In this study, we have tried to adequately approximate the singlet–triplet energy gap (ΔS−ΔT), and therefore, open-shell singlet (OSS) and close-shell singlet (CSS) geometries of Sc⁺(OH₂)_n, where n = 1–2, complexes have also been considered. Geometrical parameters showed that OSS geometries of Sc⁺(OH₂)_n, where n = 1–2, complexes are closer to the triplet-state (TS) geometry. The geometrical parameters at B3LYP/6-31G* and MP2/6-31G* are shown in Figures 2, 3 and showed good agreement with results reported by Trachtman et al., 1998, Irigoras et al., 1999b, and Russo et al., 2001. The OSS geometry is obtained to be very similar to that of TS than the CSS because of its similarity in the orbital occupancy to the TS. Additionally, analysis of the geometrical parameters showed that the Sc⁺–O bond distance increases, and the H–O–H bond angle decreases as we move from low-spin complexes to high-spin complexes. The geometrical parameters of the water molecules slightly change while forming Sc⁺(OH₂)_n, where n = 1–2, complexes. For instance, the H–O–H bond angle of the water molecule changes around 3–4°during complexation, whereas an appreciable difference is not observed in the O–H bond distance of the water molecule. The O–Sc⁺–O bond angle of Sc⁺(OH₂)₂ complexes in both the singlet state (SS) and triplet state (TS) is observed to be 180.0° (linear molecular geometry); however, a nonlinear molecular geometry (O–Sc⁺–O bond angle of 117.3°) is observed for the tricationic complex at both B3LYP/6-31G* and MP2/6-31G* levels of theory. It can be noticed from Figure 2 that the MP2 method predicts a linear molecular geometry (180.0°) for the Sc²⁺(OH₂)₂ complex, whereas B3LYP predicts a nonlinear molecular geometry (150.4°). As a result, the O–Sc²⁺–O bond angle indicates structural relaxation upon the use of the MP2 method. In the case of insertion complexes, large differences are observed in the Sc⁺–H bond distance of SS and TS complexes, as shown in Figure 3. The Sc⁺–H bond distance for HSc⁺OH complexes obtained in the SS and TS is 1.755 Å and 2.563 Å at B3LYP and 1.791 Å and 4.370 Å at MP2, respectively. However, it is 1.780 Å and 2.626 Å at B3LYP and 1.819 Å and 2.712 Å at MP2, respectively, in the case of HSc⁺OH(OH₂) complexes. In the case of the H₂Sc⁺(OH)₂ complex, long Sc⁺–H bonds, that is, 2.379 Å and 2.588 Å at B3LYP and 2.447 Å and 2.716 Å at MP2, are obtained for SS and TS complexes, respectively. Thus, the computed Sc⁺–H bond of the low-spin insertion complex was found to be substantially shorter compared to that of the high-spin complex. Hereby, long Sc⁺–H bond-containing insertion complexes may essentially correspond to H + Sc⁺OH, H + Sc⁺OH(OH₂), H₂ + Sc⁺(OH)₂, H + Sc²⁺OH, H + Sc²⁺OH(OH₂), H₂ + Sc⁺(OH)₂, and H + Sc²⁺OH rather than HSc⁺OH and HSc²⁺OH complexes. Therefore, the bond between the Sc ion and H of these insertion complexes is largely due to the electrostatic interaction, and also, the interaction is not entirely covalent in nature (Figure 3). A careful observation of the figure reveals that the trends of the structural parameters and their absolute values are very comparable, and thus, the qualitative results obtained in the study appear to be independent of the method employed. Therefore, to make a consistent choice of geometry, for further analysis, the structures optimized at the MP2/6-31G* level of theory have been undertaken.

The accurate estimation of the relative energy of spin states of transition metal ions is very important to describe the reactivity and other properties of these ions (Shaik et al., 2011; Costas et al., 2013). Since the energetics of transition metal ions are extremely difficult to predict accurately using any one computational method (Ashley et al., 2017), we have predicted the relative energy of SS and TS bare Sc⁺ ions, Sc⁺–OH₂, and HSc⁺OH complexes at 48 different levels of theory. These methods are HF, PBEPBE, B3LYP, M06, MP2, and CCSD(T) using LanL2DZ, DGDZVP, Def2TZVP, 6-31G*, 6-311+G*, 6-311G**, cc-pVTZ, and aug-cc-pVTZ basis sets, as shown in Supplementary Table S1A. To calculate an adequate approximation of ΔS–ΔT for the bare Sc⁺ ion and complexes, the procedure proposed by Yamaguchi et al., 1988 is used to correct the spin contamination problem of the OSS bare Sc⁺ ion. The calculated ΔS–ΔT values of the bare Sc⁺ ion, Sc⁺OH₂, and HSc⁺OH complexes are shown in Supplementary Tables S1, S2. A total of 14 different methods predicted a reasonably good value of ΔS−ΔT, that is, relative energy of contamination-corrected open-shell singlet (CC_OSS) and TS of the bare Sc⁺ ion, and showed an excellent agreement with the experimental value (7.38 kcal/mol) reported by Chen et al., 1994, as shown in Table 1. As expected, TS (3d¹ 4s¹) is predicted as a ground-state multiplicity for the bare Sc⁺ ion. These 14 methods, except M06, have also predicted TS (3d¹ 4s¹) as a ground-state multiplicity for the Sc⁺OH₂ complex. In contrast, M06 predicted SS (4s² 3d⁰) as a ground-state multiplicity for the Sc⁺OH₂ complex, as shown in Supplementary Table S1B. The values obtained at PBEPBE (with few exceptions), B3LYP, MP2, and CCSD(T) showed a reasonable agreement with the reported theoretical values (8.63 kcal/mol to 20.30 kcal/mol) by Irigoras et al., 1999a. However, all the considered methods predicted SS (4s² 3d⁰) as the ground-state multiplicity for insertion complexes.

Quantum theory of atoms in molecules (QTAIM) is useful to examine the interaction between atoms based on the topology of the electron density at CPs (Bader, 1998). We have examined ρ, ∇²ρ, H(r), [-(G(r)/V(r)], and percentage contribution of G(r) at the bond critical point (BCP) of the MP2/6-31G* optimized geometries of Sc ion–water and their insertion complexes to elucidate the nature of the Sc-ion bond with H₂O, OH, and H, as shown in Figure 4 and Supplementary Table S3. A high value of ρ and a negative value of ∇²ρ indicate covalent interaction, while in general, a low value of ρ and a positive value of ∇²ρ suggest a noncovalent or close-shell type bonding (Kumar et al., 2021). However, the sign of H(r) and the value of [-G(r)/V(r)] explain the covalent nature of the bond. It may be noticed that the value of ρ increases and that of ∇²ρ and H(r) decreases for the Sc–O bond of Sc^+/2+/3+–OH₂ and Sc^+/2+–OH complexes when moving from low-spin to high-spin complexes, corresponding to a covalent interaction. (Figure 4 ). However, in the case of Sc⁺(OH₂)_n, where n = 1–2, and HSc^+/2+OH(OH₂) complexes, a positive value of both ∇²ρ and H(r) with a small value of ρ (0.03–0.06 a.u.) and [−G(r)/V(r)] > 1 has been observed at the BCP of the Sc^+/2+–OH₂ bond, suggesting a noncovalent bond. A positive value of ∇²ρ and a negative value of H(r) with a significant value of ρ (∼0.10 a.u.) and 1 <[−G(r)/V(r)]> 0.5 suggest that the Sc³⁺–water bond possesses a partial covalent character. In the case of insertion complexes, a positive value of ∇²ρ and a negative H(r) with a ρ value of 0.08–0.17 a.u. have been observed for Sc⁺–OH, Sc²⁺–OH, and Sc⁺–H bonds, as shown in Figure 4 and Supplementary Table S3A. The value of [−G(r)/V(r)] is observed to be less than 1 (Figure 4), and the percentage contribution of G(r) into H(r) (Supplementary Table S3B) was less than 50% at the BCP of Sc^+/2+–OH and Sc⁺–H bonds of HSc^+/2+OH and HSc^+/2+OH(OH₂) complexes, that is, the lowering of the kinetic energy may be traced to the enhancement of the covalent nature in these bonds (Zhao et al., 2019). Besides, it may also be noted that the contribution of G(r) decreases with the increase in ρ at the BCP of the bond in most of the cases except for Sc^+/2+–H bonds, which are clearly indicated as loose bonds. Furthermore, the bond length of Sc^+/2+–H is long, clearly indicating that the Sc^+/2+–H bond is not formed in the insertion complexes on the TS and doublet-state (DS) potential energy surfaces (see Supplementary Table S3A). Thus, topological analysis of Sc ion–water and its insertion complexes indicated that Sc^+/2+–OH₂ bonds are noncovalent in nature. A closer look at the structural and topological parameters reveals that Sc³⁺–OH₂, Sc²⁺–OH, and Sc⁺–H bonds have a strong admixture of covalent nature.

The binding and reaction energies are calculated using the aforementioned 14 methods such as HF in conjugation with the LanL2DZ basis set, B3LYP in conjugation with DGDZVP, 6-31G*, and aug-cc-pVTZ basis sets, M06 in conjugation with 6-31G*, 6-311+G*, cc-pVTZ, and aug-cc-pVTZ basis sets, MP2 in conjugation with LanL2DZ, 6-311+G*, cc-pVTZ, and aug-cc-pVTZ basis sets, and CCSD(T) in conjugation with the 6-311G** basis set, as shown in Table 1 and Supplementary Table S4. The computed Sc⁺–OH₂ binding energy is in fair agreement with the experimental observations and earlier computational results for the ground state of the Sc⁺OH₂ complexes (Magnera et al., 1989) (Table 1). In comparison to the other theoretical values, our values calculated at B3LYP/aug-cc-pVTZ, MP2/cc-pVTZ, and CCSD(T)/6-311G** levels of theory showed a very good agreement with the value reported by Irigoras et al., 1999a and Sharma et al., 2016, that is, −32.52 kcal/mol and −31.18 kcal/mol at CCSD(T)/TZVP + G(3df2p) and CCSD(T)/def2-TZVP levels of theory, respectively. Moreover, our calculated values suggested that binding energy increases by moving from high spin to low spin and from lower to higher oxidation states, that is, from monocationic to tricationic, as shown in Supplementary Table S4. Figures 2 and 3 depict the optimized geometric parameters with two water molecules and provide interesting insights into the onset of microsolvation of the metal ion and the change in the propensity of competing pathways, in the presence of two water molecules. Rosi et al., 1989 have reported a similar study which is in good qualitative agreement with the results obtained here. Calculated values suggested that ΔE_seq decreases with the addition of the second water molecule to the Sc ion in all the considered methods (Supplementary Table S4). Reaction energy (ΔE_R) has been calculated using the equation Sc⁺ + OH₂→ Sc⁺O+ H₂ + ΔE, as shown in Table 1. A large inconsistency is observed among the calculated ΔE values using 14 different methods. The results clearly reveal the grossly poor performance of the bulk of methods employed in estimating the reaction energies. The calculated values at B3LYP in conjugation with DGDZVP and aug-cc-pVTZ basis sets, MP2 in conjugation with cc-pVTZ and aug-cc-pVTZ basis sets, and CCSD(T) in conjugation with the 6-311G** basis set have shown good agreement with the experimental value (Chen et al., 1994). Thus, an exothermic reaction and the formation of the low-spin Sc⁺O+ H₂ product are observed. In addition, our calculated values at MP2 in conjugation with cc-pVTZ and aug-cc-pVTZ, B3LYP in conjugation with DGDZVP and aug-cc-pVTZ, and CCSD(T) in conjugation with 6-311G** basis sets showed good agreement and seem to be better than other reported theoretical values (Tilson et al., 1991; Ye et al., 1997; Irigoras et al., 1999b). The binding and reaction energy values have also been computed at BP86, B97D, and B3LYP-D3 using 6-311G** and aug-cc-pVTZ basis sets, and the obtained values are observed to be similar to those observed using other DFT methods, as shown in Table 1 and Supplementary Table S10.

Thus, the calculated values ΔS−ΔT, ΔE seq , and ΔE_R suggested that B3LYP in conjugation with DGDZVP and aug-cc-pVTZ, MP2 in conjugation with 6–311+G*, cc-pVTZ, and aug-cc-pVTZ, and CCSD(T) in conjugation with the 6-311G** basis set predict the energetics of Sc ions accurately, and therefore, these methods can be used to delineate the interaction of the Sc ion with water molecules. We have opted the CCSD(T)/6-311G**//MP2/6-31G* method for further analysis.

The potential energy surface analysis has been carried out for interaction of the Sc^z+ ion with water molecules leading to 1) hydrate: Sc^z+(H₂O)_n, 2) hydrolyse: H–Sc^z+–OH (H₂O)_n-1orH₂–Sc^z+–(OH)₂(H₂O)_n-2, 3) dehydrogenate: Sc ^z+O+ H₂, 4) metal hydroxide: Sc^(z−1)+OH, and 5) metal hydride: Sc^(z−1)+H at CCSD(T)/6-311G**//MP2/6-31G** level of theory, as shown in Figures 5–7). No stationary point on the potential energy surface has been obtained corresponding to the OSS complex of Sc⁺OH₂, as the putative structure collapses to CSS in a barrierless fashion upon the formation of the insertion complex (HSc⁺OH), which corroborates well with the earlier observations (Irigoras et al., 1999a). The reaction starts from bare Sc⁺ ions separated from water molecules at an infinite distance and leads to the formation of the hydrated complex, insertion complex, metal oxide, metal hydroxide, and metal hydride, as shown in Supplementary Figure S1. Considering the well-known limitations of the MP2 methods, when dynamic electron correlation is substantial, in properly reproducing the geometry of small systems, we would like to benchmark the method more systematically. Thus, geometry optimization of the considered metal oxide, hydroxide, and hydride complexes has been performed at five different levels of theory. The insertion complex formed by Sc⁺ insertion into the O–H bond of water molecules, as the valance electrons of Sc⁺ ions are shared by -OH and -H atoms, making the complex a singlet, Figures 5–7, which depicts the energetics of the competing pathways, from which it can be seen that the insertion complex, that is, HSc⁺OH, is substantially stable in the singlet compared to that of the triplet, while the energetics of the spin states of the bare metal ion exhibit completely contrasting trends. The observed spin-crossover following the insertion reveals that there is a possibility for multiple electronic states crossing along the reaction paths on the potential energy surfaces. Thus, modeling of these molecules warrants careful consideration of various electronic states as there is a great potential for crossover of the spin and electronic states due to the vast variation in the energetics of these species even with small geometric changes along with the reaction coordinate. The insertion complex, on the triplet potential energy surface, is essentially a hydrogen elimination process with the formation of a low-spin Sc⁺O+ H₂ as the product, which is in agreement with the earlier study by Irigoras et al., 1999b. Thus, the metal ions under the right condition have a definitive potential to generate hydrogen, which is of course of great industrial significance. In the case of metal oxide, a relatively stable Sc⁺O complex has been observed on the SS potential energy surface than the corresponding Sc⁺OH₂ complex. However, HSc⁺OH is relatively more stable than the Sc⁺O complex on the SS potential energy surface. The energetics of the formation of a metal hydroxide ScOH and metal hydride ScH suggest that both the reactions are energetically unfavorable on both SS and TS potential energy surfaces. We have further expanded the scope of this study by examining the interaction of monohydrated, insertion, oxide, hydroxide, and hydride complexes with the second water molecule. The hydrated products of monohydrated, insertion, oxide, hydroxide, and hydride complexes are observed to be more stable than their predecessors on both SS and TS potential energy surfaces, as shown in Figure 5. Interestingly, the hydrated TS insertion complex H ^… Sc⁺OH(OH₂) is lower than both TS insertion complexes, H^…Sc⁺OH and the reactant pair Sc⁺ + 2H₂O by 41.11 kcal/mol and 27.65 kcal/mol, respectively, and nearly degenerates (∼2.96 kcal/mol) to the TS monohydrated complex Sc⁺OH₂. It can be inferred that splitting the second water molecule slightly destabilizes the TS monohydrated complex, and thus, two water molecules appeared to be required to reach a stable complex on the potential energy surface. Thus, when performing the computations with two water molecules, Sc⁺ insertion into the O–H bond of the second water molecule eventually forms metal dihydroxide, that is, Sc⁺(OH)₂ + 2H on the TS potential energy surface and Sc⁺(OH)₂ + H₂on the SS potential energy surface. It is interesting to observe that there is a great propensity exhibited for water splitting compared to solvation, with the Sc⁺(OH)₂ + H₂ product on the SS being observed well below the ground-state reactant pair Sc⁺ + 2H₂O and Sc⁺(OH₂)₂ complexes. The examination of di- and tricationic Sc-ion reactions with a water molecule indicates that the hydrated complex of Sc²⁺ and hydroxide of Sc³⁺ lie below the reactant and the corresponding complexes on the DS and SS potential energy surfaces, respectively (Figures 6, 7). Therefore, it may be inferred that the Sc²⁺ ion is likely to adopt a more feasible solvation pathway compared to the corresponding insertion alternative (Figure 6). However, the affinity of the Sc³⁺ ion to form a metal hydroxide, Sc²⁺OH, is observed to be higher than getting solvated to form Sc³⁺(OH₂) as hydroxide complexes lie above the corresponding hydrated complex (Figure 7).

In order to investigate the sequential solvation effect on the nature of Sc ion–water complexes, we have examined the microsolvation process of up to 6 water molecules and probed the relative propensity of water molecules to occupy the first and second solvation shells. The schematic depiction of the representative structure of Sc(OH₂)_n, where n = 1–6, ion complexes and their nomenclature are shown in Figure 1. We have used the nomenclature X + Y (X = number of water molecules in the first solvation shell and Y = number of water molecules in the second solvation shell) to present the number of water molecules in the solvation shells. Starting from one water molecule, various conformers of Sc(OH₂)_n, where n = 1–6, ion complexes in all possible oxidation and spin states have been explored and the lowest energy conformers are selected for further discussion, as shown in Figure 8 and Supplementary Figure S3. A cursory look of interaction energy (IE) of these complexes, obtained at the CCSD(T)/6-311G**//MP2/6-31G** level of theory, suggests that IE increases as the number of water molecules increase either in the first or both in the first and second solvation shells, as shown in Figure 9 and Supplementary Figure S2. IE has also been obtained using different methods to ensure that the obtained values are not biased (Supplementary Table S5).We have considered IE of each structure as a criterion to describe the stability of Sc(OH₂)_n, where n = 1–6, ion complexes. The sequential binding ΔE_seq energy of these stable complexes, obtained at the CCSD(T)/6-311G**//MP2/6-31G** level of theory, suggests that ΔE_seq energy decreases with the addition of every subsequent water molecule to the complexes except for the 4 + 0 structure of the triplet Sc⁺(OH₂)₄ complex, as shown in Figure 8. Our calculated values showed a good agreement with the available experimental and theoretical results reported by Rosi et al., 1989, Magnera et al., 1989, and Dalleska et al., 1994.

IE increases as the number of water molecules increases in the second solvation shell for each CN of Sc ions irrespective of oxidation and spin states, that is, it increases from 1 + 0 to 1 + 2 (CN = 1), 2 + 0 to 2 + 4 (CN = 2), 3 + 0 to 3 + 3 (CN = 3), 4 + 0 to 4 + 2 (CN = 4), and 5 + 0 to 5 + 1 (CN = 5), shown in Figure 9 and Supplementary Figure S2. Thus, this result suggests that Sc(OH₂)_n, where n = 1–6, ion complexes in each CN are more stable with the maximum number of water molecules in the second solvation shell. Furthermore, the analysis of the bond distance between the Sc ion and first solvation shell water molecules suggested that the average M–O distance of Sc(OH₂)_n, where n = 1–6, ion complexes increases with increasing number of water molecules around the Sc ion. Among the complexes with the same number of water molecules in the second solvation shell, the average M–O distance increases with the increase in the CN of the Sc ion and vice versa. In addition, it has also been observed that the average M–O distance increased by moving from low- to high-spin Sc⁺(OH₂)_n, where n = 1–6, ion complexes and decreased by moving from mono- to tricationic ground-state complexes, as shown in Supplementary Table S6. The topological analysis predicted Sc^+/2+–OH₂ bonds to be a noncovalent bond as the calculated values of ∇²ρ and H(r) are positive, and [−(G(r)/V(r)] is less than 1 irrespective of the spin states. However, the Sc³⁺–OH₂ bond is predicted as partially covalent in nature, as shown in Supplementary Table S7.

Natural population analysis has been carried out on all the considered structures of Sc(OH₂)_n, where n = 1–6, ion complexes to examine the charge transfer between the Sc ion and the water molecules. The variation in charge on the Sc ion with the increase in the CN of Sc⁺(OH₂)_n, where n = 1–6, complexes is shown in Supplementary Figure S4. The figure shows that the charge on the Sc ion decreases monotonically as the number of water molecules is increased in the complexes. Besides, the charge on the Sc ion seemed to be decreasing as the number of water molecules increases in the second solvation shell for each coordination number (i.e., CN = 1–6) and vice versa. Such differences can be explained based on the differences in the organization of water molecules around the Sc ion. The most stable structure of every CN complex showed a minimum charge on the Sc ion. A cursory view of Supplementary Table S8 shows that IE is inversely proportional to the charge on the Sc ion, which signifies that charge transfer plays an important role in the stabilization of Sc(OH₂)_n, where n = 1–6, ion complexes. Localized molecular orbital energy decomposition analysis is carried out to examine the contribution of energy components into the Sc ion–water interaction, as shown in Figure 10 and Supplementary Table S9. The figure shows that the contribution of the ∆E_ele component dominates the Sc and water interaction followed by the contribution of the ∆E_pol component, irrespective of oxidation and spin states. However, the contribution of the ∆E_disp component is found to be negligible with around a contribution of 2–6%, as shown in Figure 10.