Ab Initio Investigation of the Hydration of the Tetrahedral d 0 Transition Metal Oxoanions NbO 43 − , TaO 43 − , CrO 42 − , MoO 42 − , WO 42 − , MnO 4 − , TcO 4 − , ReO 4 − , and of FeO 4 , RuO 4 , and OsO 4

: The geometries and vibrational frequencies of various configurations of XO 4m − (H 2 O) n , X = Fe, Ru, Os, m = 0; X = Mn, Tc, Re, m = 1; X = Cr, Mo, W, m = 2; and X = Nb, Ta, m = 3; n = 0–6 are calculated at various levels up to MP2/6‑31+G* and B3LYP/6‑31+G*. These properties are studied as a function of increasing cluster size. The experimental and theoretical bond distances and vibrational spectra are compared where available, and predictions are made where they are not.


Introduction
The hydration of ions is crucial to understanding the properties of electrolyte solutions.Solution diffraction data using either X-rays or neutrons has established that water molecules arrange themselves in hydration shells around ions [1].For metal cations, the existence and nature of an aqua ion can often be proven by Raman spectroscopy and/or ab initio calculations [2][3][4][5][6][7].For oxoanions, the influence of the first hydration shell of water molecules is much smaller.One of the authors has examined the effect of the first hydration sphere on the main-group tetraoxo anions [8], some of their protonated forms [9][10][11][12][13][14], as well as several borate species [15][16][17][18] by ab initio computational methods.Therein, we have demonstrated that ab initio modeling using restricted Hartree-Fock (HF) theory with modest basis sets gives reasonable structural and vibrational properties, even if a full hydration sphere or an implicit solvation model is not employed.In this paper, we present our studies of naked and explicitly hydrated niobate, tantalate, chromate, molybdate, tungstate, permanganate, pertechnetate, perrhenate; and of iron, ruthenium, and osmium tetroxide, including optimization and frequency calculation up to the MP2/6-31+G* level and with up to twelve water molecules [19].

Materials and Methods
Calculations were carried out using Gaussian 03 [20], using the standard 6-31G* and 6-31+G* basis sets in conjunction with the standard HF, MP2, and B3LYP levels of theory.For the atoms of the second and third transition metal series, the standard SDD effective core potential and associated basis set was used in conjunction with the 6-31G(d) and 6-31+G(d) basis sets (O,H).The second-order Moller-Plesset (MP2) calculations use the frozen core approximation.A stepping stone approach was used, where the geometries and molecular orbital coefficients at the levels HF/6-31G*, HF/6-31+G*, MP2/6-31G*, MP2/6-31+G*, B3LYP/6-31G*, and B3LYP/6-31+G* were sequentially optimized (geom = allcheck guess = read).Default optimization specifications were normally used.After each level, a frequency calculation was performed at the same level, and the resulting force constants were used in the following optimization.Z-matrix coordinates constrained to the appropriate symmetry were used to speed up the optimizations and simplify the assignment of vibrational modes (FOpt = z-matrix, ReadFC).The force constants were evaluated at the first geometry as well (FOpt = CalcFC).The quadratic convergence method was applied automatically if the SCF failed to converge (SCF = XQC).Additional options were specified individually or in combination, as needed, to converge the geometry and energy (SCF: NoDIIS and/or IntRep and/or CDIIS; FOpt = CalcAll and/or GDIIS).Additional calculations with Gaussian 16 [21] were carried out to explore the BLYP and PBE functionals, often used for ab initio MD simulations and the effect of an implicit solvation model (CPCM).

Results
The XO 4 m− ion, or molecule, of T d symmetry has nine modes of internal vibration spanning the vibrational representation Γ vib = A 1 + E + 2T 2 .All modes are Raman active, whereas only the T 2 modes are IR active.The structures are analogous to those reported previously for perchlorate [8].In the following subsections, we review the geometries and vibrational frequencies of each molecule or ion, followed by our results.
The calculated bond lengths and vibrational frequencies for MnO 4 − are given in Table 1.The Hartree-Fock distances are too short compared to the experiment, but the MP2 and especially the DFT distances are in much better agreement.The Hartree-Fock frequencies are overestimated, which is a well-known problem with the theory and can be corrected by an empirical scaling factor.The DFT frequencies are reasonably close to the experiment, with the deformation modes being quite close, although the stretching frequencies are overestimated by up to 100 cm −1 .The MP2 stretching frequencies, on the other hand, are nearly twice the experimental values, and this must be regarded as an abysmal failure of the MP2 method.The difficulty that permanganate can present to computational chemistry, especially with regard to electronic transitions, is well known [35][36][37][38][39][40][41][42][43][44][45][46][47].The CPCM solvation model gives rise to slightly smaller Mn-O distances and T 2 frequencies.
The effect of water upon the Mn-O distances (MP2/6-31+G(d)) in MnO 4 − •nH 2 O (n = 0-6) is similar to that of the Cl-O distances in the analogous perchlorate [8] (Figure 1).However, the net effect is a very slight lengthening of the Mn-O distance by 0.0007 Å (n = 6), compared with the shortening observed in perchlorate.However, the individual Mn-O distances can vary by up to 0.05 Å (in perchlorate, the variation was only 0.02 Å).This might be reflected in larger bandwidths in the vibrational spectra.Although the Mn-O distances are approximately 0.1 Å longer than the corresponding Cl-O distances, the Mn…O distances are actually about 0.1 Å shorter than the corresponding Cl…O distances (Figure 2).The O…O distances are about 0.2 Å shorter (Figure 3) than the corresponding distances in perchlorate, suggesting that the hydrogen bonding is stronger in permanganate than in perchlorate.This is also seen in the O…H distances (Figure 4).Unlike perchlorate, there is only a slight increase of the hydrogen bonding indicators (O…O, O…H) on the number of water molecules, but the ranges are much larger.This also indicates that the permanganate ion has an unusual solvation dependence.In an aqueous solution, the Mn-O distance is 1.630(5) Å by LAXS, and the Mn…O distance is 4.095(8) Å [48].While the Mn-O distance is well-reproduced by the hexahydrate calculations, the Mn…O distance is too short because the double-donor water molecules are too close to the central metal.This is rectified in the dodecahydrate model, in which the Mn…O distance lies in the range 3.83-4.09Å, depending on the level of theory.The vibrational frequencies (HF/6-31+G*) as a function of hydration number are shown in Figure 5.The deformation frequencies are lower than in perchlorate and are also much closer together.In addition, the HF/6-31+G* level predicts that the asymmetric stretching mode is lower than the symmetric stretching mode for permanganate, whereas the opposite is true in the experiment and also for both the calculated and experimental perchlorate spectrum [8].

Pertechnetate
The determination of the properties of pertechnetate salts requires special care because of the radioactivity of technetium.The crystal structure of potassium pertechnetate gives a Tc-O distance of 1.711
The calculated bond lengths and vibrational frequencies for TcO 4 − are given in Table 2.The Hartree-Fock distances are too short compared to the experiment, but the MP2 distances are too long.The DFT distances are in much better agreement.In accordance with the inverse relationship between distance and vibrational frequency, the Hartree-Fock frequencies are overestimated, and the MP2 frequencies are underestimated.In addition, HF places the ν 1 -A 1 band above the ν 3 -T 2 band by 50-70 cm −1 , whereas MP2 reverses this order by 150 cm −1 .Only the DFT frequencies are reasonably close to the experiment, predicting the near degeneracy of the two modes.The CPCM solvation model gives rise to slightly larger Tc-O distances and smaller frequencies.The effect of water upon the Tc-O distances (MP2/6-31+G(d)) in TcO 4 − •nH 2 O (n = 0-6) is similar to that of the Cl-O distances in the analogous perchlorate (Figure 6) [8].The net effect is a slight shortening of the Tc-O distance by 0.005 Å (n = 6), unlike permanganate.The individual Tc-O distances vary by up to 0.01 Å (in perchlorate, the variation was larger at 0.02 Å).The Tc-O distances are approximately 0.3 Å longer than the corresponding Cl-O distances, although the Tc…O distances are about the same as the corresponding Cl…O distances (Figure S1).The O…O (Figure S2) and O…H (Figure S3) distances are about the same as in perchlorate.The vibrational frequencies (HF/6-31+G*) as a function of hydration number are shown in Figure 7.The deformation frequencies are lower than in perchlorate and are also much closer together.In addition, the HF/6-31+G* level predicts that the asymmetric stretching mode is lower than the symmetric stretching mode for pertechnetate, whereas experimentally, the modes are degenerate.The effect of hydration is similar to that in perchlorate.

Perrhenate
The crystal structure of potassium perrhenate was determined by Morrow [55]
The calculated bond lengths and vibrational frequencies for ReO 4 − are given in Table 3.The Hartree-Fock distances are slightly too short compared to the experiment, but the MP2 distances are too long.The B3LYP distances are slightly too long.In accordance with the inverse relationship between distance and vibrational frequency, the Hartree-Fock frequencies are overestimated, and the MP2 frequencies are underestimated.In addition, HF places the ν 1 -A 1 band above the ν 3 -T 2 band by 90-100 cm −1 , whereas MP2 reverses this order by 50-60 cm −1 .Only the B3LYP frequencies are reasonably close to the experiment.The CPCM solvation model gives rise to slightly larger Re-O distances and usually smaller frequencies.The effect of water upon the Re-O distances (MP2/6-31+G(d)) in ReO 4 − •nH 2 O (n = 0-6) is similar to that of the Tc-O distances in the analogous pertechnetate (Figure S4).The net effect is a slight shortening of the Re-O distance by 0.005 Å (n = 6).The individual Re-O distances vary by up to 0.01 Å (in perchlorate, the variation was larger at 0.02 Å).The Re…O, O…O, and O…H distances are about the same as the corresponding distances in the hydrated pertechnetate (Figures S5-S7).The vibrational frequencies (HF/6-31+G*) as a function of hydration number are shown in Figure 8.The deformation frequencies are lower than in perchlorate and are also much closer together.In addition, the HF/6-31+G* level correctly predicts that the asymmetric stretching mode is lower than the symmetric stretching mode for perrhenate.The effect of hydration is similar to that in perchlorate.In an aqueous solution, the Re-O distance is 1.735(2) Å by LAXS, and the Re…O distance is 4.197(7) Å [48].While the Re-O distance is well-reproduced by the calculations, the Re…O distance is too short in the hexahydrate.This is mostly rectified in the dodecahydrate model, in which the Re…O distance lies in the range 3.87-4.08Å, depending on the level of theory.

Iron, Ruthenium, and Osmium Tetroxide
Iron(VIII) oxide remains unknown.Ruthenium(VIII) oxide occurs in at least two modifications: a cubic form [67] and a monoclinic form [67,68].The Ru-O distances are 1.696(1) and 1.699(2) Å, respectively.Osmium(VIII) oxide has been studied by both X-ray [69] and (gas phase) electron [70] diffraction.The distances obtained were 1.74(2) and 1.711(3) Å, respectively.The vibrational spectra of these two molecules have been measured by several authors (Table 4).Both molecules contain a tetrahedral MO4 moiety in the gas and solution phase, consistent with the crystal structure of the solids.

Iron, Ruthenium, and Osmium Tetroxide
Iron(VIII) oxide remains unknown.Ruthenium(VIII) oxide occurs in at least two modifications: a cubic form [67] and a monoclinic form [67,68].The Ru-O distances are 1.696(1) and 1.699(2) Å, respectively.Osmium(VIII) oxide has been studied by both X-ray [69] and (gas phase) electron [70] diffraction.The distances obtained were 1.74(2) and 1.711(3) Å, respectively.The vibrational spectra of these two molecules have been measured by several authors (Table 4).Both molecules contain a tetrahedral MO 4 moiety in the gas and solution phase, consistent with the crystal structure of the solids.
The calculated bond lengths and vibrational frequencies for FeO 4 , RuO 4 , and OsO 4 are given in Table 5.In all cases, the DFT distances are greater than the HF distances.For RuO 4 and OsO 4 , the MP2 distances are longer still, whereas for the hypothetical FeO 4 , the MP2 distance is slightly shorter than HF.The DFT levels give the best agreement with the M-O distance and vibrational frequencies for RuO 4 and OsO 4 .The MP2 vibrational frequencies are poor in all cases.The CPCM solvation model gives rise to slightly larger M-O distances and usually smaller frequencies. 1For RuO 4 (CCl 4 ), combination and overtone bands were observed at 1794 cm −1 (ν 1 + ν 3 ), 1831 cm −1 (2ν 3 ). 2 For OsO 4 (g), combination and overtone bands were observed at 1919 cm −1 (ν 1 + ν 3 ), 1928 cm −1 (2ν 3 ). 3For OsO 4 (CCl 4 ), combination and overtone bands were observed at 1910 cm −1 (ν 1 + ν 3 ), 1921 cm −1 (2ν 3 ).All attempts to optimize structures of hydrated iron, ruthenium, or osmium(VIII) oxide in which the water was hydrogen-bonded to the oxygen atoms of the metal tetroxide resulted in optimized structures with imaginary modes corresponding to water wagging or partially optimized structures where the water reorients in a wagging motion to approach the metal atom more closely.In the case of iron tetroxide, other structures can form, such as peroxides or ozonides.This is consistent with the supposition that Fe VIII O 4 is not the most stable form of iron tetroxide but rather Fe VI O 2 (µ 2 -O 2 ) [77].
The calculated bond lengths and vibrational frequencies for CrO 4 2− are given in Table 6.The Hartree-Fock distances are about 0.05 Å too short compared to the experiment, but the MP2 distances are about 0.05 Å too long.The B3LYP distances are very close.The Hartree-Fock frequencies are overestimated.The MP2 stretching frequencies are also overestimated, which is somewhat unusual.In addition, HF places the ν 1 -A 1 band above the ν 3 -T 2 band by 30-50 cm −1 , whereas MP2 reverses this order by 20-70 cm −1 .Only the DFT frequencies (and MP2 deformation frequencies) are reasonably close to the experiment.The CPCM solvation model gives rise to smaller Cr-O distances.The effect of water upon the Cr-O distances (MP2/6-31+G(d)) in CrO 4 2− •nH 2 O (n = 0-6) is given in Figure 9.The net effect is a moderate shortening of the Cr-O distance by 0.01 Å (n = 6).The individual Cr-O distances vary by up to 0.05 Å (in perchlorate, the variation was smaller at 0.02 Å).The Cr…O distance is about the same as the corresponding distance in the hydrated permanganate (Figure S8), although there is less variation within a particular species.For the O…H and O…O distances, the increase of about 0.1 Å upon going from mono to hexahydrate is much more pronounced than permanganate, although the spread within a species is about half (O…H) to the same (O…O) (Figures S9 and S10).The vibrational frequencies (HF/6-31+G*) as a function of hydration number are shown in Figure 10.The frequencies are lower than in permanganate.In addition, the HF/6-31+G* level predicts that the asymmetric stretching mode is lower than the symmetric stretching mode for chromate, whereas experimentally, the modes are reversed.The effect of hydration is similar to that in permanganate, except that the splitting within degenerate modes is about twice as large, which makes it more difficult to correlate the two stretching modes because of the overlap.In an aqueous solution, the Cr-O distance is 1.660(3) Å by LAXS, and the Cr…O distance is 3.955(5) Å [48].While the Cr-O distance is well-reproduced by the calculations, the Cr…O distance is too short in the hexahydrate.This is rectified in the dodecahydrate model, in which the Cr…O distance lies in the range of 3.78-3.88Å, depending on the level of theory.
The effect of water upon the Cr-O distances (MP2/6-31+G(d)) in CrO4 2− •nH2O (n = 0-6) is given in Figure 9.The net effect is a moderate shortening of the Cr-O distance by 0.01 Å (n = 6).The individual Cr-O distances vary by up to 0.05 Å (in perchlorate, the variation was smaller at 0.02 Å).The Cr…O distance is about the same as the corresponding distance in the hydrated permanganate (Figure S8), although there is less variation within a particular species.For the O…H and O…O distances, the increase of about 0.1 Å upon going from mono to hexahydrate is much more pronounced than permanganate, although the spread within a species is about half (O…H) to the same (O…O) (Figures S9 and S10).The vibrational frequencies (HF/6-31+G*) as a function of hydration number are shown in Figure 10.The frequencies are lower than in permanganate.In addition, the HF/6-31+G* level predicts that the asymmetric stretching mode is lower than the symmetric stretching mode for chromate, whereas experimentally, the modes are reversed.The effect of hydration is similar to that in permanganate, except that the splitting within degenerate modes is about twice as large, which makes it more difficult to correlate the two stretching modes because of the overlap.In an aqueous solution, the Cr-O distance is 1.660(3) Å by LAXS, and the Cr…O distance is 3.955(5) Å [48].While the Cr-O distance is well-reproduced by the calculations, the Cr…O distance is too short in the hexahydrate.This is rectified in the dodecahydrate model, in which the Cr…O distance lies in the range of 3.78-3.88Å, depending on the level of theory.

Molybdate
Numerous authors have investigated the crystal structures of alkali metal molybdate salts.The space group of anhydrous lithium molybdate was determined to be P32 by Barinova et al. [96].The average Mo-O distance was 1.768 Å.However, Yip et al. argued [97] that the correct space group was 3 , as shown previously by Kolitsch [98] between 103 and 293 K (1.764 Å), whereas Zachariasen [99] suggested that 3 was correct.Anhydrous sodium molybdate was shown to exist in four modifications [100] between 623 and 923 K: α (Fd-3m), β (unknown), γ (Fddd), and δ (P63/mmc) by powder diffraction.The low-temperature α-form was confirmed and refined by Fortes [101] using neutron powder diffraction (Mo-O = 1.7716Å).Anhydrous potassium molybdate was found [102] to crystallize in the space group C2/m, with Mo-O distance of 1.76(1) Å.A powder diffraction study of anhydrous potassium, rubidium, and cesium molybdate [103] provided cell constants and fractional coordinates of the metal atoms, but the oxygen atoms were not located.Later, annealed anhydrous rubidium molybdate (Pnam) was found [104] to have an Mo-O distance of 1.75(2) Å. Anhydrous cesium molybdate was found [105] to crystallize in the space group Pcmn, with a Mo-O distance of 1.773 Å (corrected 1.792 Å).In addition to anhydrous salts, sodium molybdate dihydrate is also known.Mitra and Verma found [106]
The calculated bond lengths and vibrational frequencies for MoO 4 2− are given in Table 7.The Hartree-Fock distances are slightly too short (0.01 Å) compared to the experiment, but the MP2 distances are about 0.06 Å too long.The B3LYP distances are slightly too long (0.03 Å).It seems that this example is unusual compared to the others in that the Hartree-Fock distances are closest to X-ray diffraction results.The Hartree-Fock frequencies overestimate the experiment, whereas the MP2 and B3LYP frequencies underestimate it.In addition, HF and B3LYP place the ν 1 -A 1 band above the ν 3 -T 2 band, whereas MP2 reverses this order.The B3LYP frequencies are closest to the experiment.The CPCM solvation model gives rise to smaller Mo-O distances and larger A 1 frequencies.The effect of water upon the Mo-O distances 11.The net effect is a moderate shortening of the Mo-O distance by 0.013 Å (n = 6).The individual Mo-O distances vary by up to 0.025 Å (in chromate, the variation was larger at 0.05 Å).The Mo…O distance is about 0.1 Å larger/smaller than the corresponding distance in the hydrated chromate/pertechnetate (Figure S11), although there is less variation within a particular species.For the O…H and O…O distances, the increase of about 0.1 Å upon going from mono to hexahydrate is much similar to chromate, although the spread within a species is about half (Figures S12 and S13).In 2M sodium molybdate solution, the Mo-O (Mo…O) distance was found to be 1.786(8) (4.06(2)) Å by LAXS [112].The vibrational frequencies (HF/6-31+G*) as a function of hydration number are shown in Figure 12.The frequencies are lower than in pertechnetate and chromate.The effect of hydration is similar to that in chromate, except that the splitting within degenerate modes is somewhat smaller.The splitting is still large enough to obfuscate the correlation between the two deformation modes because of the overlap.In an aqueous solution, the Mo-O distance is 1.775(4) Å by LAXS, and the Mo…O distance is 4.010(3) Å [48].While the Mo-O distance is well-reproduced by the calculations, the Mo…O distance is too short in the hexahydrate.This is rectified in the dodecahydrate model, in which the Mo…O distance lies in the range of 3.82-3.94Å, depending on the level of theory.
chromate.The effect of hydration is similar to that in chromate, except that the splitting within degenerate modes is somewhat smaller.The splitting is still large enough to obfuscate the correlation between the two deformation modes because of the overlap.In an aqueous solution, the Mo-O distance is 1.775(4) Å by LAXS, and the Mo…O distance is 4.010(3) Å [48].While the Mo-O distance is well-reproduced by the calculations, the Mo…O distance is too short in the hexahydrate.This is rectified in the dodecahydrate model, in which the Mo…O distance lies in the range of 3.82-3.94Å, depending on the level of theory.

Tungstate
The crystal structures of alkali metal tungstates have been investigated.Zachariasen and Plettinger showed [113] that anhydrous lithium tungstate has space group R3, with W-O bonds of 1.79(2) Å. Okada et al. showed [114] that anhydrous sodium tungstate has space group Fd3m, with W-O bonds of 1.819(8) Å.This was revisited by Fortes [101] (Fd3m), who found a somewhat shorter length of 1.7830(2) Å using neutron diffraction.Anhydrous potassium tungstate crystallizes [115] in the space group C2/m, with a W-O distance of 1.79(2) Å. Powder diffraction on anhydrous potassium, rubidium, and cesium tungstate [103] provided cell constants and fractional coordinates of the metal atoms, but as with the molybdates, the oxygen atoms were not located.By neutron powder diffraction [104], it was found that rubidium tungstate crystallizes in the space group C2/m, W-O = 1.775(9)Å.To the best of our knowledge, the crystal structure of cesium tungstate remains unknown.
The calculated bond lengths and vibrational frequencies for WO 4 2× are given in Table 8.With the experimental distances spanning the range from 1.775-1.819Å, the Hartree-Fock distances are at the low end of this range, and the B3LYP distances are at the high end.The MP2 distances are somewhat too long.The Hartree-Fock frequencies overestimate the experiment, whereas the B3LYP frequencies underestimate it by about the same amount.The MP2 frequencies underestimate the experiment a bit more, which matches the inverse trend expected with bond length.All methods correctly place the ν 1 band higher than ν 3 and predict the near degeneracy of the ν 2 and ν 4 bands.The CPCM solvation model gives rise to slightly smaller W-O distances and larger stretching frequencies.The effect of water upon the W-O distances (MP2/6-31+G(d)) in WO 4 2− •nH 2 O (n = 0-6) is given in Figure S14.The net effect is a moderate shortening of the W-O distance by 0.012 Å (n = 6).The individual W-O distances vary by up to 0.015 Å, which is smaller than in chromate and molybdate.The W…O distance is about the same as the corresponding distance in the hydrated molybdate (Figure S15), although there is less variation within a particular species.For the O…H and O…O distances, the increase of about 0.1 Å upon going from mono to hexahydrate is much similar to molybdate, as are the actual values, although the spread within a species is about half (Figures S16 and S17).In 2M sodium tungstate solution, the W-O (W…O) distance was found to be 1.786(8) (4.06(2)) Å by LAXS [112].The vibrational frequencies (HF/6-31+G*) as a function of hydration number are shown in Figure 13.The frequencies are slightly lower than in molybdate, except for ν 1 .The effect of hydration is similar to that in molybdate, except that the splitting within degenerate modes is somewhat smaller.As with molybdate, the splitting is still large enough to obfuscate the correlation with the two deformation modes because of the overlap.In an aqueous solution, the W-O distance is 1.797(4) Å by LAXS, and the W…O distance is 4.024(4) Å [48].While the W-O distance is well-reproduced by the calculations, the W…O distance is too short in the hexahydrate.This is rectified in the dodecahydrate model, in which the W…O distance lies in the range 3.82-3.94Å, depending on the level of theory.

Niobates
Unlike the previously mentioned anions, the experimental evidence for the existence of orthoniobate (NbO4 3− ), especially in aqueous solution, is scant.We therefore review some of the literature on the complex chemistry of the Nb2O5•M2O•H2O (M = Li, Na, K, Rb, Cs) phase diagrams and the compounds therein.

Niobates
Unlike the previously mentioned anions, the experimental evidence for the existence of orthoniobate (NbO 4 3− ), especially in aqueous solution, is scant.We therefore review some of the literature on the complex chemistry of the Nb 2 O 5 •M 2 O•H 2 O (M = Li, Na, K, Rb, Cs) phase diagrams and the compounds therein.
Rubidium niobates form at ratios of 15:2 (Rb 4  [122].From these results, it appears that compounds of stoichiometry M 3 NbO 4 only exist for M = Li, Na, and K. The structures of these compounds have proven difficult to elucidate.Powder diffraction gives the intensity of the scattered waves (X-ray, neutron, electron) as a function of scattering angle 2θ, which generally uniquely characterizes the compound.If this can be indexed, then the unit cell parameters and sometimes the space group can be identified.In favorable cases, a single crystal can be grown, and the related technique of X-ray crystallography may be applied.

Niobium Oxides
Early work on the oxides of niobium suggested the existence of three polymorphs of Nb 2 O 5 , the low (T), medium (M), and high (H)-temperature forms, and the powder diffraction patterns were measured (copper-K α ) [124].The H-form of Brauer appears to be equivalent to the α-form of Reisman [116], who was able to index to a monoclinic cell, a = 21.[126], who altered the space group by delocalizing the tetrahedra over both holes.The Nb-O distance in the tetrahedra was altered to 1.826 Å. Hahn's niobium(V) oxide appears to be the low-temperature T-form [127], as do the forms examined by Frevel [128] and Nolander [129]  .We speculate that it may be possible to make in situ small amounts of transient NbO 4 3− or its protonated forms by partial dissolution of the H-form, which already contains some tetrahedral niobium(V).

Lithium Niobates
There are several known structures for lithium niobates.Lithium orthoniobate, Li 3 NbO 4 , was studied by Blasse [135], who determined a cubic form from powder X-ray diffraction with a = 8.433 Å and also located the niobium atoms.It was also studied by Grenier et al. [136], who found cubic forms for both the low (Fm3m, a = 4.212 Å) and high (I23, a = 8.429 Å) temperature modifications using powder X-ray diffraction.In the low-temperature form (I), the cations were randomly distributed, whereas, in the hightemperature form (II), powder neutron diffraction gives a tetrahedral grouping of niobium octahedra, with lithium octahedra completing the 3D network.The structure might be viewed as a Nb 4 O 4 core bound to 12 additional oxygens to form Nb 4 O 16 12− , held together by Li + ions.The positions were further refined by Grenier and Bassi [137].The unit cell was confirmed by Whiston and Smith [138] (cubic, a = 8.4300 ± 0.0008).Ukei et al. revised the space group of the high-temperature form to I-43m, a = 8.412 Å [139].
Lithium metaniobate, LiNbO 3 , is the most studied by far because of its ferroelectric properties.Bailey [140] found that LiNbO 3 crystallizes in the space group R3c with a rhombohedral unit cell (a = 5.4920 Å, α = 55.88 • ).The corresponding hexagonal unit cell has a H = 5.147 Å c H = 13.856Å.Two possible models were considered for the atomic positions.Megaw discussed the relationship of this structure to its ferroelectricity [141].Powder neutron diffraction was carried out by Shiozaki and Mitsui [142], suggesting a disordered distribution of lithium ions.Abrahams et al. carried out a single crystal Xray diffraction study at ambient temperature (R3c, a H = 5.14829(2) Å, c H = 13.8631(4)Å; a R = 5.4944, α = 55.87 • ) [143].They found that the unit cell consists of 6 planar sheets of oxygen atoms perpendicular to c, with the octahedral interstices filled by Nb, Li, and X, where X indicates a vacancy.Isotropic thermal parameters were sufficient.The following single-crystal neutron diffraction study [144] confirmed the results and cast doubt on the accuracy of the atomic positions found from powder neutron diffraction determined previously [142].The powder X-ray diffraction method was then applied at 24, 250, 500, 750, 1000, and 1200 • C [145], where the atomic arrangement essentially remains unchanged.Absorption and extinction effects gave unphysical values for the isotropic B at some temperatures.The shifts in atomic positions led Abrahams to propose that at the Curie point (1210 • C), the space group changes to the paraelectric R-3.Stoichiometric LiNbO 3 melts incongruently, and the congruently melting compound corresponds to a stoichiometry of Li 0.946 NbO 2.973 (Li 0.955 Nb 1.009 O 3 ).A small amount of Li 2 O has been lost.This form crystallizes (R3c, a H = 5.15052(6) Å, c H = 13.86496(3)Å) nearly identically to the stoichiometric compound (R3c, a H = 5.14739(8) Å, c H = 13.85614(9)Å) [146].In the congruent form, 4.2% of the niobium ions have migrated to the vacated lithium sites.It was found that the thermal vibrations of the congruent form were anharmonic.A single crystal study on the congruent form was carried out by Ohgaki et al. (R3c, a H = 5.15020(6) Å, c H = 13.8653(4)Å), who also incorporated anharmonic corrections for vibration [147].There also exists an ilmenitetype polymorph of LiNbO 3 , characterized by Kumada et al. using powder diffraction (R3, a = 5.212 Å, c = 14.356Å) [148].Neutron powder diffraction both below (R3c) and above (R-3c) the Curie point demonstrated that the high-temperature paraelectric phase of LiNbO 3 had disordered lithium on both sides of a LiO 3 pyramid [149].Synchrotron X-ray studies have also been carried out [150,151] The structure of Li 7 NbO 6 [154] was solved by Muhle using powder X-ray diffraction (P-1, a = 5.37932 8)) [175] and can be described as slight tilting of the NbO 6 octahedra from the cubic perovskite structure.Essentially, the same structure was proposed by Ishida and Honjo [176].The structure was further refined by Darlington

Potassium Niobates
The potassium niobates are described next.Potassium orthoniobate, K 3 NbO 4 , was confirmed to exist by Guerchais, who gave the d-spacings [193].It was also studied by Addison, who identified that there were two different powder patterns corresponding to two phases of the purported compound, depending on the synthesis temperature [194], in agreement with one of two scenarios suggested by the observations of Reisman [120].Stecura et al. found that a = 12.05(4) Å, b = 14.34(5)Å, c = 10.50(3)Å [195].Meyer and Hoppe found a cubic γ structure with a = 8.605 Å, and a tetragonal β structure with a = 6.14 Å, c = 8.37 Å [196].
The compound K 2 Nb 8 O 21 was found by Guerchais, who gave the d-spacings [193].It was also indexed by Whiston and Smith [138] to be primitive tetragonal, a = 27.41,c = 3.955 Å, but they expressed doubts about stoichiometry.
The compound KNb 7 O 18 was studied by electron microscopy and electron diffraction by Hu et al., who found that it was tetragonal, with a = 27.5, c = 3.94 Å [208].The x and y positions were located, but not the z positions.
The compound K 6 Nb 44 O 113 , proposed by Reisman et al. [120], was indexed as monoclinic by Whiston

Calculations on Orthoniobate
The calculated bond lengths and vibrational frequencies for NbO 4 3− are given in Table 9.The only structures containing discrete orthoniobate are the salts of potassium, rubidium, and cesium, whose structures contain a lot of disorder [196].However, with fractional coordinates for Nb (0, 0, 0) and O ± (0, 0.211, 0.01) and ±(0.211, 0, 0.01), and cubic unit cell of dimensions 8.90 Å, K 3 NbO 4 would have an Nb-O distance of 1.88 Å.The Hartree-Fock distances are slightly less than this, the B3LYP distances are slightly higher, and the MP2 distances are too long.To the best of our knowledge, the vibrational frequencies of orthoniobate have not been measured.We expect a clear separation of the symmetric and antisymmetric stretching frequencies, but it is possible that there may be an accidental degeneracy of the deformation frequencies, which are mostly predicted to be within 30 cm −1 of each other.The CPCM solvation model gives rise to smaller Nb-O distances and larger stretching frequencies.
The effect of water upon the Nb-O distances (MP2/6-31+G(d)) in NbO 4 3− •nH 2 O (n = 0-6) is given in Figure 14.The net effect is a moderate shortening of the Nb-O distance by 0.025 Å (n = 6).The individual Nb-O distances vary by up to 0.04 Å.These changes are a little more pronounced than for the divalent ions.The Nb…O distance exhibits an odd behavior, decreasing from n = 1 to n = 3 and then leveling off or even increasing slightly.(Figure S18).For the O…H and O…O distances, the increase of about 0.1 Å upon going from mono to hexahydrate is consistent with the divalent ions (Figures S19 and S20).The vibrational frequencies (HF/6-31+G*) as a function of hydration number are shown in Figure 15.The effect of hydration is to increase the stretching frequencies by about 40 cm −1 going from n = 0 to n = 6, whereas the deformation frequencies only slightly increase.The splitting is large enough to obfuscate the correlation between the two deformation modes because of the overlap, and these might not be able to be observed separately in aqueous solution.
behavior, decreasing from n = 1 to n = 3 and then leveling off or even increasing slightly.(Figure S18).For the O…H and O…O distances, the increase of about 0.1 Å upon going from mono to hexahydrate is consistent with the divalent ions (Figures S19 and S20).The vibrational frequencies (HF/6-31+G*) as a function of hydration number are shown in Figure 15.The effect of hydration is to increase the stretching frequencies by about 40 cm −1 going from n = 0 to n = 6, whereas the deformation frequencies only slightly increase.The splitting is large enough to obfuscate the correlation between the two deformation modes because of the overlap, and these might not be able to be observed separately in aqueous solution.

Tantalates
Like orthoniobate, the experimental evidence for the existence of orthotantalate (TaO4 3− ), especially in aqueous solution, is scarce.We therefore review some of the literature on the complex chemistry of the Ta2O5•M2O•H2O (M = Li, Na, K, Rb, Cs) phase diagrams and the compounds therein.
For LiTa 3 O 8 , Reisman was unable to index the powder diffraction patterns [224].Whiston proposed that this might actually have been Li 2 Ta 8 O 21 [138].Gatehouse and Leverett showed that LiTa

Potassium Tantalates
For K 3 TaO 4 , the interplanar spacings were reported but not indexed by Reisman et al. [225].Whiston also failed to index them [138].This was identified as a corrosion product of tantalum metal upon the action of potassium by comparison of the powder patterns [249].Stecura was finally able to index the pattern of the hygroscopic corrosion product and found a body-centered orthorhombic system with a = 14.19(5)Å, b = 17.04(5)Å, and c = 12.41(4) Å [195].
For KTa 5 O 13 , the interplanar spacings were reported but not indexed by Reisman et al. [222].Whiston also failed to index these [138].

Calculations on Orthotantalate
The calculated bond lengths and vibrational frequencies for TaO 4 3− are given in Table 10.It is believed that the only structures containing discrete orthotantalate are the salts of rubidium and cesium [193].To the best of our knowledge, the oxygen positions have not been precisely determined.Our predictions are quite similar for orthotantalate as for orthoniobate, with bond lengths following the trend HF < B3LYP < MP2.To the best of our knowledge, the vibrational frequencies of orthotantalate have not been measured.Like orthoniobate, we expect a clear separation of the symmetric and antisymmetric stretching frequencies but an accidental degeneracy of the deformation frequencies, which are mostly predicted to be within 30 cm −1 of each other.For the B3LYP frequencies, the degeneracy was not strictly maintained, most likely because the density matrix did not maintain full symmetry.The CPCM solvation model gives rise to smaller Ta-O distances and larger stretching frequencies.
The effect of water upon the Ta-O distances (MP2/6-31+G(d)) in TaO 4 3− •nH 2 O (n = 0-6) is given in Figure 16.The net effect is a moderate shortening of the Ta-O distance by 0.022 Å (n = 6).The individual Ta-O distances vary by up to 0.04 Å.These changes are similar to orthoniobate.The Ta…O distance exhibits an odd behavior similar to orthoniobate, decreasing from n = 1 to n = 3 and then leveling off or even increasing slightly.(Figure S21).For the O…H and O…O distances, the increase of about 0.1 Å upon going from mono to hexahydrate is consistent with orthoniobate (Figures S22 and S23).The vibrational frequencies (HF/6-31+G*) as a function of hydration number are shown in Figure 17.The effect of hydration is to increase the stretching frequencies by about 40 cm −1 going from n = 0 to n = 6, whereas the deformation frequencies only slightly in-

Discussion
For most of the tetrahedral species discussed above, the B3LYP/6-31+G* level of theory gives excellent values of the metal-oxygen bond distance and the vibrational frequencies.This result is encouraging, given the small basis set size and the lack of a solvation model.The overall trends for the anions upon hydration are, for the most part, consistent with our previous work on similar anions [8].In some cases, the MP2-FC level severely overestimates the vibrational frequencies.Our standard assumption of the structures of the hydrated anions, based on our prior calculations, can sometimes give small imaginary frequencies, usually at the MP2/6-31G* and B3LYP/6-31G* levels, which may be due to the lack of diffuse functions on these basis sets.These assumptions were shown to be incorrect for the neutral species RuO4 and OsO4.For the highly charged orthoniobate and orthotantalate anions, convergence difficulties were observed for some structures for the B3LYP levels.
Based on these calculations, we predict that it will be difficult or impossible to prepare FeO4, especially in aqueous solution, as attempts to hydrate this molecule often led to hydrated forms of FeO2(µ2-O2), FeO(µ3-O3), O + FeO3, or even O + FeO2(OH)2.The extremely basic nature of NbO4 3− and TaO4 3− suggests that these could only exist in extremely basic aqueous solutions.Ions derived from niobium(V) and tantalum(V) tend to be octahedrally coordinated and/or strongly condensed into more complex ions such as NbO5 5− , NbO6 7− , Nb4O12 4− , Nb4O16 12− , and Ta6O19 8− .We believe that the best chance to observe the NbO4 3− and TaO4 3− ions in aqueous solution would be by flowing concentrated rubidium or cesium hydroxide over the corresponding orthoniobate or tantalate salt and quickly

Discussion
For most of the tetrahedral species discussed above, the B3LYP/6-31+G* level of theory gives excellent values of the metal-oxygen bond distance and the vibrational frequencies.This result is encouraging, given the small basis set size and the lack of a solvation model.The overall trends for the anions upon hydration are, for the most part, consistent with our previous work on similar anions [8].In some cases, the MP2-FC level severely overestimates the vibrational frequencies.Our standard assumption of the structures of the hydrated anions, based on our prior calculations, can sometimes give small imaginary frequencies, usually at the MP2/6-31G* and B3LYP/6-31G* levels, which may be due to the lack of diffuse functions on these basis sets.These assumptions were shown to be incorrect for the neutral species RuO 4 and OsO 4 .For the highly charged orthoniobate and orthotantalate anions, convergence difficulties were observed for some structures for the B3LYP levels.
Based on these calculations, we predict that it will be difficult or impossible to prepare FeO 4 , especially in aqueous solution, as attempts to hydrate this molecule often led to hydrated forms of FeO 2 (µ 2 -O 2 ), FeO(µ 3 -O 3 ), O + FeO 3 , or even O + FeO 2 (OH) 2 .The extremely basic nature of NbO 4 3− and TaO 4 3− suggests that these could only exist in extremely basic aqueous solutions.Ions derived from niobium(V) and tantalum(V) tend to be octahedrally coordinated and/or strongly condensed into more complex ions such as NbO
5 5− , NbO 6 7− , Nb 4 O 12 4− , Nb 4 O 16 12− , and Ta 6 O 19 8− .We believe that the best chance to observe the NbO 4 3− and TaO 4 3− ions in aqueous solution would be by flowing concentrated rubidium or cesium hydroxide over the corresponding orthoniobate or tantalate salt and quickly observing downstream before these coordination expansion/condensation reactions can happen.