Assessing the Intrinsic Strengths of Ion–Solvent and Solvent–Solvent Interactions for Hydrated Mg 2+ Clusters

: Information resulting from a comprehensive investigation into the intrinsic strengths of hydrated divalent magnesium clusters is useful for elucidating the role of aqueous solvents on the Mg 2+ ion, which can be related to those in bulk aqueous solution. However, the intrinsic Mg–O and intermolecular hydrogen bond interactions of hydrated magnesium ion clusters have yet to be quantitatively measured. In this work, we investigated a set of 17 hydrated divalent magnesium clusters by means of local vibrational mode force constants calculated at the ω B97X-D/6-311++G(d,p) level of theory, where the nature of the ion–solvent and solvent–solvent interactions were interpreted from topological electron density analysis and natural population analysis. We found the intrinsic strength of inner shell Mg–O interactions for [Mg(H 2 O) n ] 2+ (n = 1–6) clusters to relate to the electron density at the bond critical point in Mg–O bonds. From the application of a secondary hydration shell to [Mg(H 2 O) n ] 2+ (n = 5–6) clusters, stronger Mg–O interactions were observed to correspond to larger instances of charge transfer between the lp(O) orbitals of the inner hydration shell and the unﬁlled valence shell of Mg. As the charge transfer between water molecules of the ﬁrst and second solvent shell increased, so did the strength of their intermolecular hydrogen bonds (HBs). Cumulative local vibrational mode force constants of explicitly solvated Mg 2+ , having an outer hydration shell, reveal a CN of 5, rather than a CN of 6, to yield slightly more stable conﬁgurations in some instances. However, the cumulative local mode stretching force constants of implicitly solvated Mg 2+ show the six-coordinated cluster to be the most stable. These results show that such intrinsic bond strength measures for Mg–O and HBs offer an effective way for determining the coordination number of hydrated magnesium ion clusters.


Introduction
Magnesium, being the eighth most abundant element and second most abundant alkaline earth metal, accounts for 2% of the Earth's crust [1]. As a result, magnesium (Mg 2+ ) is heavily prevalent in ground water and sea water as a result of rock and mineral dissolution (i.e., calcium magnesium carbonate CaMg(CO 3 ) 3 ). Chemical [2][3][4], automotive/aircraft [5][6][7][8][9], agricultural [10], and energy storage [11][12][13] industries have also contributed to the presence of magnesium in water. As the production and application of magnesium increase, the amount of magnesium waste increases [14], having been noted to pose dangerous environmental effects [15]. High concentrations of such mineral-based ions have resulted in hard water where the hydration of the Mg 2+ ion yields insoluble precipitates.
The structure and reactivity of hydrated divalent magnesium (i.e., Mg 2+ ) has been a topic of notable investigation, as such information is useful for understanding the role a solvent has on such physical properties. The determination of the coordination number (CN) has been highly sought after to understand the nature of hydrated magnesium ion clusters, where such information can be related to the bulk form of the metal cation in an aqueous solvent. The CN is based upon the number of inner shell water molecules (n) bounded to the Mg 2+ ion by the formation of Mg-O bonds ([Mg(H 2 O) n ] 2+ ); the Mg-O interactions are primarily electrostatic, where the dipole of these interactions are pointed from the water molecules towards the metal ion [16]. Experimental work based on X-ray diffraction (XRD) [17,18], Raman spectroscopy [19,20] X-ray Raman scattering (XRS) spectroscopy [21], infrared action spectroscopy [22], small angle X-ray scattering (SAXS) [21], black body infrared radiative dissociation (BIRD) [23], and neutron diffraction [24] have proposed a CN of 6 (i.e., [Mg(H 2 O) 6 6 ] 2+ ) may exist [23]. The XRS and SAXS work of Waluyo and co-workers estimated the Mg-O distances for the [Mg(H 2 O) 6 ] 2+ cluster to be 2.1 Å, approximately [21]. Moreover, the Infrared Action Spectroscopy work of Bush et al. [22] and the High-Pressure Mass Spectrometry work of Peschke et al. [25] reported that the inner shell holds up to a maximum of six water molecules with additional water molecules moving towards outer shells. On the other hand, the XRD work of Albright and co-workers, involving hydrated divalent magnesium clusters in concentrated chloride solutions, reported a CN of 6.8 [26]. It is widely recognized that the second hydration shell of hydrated Mg 2+ clusters accommodates up to 12 water molecules [18,[27][28][29], where, determined via X-ray diffraction, the bond distance between the metal ion and water molecules of the second solvent shell range from 4.10 to 4.28 Å and the bond distance between the inner and outer solvent shell falls between 2.75 and 2.81 Å [18,29]. The X-ray absorption spectra of 2 and 4 M concentrations of the Mg 2+ cation in chloride solution, generated by Cappa et al., revealed that a large amount of charge transfer occurs between the water molecules of the first solvent shell and the Mg 2+ ion [30].
The binding energy (BE) is a thermodynamic measure that provides structural and energetic information for molecular systems and their chemical reactions (e.g., hydrated divalent magnesium clusters). BE values are based upon the energy difference between the hydrated magnesium cluster and their fragments (i.e., Mg 2+ , H 2 O), where all systems are at a local minima. However, the BE is notably tainted with electron reorganizations and the geometrical relaxation of the fragments, causing the measure to be short of ideally describing intrinsic bond strength [53][54][55][56][57]. In this work, we used local mode stretching force constants, obtained from the local mode analysis (LMA) of Konkoli and Cremer [58][59][60][61], to directly assess the strength of a particular bond [62,63]. The local stretching force constant (k a ), characteristic of the stretching vibration of a bond, preserves the geometry and electronic structure to precisely expresses the intrinsic bond strength. The local stretching force constant has had success in describing covalent bonds [63][64][65][66][67][68] and noncovalent interactions such as tetrel bonds [69], pnicogen bonds [70], chalcogen bonds [71], halogen bonds [72][73][74], and hydrogen bonds [75][76][77][78][79][80].
Acquiring quantitative explanations for the ion-solvent and solvent-solvent interactions of hydrated Mg 2+ clusters would provide a comprehensive understanding of the hydration of divalent Mg ions and the interactions that are involved in stabilizing such clusters. In this work, we quantified, for the first time, the bond strength of the electrostatic interactions for the hydrated magnesium clusters, being Mg-O interactions (ion-solvent) and hydrogen bonds (solvent-solvent), via the local mode stretching force constants of Konkoli and Cremer [58]. The main goals of this work were: (i) to assess the intrinsic bond strengths of the ion-solvent interactions (Mg-O) and their variation as the CN, of the first hydration shell, expands from 1 to 6 ([Mg(H 2 O) n ] 2+ , where n = 1-6); (ii) to establish how the addition of a secondary hydration shell affects the intrinsic bond strength of inner ion-solvent (Mg-O) interactions ([Mg(H 2 O) n ] 2+ -m(H 2 O) with n = 5 and 6, m = 1-5 with n + m = 6-10 and where m = 12, where m represents the number of water molecules in the second hydration shell); and (iii) to quantitatively measure the intermolecular hydrogen bonding established between inner and outer solvent shells.

Computational Methods
Minimum geometries and normal vibrational modes of 17 hydrated magnesium clusters and reference molecules R1-R5 were obtained using the ωB97X-D functional [81][82][83] in combination with the Pople's 6-311++G(d,p) basis set of triple-ζ quality [84][85][86]. The ωB97X-D functional was used as it is well known for accurately describing clusters dictated by non-covalent interactions (NCIs) [43,86,87] such as hydrogen bond interactions [88][89][90]. To account for the solvation effects of the outer hydration shells, clusters 1-6 and reference molecules R1 and R2 were recalculated, at the same level of theory, using the implicit conductor-like polarizable continuum model (CPCM). The implicit CPCM model was utilized as it has been demonstrated to comprehensively account for solvent effects [43,91,92]. The initial configurations of clusters 1-6 were extracted from the work of Rao et al. [47] and those of clusters 7-17 were based off the work of Bai et al. [41]. Following geometry optimization and frequency calculations, the LMA of Konkoli and Cremer [58][59][60][61] was employed to quantify the intrinsic strength of Mg-O and HB interactions. Because a comprehensive background on LMA has been elaborated in Ref. [93], we summarize only the main essence of LMA below.
Vibrational spectroscopy provides a comprehensive overview on the electronic structure and bonding of a molecule in the form of normal vibrational modes. However, normal modes tend to delocalize over the molecule due to the presence of electronic and kinematic (mass) coupling, hampering the direct description of the intrinsic strength of chemical bonds. The fundamental equation of vibrational spectroscopy (Wilson's secular equation of molecular vibration) is used to eliminate the electronic coupling resulting from the off-diagonal elements of the force constant matrix (F x ): where F x is expressed in Cartesian coordinates and M depicts the mass matrix. The L matrix accumulates the 3N normal mode vibrational eigenvectors I µ and Λ is a diagonal matrix consisting of vibrational eigenvalues λ µ . From λ µ = 4π 2 c 2 ω 2 µ (c = speed of light) normal mode vibrational frequencies (i.e., ω µ ) are retrieved. To solve the equation, the force constant matrix F x undergoes diagonalization which requires that Cartesian coordinates be transformed into normal coordinates Q to yield in: where the force constant matrix K is expressed by Q. The electronic decoupling of vibrational modes is a general procedure for the derivation of harmonic vibrational frequencies where the mass coupling between the resultant normal modes remains. The remaining kinematic (mass) coupling can be eliminated by using a mass-decoupled equivalent of the Euler-Lagrange equation, as demonstrated by Konkoli and Cremer [58][59][60][61]. The proceeding local vibrational modes (a n ) are liberated from all mode-mode coupling (a n ): where the local mode vector corresponding to the n-th internal coordinate q n is represented by a n . Each local mode a n corresponds to a distinct local mode force constant k a n and local vibrational frequency ω a n through: and: (ω a n ) 2 = 1 4π 2 c 2 k a n G a n,n where G a n,n is the mass of the local mode a n which is a diagonal element out of the Wilson G matrix. Hereafter, k a is used instead of k a n (i.e., k a = k a n ). Via an adiabatic connection scheme (ACS), a one-to-one relation between local and normal vibrational modes is established [61,62]. Unlike normal modes, the local modes do not depend on the kind of coordinates used to describe a molecular system. Because the local mode force constants k a only rely upon electronic structure change, as they are independent of atomic masses, they capture pure electronic effects and adequately describe chemical bond strength. The measures of bond strength order can be acquired from local mode force constants k a through the generalized Badger rule [94]: From the k a values of two reference compounds of known bond orders, constants a and b can be obtained following the specification that BSO n = 0 when k a = 0. For the Mg-O interactions, reference molecules MgH 2 (R1) and Mg --O (R2) were used alongside BSO n values of 1.0 and 2.0, respectively. Using R1 and R2, the a and b, constants values of 0.7074 and 0.7968 were obtained. For HBs, references molecules F -H (R3) and [H···F···H] -(R4) were utilized and assigned BSO n values of 1.0 and 0.5, respectively. Using R3 and R4, the a and b constants of 0.5293 and 0.2781 were derived. The HB of the water dimer (R5) was also included as to assess the magnitude of intrinsic HB strengths.
All DFT calculations were carried out using the Gaussian 16 Rev. A.03 program package [95]. Geometry optimizations were conducted using a superfine grid integration and a tight convergence criterion for the forces and displacements. The local force constants were calculated using the LModeA program [93,96]. The NBO6 program was utilized to obtain natural population charges and second order perturbation stabilization energy (∆E (2) ) values through natural bond orbital (NBO) analysis [97,98]. ∆E (2) values correspond to the stabilization energy resulting from intermolecular charge transfer events (i.e., orbital interactions). Electron densities (ρ c ) and energy densities (H c ) at Mg-O and O···H bond critical points r c were determined using the AIMAll program [99]. The nature of the Mg-O and HB interactions were distinguished following the Cremer-Kraka criteria where a negative (stabilizing) value of the energy density (H c < 0) at the bond critical point r c (BCP) denotes the covalent bonding while a positive value (i.e., H c > 0) indicates electrostatic interaction [100][101][102].

Results and Discussion
First hydration shell. Table 1 7 ] 2+ results in the migration of a single water molecule from the first hydration shell to the second hydration shell, as observed in previous theoretical works [41,45,50]. Note that the initial coordinates in Ref. [45] were taken from X-ray crystallographic studies. In addition, as mentioned earlier, the experimental work of Peschke et al. and Bush et al. showed that a CN of 7 for hydrated magnesium clusters was unstable leading to a CN of 6 [22,25].   As the cluster size increases from 1-6, the Mg-O bond distance increases from 1.957 Å (1) to 2.108 Å (6). These results agree with the previous literature as the Mg-O bond length increases as n within the [Mg(H 2 O) n ] 2+ clusters increases from 1 to 6 [41,47]. We note that the Mg-O bond distances for clusters 1-6 (1.96 to 2.11 Å) agree well with experimentally determined lengths which range from 2.10 to 2.12 Å [17][18][19][20]24,26,103] Table 1). These results are in line with the work of Bai et al., which suggests that the inner Mg-O bond lengths, for clusters of similar configurations to 7-17, range between 2.045 and 2.109 Å [41]. As well, the HB distances between the solvent shells within  [104]. In the following, we evaluated clusters 7-17 to assess how the intermolecular hydrogen bonding occurring between first and second hydration shells modulates the strength of inner Mg-O interactions. NBO charges can be found for clusters 7-17 in Figures S3-S8 of the Supporting Information.    Table 2). As demonstrated in the work of Tao et al., the orbital interaction occurring between the oxygen lone pair (i.e., lp(O)) of the acceptor water molecule and the anti-bonding σ * orbital of the donor water molecule (i.e., σ * (O-H)) plays an instrumental role behind HB stabilization; the HB strength is determined by push-pull effects which are interpreted from augmented charge transfer events [105]. As the magnitude of HB charge transfer becomes greater, the HB interaction becomes stronger [105]. Average stabilization energy values (∆E (2) ), which represent the extent of charge transfer happening between the lp(O) of the second hydration shell and σ * (O-H) of the first hydration shell, unveil that cluster the 8 obtains weaker HBs in contrast to cluster 7 due to smaller charge transfers (average ∆E (2) = 15.27 (7); 14.77 kcal/mol (8)). Cluster 8 obtains, on average, shorter and stronger MgO bonds in contrast to those of 7 (see Table 1). The Mg-O bonds of 8 possess larger electrostatic contributions due to more charge being transferred from the lp(O) of inner shell H 2 O molecules to the lp * (Mg) orbital (lp * represents unfilled valence shell) (∆E (2) = 137.93 (7); 141.83 kcal/mol (8)).

Clusters with 2 nd Water Shell [Mg(H 2 O) n ] 2+ ·m(H 2 O) (n= 6 & 5; m= 1-5, where n + m = 6-10 H 2 O)
Cluster 9 contains five water molecules in the first shell and three water molecules in the second shell ([Mg(H 2 O) 5 ] 2+ -3(H 2 O)), and the six HBs that form between the two solvent shells (see Figure 1) have an average BSO n value of 0.318 (k a = 0.162 mdyn/Å). The HBs of 9 are longer and weaker in contrast to the HBs of clusters 7 and 8 (See Table 2) due to a smaller amount of stabilization energy resulting from the charge transfer between outer → σ * (O-H) and inner shell orbitals (average ∆E (2) = 13.398 kcal/mol (9)). Similarly to cluster 8, cluster 9, in comparison to cluster 7, involves a greater amount of electron charge transfer from inner lp(O) → lp * (Mg) (∆E (2) = 146.92 kcal/mol (9)) which increases the electrostatic interactions between the Mg ion and the inner water shell within 9; this strengthening of the electrostatic interactions causes the Mg-O bonds of 9 to be shorter and stronger than those of 8 (see Table 1).
Cluster 10 is composed of five water molecules in the first hydration shell and four water molecules in the second hydration shell ([Mg(H 2 O) 5 ] 2+ -4(H 2 O)), and eight HBs are formed between the two hydration shells where the average BSO n is 0.297 (k a = 0.125 mdyn/Å). In contrast to clusters 7-9, the HBs of 10 are weaker due to a smaller amount of charge transfer occurring between the two hydration shells (average ∆E (2) = 11.350 kcal/mol (10)). Furthermore, 10 possesses the weakest HBs (see Table 1). As a consequence, the strength of the electrostatic interactions between the Mg ion and inner water molecules of cluster 10 increases and in turn strengthens and shortens the Mg-O bonds beyond that of clusters 6, 7, 8, and 9 (see Table 1). Cluster 11 has a total of ten water molecules across both hydration shells ([Mg(H 2 O) 5 ] 2+ -5(H 2 O)), and nine HBs were detected between the inner and outer hydration shell where the average BSO n value is 0.309 (k a = 0.153 mdyn/Å). The HBs of 11 are stronger than those of 10 because of a greater charge transfer from the outer lp(O) to the inner σ * (O-H) orbitals (average ∆E (2) = 12.427 kcal/mol (11)). In comparison to cluster 10, the Mg-O bond length and the strength of 11, on average, are shorter and stronger (see Table 1). Furthermore, in comparison to cluster 10, the ∆E (2) based off the charge transfer between inner lp(O) and the lp * (Mg) orbital is slightly larger for cluster 11 (∆E (2) = 154.29 (10, 154.53 kcal/mol (11)).
Cluster 12 is a five-coordinated structure that is surrounded by a second solvent shell composed of 12 water molecules ([Mg(H 2 O) 5 ] 2+ -12(H 2 O)), and 22 HBs are established between the two solvent shells having an average k a (HB) value greater than that of the water dimer by 0.010 mdyn/Å (R5). Moreover, the HBs within cluster 12 are the strongest HBs amongst all clusters having a secondary hydration shell (7-17) (see Table 1), the amount of charge delocalization occurring between both hydration shells is the largest for 12 (average ∆E (2) = 14.672 kcal/mol (12)). Thus, for cluster 12, the electrostatic interactions between the Mg ion and inner H 2 O molecules become stronger and enable the charge transfer between the inner lp(O) orbitals and the lp * (Mg) orbital to be greater in magnitude in comparison to clusters 6-11 (∆E (2) = 557.56 kcal/mol (12)). This large amount of charge transfer between the Mg 2+ ion and inner shell water molecules of 12 yields Mg-O bond distances that are smaller than those of clusters 6-11 (see Table 1 (13)(14)(15)(16)(17). In comparison to cluster 6, the inner Mg-O interactions of 13 elongate and weaken slightly (See Table 1). The Mg ion within cluster 14 is coordinated to six inner shell water molecules and has a outer hydration shell holding two water molecules (  (14)). In comparison to clusters 6 and 13-16, the inner Mg-O interactions of 14 are shorter and stronger (see Table 1). The Mg-O bonds of 14 are stronger than those of 13 due to stronger electrostatic interactions as shown from the larger stabilization energy ∆E (2) value stemming from the charge transfer between inner shell lp(O) to the lp * (Mg) orbital (∆E (2) = 144.35 (13), 148.58 kcal/mol (14)).
Cluster 15 ([Mg(H 2 O) 6 ] 2+ -3 (H 2 O)) acquires six HBs which are formed between the two water shells, with an average BSO n (HB) value of 0.330. The HB bond strengths between the inner and outer hydration shells of cluster 15 are in the closest proximity to that of the water dimer with regard to [Mg(H 2 O) 6 ] 2+ -m(H 2 O) clusters (m = 1-4; 13-16) (see Table 2). The average HB strength within 15 is stronger than that of 14 due to a greater electronic charge transfer between the outer shell lp(O) orbitals and the inner shell σ * (O-H) orbitals (average ∆E (2) = 12.535 kcal/mol (15)). As a result, the decrease in the electrostatic interaction between the Mg ion and O atoms of the inner hydration shell slightly strengthen and shorten the Mg-O bonds of 15 with respect to that of 14 (see Table 1). Cluster 16 has six water molecules in the first shell and four water molecules in the second shell (  (16)).
Cluster 17 is a six-coordinated structure that is surrounded by a secondary shell comprised of 12 water molecules ([Mg(H 2 O) 6 ] 2+ -12 (H 2 O)); a HB network involving 24 HBs between the inner and outer hydration shell are established where the BSO n is nearly exact to that of the water dimer (R5) (see Table 1). Furthermore, cluster 17 has the second strongest HBs amongst clusters with a secondary hydration shell (see Table 1), where the magnitude of charge transfer between the two hydration shells is smaller in comparison to that of cluster 12 (average ∆E (2) = 14.672 (12), 13.587 kcal/mol (17). In comparison to clusters 6 and 13-16, the Mg-O bonds are shorter and stronger (see Table 1) due to an increase in the electrostatic interactions between the Mg ion and inner shell water molecules. The ∆E (2) values resulting from charge transfers from lp(O) → lp * (Mg) are the largest for 17 with regard to clusters 13-16 (∆E (2) = 652.16 kcal/mol (17)).
Utilizing cumulative local mode force constants to determine the CN for hydrated magnesium clusters. The cumulative local mode force constant k a cum , which is the sum of the local mode force constants for targeted bonds (i.e., multiple bonds) of a system, can be used to understand the stability of a molecule (i.e., cluster) in retrospect to a complementary set of molecules. In this work, the cumulative local mode force constants k a cum for clusters 1-17 are based upon the sum of all accompanying ion-solvent (i.e., Mg-O) and solvent-solvent (i.e., HBs) local mode stretching force constants. As demonstrated in Table 3, the k a cum values of explicitly solvated clusters 1-6 indicate a CN of 5 to acquire a slightly greater amount of stabilization in comparison to a CN of 6. In addition, the k a cum values for clusters 7-17, which contain explicitly modeled secondary hydration shells, demonstrate that the application of a second hydration shell to [Mg(H 2 O) n ] 2+ clusters (n = 5 and 6) further stabilizes the Mg-O interactions due to HB interactions between first and second hydration shells. Weighted BE values can be found in Table S1 of the  Supporting Information. Moreover, the k a cum values of 6 (i.e., [Mg(H 2 O) 6 ] 2+ ) and 7 (i.e., [Mg(H 2 O) 5 ] 2+ -H 2 O) suggest that cluster 7 acquires a greater stability than 6 (see Table 3 6 ] 2+ -2 (H 2 O)), revealing that, in the case where two water molecules comprise the second hydration shell, a CN of five results in a more stable configuration in contrast to a CN of 6 (see Table 3 Table 3), a CN of 5, with respect to a CN of 6, is observed to result in more stable configurations. It is noted that current ab initio molecular dynamics simulations conducted by Lynes and co-workers revealed a small percentage of the CN for Mg 2+ to be 5 [42]. , it is inferred that clusters 9-11 are more likely to be present.
However, a CN of 6 is shown to be preferred over a CN of 5 when a single water molecules occupies the second hydration shell (see values for clusters 7 and 13 in Table 3). In agreement with the results from implicit solvation and previous studies on [Mg(H 2 O) n ] 2+ -12 (H 2 O) (n = 5 and 6) clusters [41], the k a cum values for the explicit model of [Mg(H 2 O) 6 ] 2+ -12 (H 2 O) (17), in which the second hydration shell holds the maximum of 12 water molecules, reveal that cluster 17 acquires the strongest bond stabilizations in comparison to all other explicitly modeled clusters (7)(8)(9)(10)(11)(12)(13)(14)(15)(16). We note that clusters [Mg(H 2 O) 5 ] 2+ -12 (H 2 O) (12) have the second largest bond stabilizations (see Table 3 and Figure 4). Each of the six inner shell water molecules of cluster 17 form two HBs with water molecules of the second hydration shell and that second shell water molecules form a total of 12 HBs among themselves (see Figure 1) as demonstrated in previous works [16,41].
The implicit hydration of clusters 1-6 was assessed through the application of the CPCM solvation model as to determine the CN, here the water molecules of the first hydration shell were modeled explicitly while outer hydration shells are represented by a dielectric continuum. We derived k a cum values, via local mode analysis, for these clusters in order to establish the key element responsible for their chemical stability. From the k a cum values of the implicitly solvated clusters, as shown in Table 4, we observed the implicit addition of outer shells to [Mg(H 2 O)n] 2+ (n = 1-6) clusters to increase in stability alongside increasing CN, the results revealing a CN of 6 to result in the most stable configuration due to the Mg-O and HB interactions. The CN of 6, as suggested from local mode analysis, agrees with recent experimental [16,19,21,22,24] and computational works [20,32,34,39,[41][42][43][44]46,47,49,51]. In this work, via the utilization of cumulative local mode stretching force constants k a cum , we observed the divalent Mg ion to coordinate with up to six water molecules, whereas in our previous work regarding hydrated calcium clusters, the divalent Ca ion was observed to undergo coordination with up to eight water molecules [106].
The more restricted CN of the Mg 2+ has previously been attributed to the smaller ion size [39,107,108], whereas the larger size of the Ca 2+ ion has been linked to weaker inner ion-solvent interactions in retrospect to that of the hydrated Mg 2+ ion [51]. As shown by Tables 3 and 5, the k a cum values of explicitly hydrated systems, based upon inner ionsolvent interactions, are larger for [Mg(H 2 O) n ] 2+ (n = 1-6) (1-6) clusters in comparison to that of their Ca 2+ counterparts by differences of 0.601 to 1.341 mdyn/Å (i.e., 1a-6a in   Table 2). Therefore, the hydrated magnesium clusters, 15 and 16, acquire a greater stability than their Ca 2+ counterparts primarily due to stronger ion-solvent interactions between the first solvent shell and the Mg 2+ ion.

Conclusions
Herein, 17 hydrated divalent magnesium clusters were investigated at the ωB97X-D/6-311++G(d,p) level of theory to reveal how electronic effects influence the intrinsic strengths of the ion-solvent and solvent-solvent interactions. Local mode analysis was employed to obtain local stretching force constants as an intrinsic bond strength measure, where the results were complemented with NBO charges, electron and energy densities, and stabilization energies. In such a way, a deeper understanding about the nature of Mg-O and HB interactions was acquired. An increase in the NBO charges of O atoms belonging to the first hydration shell is due to a greater magnitude of charge transfer between the inner and outer hydration shell and leads to a strengthening of the ion-solvent interactions (i.e., Mg-O bonds). The ion-solvent interactions, for clusters with a secondary hydration shell, are strengthened from a larger amount of charge transfer between lp(O) → lp * (Mg) orbitals.

4.
From the explicit and implicit solvation of the Mg 2+ ion, cumulative local mode force constants k a cum , which consider both Mg-O and HB interactions, were derived and suggested a CN of 5 and 6 to be prevalent among the clusters investigated, where a CN of 6 is primarily preferred in the instance where the second shell allocates 1 and 12 water molecules.
Understanding the structure and reactivity of hydrated divalent magnesium clusters is key to elucidating the effect of aqueous solvents on the Mg 2+ ion. Shedding light upon the nature underlying Mg-O and HB intrinsic bond strengths is essential for elucidating such effects. The cumulative stretching force constant k a cum considers the strength of multiple bonds within a system, allowing one to assess the bond properties of a cluster by a single value. From k a cum values, one can obtain an immediate answer about the preferred CN among isomeric structures of hydrated divalent magnesium ion clusters.