An Ab Initio Investigation of the Hydration of Tin(II)

: The structure of tin(II) is not well known in aqueous solution. The energies, structures, and vibrational frequencies of [Sn(H 2 O) n ,] 2+ n = 0–9, 18 have been calculated at the Hartree–Fock and second order Møller–Plesset levels of theory using the CEP, LANL2, and SDD effective core potentials in combination with their associated basis sets, or with the 6-31G* and 6-31+G* basis sets. The tin–oxygen distances and totally symmetric stretching frequency of the aquatin(II) ions were compared with each other, and with solution measurements where available.


Introduction
The structure of some metal ions in solution remain elusive [1,2]. Their toxicity to man and the environment is dependent on their oxidation state and speciation, which often depends on pH and the presence of counterions that solubilize the metal by complex formation. Computational chemistry can be useful in supporting and rationalizing proposed speciation models. However, for elements of high atomic number, a drawback is that there are typically few all-electron basis sets that can be used, and relativistic effects can play an important role. One workaround is to use effective core potentials, which replace the explicit description of core electrons by a core potential, which are then paired with basis sets describing the outermost electrons. Previously, some common effective core potentials for the aqua complexes of the heavy metals mercury(II) and thallium(III) (valence electron configuration 5d 10 ) were benchmarked [3]. This work was extended to lead(II) with a valence electron configuration of 6s 2 5d 10 [4]. We expand our work now to tin(II), which has a valence electron configuration of 5s 2 4d 10 . The presence of the ns 2 subshell, as with lead(II), will be shown to have a pronounced effect on the structures compared to those without it. Hemidirected structures (which tend to be favored at lower coordination numbers) have ligands that are not symmetrically distributed around the central ion, whereas holodirected structures have a symmetrical distribution.
In the gas-phase, it was reported that tin(II), lead(II), and mercury(II) easily underwent a proton transfer reaction and that the only species observed in the mass spectra were the deprotonated MOH + (H 2 O) n−1 ions, not the M(H 2 O) n 2+ ions. These ions have anomalously high acidity in the gas phase as well as the solution phase. An ab initio study was carried out to rationalize this behavior, with a focus on the pathways to deprotonation [5]. In solution, it is believed that the relevant species are Sn 2+ (aq) (pH < 2), SnOH + (aq), Sn 2 (OH) 2 2+ (aq), Sn 3 (OH) 4 2+ (aq), Sn(OH) 2 0 (aq) (pH = 5-8), and Sn(OH) 3 − (aq) (pH > 10) [2]. The polynuclear species form at higher concentrations, and the water content cannot be determined through potentiometric means. The tin(II) ion in aqueous solution has been characterized by an X-ray study of ã 3 mol/L solution of the perchlorate salt [6]. The radial distribution curves showed peaks at 1. Some hydrolyzed solutions were also examined, and the largest variation was in the 3.6 Å and 4.2 Å peaks, which suggested a greater degree of clustering as the hydrolysis increased. Essentially the same unhydrolyzed solution was studied by EXAFS [7], in which a Sn-O distance of 2.2-2.3 Å with four water molecules was found. They also reanalyzed the data of [6]. Regarding the hydrolysis products, potentiometric titrations suggested the existence of the species Sn 3 (OH) 4 2+ (aq), in addition to Sn 2 (OH) 2 2+ (aq) and SnOH + (aq) [8]. The crystal structure of the hydrolysis product Sn 3 O(OH) 2 SO 4 , which is potentiometrically equivalent to Sn 3 (OH) 4 2+ (aq), has been determined and shown to contain the discrete [Sn 3 O(OH) 2 ] 2+ ion [9,10].
A QM/MM-MD study has also been carried out on the tin(II) ion in aqueous solution [11]. It was found that the Sn-O distance peaked at 2.5 Å, with a shoulder at 2.65 Å. Gaussian fitting indicated peaks at 2.45 Å and 2.75 Å. A coordination number of eight was found. The power spectrum of the Sn-O stretching suggested peaks at 85 and 208 cm −1 .

Materials and Methods
Calculations were performed using Gaussian 98 [12]. This program version was the first to allow the analytical frequency calculation of molecules in which core electrons are described by effective core potentials (ECPs), and thus, many variants of these were tried. The MP2 calculations use the frozen core approximation. A stepping-stone approach was used for geometry optimization, in which the geometries at the levels HF/CEP-4G, HF/CEP-31G*, HF/CEP-121G*, HF/LANL2MB, HF/LANL2DZ, and HF/SDD were sequentially optimized. For minimum energy structures, the MP2/CEP-31G* and MP2/CEP-121G* calculations were also performed. Calculations were also carried out using the 6-31G* and 6-31+G* basis sets on the atoms of the water molecules (5d) with an effective core potential and basis set on the metal ion (denoted as ECP+6-31G* or 6-31+G*). For shorthand, we denoted the mixed basis sets as follows: CEP-121G* on Sn and 6-31G* on O, H, as basis set A; LANL2DZ on Sn and 6-31G* on O, H, as basis set B; and SDD on Sn and 6-31G* on O, H, as basis set C. The corresponding basis sets with diffuse functions were indicated by adding a "+" to the basis set name. Default optimization specifications were normally used. After each level, where possible, a frequency calculation was performed at the same level, and the resulting Hessian was used in the following optimization. Z-matrix coordinates constrained to the appropriate symmetry were used as required to speed up the optimizations. Since frequency calculations are done at each level, any problems with the Z-matrix coordinates would manifest themselves by giving imaginary frequencies corresponding to modes orthogonal to the spanned Z-matrix space. The Hessian was evaluated at the first geometry (opt = CalcFC) for the first level in a series in order to aid geometry convergence. We note that, for the heavy elements only, the three different CEP basis sets are equivalent (CEP-121G*) but differ for the oxygen and hydrogen atoms. We also note that the choice of core electrons defining the pseudopotential depends on the specific core potential (CEP and LANL2, [Kr]4d 10 ; SDD, [Ar]3d 10 ). In some cases, Gaussian 03 [13] and Gaussian 16 [14] were used to correct errors and omissions.
In many cases to follow, the symmetry of the minimum-energy complexes was the same as those previously found for bismuth [15]. To confirm these results, starting with high symmetry structures, systematic desymmetrization along the various irreducible representations was carried out [16,17]. We did not employ an implicit solvation model or additional electron correlation treatments for reasons described previously in [4]. The energies of all structures are found in Table S1.

A Survey of Structures
Tin(II), as a lighter element in the same group as lead(II), might be expected to show similar properties. The point group symmetry for mono-through hexaaquatin(II) was usually found to be C 2v , C 2 , C 3 , C 2 , C s , and C 3 . The diaquatin(II) species, like lead, ascends in symmetry to a planar C 2v structure at HF/LANL2MB. The tetraaquatin(II) species has C 2v symmetry at HF/LANL2DZ and C s symmetry at HF/LANL2MB. The pentaaqua species has C 2 symmetry at HF/LANL2MB. At all levels for the pentaaqua species, if the pentacoordinate [5+0] species exists, it is competitive in energy with the [4+1], and the most stable form is dependent on the level of theory. For the hexaaquatin(II), the C 3 [3+3] form was always more stable than the C 3 [6+0] form, which occasionally had imaginary frequencies or reverted to the [3+3] form. We did not find stable hepta-, octa-, or enneaaquatin(II) structures. At the HF levels, the [6+12] form always contained at least an imaginary T mode.
The results of the systematic desymmetrization procedure [16] for aquatin(II) are as follows (see Figures 1 and S1):   • The monoaquatin(II) remained as C 2v at all levels; • The most stable diaquatin(II) remained as the bent C 2 at all levels except HF/LANL2MB (C 2v planar). The linear holodirected D 2d structure was approximately 50 kJ/mol higher in energy, but the unstable bent C s structure was only slightly higher in energy (<1 kJ/mol for nonminimal basis sets). All attempts to generate a [1+1] structure instead resulted in proton transfer to give a SnOH + + H 3 O + complex, which was 25-40 kJ/mol higher in energy; • The most stable triaquatin(II) remained as the pyramidal C 3 at all levels. The two pyramidal C 3v structures were 12-25 kJ/mol higher in energy, whereas the planar holodirected D 3h and D 3 structures were 60-90 kJ/mol higher in energy. The stable [2+1] C 2v structure was 25-50 kJ/mol higher in energy; • The most stable tetraaquatin(II) was usually the see-saw C 2 , but it could be the C 2v #3 (HF/LANL2DZ, HF/B+) or C s (HF/LANL2MB). The C 2v #3 was slightly higher in energy (<2 kJ/mol), with the other C 2v structures being higher (15- For the D 4d #1 and #2 structures, desymmetrization along the A 2 imaginary mode gave the same S 8 structure; along the B 1 imaginary mode, they gave the same D 4 #2 structure; along the B 2 imaginary mode, they gave the C 4v #1 and #2 structures; and along the imaginary E 1 mode, they gave the same C 2v #1 structure (via C s ); For the D 4h #1 and #2 structures, desymmetrization along the A 1u imaginary mode gave the D 4 #2 structure found before; along the A 2g imaginary mode, they gave the same C 4h #1 structure; along the A 2u imaginary mode, they gave the C 4v #3 [4+4] and #4 structures; along the B 2g imaginary mode (D 4h #1), the D 2h #1 structure ascended in symmetry to D 4h #2; along the B 1g (D 4h #2) imaginary mode, they gave the D 2h #2 structure; along the B 1u imaginary mode, they gave the same D 2d #1 structure; along the B 2u mode, they gave the D 2d #2 and #4 structures, respectively; and along the E g and E u modes, they gave D 2h #3 and #4 (via C 2h and C 2v ); Examination of lower symmetry structures (h = 8) gave the following results: For the S 8 #1 structure, all E 1 and B imaginary modes corresponded to the expulsion of water molecules to the second hydration shell. Desymmetrization along the B mode gave the [4+4] C 4 ; For the D 4 #2 structure, the E mode corresponded to the expulsion of water molecules to the second hydration shell. Desymmetrization along the A 2 mode gave either a [4+4] C 4 structure or a C 4 #2 structure, whereas along the B 1 mode, it gave the D 2 #1 structure; For the C 4h #1 structure, the imaginary E u mode corresponded to the expulsion of two water molecules. Desymmetrization along the imaginary A u mode for the most part gave either a [4+4] C 4 structure or ascended in symmetry to S 8 ; along the imaginary B u mode at HF/CEP-4G and HF/LANL2MB, it gave a D 2d #5 and S 4 structure, respectively. This D 2d #5 structure was then rerun at all levels; For the D 2d structures, desymmetrization along an A 2 imaginary mode would give an S 4 structure; along a B 1 imaginary mode, they gave a D 2 structure; along a B 2 imaginary mode, they gave a C 2v structure; along an E mode, they gave either a C 2 or C s structure. Along the A 2 mode, an S 4 #2 or #4 structure typically resulted, or ascension in symmetry to the D 2d #5; along the B 1 mode, there was usually ascension to D 4 #2; along the B 2 mode, there was dissociation to a [6+2] or [4+4] structure; and along an E mode, dissociation would occur; For the D 2h structures, desymmetrization along the imaginary A u mode would give a D 2 structure, and along the imaginary B ng modes, a C 2h structure was given. In all cases, these desymmetrized, and most ascended in symmetry to structures already found (D 2 #5, D 4 #2, C 4h #1). The B nu modes corresponded to the expulsion of water molecules from the first hydration sphere; Examination of lower symmetry structures (h = 4) gave the following results: For the C 2v structures, desymmetrization along the A 2 mode would give a C 2 structure, and along the B 1 or B 2 mode, different C s structures were given. For the C 2v structures, at least one of the imaginary B modes in each structure corresponded to dissociation to a [6+2] structure, whereas desymmetrization along the A 2 mode led to a [4+4] or [4+2+2] structure; For the C 4 and S 4 structures, the imaginary E mode corresponded to dissociation to a [6+2] structure, whereas desymmetrization along the B mode to give a C 2 structure resulted in dissociation; For the D 2 structures, at least one of the imaginary B 2 or B 3 modes corresponded to dissociation to a [6+2] structure, whereas desymmetrization along the B 1 mode to give a C 2 structure resulted in dissociation to a [6+2] or [4+4]; Based on these results, we must conclude that a stable 8-coordinate octaaquatin(II) ion cannot exist.

•
Of the enneaaquatin(II) structures, four D 3h structures (point group order h = 12) were first examined. Multiple imaginary modes were present. Desymmetrization along the A 1 " mode would yield D 3 structures; along A 2 , C 3h structures were given; and along A 2 ", C 3v structures were given. A common D 3 #1 structure was found for most, and in some cases gave an additional [6+3] structure. Two possible C 3h structures were found, and in some cases gave an additional To summarize these results, by using the systematic desymmetrization procedure, we have found stable structures for the mono-through hexaaquatin(II) complexes, and we have shown that hepta-, octa-, and enneaaqua structures do not exist on the potential energy surface. The hexaaquatin(II) C 3 structure is only stable at HF/CEP-31G*, HF/CEP-121G*, and MP2/CEP-31G*. The pentaaquatin(II) C s structure is not stable at HF/CEP-4G, HF/LANL2DZ, and HF/SDD. In most cases, for systems with more than three water molecules, the most stable structure on the potential energy surface is tricoordinate, with the remaining water molecules in the second hydration sphere (the main exceptions being HF/CEP-31G* and HF/CEP-121G*). These results suggest that tin(II) would be tricoordinate trigonal pyramidal in an aqueous solution.

The Sn-O Distance
The average Sn-O distance as a function of coordination number is plotted in Figure 2 for all of the levels studied. The Sn-O distance always lengthened following an increase in the coordination number. We can see some gaps for n = 4-6 at some levels where no local minimum existed. The Sn-O distance using the minimal basis HF/LANL2MB was shorter than the other levels at the same hydration number by 0.1-0.3 Å, which tended to cluster together. The results using CEP-31G* were nearly coincidental with those of CEP-121G*. For all levels, there was a pronounced change in slope at n = 3. Within the cluster noted above, the Sn-O distance using the SDD basis set/pseudopotential on Sn (HF/SDD, HF/C, HF/C+) tended to be the longest (n = 1-5, 2.20-2.50 Å), whereas those using the LANL2DZ basis set/pseudopotential on Sn (HF/LANL2DZ, HF/B, HF/B+) were the shortest. This differed for lead, where the CEP basis set/pseudopotential tended to be the shortest [4]. The effect of the basis set/pseudopotential combination was more important than the presence or absence of correlation (HF vs. MP2). Metal-oxygen bond lengths to those oxygens making a smaller angle to the principal symmetry axis were longer, as was noted previously for aqualead(II) [4].

The Sn-O Vibrational Frequency
The vibrational frequency of the totally symmetric Sn-O stretch as a function of coordination number is plotted in Figure 3. As expected, the frequencies at the minimal basis HF/LANL2MB were much higher than the other levels which clustered together. For the

The Sn-O Vibrational Frequency
The vibrational frequency of the totally symmetric Sn-O stretch as a function of coordination number is plotted in Figure 3. As expected, the frequencies at the minimal basis HF/LANL2MB were much higher than the other levels which clustered together. For the most part, the vibrational frequency decreased as a function of hydration number. There was a levelling effect upon going from n = 4 to n = 5, because, for the square pyramidal n = 5, the character of the mode changed to be predominantly a Sn-O apex stretch. The results using CEP-31G* were nearly coincidental with those of the CEP-121G*, and the MP2 values were~15 cm −1 higher than the HF values. The addition of diffuse functions (A+ vs. A) lowered the vibrational frequency by~30 cm −1 , and correlation increased it by~5-15 cm −1 . This was also true for the mixed basis set calculations. All other things being equal, the LANL2DZ frequencies were the highest, and the CEP frequencies were the lowest, with the SDD frequencies falling in the middle. The inverse relationship between average bond length and symmetric stretching frequency can be clearly seen.

Discussion
Because of the lack of experimental data on aquatin(II) complexes, certainty regarding the structure and vibrational frequencies of the aquatin(II) is lacking. We may compare a series of structures, such as , except for HF/CEP-31G* and HF/CEP-121G*, but all are still competitive in energy. For a hydration number of six, the [4+2] structure seems to be the most stable, with the [3+3] structure being competitive in energy. The picture emerging is that of a variable coordination number between three and six, with three (trigonal pyramidal) and four (see-saw) being the most likely.
We recently became aware [18] of a crystal structure determination of an aquatin(II) ion in the compound tin(II) perchlorate trihydrate [19]. The Sn-O distance was reported to be 2.201(7) Å. Examination of Figure 2, at n = 3, and Table 1 suggest that, if crystal packing forces are negligible, then the MP2/A+, HF/LANL2DZ, and HF/B+ levels give excellent agreement with the experiment. This result was confirmed by Persson and coworkers, who obtained 2.208(9) Å and obtained EXAFS results for the crystal and solution Sn-O distance of 2.209(3) and 2.219(3) Å, respectively, which quite importantly shows that the solution structure of tin(II) is the same as the solid. The LAXS measurement gives, at 2.206(2) Å, nearly the same value for the Sn-O distance [20].

Discussion
Because of the lack of experimental data on aquatin(II) complexes, certainty regarding the structure and vibrational frequencies of the aquatin(II) is lacking. , except for HF/CEP-31G* and HF/CEP-121G*, but all are still competitive in energy. For a hydration number of six, the [4+2] structure seems to be the most stable, with the [3+3] structure being competitive in energy. The picture emerging is that of a variable coordination number between three and six, with three (trigonal pyramidal) and four (see-saw) being the most likely.
We recently became aware [18] of a crystal structure determination of an aquatin(II) ion in the compound tin(II) perchlorate trihydrate [19]. The Sn-O distance was reported to be 2.201(7) Å. Examination of Figure 2, at n = 3, and Table 1 suggest that, if crystal packing forces are negligible, then the MP2/A+, HF/LANL2DZ, and HF/B+ levels give excellent agreement with the experiment. This result was confirmed by Persson and co-workers, who obtained 2.208(9) Å and obtained EXAFS results for the crystal and solution Sn-O distance of 2.209(3) and 2.219(3) Å, respectively, which quite importantly shows that the solution structure of tin(II) is the same as the solid. The LAXS measurement gives, at 2.206(2) Å, nearly the same value for the Sn-O distance [20]. Table 1. Bond lengths (Å) of triaquatin(II). The theoretical levels A, B, and C are described in the text. HF = Hartree-Fock, MP2 = second-order Møller-Plesset, Expt. = experiment, XRD = X-ray diffraction, EXAFS = Extended X-ray absorption fine structure, LAXS = Large angle X-ray scattering.  [19] 2.201(7) Expt. XRD [20] 2.208(9) Expt. EXAFS xtal. [20] 2.209(3) Expt. EXAFS soln. [20] 2.219(3) Expt. LAXS soln. [20] 2.206 (2) The vibrational frequency for the n = 3 structure lay in the range 320-370 cm −1 . The effect of a second hydration sphere should be to increase this value somewhat. For most octahedral metal ions that we have previously examined, a rough rule of thumb is that, upon including a second hydration sphere, the vibrational frequency increases by 20q cm −1 , where q is the total charge on the octahedral ion. For tin(II), comparison of the vibrational frequencies of [Sn(H 2 O) 3 ] 2+ C 3 , and [Sn(H 2 O) 3 ] 2+ ·(H 2 O) 3 C 3v revealed a large~85 cm −1 increase in the vibrational frequency. This suggests that tin(II) should have an observable band in the isotropic Raman spectra corresponding to the totally symmetric stretching motion in the range of 400-450 cm −1 .

Conclusions
The common CEP, LANL2DZ and SDD pseudopotentials were paired with various basis sets to study the hydrated tin(II) ion. The calculations using minimal basis sets performed poorly. For the most part, the calculated structures were consistent with recent experimental results of a tricoordinate trigonal pyramidal hemidirected aqua complex. The careful use of symmetry can be used to both guide the search for new structures and also to rule out structures.