DFT Investigation of CO₂ Adsorption on Cu₄ and Sc₄ Clusters: Effects of Functional Choice, Spin State, and Vibrational Stability

Abstract

CO₂ adsorption on subnanometric metal clusters is highly sensitive to the computational protocol used to describe the potential energy surface, particularly when several low-lying geometries and spin states are accessible. In this work, CO₂ adsorption on Cu₄ and Sc₄ clusters was investigated using density functional theory (DFT) to evaluate how the choice of functional/basis-set protocol, spin multiplicity, initial geometry, and vibrational stability affects the predicted adsorption behavior. Four representative computational protocols (TPSSh, r²SCAN-3c, PBE-D4/def2-TZVP, and PBE0-SDD) were assessed for isolated clusters and cluster–CO₂ complexes. The lowest harmonic vibrational frequency, ω_min, was used as a diagnostic criterion to distinguish true minima from unstable or weakly defined stationary points. Selected cases were also cross-checked using the ORCA and Gaussian quantum-chemistry packages to assess whether comparable computational settings yielded consistent stationary-point character. The results show that Cu₄ generally exhibits weak CO₂ binding, whereas Sc₄ displays stronger but more protocol-dependent adsorption, consistent with its higher structural flexibility and more pronounced Lewis-acid character. Low-frequency and imaginary modes were found in several optimized structures, indicating that adsorption energies should not be interpreted without prior vibrational validation. The comparison also shows that variations in functional/basis-set treatment and spin multiplicity can alter both the optimized geometry and the predicted adsorption strength. Therefore, CO₂ adsorption on small metal clusters should be discussed using combined structural, vibrational, and energetic criteria rather than electronic adsorption energies alone. Overall, this study provides a protocol-oriented framework for evaluating the reliability of DFT predictions in CO₂ adsorption on Cu₄ and Sc₄ clusters.

1. Introduction

CO₂ capture and utilization are central topics in materials chemistry and catalysis because they address both emission mitigation and the transformation of CO₂ into value-added carbon feedstocks [1,2,3]. Beyond its environmental relevance, CO₂ conversion is directly connected with the development of circular chemical processes in which carbon-containing waste streams can be reintegrated into productive routes [4,5,6,7]. At the molecular level, the first step in many capture and activation processes is the interaction of CO₂ with a metallic site. This interaction controls the adsorption strength and may induce structural perturbations in the molecule, including variation in the O–C–O angle and elongation of the C–O bonds [8,9,10]. These geometric changes are commonly interpreted as early indicators of CO₂ activation. However, their magnitude and chemical meaning depend strongly on the nature of the metal center and on the theoretical description used to model the system. In metal–CO₂ interactions, electrostatics, polarization, charge transfer, metal-to-CO₂ back-donation into π* orbitals, and dispersion forces can contribute simultaneously. Therefore, the predicted adsorption mode and adsorption energy are not purely structural outputs, but are directly affected by the computational protocol [11].

Small metal clusters provide molecular-scale models of undercoordinated active sites and can bridge the gap between isolated coordination complexes and extended metallic surfaces. Their low nuclearity makes them computationally accessible, while their electronic structure remains sufficiently complex to exhibit several relevant phenomena, including multiple low-lying isomers, soft vibrational modes, and closely spaced spin states [12,13,14]. In this context, Cu₄ and Sc₄ were selected because they represent two chemically contrasting cases for CO₂ adsorption. Copper clusters are expected to show relatively weak and often reversible interactions with CO₂ due to the more filled character of the Cu d manifold and the limited tendency of Cu to polarize the adsorbate strongly. In contrast, scandium clusters are more electropositive and can behave as stronger Lewis-acidic sites, favoring CO₂ polarization through oxygen coordination and stronger metal–adsorbate electrostatic interactions. Therefore, the Cu₄/Sc₄ comparison is not arbitrary; it allows the evaluation of two distinct adsorption regimes: weak adsorption dominated by modest polarization in Cu₄ and stronger, more structurally adaptive adsorption in Sc₄.

Tetrameric clusters were chosen as minimal but chemically meaningful models. Compared with dimers or trimers, M₄ clusters can support planar and three-dimensional arrangements, symmetry changes, and spin-dependent structural reorganization. At the same time, they remain small enough to permit systematic testing of several density functional theory (DFT) protocols, spin multiplicities, and initial geometries. This balance is important because subnanometric clusters often have relatively flat potential energy surfaces, where small changes in the exchange–correlation functional, basis set, effective core potential (ECP), dispersion correction, integration grid, convergence criteria, or optimization algorithm can lead to different stationary points [15]. Consequently, a single optimized structure or a single electronic adsorption energy may not be sufficient to define a reliable adsorption trend.

From a methodological perspective, benchmarking is necessary to avoid overinterpreting results arising from a particular functional/basis-set combination rather than from the intrinsic chemistry of the cluster–CO₂ system [16,17]. This is especially relevant for small transition-metal clusters, where geometry, spin state, and vibrational stability are strongly coupled. In this work, the term “computational protocol” refers to the combined use of the exchange–correlation functional, basis set, ECP when applicable, dispersion treatment, and numerical settings. Four representative protocols—TPSSh, r²SCAN-3c, PBE-D4/def2-TZVP, and PBE0-SDD—were evaluated to assess their influence on the optimized geometries and adsorption energies of Cu₄–CO₂ and Sc₄–CO₂ complexes. The use of r²SCAN-3c provides a compact composite DFT approach, PBE-D4/def2-TZVP allows evaluation of a generalized gradient approximation with explicit dispersion and a triple-ζ basis set, TPSSh provides a meta-hybrid reference commonly used in transition-metal chemistry, and PBE0-SDD allows the effect of a hybrid functional combined with an ECP-based description of the metal centers to be assessed [18,19,20].

The present study aims to evaluate the sensitivity of CO₂ adsorption on Cu₄ and Sc₄ clusters to the choice of DFT protocol, spin multiplicity, initial geometry, and vibrational stability. The lowest harmonic frequency, ω_min, was used as an operational descriptor to distinguish true minima from unstable stationary points or shallow regions of the potential energy surface. In addition, selected calculations were compared between the ORCA and Gaussian quantum-chemistry packages. This comparison was not intended as a general benchmark of software performance, but as a limited cross-validation of whether comparable computational settings produce consistent stationary-point character. To support the methodological assessment, Cu₂ was used as an auxiliary reference system through experimentally comparable observables, including r(Cu–Cu), vibrational frequency, ionization potential, and electron affinity. Overall, this work seeks to establish a reliability-oriented framework for interpreting CO₂ adsorption on small metal clusters, emphasizing that adsorption energies should be evaluated together with geometric descriptors, vibrational validation, and protocol-dependent uncertainty [21].

2. Results and Discussion

2.1. Geometric Reorganization of Cu₄ After Optimization

The initial and optimized Cu₄ structures were compared for three starting symmetries and two spin multiplicities to evaluate structural reorganization and protocol-dependent geometric sensitivity, as summarized in Figure 1.

Figure 1 indicates that Cu₄ cannot be treated as a rigid tetrameric unit, even before CO₂ adsorption. The optimized geometry depends simultaneously on the initial symmetry, spin multiplicity, and computational protocol, confirming that the Cu₄ potential energy surface contains several closely related structural basins. For the D₂h starting geometry, the rhombohedral framework is generally preserved after optimization; however, the magnitude of the Cu–Cu contraction or elongation varies across the evaluated methods. This behavior indicates that each protocol describes the balance between metal–metal bonding, exchange–correlation effects, and spin-dependent electron redistribution differently.

In contrast, Figure 1 shows that the D₃h starting geometry is more likely to lose its initial symmetry during optimization. Several optimized structures evolve toward lower-symmetry arrangements, indicating that not all protocols stabilize the D₃h motif equally. This structural instability is consistent with a shallow region of the potential energy surface, where small changes in the functional treatment or spin state can redirect the optimization toward neighboring minima. Therefore, adsorption energies obtained from D₃h-derived structures must be interpreted cautiously, because differences in Eads may arise not only from the metal–CO₂ interaction itself, but also from changes in the underlying Cu₄ isomer [12,22,23].

For the Td starting geometry, the three-dimensional character is more clearly retained, although method-dependent distortions remain. This result is relevant because three-dimensional Cu₄ motifs may expose coordination environments for CO₂ adsorption that differ from those available in planar geometries. Overall, Figure 1 establishes the structural basis for the subsequent adsorption analysis. Before comparing Eads values, it is necessary to verify whether each protocol preserves the same cluster topology or drives the system toward a different optimized minimum. This justifies the later use of vibrational validation and stationary-point analysis as mandatory filters for reliable interpretation of CO₂ adsorption on Cu₄ [19,22,24].

2.2. Geometric Metrics for Cu₄ D₂h and Method and Spin Effect

The D₂h Cu₄ structures were analyzed to determine whether the optimized metal framework preserves the bond equivalence of the initial geometry or undergoes protocol-dependent contraction. This comparison, summarized in

Table 1. Cu–Cu bond distances for the initial and optimized D₂h Cu₄ clusters calculated using different DFT protocols and spin multiplicities.

Cu4	R (1-3)	R (1-2)	R (2-4)	R (4-3)
D2h/singlet (Without opt)	2.40348	2.40348	2.40348	2.40348
D2h/triplet (Without opt)	2.40348	2.40348	2.40348	2.40348
D2h/PBE0-SDD/singlet (opt)	2.38499	2.38455	2.38499	2.38453
D2h/PBE0-SDD/triplet (opt)	2.33511	2.33478	2.33504	2.33488
D2h/PBE-D4-/singlet (opt)	2.41662	2.41613	2.41662	2.41613
D2h/PBE-D4/triplet (opt)	2.36868	2.36863	2.36868	2.36864
D2h/r2SCAN-3C/singlet (opt)	2.36606	2.36475	2.36605	2.36476
D2h/r2SCAN-3C/triplet (opt)	2.32235	2.30765	2.31506	2.31500
D2h/TPSSh/singlet (opt)	2.41022	2.40927	2.41022	2.40927
D2h/TPSSh/triplet (opt)	2.36061	2.36045	2.36060	2.36046

, is necessary before interpreting CO₂ adsorption because changes in the isolated cluster geometry modify the electronic structure and the local coordination environment available for adsorbate binding.

As shown in Table 1, the optimized D₂h Cu₄ structures exhibit a clear dependence on both the computational protocol and spin multiplicity. The initial geometry contains four equivalent Cu–Cu distances of 2.40348 Å, whereas geometry optimization produces method-dependent contraction or slight elongation of the metallic framework. In the singlet state, PBE-D4 and TPSSh preserve relatively long Cu–Cu distances of ~2.41 Å, indicating a less compact metal framework. In contrast, r²SCAN-3c and PBE0-SDD predict shorter Cu–Cu contacts. This contraction becomes more pronounced under triplet conditions, where selected Cu–Cu distances decrease to approximately 2.31–2.33 Å.

This trend indicates that the description of Cu–Cu bonding is sensitive to the exchange–correlation functional, basis/ECP treatment, and spin configuration. The stronger contraction observed for the triplet structures suggests that spin-state changes can modify the distribution of electron density within the Cu₄ framework and, consequently, the availability of frontier metal states involved in CO₂ polarization and metal-to-CO₂ donation/back-donation interactions. Therefore, Table 1 provides more than a geometric comparison; it establishes the structural reference required to interpret subsequent adsorption energies. If the isolated cluster is described with different degrees of compactness depending on the method, variations in Eads may reflect not only the metal–CO₂ interaction, but also differences in the initial electronic structure and relaxation capacity of the Cu₄ cluster [10,19,25,26].

2.3. Geometric Instability of Cu₄ D_3h and Loss of Symmetry

The D₃h Cu₄ topology was examined to evaluate whether this higher-symmetry arrangement corresponds to a stable structural motif or to a shallow region of the potential energy surface that relaxes toward lower-symmetry configurations. The corresponding Cu–Cu structural descriptors are summarized in

Table 2. Cu–Cu structural descriptors for the initial and optimized D₃h Cu₄ clusters calculated using different DFT protocols and spin multiplicities.

Cu4	R (1-2)	R (1-4)	R (2-4)	R (4-3)
D3h/singlet (Without opt)	−	2.34000	2.34000	2.34000
D3h/triplet (Without opt)	−	2.34000	2.34000	2.34000
D3h/PBE0-SDD/singlet (opt)	2.27066	2.37909	2.41463	2.26833
D3h/PBE0-SDD/triplet (opt)	−	2.33630	2.33522	2.53501
D3h/PBE-D4-/singlet (opt)	−	2.41683	2.41683	2.30011
D3h/PBE-D4/triplet (opt)	−	2.36754	2.36743	2.59440
D3h/r2SCAN-3C/singlet (opt)	−	2.36616	2.36617	2.26137
D3h/r2SCAN-3C/triplet (opt)	−	2.31424	2.31423	2.51699
D3h/TPSSh/singlet (opt)	−	2.40961	2.40961	2.27720
D3h/TPSSh/triplet (opt)	−	2.36007	2.35973	2.54220

.

As shown in Table 2, the D₃h arrangement is not consistently preserved after geometry optimization. Several optimized structures exhibit asymmetric Cu–Cu distances or missing equivalent contacts, indicating that the initial D₃h motif relaxes toward lower-symmetry geometries depending on the computational protocol and spin state. This behavior suggests that D₃h Cu₄ lies in a shallow region of the potential energy surface, where small changes in exchange–correlation treatment, basis-set representation, or spin configuration can drive the system toward different local minima.

This result is important for the adsorption analysis because, as indicated by Table 2, Eads values obtained across different final cluster geometries cannot be interpreted solely in terms of electronic differences. They may also include contributions from isomeric reorganization. Therefore, Table 2 provides a structural filter for assessing whether adsorption energies are being compared for equivalent or non-equivalent Cu₄ frameworks [12,22,25].

2.4. Cross-Validation Between Codes Using ω_min as Minimum True Control

To evaluate whether the optimized Cu₄ structures correspond to true minima or unstable stationary points, the lowest harmonic frequency, ωmin, was compared between selected Gaussian reference data and ORCA calculations. The comparison, summarized in

Table 3. Inter-code comparison of Cu₄ stationary-point character using ωmin and final symmetry as vibrational-stability descriptors.

Data Source	Software	Initial Symmetry	Final Symmetry	Multiplicity	Energy (kcal mol−1)	ωmin (cm−1)
[22]	Gaussian 16 Rev.C.01	D2h	D2h	1	−495,182.79	59.32
[22]	Gaussian	D2h	D2h	3	−495,166.47	−23.22
This work	ORCA 6.1.0	D2h	D2h	1	−514,186.91	63.04
This work	ORCA 6.1.0	D2h	D2h	3	−514,169.97	33.76
[22]	Gaussian	D3h	D2h	1	−495,182.79	59.13
[22]	Gaussian	D3h	D3h	3	−495,146.39	−31.92
This work	ORCA 6.1.0	D3h	C2v	1	−514,187.54	24.66
Our work	ORCA 6.1.0	D3h	D2h	3	−514,169.97	20.48
[22]	Gaussian	Td	D2d	1	−495,158.31	−88.92
[22]	Gaussian	Td	D2d	3	−495,165.84	−23.92
This work	ORCA 6.1.0	Td	D2d	1	−514,187.54	−40.83
This work	ORCA 6.1.0	Td	D2d	3	−514,174.36	75.21

, focuses on the final symmetry, relative stationary-point character, and presence or absence of imaginary frequencies.

As shown in Table 3, nominally comparable computational protocols do not always lead to the same stationary-point character. For the D₂h singlet, both Gaussian and ORCA produce positive ωmin values close to 60 cm⁻¹, supporting the assignment of this structure as a reproducible true minimum. In contrast, for the D₂h triplet, the Gaussian reference reports an imaginary frequency, whereas the ORCA calculation gives a positive ωmin. This difference was interpreted from the sign of the lowest harmonic frequency and the final optimized geometry obtained with each code. Within the harmonic approximation, an imaginary frequency indicates negative curvature along at least one normal coordinate, while a positive ωmin indicates the absence of negative curvature in the local Hessian [27,28].

Shows that this behavior is not restricted to one geometry. For D₃h-derived structures, changes in final symmetry and ωmin indicate that the optimization pathway may lead to different local regions of the potential energy surface depending on the code implementation. Similarly, Td-derived structures include cases with imaginary frequencies and cases with positive ωmin, confirming that the Cu₄ potential energy surface contains soft distortion coordinates and nearby stationary points.

Table 3 should not be interpreted as a simple comparison of total energies. Its main purpose is to ensure reproducibility and vibrational stability before using optimized structures in the adsorption-energy analysis. When nominally similar calculations produce different ωmin signs or different final symmetries, the corresponding Eads values must be interpreted cautiously, because they may reflect differences in the stationary-point character rather than only differences in the metal–CO₂ interaction strength.

2.5. Ωmin Statistics and Quantitative Criterion of Vibrational Stability

The distribution of ωmin values was summarized by separating stable and unstable cases to quantify the frequency with which real and imaginary frequencies occur in the benchmark. The resulting statistical descriptors are presented in

Table 4. Statistical summary of ωmin values grouped according to real frequencies (ω > 0) and imaginary frequencies (ω < 0).

Group	N	Media (cm−1)	Desv. Standard (cm−1)	MAD (cm−1)	Minimum (cm−1)	Maximum (cm−1)
True frequencies (ω > 0)	7	47.94	21.31	18.55	20.48	75.21
Imaginary frequencies (ω < 0)	5	–41.76	27.32	18.86	–88.92	–23.22

.

As shown in Table 4, the ωmin distribution reflects a characteristic feature of small metal clusters: the presence of structurally soft stationary points with low-frequency vibrational modes. The group of real frequencies (ω > 0) shows a mean value close to 48 cm⁻¹, with moderate dispersion, indicating that even the verified minima are associated with relatively shallow potential-energy surfaces. These results are consistent with true minima that remain vibrationally soft and therefore sensitive to small structural distortions.

In contrast, the group of imaginary frequencies (ω < 0) has a mean value of approximately −42 cm⁻¹ and a broader dispersion, with a minimum as negative as −88.92 cm⁻¹. As indicated by Table 4, these values are not compatible with merely negligible numerical noise, but instead suggest the presence of genuine instabilities associated with low-barrier distortion coordinates. Such coordinates may involve out-of-plane deformations, incipient isomerization, or structural relaxations coupled to electronic reorganization [29,30].

Table 4 supports the use of ωmin as an operational quality descriptor rather than as secondary vibrational information. By statistically separating real from imaginary frequencies, the table defines which optimized structures can be treated as genuinely comparable minima for discussing CO₂ adsorption and activation, and which should be excluded from direct energetic interpretation.

2.6. Protocol Validation with Cu₂ Using IP and EA as Sensitive Observables

The electronic properties of the Cu₂ dimer were compared with experimental reference values to provide an external validation criterion for selecting a computational protocol with lower risk of methodological artifacts, as shown in Figure 2.

As shown in Figure 2, this comparison allows evaluation of whether each method reproduces the energy differences between charged states, which are highly sensitive to exchange–correlation treatment, self-interaction error, basis-set quality, and ECP treatment. The IP predicted by r²SCAN-3c is close to the experimental value, and TPSSh also shows good agreement, suggesting that both methods reasonably describe the HOMO energy and the balance of stabilization upon ionization.

For EA, Figure 2 shows that PBE-D4 approaches the experimental value, whereas r²SCAN-3c underestimates it, and PBE0-SDD even gives a negative EA. This behavior indicates a poor description of the anionic state, likely due to limited diffuse character in the basis set, ECP imbalance, or an excessive fraction of exact exchange for the available basis-space description [31]. The useful conclusion for the benchmark is that r²SCAN-3c is not necessarily the best method for each isolated observable, but it maintains global coherence without catastrophic failures. For this reason, it is defensible as a practical reference protocol for studying adsorption, where metal–CO₂ charge-transfer effects are central.

2.7. Geometric Validation of Cu₂ Using Cu–Cu Distance

The Cu–Cu equilibrium distance of the Cu₂ dimer was compared with the experimental reference value to evaluate how each protocol describes the balance of the metal–metal bond, as summarized in Figure 3.

As shown in Figure 3, r²SCAN-3c reproduces the experimental Cu–Cu distance very closely, while PBE0-SDD also provides a competitive description of the metal–metal bond. In contrast, TPSSh and PBE-D4 predict longer Cu–Cu distances, indicating a slight expansion of the dimer relative to the experimental reference [31]. This elongation suggests that these protocols describe a less compact metal–metal interaction, probably due to differences in the exchange–correlation balance, basis-set treatment, or effective repulsion in the Cu–Cu bonding region.

This geometric validation is relevant for the Cu₄ adsorption study because the electron-density distribution and polarizability of the metallic framework influence CO₂ binding and possible metal-to-CO₂ donation/back-donation interactions. A method that already overestimates the Cu–Cu distance in Cu₂ may alter the electronic structure used to describe CO₂ adsorption in larger Cu clusters. Therefore, Figure 3 supports the use of r²SCAN-3c as a practical reference protocol for subsequent adsorption analysis, while also emphasizing that geometric accuracy should be considered alongside vibrational and energetic criteria.

2.8. Vibrational Validation of Cu₂ Using Fundamental Frequency

The Cu–Cu stretching frequency of the Cu₂ dimer was compared with the experimental reference value to evaluate how each protocol describes the local curvature and stiffness of the metal–metal bond potential, as summarized in Figure 4.

As shown in Figure 4, PBE0-SDD overestimates the experimental Cu–Cu stretching frequency, indicating an excessively rigid potential near the equilibrium geometry [31]. In contrast, TPSSh and PBE-D4 underestimate this frequency, suggesting a softer potential and a less stiff description of the metal–metal interaction. In this comparison, r²SCAN-3c yields an intermediate result and more closely reproduces the experimental vibrational curvature, consistent with its balanced performance across structural and energetic properties.

This result is relevant for the adsorption study because low-frequency modes and soft potential-energy surfaces play a central role in the structural rearrangement of small metal clusters during CO₂ binding. A protocol that stiffens the Cu–Cu interaction excessively may overestimate distortion barriers and overstabilize specific optimized geometries, whereas an overly soft description may exaggerate structural reorganization. Therefore, the vibrational agreement shown in Figure 4 supports the use of r²SCAN-3c as a practical reference for the Cu₄ and Sc₄ adsorption analysis, without introducing extreme rigidity or softness biases in the metal framework.

2.9. Structural Definition of CO₂ Adsorption Modes on Cu

The CO₂ coordination modes considered in the adsorption study are introduced in Figure 5, and these motifs are subsequently used to interpret differences in optimized geometry and adsorption energy.

As defined in Figure 5, the main CO₂ adsorption motifs considered for the Cu₄ clusters are not only structural labels, but also distinct interaction scenarios between the adsorbate and the metallic framework. In the η¹-C mode, the carbon atom of CO₂ interacts directly with a Cu center, which may favor polarization of the C–O bonds and possible metal-to-CO₂ donation/back-donation contributions involving the π* orbitals of CO₂. In the μ₂-O mode, one oxygen atom coordinates to the metallic framework, reflecting a more Lewis acid–base type interaction. In turn, the μ₃ and μ₄ arrangements involve a larger number of Cu atoms interacting with the adsorbate, which may distribute the metal–CO₂ interaction over several centers and promote stronger geometric distortion of the molecule.

Therefore, Figure 5 provides the structural basis for interpreting the optimized Cu₄–CO₂ complexes discussed in the following section. It also serves as a reference for distinguishing weak molecular adsorption, in which CO₂ remains only slightly perturbed, from configurations in which CO₂ may undergo stronger geometric and electronic activation through bending of the O–C–O angle, elongation of the C–O bonds, or charge transfer from the metallic cluster.

2.10. Optimized Cu₄ CO₂ Geometries and Dependence on Symmetry and Mode

The optimized structures of the Cu₄–CO₂ complexes were analyzed to relate the adsorption geometry to the type of coordination mode and its possible influence on the relative stability of the adsorbed configurations, as summarized in Figure 6.

As shown in Figure 6, Cu₄ exhibits several coordination patterns for CO₂ adsorption, even within the same initial symmetry. In several D₂h-derived structures, bridge-type configurations are observed in which the carbon and oxygen atoms interact with different Cu centers. This type of arrangement is usually associated with greater stabilization through cooperative coupling and with a higher probability of O–C–O bending. In other D₂h cases, oxygen binds more locally to a single Cu atom, corresponding to a more direct coordination mode with greater rotational freedom. Such configurations may be more enthalpically stable in some protocols, but not necessarily more activating, because charge transfer toward the C–O axis may be reduced if the carbon atom is not simultaneously involved in the interaction [32].

Figure 6 also shows that the Td-derived structures display variations in inclination and cases of minimal interaction, indicating that the three-dimensional geometry may present less accessible adsorption sites or an electronic distribution that does not favor CO₂ anchoring under certain protocols. This observation prepares the energetic interpretation, because if a method favors bridge-type geometries, it may also predict more negative Eads values. At the same time, such stabilization may increase the risk of overadsorption when dispersion contributions dominate or when the protocol underestimates short-range repulsion.

As shown in

Table 5. NPA charge-transfer analysis for CO₂ adsorption on Cu₄ singlet complexes. qCO₂ was calculated as the sum of the NPA charges of the atoms belonging to the adsorbed CO₂ molecule, qCO₂ = qC + qO1 + qO2, whereas qCu₄ corresponds to the net charge of the metallic fragment. Negative qCO₂ values indicate net electron accumulation on CO₂, while values close to zero indicate negligible charge transfer. Rows affected by ECP/NPA charge-convention inconsistencies were retained only as qualitative records and were not used for direct quantitative comparison.

Catalyst	Symmetry	Mult.	Method	qO1	qC	qO2	qCO2	qCu4	Eads (eV)
Cu4	D2h	1	r2SCAN-3c	−0.631	0.575	−0.560	−0.616	0.616	−0.387
Cu4	D2h	1	PBE-D4/def2-TZVP	−0.538	0.948	−0.415	−0.005	0.005	0.073
Cu4	D2h	1	TPSSh/def2-TZVP	−0.555	0.997	−0.440	0.002	−0.002	0.191
Cu4	D2h	1	PBE0-SDD	1.399	3.171	1.484	6.053	39.947	−0.198
Cu4	D3h	1	r2SCAN-3c	−0.630	0.574	−0.561	−0.617	0.617	−0.385
Cu4	D3h	1	PBE-D4/def2-TZVP	−0.405	0.939	−0.535	−0.001	0.001	−0.031
Cu4	D3h	1	TPSSh/def2-TZVP	−0.430	0.996	−0.554	0.012	−0.012	0.118
Cu4	D3h	1	PBE0-SDD	1.493	3.133	1.399	6.025	39.975	−0.701
Cu4	Td	1	r2SCAN-3c	−0.560	0.575	−0.631	−0.616	0.616	−1.402
Cu4	Td	1	PBE-D4/def2-TZVP	−0.614	0.604	−0.513	−0.524	0.524	−1.146
Cu4	Td	1	TPSSh/def2-TZVP	−0.430	0.997	−0.554	0.013	−0.013	−0.934
Cu4	Td	1	PBE0-SDD	1.494	3.132	1.399	6.024	39.976	−1.084

, the charge-transfer response of Cu₄–CO₂ singlet complexes depends strongly on both the initial symmetry and the computational protocol. For the D₂h and D₃h arrangements, r²SCAN-3c predicts substantial electron accumulation on CO₂, with qCO₂ values close to −0.62 e. In contrast, PBE-D4/def2-TZVP and TPSSh/def2-TZVP give qCO₂ values close to zero for the same planar arrangements, indicating weak adsorption with limited electronic activation. Therefore, in these cases, favorable or near-thermoneutral Eads values should not be interpreted automatically as evidence of CO₂ activation.

The Td singlet configurations in Table 5 show a more pronounced electronic response. Both r²SCAN-3c and PBE-D4/def2-TZVP predict clearly negative qCO₂ values, indicating significant metal-to-CO₂ charge transfer. This trend is consistent with the more negative adsorption energies obtained for Td-derived Cu₄ complexes and suggests that the three-dimensional arrangement provides a more favorable electronic environment for CO₂ activation. However, TPSSh/def2-TZVP predicts a negative Eads for Td Cu₄ while qCO₂ remains close to zero. This discrepancy indicates that the stabilization predicted by TPSSh is not primarily due to net charge transfer to CO₂. Still, it may instead arise from geometry-dependent polarization or other method-dependent energetic contributions [25,33,34].

2.11. Sc₄ Structural Reorganization in D_2h and Expansion of the Metal Network

The initial and optimized geometries of Sc₄ with D₂h symmetry were compared to assess changes in network size and structural flexibility relative to those of Cu₄, as summarized in Figure 7.

Figure 7 evidences a marked expansion of the Sc–Sc framework after optimization of the D₂h Sc₄ structures, in agreement with the larger metal–metal distances later quantified by the corresponding structural descriptors. This expansion suggests weaker Sc–Sc bonding and a more flexible metallic network than that observed for Cu₄. Such behavior is chemically consistent with the greater electropositivity of scandium and its lower tendency to form compact metal–metal frameworks.

This structural flexibility is relevant for CO₂ adsorption because a more open Sc₄ framework may facilitate the formation of accessible binding environments and accommodate multidentate coordination patterns. However, the same flexibility also increases methodological sensitivity: a softer cluster can reorganize more easily toward nearby minima, amplifying differences associated with the functional, basis/ECP treatment, spin state, or numerical implementation [13,35].

For Sc₄, this creates a dual scenario. Its structural adaptability and Lewis-acidic character may favor CO₂ coordination, but the predicted adsorption behavior must be filtered carefully through vibrational validation and protocol comparison. Otherwise, the apparent accessibility of favorable adsorption geometries could reflect method-dependent stabilization rather than a robust structural feature.

2.12. Geometric Metrics for Sc₄ D_2h and Method and Spin Effect

The D₂h-optimized Sc₄ structures were analyzed through their Sc–Sc bond distances to quantify protocol-induced expansion and asymmetry relative to the initial geometry. These structural descriptors are summarized in

Table 6. Sc–Sc bond distances for the initial and optimized D₂h Sc₄ clusters calculated using different DFT protocols and spin multiplicities.

Sc4	R (1-3)	R (1-2)	R (1-4)	R (2-4)
D2h/singlet (Without opt)	2.40348	2.24956	2.40348	2.40348
D2h/triplet (Without opt)	2.40348	2.24956	2.40348	2.40348
D2h/PBE0-SDD/singlet (opt)	2.78934	2.42895	2.78910	2.78928
D2h/PBE0-SDD/triplet (opt)	2.86430	2.89278	2.78107	2.69984
D2h/PBE-D4-/singlet (opt)	2.80578	2.48797	2.80484	2.80558
D2h/PBE-D4/triplet (opt)	2.83223	2.71688	2.83249	2.83172
D2h/r2SCAN-3C/singlet (opt)	2.80858	2.48219	2.80703	2.80836
D2h/r2SCAN-3C/triplet (opt)	2.83778	2.77437	2.90099	2.83756
D2h/TPSSh/singlet (opt)	2.79371	2.46511	2.79275	2.79352
D2h/TPSSh/triplet (opt)	2.85617	2.96399	2.83750	2.85697

.

As reported in Table 6, the initial D₂h Sc₄ structure contains Sc–Sc distances in the range of 2.25–2.40 Å, whereas the optimized structures shift toward substantially longer metal–metal separations. In the singlet state, the optimized distances are mainly located around 2.79–2.81 Å, with moderate differences among the evaluated protocols. This systematic expansion indicates that the Sc₄ framework is less compact than the corresponding Cu₄ structures and that Sc–Sc bonding is more susceptible to relaxation during optimization.

The triplet structures exhibit greater asymmetry. Several Sc–Sc contacts approach or exceed 2.9 Å, particularly for r²SCAN-3c and TPSSh, indicating a more pronounced reorganization of the metal framework. This behavior suggests that spin-state changes affect the balance of Sc–Sc interactions and can reduce the cluster’s structural cohesion. Rather than preserving a rigid D₂h arrangement, the triplet Sc₄ structures tend to open or distort, generating a more flexible coordination environment.

This structural response is directly relevant for CO₂ adsorption. An expanded Sc₄ network can facilitate oxygen coordination and stabilize multidentate adsorption geometries, which is consistent with a polarization-dominated Lewis acid–base interaction mechanism. At the same time, the greater structural flexibility may increase Eads’s sensitivity to the computational protocol, because adsorption can be coupled to cluster reorganization. In this sense, Table 6 links the intrinsic geometry of Sc₄ to the expected adsorption behavior and supports the need to interpret adsorption energies in conjunction with structural and vibrational validation [36,37,38].

2.13. Loss of Symmetry in Sc₄ D_3h as a Soft Surface Indicator

The initial and optimized D₃h Sc₄ geometries were compared to evaluate symmetry breaking and possible relaxation toward lower-symmetry structures, as summarized in Figure 8.

Figure 8 evidences that Sc₄ does not consistently retain the D₃h motif after structural relaxation. Several optimized geometries depart from the initial high-symmetry arrangement and evolve toward distorted or lower-symmetry structures. This behavior is consistent with a flexible metal framework and with the presence of nearby minima on a relatively shallow potential energy surface.

The loss of symmetry can be interpreted as a structural response to electron redistribution within the Sc₄ cluster. Instead of preserving equivalent Sc–Sc interactions, the optimized structures tend to develop uneven metal–metal contacts, suggesting multicenter stabilization and partial opening of the framework. This behavior is characteristic of small early-transition-metal aggregates, where the balance between metal–metal cohesion, spin state, and geometric relaxation can be strongly method-dependent.

For CO₂ adsorption, the structural plasticity observed in Figure 8 has two implications. On the one hand, a flexible Sc₄ framework may accommodate diverse adsorption motifs and facilitate oxygen-centered or multidentate coordination. On the other side, this same flexibility increases the risk that adsorption trends depend on the optimized cluster basin selected by each computational protocol. For this reason, the D₃h-derived Sc₄ structures require careful vibrational validation before their adsorption energies are compared or interpreted as evidence of catalytic activation [13,35,39].

2.14. Geometric Metrics for Sc₄ D_3h and Asymmetry Quantification

The Sc–Sc distances of the D₃h-derived Sc₄ structures were analyzed to quantify the extent of symmetry breaking and structural rearrangement after optimization. The corresponding structural descriptors are summarized in

Table 7. Sc–Sc bond distances for the initial and optimized D₃h Sc₄ clusters calculated using different DFT protocols and spin multiplicities.

Sc4	R (1-3)	R (1-4)	R (4-3)	R (3-2)
D3h/singlet (Without opt)	-	2.34000	2.34000	-
D3h/triplet (Without opt)	-	2.34000	2.34000	-
D3h/PBE0-SDD/singlet (opt)	2.99810	2.63547	2.51146	2.99802
D3h/PBE0-SDD/triplet (opt)	-	2.74343	2.79538	-
D3h/PBE-D4-/singlet (opt)	-	2.78165	2.55876	-
D3h/PBE-D4/triplet (opt)	2.85485	2.85465	2.99546	2.85586
D3h/r2SCAN-3C/singlet (opt)	-	2.63214	2.63186	3.01968
D3h/r2SCAN-3C/triplet (opt)	-	2.83736	2.80251	-
D3h/TPSSh/singlet (opt)	-	2.64399	2.64449	2.97618
D3h/TPSSh/triplet (opt)	-	2.84160	2.90591	-

.

The data collected in Table 7 show that the initial D₃h pattern is not preserved after optimization. Several protocols generate markedly uneven Sc–Sc distances, and in some cases, the longest contacts approach 3.0 Å. In contrast, other Sc–Sc separations remain in the approximate range of 2.63–2.85 Å, indicating that the optimized structures undergo nonuniform reorganization rather than simple isotropic expansion.

Within this set, the PBE0-SDD singlet structure exhibits one of the clearest distance separations, consistent with pronounced distortion of the original D₃h motif. A similar trend is observed for the PBE-D4 triplet case, where multiple Sc–Sc contacts fall in the 2.85–3.00 Å range. The r²SCAN-3c and TPSSh results also display two distance scales, suggesting partially open frameworks and incomplete retention of the starting connectivity pattern.

From an adsorption perspective, these structural changes are relevant because they may generate more exposed or coordinatively unsaturated regions, which can favor Lewis-type interactions with CO₂. At the same time, Table 7 indicates that some entries appear as hyphens, reflecting that the effective symmetry or connectivity of the optimized structure has changed. For this reason, the adsorption analysis should not rely only on metal–metal distances. A more reliable interpretation requires additional descriptors of the adsorbed CO₂ geometry, particularly the O–C–O angle and the C–O bond lengths, together with the corresponding energetic and vibrational criteria.

2.15. Sc₄ CO₂ Geometries and Conceptual Comparison of Interaction Mechanism

The optimized Sc₄–CO₂ structures were examined across different adsorption modes to relate the electropositive character of scandium to the observed coordination patterns and possible CO₂ activation pathways, as summarized in Figure 9.

The set of structures collected in Figure 9 reveals a broader diversity of adsorption motifs for Sc₄–CO₂ than that observed for the Cu₄ analogs. In these complexes, the interaction is expected to be governed primarily by the Lewis-acid character of scandium and by polarization of the adsorbate, rather than by strong metal-to-CO₂ back-donation. This behavior is consistent with the preference for oxygen-centered coordination and with the appearance of geometries in which CO₂ undergoes bending or selective C–O elongation.

Another relevant feature of Figure 9 is the Sc₄ framework’s structural adaptability. Several optimized complexes show that the cluster can reorganize substantially to accommodate the adsorbate, thereby generating a wide variety of minima and coordination environments. Such flexibility may be advantageous for CO₂ binding because it facilitates multidentate interactions and stabilizes activated configurations. At the same time, this same property makes the predicted adsorption mode more sensitive to the computational protocol. It increases the need for careful discrimination between genuine activation and method-dependent overbinding [13,15,39].

For that reason, Figure 9 should be interpreted together with Table 7 and

Table 8. NPA charge-transfer analysis for CO₂ adsorption on Sc₄ singlet complexes. qCO₂ was obtained by summing the NPA charges of the carbon and oxygen atoms of the adsorbed CO₂ molecule, whereas qSc₄ corresponds to the net charge of the Sc₄ metallic fragment. Negative qCO₂ values indicate electron accumulation on CO₂, while values close to zero suggest weak adsorption with limited electronic activation. Rows showing non-neutral fragment-charge balance due to ECP/NPA charge conventions were retained only as qualitative records and were not directly compared with all-electron/def2-based protocols.

Catalyst	Symmetry	Mult.	Method	qO1	qC	qO2	qCO2	qSc4	Eads (eV)
Sc4	D2h	1	r2SCAN-3c	−0.554	0.995	−0.423	0.018	−0.018	0.122
Sc4	D2h	1	PBE-D4/def2-TZVP	−0.517	0.954	−0.416	0.021	−0.021	0.010
Sc4	D2h	1	PBE0-SDD	1.422	3.173	1.487	6.082	39.918	−0.223
Sc4	D2h	1	TPSSh/def2-TZVP	1.422	3.173	1.487	6.082	39.918	−2.752
Sc4	D3h	1	r2SCAN-3c	−0.629	0.250	−0.697	−1.076	1.076	−2.819
Sc4	D3h	1	PBE-D4/def2-TZVP	−0.764	−0.104	−0.764	−1.632	1.632	−6.252
Sc4	D3h	1	TPSSh/def2-TZVP	−0.629	0.250	−0.697	−1.076	1.076	5.885
Sc4	D3h	1	PBE0-SDD	1.356	2.434	1.285	5.074	40.926	−2.819
Sc4	Td	1	PBE0-SDD	1.327	2.431	1.327	5.084	40.916	−2.973

. The structural motifs alone are insufficient to distinguish weak molecular adsorption from highly activated configurations. A reliable interpretation requires combining the optimized geometry with charge-transfer analysis and energetic data, especially in cases where very negative adsorption energies could reflect either strong Lewis-type stabilization or protocol-induced bias. From an applied perspective, Sc₄ offers a useful model for identifying active-site motifs that can stabilize moderately activated CO₂, which may be relevant to the design of lightweight or doped materials for capture and pre-activation in CCU-related systems.

The NPA results summarized in Table 8 indicate that the charge-transfer response of Sc₄–CO₂ singlet complexes depends strongly on the initial symmetry and the computational protocol. For the D₂h structures, r²SCAN-3c and PBE-D4/def2-TZVP produce qCO₂ values close to zero, indicating negligible net electron transfer from the Sc₄ framework to the adsorbed CO₂ molecule. This behavior is consistent with their weak or nearly thermoneutral Eads values and suggests that the D₂h singlet arrangement does not provide clear electronic activation of CO₂ under these protocols.

A different behavior is observed for the D₃h singlet structures. In Table 8, r²SCAN-3c gives qCO₂ = −1.076 e, whereas PBE-D4/def2-TZVP gives qCO₂ = −1.632 e. These strongly negative values indicate substantial electron accumulation on CO₂, supporting the formation of electronically activated adsorption states. This interpretation agrees with the more negative Eads values calculated for these configurations, particularly for PBE-D4/def2-TZVP. However, the simultaneous presence of very negative qCO₂ and strongly exergonic Eads should be treated cautiously, because it may reflect genuine CO₂ activation but may also indicate protocol-dependent overbinding or excessive stabilization of a highly reorganized adsorption geometry [33,34,40].

The TPSSh D₃h singlet case provides an important counterexample. Although Table 8 indicates charge transfer toward CO₂, the corresponding Eads value is positive. This result shows that charge accumulation on CO₂ alone is not sufficient to define a thermodynamically favorable adsorption process. A reliable assignment of CO₂ activation must combine qCO₂ with Eads, vibrational stability, and geometric descriptors such as O–C–O bending and C–O bond elongation.

The available Td singlet result corresponds only to PBE0-SDD, whose NPA charges are not directly comparable because of the ECP/NPA charge convention. This row was therefore retained only as a qualitative record and was not used to establish quantitative charge-transfer trends. Completing the Td comparison with all-electron/def2-based protocols would be necessary to evaluate whether the charge-transfer behavior observed for D₃h also extends to three-dimensional Sc₄-derived adsorption geometries.

2.16. Gibbs Free Energy Analysis and Thermodynamic Consistency of CO₂ Adsorption

The adsorption energies of CO₂ on Cu₄ and Sc₄ were compared within a single energetic profile to assess how catalyst identity, initial symmetry, computational protocol, and spin multiplicity affect the predicted adsorption strength. The complete comparison is presented in Figure 10.

The energetic profile in Figure 10 confirms that Cu₄ generally remains within a weak or marginal adsorption regime. This behavior is especially evident for the D₂h- and D₃h-derived structures, where both singlet and triplet states cluster near zero and may shift from slightly exergonic to slightly endergonic depending on the computational protocol. Such small energetic variations are consistent with weak adsorption or incipient chemisorption, where the final Eads value is sensitive to the balance between dispersion, exchange–correlation treatment, and structural relaxation.

A different trend is observed for Td-derived Cu₄ structures. In this case, the singlet state tends to produce more negative adsorption energies than the corresponding planar arrangements, suggesting that the three-dimensional topology may provide a more favorable local coordination environment for CO₂ stabilization. This could arise from stronger metal–adsorbate cooperation or from a more suitable orbital arrangement for adsorbate polarization. The triplet state, however, remains closer to the weak-adsorption regime in several protocols, indicating that spin multiplicity modifies the electronic availability of the Cu₄ framework toward CO₂ binding [22,25,33].

For Sc₄, Figure 10 reveals a broader and more irregular energetic distribution. Several D₃h- and Td-derived cases show markedly exergonic adsorption, but the magnitude and even the sign of Eads depend strongly on the method and spin state. The presence of outliers, including a sign inversion for one D₃h singlet case, indicates that Sc₄ adsorption is highly sensitive to the selected computational protocol and to the optimized structural basin reached during relaxation.

Chemically, this larger dispersion is consistent with the greater electropositivity and structural deformability of scandium. These features can enhance Lewis acid–base interactions and CO₂ polarization, but they also make the computed adsorption energy more sensitive to the cluster’s deformation energy and to how each functional treats polarization, dispersion, and metal–adsorbate stabilization. For this reason, the most negative Sc₄ adsorption energies in Figure 10 should not be interpreted automatically as evidence of superior CO₂ capture or activation. Within a benchmarking framework, they require additional validation using vibrational stability, optimized CO₂ geometry, and charge-transfer descriptors to rule out protocol-dependent overbinding or convergence toward a different structural regime [38,41,42].

Cu₄ with weak adsorption suggests potential for more reversible capture and lower regeneration penalty, while Sc₄ may offer more intense stabilization compatible with pre-activation, but only if the prediction is reproducible and filtered by vibrational stability. This point is also relevant from methodological sustainability, because early identification of sensitivity by method and spin reduces computational false positives and avoids rework in simulation campaigns.

Figure 11 provides a consistency check for the thermochemical dataset by comparing the absolute Gibbs free energies of the isolated clusters and the corresponding cluster–CO₂ complexes for both spin multiplicities. In both panels, the catalyst–CO₂ complexes exhibit more negative absolute Gibbs free energies than the isolated catalysts, as expected, because they include an additional CO₂ fragment. This trend confirms the dataset’s internal construction within each computational protocol.

The comparison between panels (a) and (b) also evidences the effect of spin multiplicity on the absolute Gibbs free-energy scale. Variations between singlet and triplet states reflect the spin–structure coupling characteristic of low-nuclearity metal clusters, where changes in electron occupancy can modify both relative stability and the degree of geometric reorganization. However, Figure 11 should not be interpreted as a direct comparison of stability across different DFT protocols. The large offsets in absolute G values are mainly determined by the electronic-structure model, basis set, and ECP treatment, particularly in PBE0-SDD, where the energy reference is not directly comparable with all-electron/def2-based calculations [13,19,35].

For this reason, the chemical interpretation must rely on internally derived quantities, such as Eads and ΔGads, where systematic contributions partially cancel within the same protocol. In this context, Figure 11 functions as a traceability control rather than as a standalone adsorption-stability descriptor. It supports the subsequent use of adsorption energies, Gibbs-corrected adsorption quantities, vibrational filters, and dispersion metrics as more appropriate criteria for evaluating CO₂ binding and activation.

2.17. Adsorption Energies and Direct Comparison Between Cu₄ and Sc₄

To complement the electronic adsorption-energy profile and include a thermodynamic layer in the analysis,

Table 9. Gibbs free energies of the isolated catalysts and catalyst–CO₂ complexes, together with the corresponding CO₂ adsorption energies calculated for Cu₄ and Sc₄ clusters using different initial symmetries, spin multiplicities, and DFT protocols.

#	Catalyst	Symmetry	Mult.	Method	Gcat (kcal mol−1)	Gcat + CO2 (kcal mol−1)	Eads (kcal mol−1)	Eads (eV)
1	Cu4	D2h	1	r2SCAN-3c	−4,117,778.04	−4,236,120.68	−8.79	−0.387
2	Cu4	D2h	1	PBE0-SDD	−490,778.30	−514,219.54	−4.39	−0.198
3	Cu4	D2h	1	PBE-D4/def2-TZVP	−4,117,236.50	−4,235,511.99	1.88	0.073
4	Cu4	D2h	1	TPSSh/def2-TZVP	−4,117,839.53	−4,236,230.49	4.39	0.191
5	Cu4	D3h	1	r2SCAN-3c	−4,117,778.04	−4,236,120.68	−8.79	−0.385
6	Cu4	D3h	1	PBE0-SDD	−490,766.37	−514,218.91	−16.32	−0.701
7	Cu4	D3h	1	PBE-D4/def2-TZVP	−4,117,236.50	−4,235,514.50	−0.63	−0.031
8	Cu4	D3h	1	TPSSh/def2-TZVP	−4,117,839.53	−4,236,232.37	2.51	0.118
9	Cu4	Td	1	r2SCAN-3c	−4,117,754.82	−4,236,120.68	−32.63	−1.402
10	Cu4	Td	1	PBE0-SDD	−490,756.33	−514,217.66	−25.10	−1.084
11	Cu4	Td	1	PBE-D4/def2-TZVP	−4,117,212.65	−4,235,516.38	−26.36	−1.146
12	Cu4	Td	1	TPSSh/def2-TZVP	−4,117,815.06	−4,236,232.37	−21.96	−0.934
13	Sc4	D2h	1	r2SCAN-3c	−1,909,166.20	−2,027,496.92	3.14	0.122
14	Sc4	D2h	1	PBE0-SDD	−115,294.82	−138,736.98	−5.02	−0.223
15	Sc4	D2h	1	PBE-D4/def2-TZVP	−1,908,737.00	−2,027,014.22	0.00	0.010
16	Sc4	D2h	1	TPSSh/def2-TZVP	−1,909,306.76	−2,027,766.12	−64.01	−2.752
17	Sc4	D3h	1	r2SCAN-3c	−1,909,167.45	−2,027,566.57	−65.26	−2.819
18	Sc4	D3h	1	PBE0-SDD	−115,296.70	−138,799.73	−65.26	−2.819
19	Sc4	D3h	1	PBE-D4/def2-TZVP	−1,908,701.23	−2,027,123.55	−145.58	−6.252
20	Sc4	D3h	1	TPSSh/def2-TZVP	−1,909,308.02	−2,027,566.57	136.80	5.885
21	Sc4	Td	1	r2SCAN-3c	−1,909,190.04	−2,027,483.11	−94.13	−4.051
22	Sc4	Td	1	PBE0-SDD	−115,324.94	−138,831.45	−69.03	−2.973
23	Sc4	Td	1	PBE-D4/def2-TZVP	−1,908,760.83	−2,026,843.60	−85.77	−3.692
24	Sc4	Td	1	TPSSh/def2-TZVP	−1,909,330.61	−2,028,109.42	−154.37	−6.629
25	Cu4	D2h	3	r2SCAN-3c	−4,117,766.74	−4,236,316.78	1.26	0.051
26	Cu4	D2h	3	PBE0-SDD	−490,765.12	−514,207.00	−5.65	−0.244
27	Cu4	D2h	3	PBE-D4/def2-TZVP	−4,117,221.44	−4,235,495.05	3.77	0.169
28	Cu4	D2h	3	TPSSh/def2-TZVP	−4,117,825.10	−4,236,214.80	5.65	0.256
29	Cu4	D3h	3	r2SCAN-3c	−4,117,766.11	−4,236,096.83	3.14	0.125
30	Cu4	D3h	3	PBE0-SDD	−490,765.12	−514,203.23	−1.26	−0.051
31	Cu4	D3h	3	PBE-D4/def2-TZVP	−4,117,220.82	−4,235,496.94	1.26	0.052
32	Cu4	D3h	3	TPSSh/def2-TZVP	−4,117,824.47	−4,236,216.69	3.77	0.155
33	Cu4	Td	3	r2SCAN-3c	−4,117,765.49	−4,236,101.85	−3.14	−0.129
34	Cu4	Td	3	PBE0-SDD	−490,765.12	−514,207.00	−5.65	−0.239
35	Cu4	Td	3	PBE-D4/def2-TZVP	−4,117,219.56	−4,235,496.31	0.00	0.012
36	Cu4	Td	3	TPSSh/def2-TZVP	−4,117,823.84	−4,236,216.69	3.14	0.145
37	Sc4	D2h	3	r2SCAN-3c	−1,909,181.26	−2,027,602.96	−87.85	−3.783
38	Sc4	D2h	3	PBE0-SDD	−115,313.02	−138,837.09	−87.22	−3.742
39	Sc4	D2h	3	PBE-D4/def2-TZVP	−1,908,747.04	−2,027,040.57	−16.94	−0.726
40	Sc4	D2h	3	TPSSh/def2-TZVP	−1,909,338.76	−2,027,820.09	−85.97	−3.694
41	Sc4	D3h	3	r2SCAN-3c	−1,909,146.75	−2,027,602.96	−122.36	−5.258
42	Sc4	D3h	3	PBE0-SDD	−115,278.51	−138,817.63	−102.08	−4.401
43	Sc4	D3h	3	PBE-D4/def2-TZVP	−1,908,767.10	−2,027,105.98	−60.87	−2.629
44	Sc4	D3h	3	TPSSh/def2-TZVP	−1,909,284.17	−2,027,798.12	−118.60	−5.102
45	Sc4	Td	3	r2SCAN-3c	−1,909,202.59	−2,027,603.59	−67.77	−2.906
46	Sc4	Td	3	PBE0-SDD	−115,331.85	−138,837.09	−68.40	−2.933
47	Sc4	Td	3	PBE-D4/def2-TZVP	−1,908,766.48	−2,027,124.80	−80.95	−3.493
48	Sc4	Td	3	TPSSh/def2-TZVP	−1,909,336.88	−2,027,798.75	−67.14	−2.883

reports the Gibbs free energies of the isolated catalysts, the corresponding catalyst–CO₂ complexes, and the calculated CO₂ adsorption energies. This comparison allows the adsorption trends to be evaluated while keeping the energy quantities internally consistent within each computational protocol.

The thermochemical data summarized in Table 9 provide an additional basis for interpreting CO₂ adsorption beyond the structural and charge-transfer descriptors discussed above. The Gibbs free energies of the isolated catalyst, Gcat, and the catalyst–CO₂ complex, Gcat + CO₂, follow the expected internal trend within each computational protocol, with the complex showing a more negative total free energy due to the incorporation of CO₂. However, these absolute Gibbs free-energy values should not be compared directly across different DFT protocols, because each functional/basis-set/ECP combination defines a different electronic-energy reference. The chemically meaningful comparison is therefore obtained from derived adsorption quantities, where systematic contributions partially cancel within the same protocol.

For Cu₄, Table 9 confirms that most planar D₂h and D₃h cases remain within a weak adsorption regime. Several adsorption energies are close to thermoneutrality or only slightly exergonic/endergonic, indicating that the Cu₄–CO₂ interaction is generally weak and strongly dependent on the computational protocol [22,24,25]. This behavior is consistent with the structural and NPA analyses, where Cu₄ does not systematically induce strong CO₂ stabilization or charge-transfer-driven activation [33,34]. The T_d singlet configurations show more negative adsorption values than the planar analogs, suggesting that the three-dimensional arrangement can provide a more favorable coordination environment for CO₂ binding [22,25].

For Sc₄, the adsorption energies listed in Table 9 span a much broader range. Several D₃h and Td configurations show markedly negative values, especially for selected PBE-D4/def2-TZVP and TPSSh/def2-TZVP cases. This trend is chemically consistent with the greater electropositivity and Lewis acidity of scandium, which can enhance CO₂ polarization and stabilize oxygen-coordinated or multidentate adsorption motifs. The interpretation, however, must remain cautious. Highly negative adsorption energies may reflect stronger CO₂ binding. Still, they may also include contributions from extensive cluster reorganization, changes in the final optimized geometry, or protocol-dependent stabilization of a particular adsorption basin.

The multiplicity comparison in Table 9 further shows that the spin state must be included in the adsorption model. In Cu₄, triplet cases remain mostly close to weak adsorption, whereas singlet Td configurations become more exergonic. In Sc₄, several triplet configurations preserve negative adsorption values across the evaluated symmetries, while the singlet cases show larger method-dependent variation, including both strongly favorable and unfavorable adsorption values. This reinforces that adsorption thermodynamics in low-nuclearity clusters cannot be separated from the spin-state landscape or from the structural relaxation pathway.

Taken together with the NPA results, Table 9 also shows why adsorption energy alone is insufficient to assign CO₂ activation. In several Cu₄ cases, Eads is favorable or near thermoneutrality while qCO₂ remains close to zero, indicating weak adsorption or polarization-dominated stabilization rather than charge-transfer-driven activation. Conversely, selected Sc₄ D₃h cases combine strongly negative qCO₂ with exergonic Eads, supporting stronger electronic activation. However, the largest values should still be examined carefully because they may also reflect protocol-dependent overbinding or excessive structural reorganization. Thus, the revised analysis distinguishes CO₂ activation from possible overbinding artifacts by requiring consistency among qCO₂, Eads, vibrational stability, and optimized adsorption geometry.

2.18. Global Benchmark Uncertainty at E_ads and Practical Recommendation

The global dispersion of Eads was quantified to estimate the methodological variability associated with the use of different DFT protocols across the complete Cu₄/Sc₄ adsorption dataset. The resulting statistical descriptors are summarized in

Table 10. Global statistical dispersion of CO₂ adsorption energy values, Eads, calculated for the complete Cu₄/Sc₄ dataset. The table reports the mean absolute deviation, standard deviation, and maximum absolute error in eV.

Statistical Calculation	Value
Mean Absolute Deviation (MAD)	1.82
Max Error	7.41
Standard Deviation (STD)	2.21

.

As summarized in Table 10, the CO₂ adsorption-energy dataset exhibits substantial global dispersion across the evaluated computational protocols. The mean absolute deviation of 1.82 eV indicates that, on average, the calculated Eads values deviate considerably from the selected reference level. Likewise, the standard deviation of 2.21 eV confirms a broad spread in the predicted adsorption energies. In comparison, the maximum absolute error of 7.41 eV evidences the presence of extreme deviations within the dataset. These deviations may originate from differences in optimized geometry, spin-state stabilization, or convergence toward different stationary points on shallow potential-energy surfaces [43].

Because the values in Table 10 correspond to global statistics for the complete Cu₄/Sc₄ dataset, they should not be used to assign the dispersion specifically to Cu₄ or Sc₄, nor to separate the individual effects of initial symmetry or spin multiplicity. The appropriate conclusion is therefore methodological: when all Cu₄–CO₂ and Sc₄–CO₂ calculations are considered together, Eads is strongly dependent on the computational protocol. Consequently, adsorption energy should not be interpreted as a standalone descriptor of adsorption strength without complementary structural, vibrational, and charge-transfer validation.

The main value of Table 10 is diagnostic. It quantifies the global uncertainty associated with the choice of DFT protocol and supports the use of additional acceptance criteria before comparing adsorption trends. In this context, optimized-geometry consistency, positive ωmin values, Gibbs free-energy corrections, and NPA-derived qCO₂ descriptors are required to reduce the risk of overinterpreting isolated adsorption-energy values. This approach provides a more cautious and reproducible basis for evaluating CO₂ adsorption and activation in low-nuclearity metal clusters.

3. Materials and Methods

3.1. Benchmark Design and Case Set

A benchmark set was designed to evaluate the sensitivity of CO₂ adsorption on small metal clusters to the computational protocol, spin state, initial geometry, and vibrational stability. Cu₄ and Sc₄ were selected as model systems because they represent electronically contrasting metal clusters and exhibit structural and spin-state sensitivity during geometry optimization. The benchmark included the isolated Cu₄ and Sc₄ clusters, as well as the corresponding Cu₄–CO₂ and Sc₄–CO₂ adsorption complexes [22,25,35].

For each metal cluster, representative low-nuclearity topologies were considered, including different initial symmetry arrangements and plausible adsorption orientations of CO₂. The initial CO₂ configurations were selected to sample distinct interaction regions of the cluster and to avoid biasing the optimization toward a single adsorption mode [33]. The calculations were performed without imposing symmetry constraints during structural relaxation, allowing the clusters and cluster–CO₂ complexes to reorganize freely toward the nearest stationary point. Singlet and triplet multiplicities were explored to account for the possible competition between low-lying spin states in these small transition-metal aggregates. The complete matrix of systems, initial geometries, spin multiplicities, and identifiers is summarized in Table 1.

3.2. DFT Protocols Evaluated

Four density functional theory (DFT) protocols were evaluated: TPSSh, r²SCAN-3c, PBE-D4/def2-TZVP, and PBE0-SDD. In this work, the term “DFT protocol” refers not only to the exchange–correlation functional but also to the associated basis set, effective core potential (ECP), dispersion correction, and, when applicable, internal numerical corrections. This definition was adopted because the functional and basis-set treatments are inseparable components of the computational description of metal clusters and adsorption complexes [16,21,44,45].

The r²SCAN-3c protocol was included as a low-cost composite DFT method that incorporates internal corrections designed to improve the robustness of geometries and relative energies. PBE-D4/def2-TZVP was selected as a generalized gradient approximation protocol with explicit D4 dispersion correction and a triple-zeta basis set. TPSSh was included as a meta-hybrid functional commonly used in transition-metal chemistry. In contrast, PBE0-SDD was used to evaluate the influence of a hybrid functional combined with an ECP-based description of the metal centers. This set of protocols was chosen to compare different levels of exchange–correlation treatment, dispersion description, and basis/ECP representation in the prediction of metal–CO₂ interactions [46].

3.3. Compute Packets and Numerical Configuration

The calculations were performed using ORCA and Gaussian to evaluate the consistency of selected results between two quantum-chemistry packages. The purpose of this comparison was not to benchmark the overall capabilities of both programs, but to assess whether selected stationary points obtained under comparable computational settings showed consistent geometries, vibrational character, and minimum/saddle-point assignments [47].

In ORCA, geometry optimizations and frequency calculations were carried out using convergence criteria suitable for transition-metal clusters and metal–adsorbate complexes. Fine integration grids were used for the evaluated meta-GGA, hybrid, and dispersion-corrected protocols. When required by the specific method, auxiliary-basis or resolution-of-identity approximations were applied in accordance with the standard implementation of the corresponding protocol. In Gaussian, analogous SCF convergence criteria, geometry optimization thresholds, and integration-grid settings were used whenever possible to minimize numerical differences between implementations [48,49]. The ORCA–Gaussian comparison was restricted to protocols and cases that could be defined in a comparable manner in both codes. For cases where an exact one-to-one correspondence was not possible, the comparison was limited to stationary-point character, final geometry, and vibrational stability.

3.4. Geometric Optimization and Multiplicity Exploration

For each system in the benchmark set, full geometry optimizations were performed in the gas phase without symmetry restrictions. Different initial geometries and spin multiplicities were explored to account for the presence of multiple low-lying minima, which is characteristic of small metal clusters. For each protocol, optimized structures were evaluated according to their electronic energy, final geometry, and vibrational stability.

The lowest-energy structure was not accepted automatically as the representative candidate unless it also corresponded to a physically meaningful stationary point. Structures showing nonphysical dissociation of CO₂, collapse of the metallic framework, or reorganization toward a topology unrelated to the intended starting motif were documented and, when appropriate, reoptimized from alternative initial geometries. In this way, the analysis considered not only the energetic ordering but also the structural consistency and physical validity of the optimized configurations [19].

3.5. Vibrational Validation and Minimum Criterion Using ω_min

The nature of each optimized structure was evaluated by harmonic vibrational frequency analysis. The lowest harmonic frequency of each structure, ω_min, was used as an operational descriptor of stationary-point stability. Structures with ω_min > 0 were assigned as vibrational minima within the harmonic approximation, indicating that all calculated frequencies were real. In contrast, structures with ω_min < 0 were considered to contain at least one imaginary frequency and were therefore assigned as unstable stationary points or possible saddle points [50,51].

For cases with small-magnitude imaginary frequencies, additional checks were performed to distinguish between numerical artifacts and genuine instabilities on the potential energy surface. These checks included reoptimization with stricter SCF, geometry convergence, and integration-grid criteria when necessary. The use of ω_min was particularly relevant because low-nuclearity metal clusters often display soft vibrational modes and shallow potential energy surfaces [52]. Therefore, vibrational validation was used as a mandatory criterion before comparing adsorption energies. The results were summarized in a classification table and visualized by group in the corresponding figure.

3.6. Adsorption Energy and Thermodynamic Conventions

Adsorption energy was defined as:

Eads = E(cluster–CO₂) − E(cluster) − E(CO₂)

where E(cluster–CO₂), E(cluster), and E(CO₂) correspond to the electronic energies of the optimized adsorption complex, isolated metal cluster, and isolated CO₂ molecule, respectively. All energies used in a given Eads calculation were obtained using the same DFT protocol and, when applicable, the same numerical treatment. This ensured internal consistency between the isolated fragments and the adsorption complex [22,39].

In addition to electronic adsorption energies, zero-point energy (ZPE) and thermal Gibbs free-energy corrections were calculated for the vibrationally stable minima. These corrections were obtained from harmonic vibrational frequency calculations performed at the same level of theory as the corresponding geometry optimization. The Gibbs free energy of adsorption at 298.15 K was calculated as:

ΔGads = G(cluster–CO₂) − G(cluster) − G(CO₂)

where G(cluster–CO₂), G(cluster), and G(CO₂) correspond to the Gibbs free energies at 298.15 K of the optimized adsorption complex, isolated metal cluster, and isolated CO₂ molecule, respectively. Only structures confirmed as vibrational minima were considered for the thermodynamic comparison between Eads and ΔGads [40].

The basis set superposition error (BSSE) was also considered because it can artificially stabilize weakly bound cluster–adsorbate complexes when finite basis sets are used. In the r²SCAN-3c protocol, BSSE is not removed through an explicit counterpoise calculation, but is reduced by the geometrical counterpoise correction included in the composite scheme. In contrast, TPSSh, PBE-D4/def2-TZVP, and PBE0-SDD do not include an intrinsic BSSE correction; therefore, their adsorption energies were interpreted consistently within each protocol [15,19]. When necessary, representative counterpoise calculations can be used to estimate the residual BSSE contribution and evaluate its effect on adsorption-energy trends.

3.7. NPA Charge-Transfer Analysis

Natural population analysis (NPA) was performed for the selected Cu₄–CO₂ and Sc₄–CO₂ adsorption complexes to evaluate the extent of charge transfer between the metal cluster and the adsorbed CO₂ molecule. The net charge on CO₂ was calculated as qCO₂ = qC + qO1 + qO2, where qC and qO correspond to the NPA charges of the carbon and oxygen atoms of the adsorbed molecule [34]. The net charge of the metallic fragment was calculated as the sum of the NPA charges of the four metal atoms. Negative qCO₂ values were interpreted as electron accumulation on CO₂, whereas values close to zero were interpreted as negligible net charge transfer. These descriptors were analyzed together with Eads and the optimized adsorption geometry to distinguish electronically activated CO₂ from weakly adsorbed or possibly overbound configurations.

3.8. Validation with Cu₂ as a Physical-Chemical Reference

The Cu₂ dimer was used as an auxiliary validation system to support the selection of a reference protocol through experimentally comparable observables. This validation was based on the Cu–Cu equilibrium distance, the harmonic vibrational frequency associated with the Cu–Cu stretching mode, the ionization potential (IP), and the electron affinity (EA). These properties were selected because they probe complementary aspects of the electronic and structural description of copper clusters, including metal–metal bonding, vibrational curvature, and charge-sensitive behavior.

For each evaluated protocol, the neutral Cu₂ dimer was optimized, and its vibrational frequency was calculated. The IP and EA were obtained using an adiabatic approach, in which the neutral, cationic, and anionic species were independently optimized for their corresponding charge and spin states. The IP was calculated as the energy difference between the optimized cation and neutral species. In contrast, the EA was calculated as the energy difference between the optimized neutral and anionic species. The calculated values were compared with the available experimental references summarized in Table 3. The protocol that showed the best overall compromise among Cu–Cu distance, vibrational frequency, IP, and EA was used as an internal reference to evaluate the dispersion of adsorption energies in the Cu₄–CO₂ and Sc₄–CO₂ systems [53,54].

3.9. ORCA vs. Gaussian Cross-Validation and Discrepancy Diagnosis

Selected cases were cross-validated between ORCA and Gaussian to assess the reproducibility of the optimized stationary points. The comparison focused on final geometry, final symmetry, the lowest vibrational frequency, and the character of the stationary point, distinguishing between true minima and structures with imaginary frequencies. This analysis was restricted to selected representative cases and was not intended as a general comparison of the two software packages.

When discrepancies were observed, they were interpreted as possible differences in numerical implementation, including integration grids, SCF convergence behavior, optimization algorithms, treatment of exact exchange, auxiliary-basis approximations, and initial SCF guesses. Cases in which one code produced a positive ω_min while the other produced ω_min < 0 were analyzed as examples of sensitivity in shallow regions of the potential energy surface. These differences indicate that nominally similar computational protocols may converge to different local regions of the potential energy surface or may yield different Hessian curvatures for low-frequency modes. A representative comparison is shown in Figure 3.

3.10. Statistical Analysis of Dispersion and Uncertainty in E_ads

The uncertainty associated with the choice of computational protocol was quantified using dispersion metrics applied to Eads. Deviations were calculated relative to the selected reference protocol according to:

ΔE_ads = E_ads (method) − E_adsreference)

where Eads(method) is the adsorption energy obtained with a given protocol and Eads(reference) is the adsorption energy obtained with the selected reference protocol for the same system, geometry class, and spin state. From the ΔEads values, the mean absolute deviation (MAD), standard deviation (STD), and maximum absolute error, max |ΔEads|, were calculated. These metrics were evaluated globally and, when appropriate, separately for Cu₄ and Sc₄ to identify whether the protocol-dependent dispersion was system-specific [55,56].

This statistical analysis was used to quantify the extent to which adsorption energies depend on the computational protocol. In this way, the comparison does not rely only on qualitative differences between optimized structures but also provides a numerical estimate of methodological uncertainty. The resulting dispersion metrics support the selection of more reliable computational protocols for subsequent studies of CO₂ adsorption and activation in small metal clusters.

4. Conclusions

This study evaluated the sensitivity of CO₂ adsorption on Cu₄ and Sc₄ clusters to the computational protocol, spin multiplicity, initial geometry, and vibrational stability. The results confirm that low-nuclearity metal clusters should not be treated as rigid adsorption sites, because structural relaxation may induce symmetry breaking, convergence toward nearby minima, or stationary points with imaginary frequencies. For this reason, ωmin was used as a necessary acceptance criterion before comparing adsorption energies.

The Cu₄ and Sc₄ systems exhibited distinct adsorption behaviors. Cu₄ generally showed weak or moderate CO₂ interactions, especially for D₂h- and D₃h-derived structures, whereas more favorable adsorption was observed for selected Td configurations. In contrast, Sc₄ displayed broader energetic dispersion and stronger structural reorganization, reflecting the greater sensitivity of its adsorption behavior to cluster deformation, electrostatic polarization, spin state, and exchange–correlation treatment. These findings indicate that Eads alone is insufficient to assign CO₂ activation in small clusters and must be interpreted together with structural, vibrational, thermodynamic, and charge-transfer descriptors.

The Cu₂ reference calculations provided an auxiliary experimental anchor for evaluating the computational protocols using Cu–Cu bond distances, vibrational frequencies, ionization potentials, and electron affinities. However, this validation was used only as a physical–chemical reference and not as a direct substitute for tetramer-level experimental benchmarking. The ORCA–Gaussian comparison further showed that, even under comparable nominal settings, some optimized structures may differ in final symmetry or stationary-point character. This reinforces the need for explicit reporting of numerical settings, convergence behavior, and vibrational validation when studying soft potential-energy surfaces.

The NPA charge-transfer analysis introduced an electronic criterion to distinguish genuine CO₂ activation from possible protocol-dependent overbinding. Several Cu₄ cases showed favorable or near-thermoneutral adsorption energies with qCO₂ values close to zero, indicating weak adsorption or polarization-dominated stabilization rather than charge-transfer-driven activation. Conversely, selected Sc₄ configurations exhibited both negative qCO₂ and exergonic adsorption energies, supporting stronger electronic activation. However, the most extreme values require cautious interpretation because they may also involve significant cluster reorganization or method-dependent stabilization.