially compensated by substantial electrostatic and orbital stabilization, while electronic correlation provides the additional attractive contribution required to transform an otherwise repulsive interaction into a bound metallophilic contact. The Pauli repulsion decreases when an Hg complex is incorporated. The total interaction energies for Au and Ag with Hg are very similar in magnitude, whereas for the systems in Table , greater sensitivity to the change in metal is observed at the MP2 and SCS-MP2 levels. Again, in these bimetallic systems, the electrostatic contribution is greater with Au. In summary, the electron correlation/dispersion term decreases with respect to the systems in Table , and the orbital term increases marginally with Hg.

The Pd-containing models in Table represent an even more stringent test for theoretical descriptions, as they involve d¹⁰–d⁸ interactions with significantly longer metal–metal distances (between 350 and 370 pm) and weaker overall stabilization. At the Hartree–Fock level, all Pd–M complexes are predicted to be repulsive, reflecting the reduced electrostatic attraction and limited orbital overlap at these separations. The inclusion of electronic correlation through MP2 and SCS-MP2 yields total interaction energies (ΔE _int) ranging from approximately −7 to −11 kcal/mol, with Pd–Au interactions remaining more strongly bound than their Pd–Ag counterparts. A notable feature of the Pd–M systems is the reduced relative contribution of orbital interactions, which typically accounts for less than 20% of the total stabilization, compared to the homobimetallic and Hg-containing dimers. Electrostatic contributions remain significant but are counteracted by substantial Pauli repulsion, particularly for Pd–Au, where steric effects are enhanced by the involvement of filled d orbitals on both centers. Electronic correlation and dispersion terms, therefore, play an even more critical role in stabilizing these interactions, consistent with previous analyses reported in Table and recent studies on d¹⁰–d⁸ metallophilic contacts. −

Tables and show a consistent trend: as one moves from homobimetallic to heterobimetallic systems, the balance of stabilizing forces shifts progressively toward correlation/dispersion and electrostatic terms, while orbital contributions become more system-specific and sensitive to metal identity and electronic configuration. In the Supporting Information, Figure S2 shows the EDA for each method for models 4–7. Importantly, Pauli repulsion remains a dominant opposing factor in all cases, emphasizing that metallophilic interactions arise not from a single attractive mechanism but from a finely tuned compensation among multiple physical effects.

At the Hartree–Fock level, all complexes are predicted to be weakly repulsive, regardless of metal composition. This uniform behavior highlights the inability of mean-field approaches to describe metallophilic interactions and underscores the essential role of electronic correlation. Upon inclusion of electronic correlation effects using MP2 and SCS-MP2, all systems stabilize, with interaction energies systematically following the trend Ag > Au > Cu for homobimetallic dimers. Thus, in all models, electrostatic interactions constitute a major fraction of the total stabilization, while orbital interactions provide a non-negligible contribution that reflects partial covalent character. These orbital contributions are most pronounced in Au-containing systems, consistent with enhanced relativistic effects and greater orbital flexibility, and progressively diminish in lighter coinage metals and in d¹⁰–d⁸ Pd-containing complexes, where larger metal–metal separations limit effective orbital overlap. Nevertheless, even in these cases, orbital interactions remain essential for shaping the equilibrium geometry and directional character of the metal–metal contact. In the origin of the EDA scheme, the term orbital can also be considered the origin of polarization. That is, an attraction is present between neighboring molecules due to their geometry.

Electronic correlation and dispersion are essential for overcoming the strong Pauli repulsion present in all systems, particularly in heterobimetallic complexes, where steric repulsion remains significant, and dispersion becomes the key stabilizing factor. Dispersion with DFT methods reproduces the qualitative trends obtained from post-Hartree–Fock approaches but shows a functional-dependent redistribution of energy components, generally underestimating Pauli repulsion and overestimating orbital contributions. These differences arise from the distinct treatment of exchange and electron density: post-Hartree–Fock methods provide a more rigorous description of exchange and steric effects, whereas approximate DFT functionals suffer from self-interaction and delocalization errors that soften Pauli repulsion and redistribute interaction energy into orbital terms. Consequently, DFT often predicts reduced steric penalties and enhanced covalent character. These discrepancies do not imply different bonding mechanisms but rather reflect the sensitivity of energy decomposition analyses to the underlying theoretical framework, emphasizing the importance of benchmarking DFT results against wave function-based methods.

Taken together, the results in Tables – provide a unified interpretation of metallophilic interactions. Rather than being governed by a single factor, metal–metal attraction arises from a balance among stabilizing electrostatic, orbital, and electronic correlation/dispersion contributions, offset by substantial Pauli repulsion. The analysis of both homo- and heterobimetallic systems demonstrates that electronic correlation/dispersion term is necessary but not sufficient, while electrostatic and orbital interactions are equally important and strongly system dependent. Pauli repulsion remains a decisive factor controlling both interaction strength and equilibrium geometry. The differences between post-Hartree–Fock and DFT descriptions mainly reflect their treatment of exchange and electron density, with DFT generally reducing steric penalties by softening exchange-driven repulsion and increasing orbital contributions.

Steric Interaction as the Combined Effect of Electrostatics and Pauli Repulsion

Within the EDA framework, the so-called steric interaction term (ΔE _steric) is defined as the sum of the electrostatic interaction (ΔE _elst) and the Pauli repulsion (ΔE _Pauli), i.e.,

$${\Delta E}_{\mathrm{steric}}={\Delta E}_{\mathrm{elst}}+{\Delta E}_{\mathrm{Pauli}}$$ 2

Although these two contributions are often discussed separately, their combination provides a more physically direct description of the short-range interactions governing equilibrium geometries in closed-shell metal–metal systems. The electrostatic term represents the classical Coulomb interaction between the unperturbed charge densities of the interacting fragments, and is generally attractive for metallophilic systems. In contrast, the Pauli repulsion arises from the antisymmetrization of the total wave function and reflects the destabilizing exchange repulsion between occupied orbitals on different fragments. Both terms originate from the Hartree–Fock level of theory and together define the noncovalent, noncorrelation part of the interaction prior to orbital relaxation and electronic correlation effects. By grouping ΔE _elst and ΔE _Pauli into a single steric term, EDA emphasizes that at short intermetallic distances these two effects cannot be interpreted independently. In metallophilic interactions, the attractive electrostatic contribution is systematically overcompensated by the rapidly increasing Pauli repulsion as metal centers approach each other. As a result, ΔE _steric is positive (repulsive) at equilibrium distances, even in systems that exhibit a net attractive interaction energy once orbital relaxation and correlation effects are included.

From a structural perspective, ΔE _steric plays a decisive role in determining equilibrium metal–metal distances. While electrostatics favors shorter separations, Pauli repulsion imposes a strict spatial constraint that limits how close two closed-shell metal centers can approach. The equilibrium geometry, therefore, corresponds to a compromise in which steric repulsion is minimized while allowing sufficient stabilization from orbital interactions and electronic correlation/dispersion terms. In this sense, ΔE _steric acts as a geometric regulator, defining the region of configuration space in which metallophilic interactions can occur. In the context of closed-shell metal–metal interactions, interpreting electrostatics and Pauli repulsion as a combined steric term avoids misleading conclusions that might arise from considering only attractive contributions. A large electrostatic stabilization does not necessarily imply strong bonding if it is accompanied by an even larger Pauli penalty. Thus, ΔE _steric provides a physically meaningful measure of the intrinsic resistance of the system to metal–metal contact, against which orbital and electronic correlation/dispersion interactions must compete to produce net stabilization.

For the dimers in Table , ΔE _steric is consistently positive at equilibrium geometries for Au–Au, Ag–Ag, and Cu–Cu interactions, as depicted in Figure S3 in Supporting Information. Although the electrostatic component is attractive, it is systematically overcompensated by the rapidly increasing Pauli repulsion, leading to a net steric penalty. This shows that, from a purely steric perspective, none of the coinage-metal dimers would be bound. Consequently, the existence of metallophilic attraction in these systems necessarily requires stabilization from terms beyond ΔE _steric. A similar situation is observed in the systems from Tables and . For the Hg–M and Pd–M complexes, the electrostatic contribution is generally comparable to or smaller than that found for the homobimetallic dimers. Nevertheless, when combined with Pauli repulsion, the resulting steric term (ΔE _steric = ΔE _elst + ΔE _Pauli) remains positive in all cases, indicating that short-range metal–metal contact is intrinsically repulsive before orbital relaxation and electronic correlation are included. See Figure S4 in Supporting Information. In the Pd–M systems, where metal–metal distances are longer and orbital overlap is reduced, the steric term still imposes a substantial energetic penalty, underscoring that Pauli repulsion remains effective even at relatively large separations. Thus, across all systems studied, ΔE _steric defines an intrinsic resistance to metal–metal contact that is largely independent of whether the interaction is d¹⁰–d¹⁰ or d¹⁰–d⁸ in nature.

The use of a unified steric term highlights that metallophilic interactions are not driven solely by favorable short-range forces but rather arises from the compensation between strong steric repulsion and multiple longer-range attractive contributions. This perspective is particularly valuable for rationalizing why metallophilic interactions remain weak, highly distance-sensitive, and strongly method-dependent, despite the presence of significant electrostatic and orbital stabilization.

Steric Term versus Orbital and Electronic Correlation/Dispersion Terms

When compared with the other two EDA contributions, the role of the steric term becomes even clearer. The orbital interaction term (ΔE _orb) partially compensates the steric penalty, particularly in Au systems where relativistic effects enhance orbital flexibility and polarization. However, ΔE _orb alone is never sufficient to overcome ΔE _steric, especially in Ag- and Cu-based complexes, or in heterobimetallic systems with reduced orbital overlap. Orbital interactions therefore act as a secondary, modulating contribution, rather than the primary driving force of metallophilicity. Figure shows the EDA for each method in steric (ΔE _steric), orbital (ΔE _orb), and electronic correlation/dispersion (ΔE _corr/disp) terms for the models 1–3. In contrast, the electronic correlation/dispersion term (ΔE _corr/disp) is the only contribution that consistently exceeds the steric penalty across Tables –. Figure shows the EDA for each method in steric (ΔE _steric), orbital (ΔE _orb), and correlation/dispersion (ΔE _corr/disp) terms for the models 4–7. For all systems, net stabilization is achieved only when ΔE _corr/disp is added to the interaction energy, demonstrating that electronic correlation and dispersion are indispensable for close-shell contact. However, electronic correlation/dispersion does not operate in isolation: its stabilizing effect must first compensate for the steric repulsion before any additional stabilization from orbital interactions can become effective.

2.

Energy Decomposition Analysis (EDA) of the homobimetallic dimers [MCl­(PH₃)]₂ (M = Au, Ag, Cu; models 1–3). Shown are the steric (ΔE _steric), orbital (ΔE _orb), and correlation/dispersion (ΔE _corr/disp) contributions (kcal/mol) obtained at different theoretical levels. The steric term is repulsive in all cases, while correlation/dispersion provides the dominant stabilization responsible for metallophilic attraction.

3.

Comparative Energy Decomposition Analysis (EDA) for the heterobimetallic complexes [HgCl₂]­[MCl­(PH₃)] (M = Au, Ag; models 4–5, top) and [PdCl₂(PH₃)₂]­[MCl­(PH₃)] (M = Au, Ag; models 6–7, bottom). Shown are the steric (ΔE _steric), orbital (ΔE _orb), and correlation/dispersion (ΔE _corr/disp) contributions (kcal/mol) obtained at different theoretical levels. In all cases, the steric term is repulsive, while correlation/dispersion provides the dominant stabilization of the metal–metal interaction.

Thus, the comparison of ΔE _steric with ΔE _orb and ΔE _corr/disp reveals a hierarchy of physical effects governing metallophilic interactions. The steric term acts as the baseline constraint, defining how unfavorable metal–metal contact is at short-range. Orbital interactions provide a limited and system-dependent correction to this penalty, while electronic correlation and dispersion supply the decisive stabilization required to overcome it. This balance is consistently observed across all models in Tables –, and explains why metallophilic interactions remain weak, distance-sensitive, and highly dependent on electronic correlation, despite the presence of attractive electrostatic and orbital contributions.

In this sense, interpreting the EDA results solely through the steric term highlights that metallophilicity is not driven by favorable short-range interactions but rather emerges from the compensation of a fundamentally repulsive steric interaction by longer-range electronic correlation and, to a lesser extent, orbital effects. This perspective provides a unified, physically direct interpretation of the source of intermolecular stabilization between metal centers derived from the EDA scheme, thereby providing a rationale for the trends observed across homo- and heterobimetallic systems.

Topological Description of the Forces Governing the Metallophilic Interactions

To generate a qualitative picture of the energetic contributions to the metallophilic interaction within the intermolecular complexes from the three sets of models, we carried out the Independent Gradient Model based on Hirshfeld partition (IGMH) and noncovalent interactions (NCI) analyses. The IGMH enables the characterization of intermolecular interactions by partitioning the real electron density into atomic fragments using the Hirshfeld scheme. To this end, the IGMH analysis defines regions of intermolecular interactions to determine the independent gradient function (δg), and the inter- and intra-fragments terms (δg ^inter and δg ^intra). Thus, the IGMH analysis provides a more (than NCI index) accurate description of the interactions at the interface between the homo- and heterometallic complexes. These approaches allow the generation of a real-space description of noncovalent interactions, represented as surfaces that, in this case, are taken as a measure of dispersion and correlation effects. To characterize the weak interactions between the metal centers, we integrate the reduced density gradient (RDG) over the corresponding spatial domains. RDG domains correspond to the regions enclosed by an RDG isosurface of 0.5. In this case, the domains of interest are those located between the interfaces of the metal centers. From these surfaces, it is possible to integrate the RDG domains located between the metal centers, thereby providing the number of electrons (q _NCI) in the interatomic space. The reduced density gradient (RDG) is a dimensionless quantity derived from the electron density and its spatial derivatives that highlights regions where the electron density is low but changes slowly in space. Such regions are typically associated with weak intermolecular interactions, including dispersion, metallophilic contacts, hydrogen bonds, and van der Waals interactions. Integration of the RDG domain provides the q _NCI descriptor, which measures the amount of electron density enclosed within the interaction region. Chemically, larger q _NCI values indicate greater accumulation of electron density within the intermolecular region and generally reflect a more pronounced interfacial interaction. Conversely, smaller q _NCI values correspond to weaker electron density accumulation and a less extensive interaction region. RDG based quantities describe the spatial distribution and extent of the interaction throughout the entire intermolecular region.

As a complement, the quantum theory of atoms in molecules (QTAIM) and electron localization function (ELF) were calculated to evaluate the nature of the metallophilic interaction of each dimer. Then, the Wiberg bond index was used to estimate the degree of covalency of the metal–metal interactions. The results from these analyses for the homobimetallic complexes at the SCS-MP2 level of theory are shown in Figure . In IGMH analysis, blue indicates strong attractive interactions, green indicates electronic correlation interactions, and red indicates steric repulsive interactions. Thus, the intermolecular region is dominated by weak attractive forces between the subunits of the dimer, whereas the intermetallic space demonstrates the strong attractive character of this non–covalent interaction, particularly in systems with Au. The integration of the RDG domains (q _NCI) yielded values of 0.0096, 0.0089, and 0.0077 for Au, Ag, and Cu, respectively. The AIM provides a framework for relating the magnitudes of the densities at the bond critical points (BCPs) to certain types of interactions. The magnitudes of the BCPs are in units of (|e̅|·bohr^–3), which are presented as (ρ × 100) to facilitate the comparative analysis. A BCP value between 4 > (ρ × 100) > 1 is indicative of a strong noncovalent interaction, while below (ρ × 100) < 1 points to weak noncovalent interactions. The surfaces are colored according to the sign function (ρ) on the same scale used with the IGMH analysis.

4.

Graphical representation of the metallophilic interactions for the homobimetallic dimers [MCl­(PH₃)]₂ (M = Au, Ag, Cu). Each row corresponds to a homometallic dimer (Au–Au, Ag–Ag, and Cu–Cu). (a) Intermolecular interactions shown as IGMH isosurfaces (isovalue of 0.0005 au). (b) Noncovalent metallophilic interactions through NCI surfaces. The values for the BCP, Wiberg bond order, and the integrated RDG domain in the intermetallic space (q _NCI) are included. (c) ELF contour maps along the planes containing the metal–metal interface, the color bar. The ELF color scale ranges from 0 to 1. In IGMH analysis, blue indicates strong attractive interactions, green indicates electronic correlation interactions, and red indicates steric repulsive interactions. WBO is Wiberg Bond Order.

In our case, for the homobimetallic complexes, all values correspond to strong noncovalent interactions, with the Au–Au system showing a magnitude of 3.25, while the Cu–Cu model yielded a BCP of 1.90. The ELF counter map was performed to gain insight into the degree of electron pairing and localization in the regions between the metal centers. In the homobimetallic complex, the electrons are localized at the metal–metal interface, with more pronounced localization in the Au–Au complex than in the Ag–Ag and Cu–Cu complexes. The bond order evidence a minor orbital contribution as exposed by the EDA analysis, with values below the magnitude associated with a single bond, specifically, 0.72, 0.50, and 0.39 for Au–Au, Ag–Ag, and Cu–Cu, respectively. The BCP values follow the order Au–Au > Ag–Ag > Cu–Cu, in agreement with the q _NCI values and Wiberg bond indices, indicating greater electron-density accumulation and a stronger metal–metal contact in the Au-containing dimer. However, this trend should not be directly correlated with the steric contribution, since ΔE _steric reflects the balance between electrostatic attraction and Pauli repulsion rather than the electron density localized at the bond critical point. At the SCS-MP2 level, ΔE _steric follows the order Au–Au > Cu–Cu > Ag–Ag. Thus, the stronger net stabilization of the Ag–Ag dimer arises from the overall balance among steric, orbital, and correlation contributions, rather than from having the lowest steric term in the Cu–Cu system.

The dimers modeling heterobimetallic interactions between Hg–M (M = Au, Ag) metal centers showed a strong noncovalent interaction, represented by the blue IGMH surface in the intermetallic space shown in Figure . The integration of the RDG domain (q _NCI) determines the presence of electronic density in the intermetallic space, with values of 0.0082 and 0.0055 for Au and Ag, respectively. Meanwhile, the bond order showed a slight decrease relative to the homobimetallic counterparts, although the low covalent character is a general feature of the dimers. The BCPs show values similar to those obtained for the homobimetallic complexes, thereby confirming the strong character of the noncovalent interaction. The ELF contour map shows that, in a similar way to the homobilmetallic, the complex that contains Au in its structure has slightly more localized electrons than the Hg–Ag complex. The q _NCI and BCP values follow trends similar to those observed for the Wiberg bond indices and reflect the extent of electron density accumulation in the intermetallic region. These descriptors provide qualitative evidence for the relative strength of the metal–metal contacts and the degree of electron sharing between the interacting centers. However, they should not be directly correlated with the steric contribution obtained from EDA, since ΔE _steric reflects the energetic balance between electrostatic attraction and Pauli repulsion rather than the local electron density distribution. Despite the interaction energies between these two dimers being lower than 1.0 kcal/mol, the magnitudes follow distinct behavior depending on the metal. At the SCS-MP2 level, the dimer with Au shows a higher orbital contribution compared to the dimer with Ag, in line with the slight increase in the bond order with Au. However, the Ag dimer exhibits a lower steric contribution to the interaction than its counterpart.

5.

Graphical representation of the metallophilic interactions for the heterobimetallic dimers [HgCl₂]­[MCl­(PH₃)] (M = Au, Ag). Each row corresponds to a heterometallic dimer (Hg–Au and Hg–Ag). (a) Intermolecular interactions shown as IGMH isosurfaces (isovalue of 0.0005 au). (b) Noncovalent metallophilic interactions through NCI surfaces. The values for the BCP, Wiberg bond order, and the integrated RDG domain in the intermetallic space (q _NCI) are included. (c) ELF contour maps along the planes containing the metal–metal interface. The ELF color scale ranges from 0 to 1. In IGMH analysis, blue indicates strong attractive interactions, green indicates electronic correlation interactions, and red indicates steric repulsive interactions. WBO is Wiberg Bond Order.

According to the IGMH and NCI analyses, the models for the heterobimetallic interaction between [MCl­(PH₃)] (M = Au, Ag) and [PdCl₂(PH₃)₂] showed a dominant nature based on attractive non-covalent interactions, as graphically depicted by the green surfaces in Figure . The Pd–M systems display weaker topological signatures in the intermetallic region, as reflected by their lower BCP and q _NCI values compared with the Hg–M and homobimetallic systems. However, this does not imply that their total interaction energies are necessarily weaker. In fact, the Pd–Au complex exhibits a SCS-MP2 interaction energy comparable to, and slightly more stabilizing than, that of the Hg–Au complex. Therefore, the Pd–M contacts should be described as metallophilic interactions with reduced local electron-density accumulation but with significant energetic stabilization arising from combined electrostatic, orbital, and correlation contributions.

6.

Graphical representation of the metallophilic interactions for the heterobimetallic dimers [PdCl₂(PH₃)₂]­[MCl­(PH₃)] (M = Au, Ag). Each row corresponds to a heterometallic dimer (Pd–Au and Pd–Ag). (a) Intermolecular interactions shown as IGMH isosurfaces (isovalue of 0.0005 au). (b) Noncovalent metallophilic interactions through NCI surfaces. The values for the BCP, Wiberg bond order, and the integrated RDG domain in the intermetallic space (q _NCI) are included. (c) ELF contour maps along the planes containing the metal–metal interface. The ELF color scale ranges from 0 to 1. In IGMH analysis, blue indicates strong attractive interactions, green indicates electronic correlation interactions, and red indicates steric repulsive interactions. WBO is Wiberg Bond Order.

The integration of the RDG domains (q _NCI) reveals a decrease in the strength of the electronic correlation contribution to the metal–metal interaction compared with other models for heterobimetallic and homobimetallic complexes. In terms of covalency, the low bond order values indicate a lack of covalent bonding in these complexes, with 0.26 for Au and 0.18 for Ag. This is also evident in the ELF contour map. The magnitudes of the BCPs at the interatomic region are 0.94 for Au and 0.68 for Ag, indicative of a weak non–covalent interaction, in the same line as the IGMH and NCI analyses. For these dimers, the difference in interaction energy at the SCS-MP2 level is close to 1.0 kcal/mol, due to a systematic decrease across all contributions. In general, the magnitudes of the interaction energies calculated for these dimers are similar to those calculated for the other heterobimetallic dimers with Hg, despite the decrease in the orbital contribution in the dimers with Pd. This is a consequence of the decrease in the penalty caused by the steric contribution in the latter dimers, which compensates for the decrease in ΔE _orb.

Conclusion

This study demonstrates that metallophilic interactions in closed-shell homo- and heterobimetallic systems arise from a delicate balance of competing physical effects rather than from a single bonding mechanism. Correlated methods (MP2 and SCS-MP2) show that dynamic electronic correlation is essential to obtain stable metal–metal contacts, whereas Hartree–Fock predicts repulsive interactions. Energy Decomposition Analysis reveals that the electronic correlation/dispersion term provides the dominant stabilizing contribution, but the electrostatic and orbital terms remain significant and strongly system- dependent. In particular, gold complexes exhibit enhanced orbital stabilization and weaker covalent character due to relativistic effects, which explains their stronger interactions relative to Ag- and Cu-containing analogs. The results presented in this study allow us to understand the contribution of hybridization to its orbital component as originally proposed by R. Hoffmann’s group. A comparison between post-Hartree–Fock and DFT EDA further reveals that approximate density functionals often soften Pauli repulsion and redistribute energy among orbital terms due to self-interaction and delocalization errors. Although many dispersion functionals reproduce qualitative trends, their quantitative partitioning of the interaction energy remains a method. Benchmarking against correlated wave function methods is therefore essential when discussing the physical origin of metal–metal attraction.

A key conceptual result is that the steric term is repulsive at equilibrium geometries in all models, indicating that metallophilicity emerges only when this intrinsic resistance to closed-shell contact is compensated by longer-range attractive contributions, mainly the electronic correlation/dispersion term and, to a lesser extent, orbital relaxation. Although dispersion with DFT methods reproduces general trends, their energy partitioning remains functionally dependent because approximate exchange often underestimates Pauli repulsion. Overall, these findings unify previous dispersion-, electrostatic-, and covalency-based interpretations into a consistent framework for understanding and designing metal–metal interactions in the complexes. Beyond elucidating the physical origin of metallophilic interactions, the present results provide a benchmark data set for evaluating computational methods and interpreting closed-shell metal–metal contacts in more complex systems. The interaction energy–distance relationships and energy of the decomposition data reported herein may also serve as valuable references for future studies of coordination materials and data-driven models of metallophilic interactions.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.inorgchem.6c02312.

• Figure S1: Energy Decomposition Analysis (EDA) of the homobimetallic dimers [MCl­(PH₃)]₂ (M = Au, Ag, Cu; models 1–3). Shown are the electrostatic (ΔE _elst), Pauli repulsion (ΔE _Pauli), orbital (ΔE _orb), correlation/dispersion (ΔE _corr/disp), and interaction energy (ΔE _int) contributions (kcal/mol) obtained at different theoretical levels; Figure S2: comparative Energy Decomposition Analysis (EDA) for the heterobimetallic complexes [HgCl₂]­[MCl­(PH₃)] (M = Au, Ag; models 4–5, top) and [PdCl₂(PH₃)₂]­[MCl­(PH₃)] (M = Au, Ag; models 6–7, bottom) and shown are the electrostatic (ΔE _elst), Pauli repulsion (ΔE _Pauli), orbital (ΔE _orb), correlation/dispersion (ΔE _corr/disp), and interaction energy (ΔE _int) contributions (kcal/mol) obtained at different theoretical levels; Figure S3: Energy Decomposition Analysis (EDA) of the steric components for the homobimetallic dimers [MCl­(PH₃)]₂ (M = Au, Ag, Cu; models 1–3) and shown are the electrostatic (ΔE _elst), Pauli repulsion (ΔE _Pauli), and steric (ΔE _steric) contributions (kcal/mol) obtained at different theoretical levels; Figure S4: Comparative Energy Decomposition Analysis (EDA) for the heterobimetallic complexes [HgCl₂]­[MCl­(PH₃)] and [PdCl₂(PH₃)₂]­[MCl­(PH₃)] (M = Au, Ag; models 4–7) and shown are the electrostatic (ΔE _elst), Pauli repulsion (ΔE _Pauli), and steric (ΔE _steric) contributions (kcal/mol) obtained at different theoretical levels; Table S1: Cartesian coordinates (in Angstroms) for the optimized geometries of the models 1–7 studied in this work at the different methods (PDF)

S.M.-R., N.G.-S., and F.M. designed research, analyzed data, performed research, and wrote the manuscript.

Financial support of this work was provided under Fondecyt projects 1260067and1262462. Powered@NLHPC: This research was partially supported by the supercomputing infrastructure of the NLHPC (ECM-02).

The authors declare no competing financial interest.