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Ab Initio Investigation of the Hydration of the Tetrahedral Perchlorate, Perbromate, Selenate, Arsenate, and Vanadate Anions Cory C. Pye* and Victoria E. J. Walker Department of Chemistry, Saint Mary’s University, Halifax, Nova Scotia, Canada B3H 3C3
bS Supporting Information ABSTRACT: The geometries, energies, and vibrational frequencies of various isomers of XO4m (H2O)n, x = Cl, Br, Se, As, V; n = 0 6, m = 1 3 are calculated at various levels up to MP2/6-31+G*. These properties are studied as a function of increasing cluster size. The experimental and theoretical vibrational spectra are compared where available.
1. INTRODUCTION An understanding of the hydration of ions is an important first step to predicting the properties of electrolyte solutions. Analysis of diffraction data, as obtained by the use of either X-rays or neutrons, suggest that water molecules can be viewed as a series of hydration shells.1 When the analysis is applied to metal cations, vibrations characteristic of the first hydration shell can often be characterized by Raman spectroscopy and ab initio calculations.2 8 The influence of the first hydration shell on oxoanions (which may in principle be regarded as a highly charged cation surrounded by oxo ligands) is much smaller and can be compared to second-sphere interactions for metal cations. One of the authors has examined the influence of the first hydration sphere on the sulfate,9 phosphate,10 hydrogen phosphate,11 and dihydrogen phosphate ions.12 Although small highly charged anions often require several water molecules to stabilize the charge, ab initio modeling using restricted Hartree Fock (HF) theory with modest basis sets, for the purposes of modeling the solution, gives reasonable values for structural and vibrational properties even without sufficient water.9 In this paper, we present our studies of both naked and hydrated perchlorate, perbromate, selenate, arsenate, and vanadate ions, including optimization and frequency calculation up to the MP2/6-31+G* level and with up to six water molecules. 2. METHOD Calculations were performed using Gaussian 92,13 98,14 or 03,15 utilizing the 6-31G* 16 and 6-31+G* 17 basis sets. The second-order Moller Plesset (MP2) calculations utilize the frozen core approximation. The geometries were optimized using a stepping stone approach, in which the geometries at the levels HF/STO3G, HF/3-21G, HF/6-31G*, HF/6-31+G*, MP2/6-31G*, and r 2011 American Chemical Society
MP2/6-31+G* were sequentially optimized. For perchlorate, the MP2/6-31G* and HF/6-31+G* order was reversed. For the related sulfate ion, larger basis sets did not lead to significant improvement,9 and the same is to be expected here. Default optimization specifications (rms/maximum displacement smaller than 0.0003/0.00045 au, rms/maximum force smaller than 0.0012/ 0.0018 au) 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 to speed up the optimizations. Because 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 to aid geometry convergence. For brevity we refer to 6-31G* as basis set A and 6-31+G* as B. Only the results for basis set B are reported herein, the others are given in the Supporting Information. The initial geometries were taken from the corresponding sulfate structures.9
3. RESULTS AND DISCUSSION The “free”, undistorted XO4m ion, of Td symmetry, has nine modes of internal vibration and spans the vibrational representation Γvib = A1 + E + 2T2. All modes are Raman active but only the Special Issue: Richard F. W. Bader Festschrift Received: May 23, 2011 Revised: August 25, 2011 Published: September 16, 2011 13007
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Figure 1. Structures of all perchlorate water clusters investigated in this study.
T modes are IR active. In a few cases, for some levels, small imaginary frequencies are present, which prevents a proper evaluation of zero-point energy, thermal, and entropy corrections to the energy. 3.1. Perchlorate. The structures of the perchlorate water clusters are given in Figure 1. Essentially, these are analogous to the sulfate and phosphate structures investigated previously.9,10 We note that the STO-3G and 3-21G basis sets gave much larger chlorine oxygen bond distances (and smaller vibrational frequencies) than calculations at the other levels. For these reasons, we only report these low-level results for the naked perchlorate ion. Our naked perchlorate MP2/B Cl O distance of 1.489 Å is slightly shorter and longer, respectively, than the B3LYP and MP2/6-311+G* distance of 1.501 and 1.476 Å reported by Weber and Tao.18 We confirm the MP2 result of Weber and also report the MP2/6-311+G(2d) Cl O distance of 1.466 Å. 3.1.1. Discussion of Geometries. In Figure 2 we present the Cl O distances in the perchlorate ion as surrounded by various numbers of water molecules in various configurations. As with the sulfate and phosphate, we can distinguish the Cl O bonds by the number of hydrogen bonds that each oxygen accepts and connect them with lines. The more hydrogen bonds to an oxygen of perchlorate, the longer the Cl O bond distance will become. However, adding water molecules to the cluster shortens all other Cl O bonds and a partial cancellation results. Overall, the net effect is a shortening of the Cl O distances between naked perchlorate and its hexahydrate. The trends for the hydrogenbonding indicators O 3 3 3 H (Figure 3), O 3 3 3 O (Figure 4), and Cl 3 3 3 O (Figure 5) are very clear: these distances lengthen as more waters hydrogen bond to the perchlorate. This suggests that the hydrogen bonding becomes weaker as more waters bind to perchlorate. The same trends can be observed for the other anions, as will be seen later. 3.1.2. Discussion of Vibrational Frequencies. Figure 6 presents the vibrational frequencies of the clusters investigated at the HF/B level. We note that, upon hydrating the perchlorate ion with six waters, the frequencies increase by 10 cm 1. This is much smaller than the effect of water on the sulfate anion. The experimental values of the frequencies of perchlorate in water (as the magnesium perchlorate solution) are 464, 631, 934, and 1114 cm 1. The Hartree Fock calculation of both the naked and hydrated perchlorate overestimates the vibrational frequency
Figure 2. MP2/6-31+G* Cl O distance in perchlorate water clusters.
Figure 3. MP2/6-31+G* O 3 3 3 H distance in perchlorate water clusters.
as compared to experiment. However, it is well-known that Hartree Fock theory overestimates the frequencies and usually need to be scaled to account for both deficiencies in theory and neglect of anharmonic contributions. For the naked ion, we derived scaling factors for frequencies by linear regression with zero intercept. The Hartree Fock values for basis sets A and B are 0.947 and 0.971. The corresponding MP2 values are 0.978 and 1.021, respectively. The scatter for the MP2 values is much higher but systematically improved by allowing the intercept to deviate (by up to +100 cm 1). For the hexahydrate, the HF/A and HF/B values are 0.954 and 0.960; the MP2 values are 0.996 and 1.014. These suffer from the same problems as the naked perchlorate. The experimental full-widths at half-height (fwhh) were reported as 26, 24, 9, and 98 cm 1 for the bands at 462, 630, 935, and 1111 cm 1 of an aqueous perchloric acid solution (water to acid ratio of 11.04:1, 276 K).19 From visual inspection of Figure 6 and the corresponding Figure 1 in ref 19, we see that the theoretical 13008
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Figure 4. MP2/6-31+G* O 3 3 3 O distance in perchlorate water clusters.
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Figure 6. HF/6-31+G* perchlorate mode frequencies in clusters.
Figure 7. Perchlorate incremental binding energies (uncorrected).
Figure 5. MP2/6-31+G* Cl 3 3 3 O distance in perchlorate water clusters.
degenerate mode splitting agrees with this observation, although the theoretical splitting of ν4(T2) seems to be anomalously small. We may take all of the complexes from the monoaqua through to the pentaaqua clusters as a crude “sampling” of the perchlorate solution environment, at the HF/6-31+G* level. The average splitting is 10.3, 3.4, 0, and 29 cm 1. The maximum splitting is 21, 6, 0, and 32 cm 1. From this, we would expect that the halfwidths of the degenerate modes of perchlorate would all be smaller than in sulfate. Experimentally, this is true. The predicted half-width of the ν2(E) mode is comparable to that predicted for sulfate. The half-width of the ν4(T2) mode is predicted to be very small, whereas experimentally it is comparable to the ν2(E) mode. Qualitatively, we have predicted that the half-width of ν3 should be the greatest. The justification and interpretation of these half-widths are given in ref 9. For a Td model of tetrahedral anions, the depolarization ratio is zero for the ν1(A1) mode. In practice, totally symmetric modes
give a small finite value. Any of the nonsymmetric model structures can give small nonzero depolarization ratios. The average and maximum values for the considered hydrated perchlorate ions are 0.0015 and 0.0035, respectively, which are of the same order of magnitude as those of sulfate and phosphate.10 3.1.3. Discussion of Binding Energies. The incremental binding energies of perchlorate with water, uncorrected for BSSE effects, are given in Figure 7. A comparison of the results for basis sets A and B indicate a small effect of the diffuse functions on the binding energy, with a larger effect at the correlated level (
