An Ab Initio Investigation of the Hydration of Iron(III)

Abstract

The energies, structures, and vibrational frequencies of [Fe(H₂O)_n]³⁺, n = 0–6, 18 have been calculated at the Hartree–Fock, second-order Møller–Plesset, and density functional (B3LYP) levels of theory using the 6−31G* and 6−31+G* basis sets. The metal–oxygen distances and stretching frequencies were compared with each other, with related crystal structure and solution measurements and with previous calculations. The Fe-O distances and stretching vibrational frequencies were well reproduced with an explicit model for the second hydration shell.

1. Introduction

Planning for the next generation of nuclear reactors is being guided by the Generation-IV International Forum (GIF) [1]. Six reactor systems have been selected as meeting the sustainability and proliferation resistance goals: the very high-temperature gas-cooled reactor (VHTR), the gas-cooled fast reactor (GFR), the sodium-cooled fast reactor (SFR), the lead-cooled fast reactor (LFR), the molten salt reactor (MSR), and the supercritical water-cooled reactor (SCWR) [2]. Canada is a signatory to the GIF Framework Agreement and has committed to research the supercritical water-cooled reactor concept (Figure 1), with some experience operating supercritical water-cooled fossil fuel power plants. The approach Canada has taken is to continually advance the existing Canadian Deuterium Uranium (CANDU) reactors [3], which were originally developed by Atomic Energy of Canada Ltd. (AECL). The same evolutionary approach will be used to develop the SCWR.

The SCWR is a pressure tube reactor (Figure 2) using H₂O as a coolant, with a low-pressure D₂O moderator vessel that produces supercritical water to be used to generate electricity, hydrogen, and district heating. The critical point of water (647 K, 22 MPa) is the point above which water can no longer be distinguished as a liquid or gas, only a supercritical fluid. The viability of the SCWR depends on the ability of operators to maintain water chemistry to minimize material degradation and the transport of corrosion products and radionuclides, because supercritical water is known to be rather corrosive [4]. Because the coolant is in contact with both the fuel bundle cladding and the condenser in this single-loop system (unlike the current two-loop CANDU reactors), any contaminants introduced via condenser leaks or makeup water could become concentrated because of temperature-dependent solubility differences and interact with corrosion products and, more rarely, with actinides and their fission products from fuel failures. If these prove to be volatile, then high radiation fields could exist outside the core of the reactor, posing a safety risk.

The SCWR, operating at pressures of 25–30 MPa, would require careful consideration of both the water chemistry and the materials used (typically steels and zirconium alloys). Steel is mostly composed of iron metal. There are several temperature- and pressure-dependent allotropes of iron [5]. Under ambient conditions, iron exists as the ferromagnetic body-centered cubic (bcc) α-iron (the mineral ferrite). Above the Curie point of 771 °C, it transforms to the paramagnetic form, known as β-iron in the older literature. Above 912 °C, it transforms to the face-centered cubic (fcc) γ-iron (the mineral austenite). At 1394 °C, it transforms again to a bcc δ-iron before melting at 1538 °C. At pressures of ~10 GPa, iron can transform to the hexagonal close-packed ε-iron (hexaferrum), which may be relevant in the inner core of earth. The triple point of ferrite, austenite, and hexaferrum exists at 10.5 GPa and 477 °C.

In the presence of oxygen under ambient conditions, iron is in equilibrium with Fe₃O₄ (the mineral magnetite), a mixed iron(II/III) oxide [6,7]. For higher levels of oxygen, Fe₃O₄ is in equilibrium with Fe₂O₃ (the mineral hematite). Between about 560 and 1371 °C, the nonstoichiometric “FeO” (the mineral wüstite) appears. Solid solutions with varying oxygen content dominate the higher temperature range. Even when exposed to water, iron will form magnetite and hydrogen gas, especially at higher steam temperatures [8]. These iron oxides were some of the first compounds characterized by X-ray diffraction. Magnetite was thought to have the cubic spinel structure (Fe²⁺)(Fe³⁺)₂O₄, in which the divalent iron forms the center of a FeO₄ tetrahedron and the trivalent iron the center of an FeO₆ octahedron (a = 8.30 Å) [9]. The unit cell is Fd-3m, a = 8.400 Å [10]. A modern redetermination gave an inverse spinel structure, Fd3m, a = 8.3941(7) Å, with Fe-O distances of 1.8883(17) Å (tetrahedral, Fe³⁺) and 2.0584(9) Å (octahedral, randomly distributed Fe²⁺ and Fe³⁺) [11]. Hematite was found to be rhombohedral, with space group R-3c, a = 5.420 Å, α = 55.27° [12]. The Fe-O distances were found to be 2.060(35) and 1.985(25) Å in an approximate octahedron. A later refinement gave space group R3c, a = 5.038(2), c = 13.772(12) Å, with Fe-O distances of 2.116 and 1.945 Å [13]. Wüstite was found to be cubic, Fm-3m, a = 4.29(4) Å, with the rock salt structure, implying an Fe-O distance of 2.145 Å [14].

In CANDU reactors, the apparent pH is typically maintained in the range of 10.2–10.4 using LiOD and adding H₂ to maintain a reducing atmosphere and suppress radiolysis products. Under these conditions, the primary corrosion product of iron is magnetite, which may undergo further dissolution [15,16]. Major aqueous species produced include Fe²⁺, FeOH⁺, Fe(OH)₂, Fe(OH)₃⁻, Fe(OH)₃, and Fe(OH)₄⁻. Under normal operation, Fe³⁺ is not expected to be formed. Condenser leaks could potentially introduce species such as chloride and ammonia. The latter is sometimes used as a pH control agent. The soluble iron(II) and iron(III) could form complexes with chloride, hydroxide, and ammonia. Power stations, including nuclear ones, experience chemical hideout, where contaminants and additives are concentrated by boiling or precipitation at local hot spots under deposits and in crevices containing large thermal and concentration gradients under full power. At low power, temperatures drop and these products re-dissolve. One widely accepted treatment is the oxygenated feed-water treatment, which uses stringently pure water and carefully controlled oxygen levels to maintain protective oxide layers on steel alloys. If too much oxygen was inadvertently introduced, the ratio of iron(III) to iron(II) would increase as oxidation occurs and more Fe³⁺ would form, which would lower the pH. These ions at low pH typically exist as the aqua ion, and the formation of these other complexes would require the displacement of water molecules from the inner solvation shell. Therefore, the study of aquairon(III), the most common form of iron(III) at low pH, as a foundation from which ligand displacement might occur, is warranted. Another application would be the study of acid mine drainage, where pyrite FeS₂ is oxidized to give sulfate and iron(II), which is further oxidized to iron(III), producing a large amount of H⁺, which solubilizes the iron.

The electron configuration of the iron atom is 4s²3d⁶. Upon losing three electrons to form the iron 3+ cation, the electron configuration becomes 3d⁵. For high-field ligands, a doublet would be expected, whereas for low-field ligands, a sextet is expected. In an aqueous solution, iron(III) is believed to form the hexaaquairon(III) ion. With water being a low-field ligand, the hexaaquairon(III) would be a high-spin complex. Typically, hexaaquairon(III), in solution and its salts, is pale violet in color. In the mineral paracoquimbite, Fe₂(SO₄)₃•9H₂O contains octahedral iron(III), with some iron ions only bound to water molecules, others bound only to sulfate oxygens, and others bound to both (space group R-3) [17]. For the hexaaquairon(III) component, the Fe-O distance is 1.98(2) Å. but with both water and sulfate ions in the inner hydration shell. The hygroscopic iron(III) nitrate crystallizes from dilute nitric acid solutions as [Fe(H₂O)₆]³⁺(NO₃⁻)₃•3H₂O, containing the hexaaquairon(III) ion in the space group P2₁/c [18]. The two crystallographically distinct hexaaquairon(III) entities have a mean Fe-O distance of 1.986(7) Å, with a range of 1.966(3)–2.014(3) Å. Commercially available iron perchlorate “hexahydrate” has been shown to actually be [Fe(H₂O)₆]³⁺(ClO₄⁻)₃•3H₂O (space group R-3c, 250 K), with an Fe-O distance of 1.988(2) Å [19]. Other determinations resulted in space group R-3 (293 K, Fe-O distance of 1.997(1) Å [20]; 100 K, Fe-O distance of 2.007 Å [21]), with two distinct iron atoms and disordered perchlorates. In unheated iron(III) nitrate and perchlorate heavy water solutions, the Fe-O distance was found to be 2.01(2) Å by neutron diffraction [22]. The Fe-O distance in iron(III) perchlorate hexahydrate and its solution was found by EXAFS to be 2.003(3) and 1.994(3) Å, respectively [23]. The solution measurements are in good agreement with the solid-state measurements, strongly suggesting that the iron remains hexacoordinate in solution. In addition to the usual perchlorate Raman bands, Sharma [24] found that a 1.98 M Fe(ClO₄)₃ solution in 1 M HClO₄ contains a polarized band at ~509 cm⁻¹ and a very weak depolarized band at 308 cm⁻¹. Similarly, a 2.2 M Fe(NO₃)₃ solution in 1 M HNO₃ contains, in addition to the nitrate bands, the same bands. These have been assigned to the A_g and E_g Fe-O stretching modes of [Fe(H₂O)₆]³⁺. Sharma also found that there must be two distinct hexaaquairon(III) ions in the nitrate crystal (verified later in [17]) as the nondegenerate Raman A_g mode appeared at both 512 and 520 cm⁻¹ in the crystal but became a single band at 509 cm⁻¹ upon melting [25]. Kanno assigned bands of glassy iron(III) nitrate solutions at 325 (dp), 450 (dp), and 506 cm⁻¹ (p) to hexaaquairon(III) but argued that the band at 325 was due to the T_g mode [26]. Murata et al. also observed the hexaaquairon(III) band at 510 cm⁻¹ [27]. A small concentration dependence was noted by Biswas et al., who found a 2.5 cm⁻¹ decrease in the totally symmetric stretching frequency when the concentration was increased from 0.75 to 3.0 mol/kg [28].

2. Materials and Methods

Calculations were carried out using Gaussian 03 [29], using the standard STO−3G, 3−21G, 6−31G*, and 6−31+G* basis sets in conjunction with the standard HF and MP2 levels of theory. 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/STO−3G, HF/3−21G, HF/6−31G*, HF/6−31+G*, MP2/6−31G*, and MP2/6−31+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 subsequent 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). Spin contamination was found to be minor. The calculations were repeated and expanded with Gaussian 16 [30], adding in the B3LYP/6−31G* and B3LYP/6−31+G* levels and removing the HF/STO−3G and HF/3−21G levels, taking care to carefully constrain the symmetry of the wave function by use of SCF = Symm. The use of Guess = Only and Guess = Alter was not needed. A systematic desymmetrization procedure was followed [31,32].

3. Results

We first describe the structures found and then compare our results to the experimental Fe-O distances and vibrational stretching frequencies.

3.1. A Survey of Structures

3.1.1. Monoaquairon(III)

In all cases, monoaquairon(III) possesses C_2v symmetry (Figure 3). The stable sextet configuration is ⁶A₁. The Fe-O distance and Fe-O stretching vibrational frequency are given in

Table 1. Average Fe-O distances (Å) of aquairon(III) complexes. n/a = not applicable. () value corresponds to the lower symmetry indicated.

Species	Sym.	Config.	HF		MP2		B3LYP
			6−31G*	6−31+G*	6−31G*	6−31+G*	6−31G*	6−31+G*
Fe(H2O)3+	C2v	6A1	1.8362	1.8394	1.8159	1.8218	n/a	n/a
Fe(H2O)23+	D2d(C2)	6A2	1.8746	1.8787	(1.8579)	1.8632	1.8947	1.9102
Fe(H2O)33+	D3	6A1	1.9129	1.9168	1.8977	1.9017	1.9062	1.9151
Fe(H2O)43+	S4	6A	1.9505	1.9556	1.9351	1.9393	1.9375	1.9462
Fe(H2O)53+	C2v	6A1	2.0071	2.0116	1.9912	1.9969	1.9954	2.0054
Fe(H2O)63+	Th	6Ag	2.0483	2.0535	2.0315	2.0377	2.0371	2.0491
Fe(H2O)183+	Th #1	6Ag	2.0169	2.0231	1.9989	2.0043	2.0065	2.0191
Fe(H2O)183+	Th #2	6Ag	2.0183	2.0241	2.0004	2.0044	2.0086	2.0201
Fe(H2O)183+	T #1	6A	2.0230	2.0275	2.0073	2.0105	2.0161	2.0265
Fe(H2O)183+	T #2	6A	2.0246	2.0278	2.0099	2.0113	2.0188	2.0279
Fe(H2O)183+	S6 #1	6Ag	2.0229	2.0287	2.0070	2.0116	2.0134	2.0252
Fe(H2O)183+	(S6 #2)	6Ag	2.0204	2.0259	2.0041	2.0084	2.0131	2.0243

and

Table 2. Totally symmetric Fe-O vibrational frequencies (cm⁻¹) of aquairon(III) complexes.

Species	Sym.	Config.	HF		MP2		B3LYP
			6−31G*	6−31+G*	6−31G*	6−31+G*	6−31G*	6−31+G*
Fe(H2O)3+	C2v	6A1	696	687	715	697	n/a	n/a
Fe(H2O)23+	D2d(C2)	6A2	572	563	(582)	563	462	430
Fe(H2O)33+	D3	6A1	537	527	541	526	487	468
Fe(H2O)43+	S4	6A	504	493	508	494	478	462
Fe(H2O)53+	C2v	6A1	472	461	477	463	453	432
Fe(H2O)63+	Th	6Ag	445	435	451	438	431	413

. For the B3LYP levels, no stable configuration exists because the molecule dissociated into Fe and H₂O fragments. Thermodynamically, the Fe²⁺ + H₂O⁺ pair at infinite separation lies significantly lower (900–1000 kJ/mol) in energy than the Fe³⁺ + H₂O pair at the minimum distance of around 1.8 Å but about 750–1000 kJ/mol lower than the Fe³⁺ + H₂O pair at infinite separation (Figure S1). For the B3LYP levels, there is no minimum in energy. The crossing between the two curves for the HF and MP2 levels occurs at around 2.3 Å.

3.1.2. Diaquairon(III)

In most cases, the lowest-energy structure of diaquairon(III) possesses D_2d symmetry (Figure 3). However, at the HF/3−21G and MP2/6−31G* levels, the C₂ structure is preferred, in which the linearity is lost. The most stable sextet configuration is ⁶A₂/⁶A. The D_2h-⁶A_g structure is slightly higher in energy.

3.1.3. Triaquairon(III)

The two D_3h structures were first examined (Figure 3). In both cases, the configuration was ⁶A₁′, but neither was a minimum on their respective potential energy surfaces. Both structures desymmetrize to the stable D₃ form (⁶A₁).

3.1.4. Tetraaquairon(III)

The two D_4h structures (⁶A_2g) were first examined (Figure 4), but neither was an energy minimum. Desymmetrization along the water twisting modes gave either a D₄ (⁶A₂) or a D_2h (⁶A_g) structure, which were comparable in energy to each other and about 30 kJ/mol lower in energy than D_4h #2. More energy was gained by desymmetrization from a square planar to two tetrahedral D_2d (⁶A₂) structures. These nearly isoenergetic structures were 75–100 kJ/mol lower in energy than D_4h #2. Both desymmetrize to the stable S₄ (⁶A) structure to lower the energy up to an additional 10 kJ/mol.

3.1.5. Pentaaquairon(III)

The planar D_5h structure was first examined (Figure 5), but this possessed numerous imaginary frequencies, most of which destroyed the σ_h plane of symmetry perpendicular to the principal C₅ rotation axis. Eight C_2v structures were next examined. These occur as pairs that could potentially interconvert via a Berry pseudorotation: C_2v #1 and C_2v #8, C_2v #2 and C_2v #6, C_2v #3 and C_2v #7, and C_2v #4 and C_2v #5. Many of these did undergo this, so they were reoptimized to ensure this was not an artifact from the initial geometry being from the previous level’s optimized geometry. The initial OFeO angles were reset to 90.0 and 120.0 degrees, and the initial Hessian value was recalculated, but the molecular orbital estimates were retained to the desired symmetry. In most cases, the pseudorotation persisted. The C_2v #1 structure always converted to C_2v #8, and the C_2v #5 structure always converted to C_2v #4. The C_2v #8 structure was lowest in energy and was always a minimum. An additional square pyramidal C_2v #9 structure was also examined, but it was not a minimum.

3.1.6. Hexaaquaaquairon(III)

First, the T_h structure (A_g) was examined (Figure 6). As this highly symmetrical structure was a minimum at all levels, no further structures were examined.

3.1.7. Octadecaaquairon(III)

We now add a second hydration shell consisting of 12 water molecules. First, two T_h structures were calculated. We were able to calculate the vibrational frequencies at the MP2-FC levels for the first time for such a large number of water molecules, which required special care not to exceed computational resources. These would desymmetrize along imaginary E_g, T_g, A_u, E_u, and T_u modes to give D_2h, S₆, T, D₂, and C₃ structures, respectively [31,32]. Neither structure had imaginary E_g modes. Both structures had imaginary A_u, E_u, and T_g modes at all levels, which would give T #1 and #2, D₂ #1 and #2, and S₆ #1 and #2, respectively. The T_h #1 had an imaginary T_u mode at one level, giving rise to C₃ #1, whereas T_h #2 only had an imaginary T_u mode at the two MP2 levels and B3LYP/6−31G*, giving C₃ #2. Both distinct T structures were found, which differ in the relative orientation of the water trimers of the second shell relative to the first shell. These were stable at all levels of theory. Similarly, two distinct S₆ structures were found, the disk (S₆ #1) and the pinwheel (S₆ #2), of which only the disk is stable. Of the two possible D₂ structures, these usually converged to the corresponding T structures (D₂ is a subgroup of T), but there are stable D₂ structures in which only two of the three hydrogen bonds in the second-sphere water trimers have formed. These are somewhat higher in energy than the T structures. The S₆ #1 structure is the lowest-energy structure of those examined. The S₆ #2 structure desymmetrizes to the stable C₃ #3 structure, which contains two water trimers and three water dimers in the second hydration shell. In some cases, the C₃ #1 structure ascends in symmetry to give the cog structure (S₆ #3).

3.2. Comparison of Fe-O Distance in Hexaaquairon(III) with Experiment

The calculated iron(III)–oxygen distances increase as the coordination number increases from one to six. The Fe-O distances using the 6−31+G* basis set are slightly longer than those using the 6−31G* basis set, and the level trend follows MP2 < B3LYP < HF (Table 1). The calculated iron(III)–oxygen distances of hexaaquairon(III) range from 2.032 to 2.054 Å, which are slightly too long compared to the experimental solid-phase results of 1.98–2.01 Å. However, including a second hydration shell reduces the Fe-O distance by around 0.025 Å to the range 1.999–2.029 Å. The choice of model for the second hydration shell does not seem to matter too much in this case. This level of agreement is excellent.

3.3. Comparison of Skeletal FeO₆ Vibrational Frequencies with Experiment

The calculated Fe-O totally symmetric vibrational frequency decreases with coordination number (Table 2). The totally symmetric vibrational frequency using the 6−31+G* basis set is about 10–20 cm⁻¹ lower than for the 6−31G* basis set. Compared to Hartree–Fock, this frequency is typically slightly higher at the MP2 levels but moderately lower at the B3LYP levels. Of some concern is the underestimation of the totally symmetric A_1g stretching vibrational frequency (

Table 3. Skeletal Fe-O vibrational frequencies (cm⁻¹) of hexaaquairon(III). Treating water as a point mass, the point group symmetry is O_h, with ${\mathsf{\Gamma }}_{\mathrm{vib}}=A_{1g}+E_g+2T_{1u}+T_{2g}+T_{2u}$. R = Raman active; IR = infrared active; n.a. = not active; p = polarized; dp = depolarized. The ? indicates an uncertain assignment.

Mode	Activity	Expt.	HF		MP2		B3LYP
			6−31G*	6−31+G*	6−31G*	6−31+G*	6−31G*	6−31+G*
ν1(A1g)	R,p	510	445	435	451	438	431	413
ν2(Eg)	R,dp	308?	351	339	360	346	341	323
ν3(T1u)	IR	450?	438	428	440	427	417	399
ν4(T1u)	IR		186	186	178	184	171	175
ν5(T2g)	R,dp		180	179	170	166	164	169
ν6(T2u)	n.a.		124	123	120	122	113	115

), which is predicted to lie between 413 and 451 cm⁻¹. Experimentally, this band appears at ~510 cm⁻¹. It is common practice to introduce frequency correction factors of ca. 0.9 (HF) to 0.95 (MP2) to improve the agreement with experiments [33]. However, this process would lower the predicted frequencies, making the agreement with the experiment worse. In accordance with numerous other calculations by one of the authors [34], this was correctly hypothesized to be due to the lack of a second hydration shell (

Table 4. Fe-O stretching vibrational frequencies (cm⁻¹) of hexaaquairon(III) in models containing explicit second hydration shell. * = Contains imaginary frequencies. The actual irreducible representations in the lower symmetry corresponding to those assuming a point mass (O_h) can be found through correlation diagrams.

Mode	[6+0] Th	Th #1 *	Th #2 *	T #1	T #2	S6 #1
	HF/6−31G*
ν1(A1g)	445	534	521	494	504/14	527
ν2(Eg)	351	467	448	426	428	430
ν3(T1u)	438	535	528	494	514	506,519
	HF/6−31+G*
ν1(A1g)	435	520	509	525	494	513
ν2(Eg)	339	453	435	414	418	414
ν3(T1u)	428	520	514	512	501	490,499
	MP2/6−31G*
ν1(A1g)	451	546	531	516	518	529
ν2(Eg)	360	481	463	444	441	447
ν3(T1u)	440	548	540	510	536	528,543
	MP2/6−31+G*
ν1(A1g)	438	532	520	505	506	517
ν2(Eg)	346	467	451	433	433	429
ν3(T1u)	427	533	526	511	520	512,525
	B3LYP/6−31G*
ν1(A1g)	431	529	512	506	503	524
ν2(Eg)	341	466	441	431	426	404
ν3(T1u)	417	526	516	497	491	475,497
	B3LYP/6−31+G*
ν1(A1g)	413	507	492	459	479	457
ν2(Eg)	323	446	422	409	406	418
ν3(T1u)	399	502	494	461	472	457,459

). In this case, the choice of model matters more. Using one of the two T_h models increases the vibrational frequencies somewhat more than the two T and S₆ #1 models. In the S₆ model, the splitting of the T modes into the A and E components are noted, whereas the degeneracy is preserved for the T models. The minimum-energy models give good predictions for the totally symmetric stretch (except B3LYP/6−31+G*, for unknown reasons). The excellent performance of MP2/6−31+G* is noted here. Based on these results, the depolarized mode at ~450 cm⁻¹ is probably the E_g mode, in agreement with Kanno. The T_1u stretching mode is predicted to be nearly degenerate with the A_1g mode. While the deformation modes can be easily assigned in the hexaaqua complex, the same cannot be said of the 18-water complexes, as there is extensive coupling with higher-frequency water librational modes of the second hydration shell. While other options to model the second hydration shell exist, such as using a polarizable continuum model on either the hexaaqua- or octadecaaquairon(III) ion, some disadvantages can be introduced, such as artificial symmetry reduction or poorer convergence of the geometry optimization.

4. Discussion

In their study of the many-body effects on iron(III)–water interactions in [Fe(H₂O)_n]³⁺, n = 4, 6, 8, Curtiss et al. showed that the use of two-body potentials would tend to favor higher coordination numbers [35]. Their Hartree–Fock calculations with a double-zeta basis set assumed an Fe-O distance of 2.0 Å. Chang et al. used VWN/DNP, BP/DNP, and BLYP/DNP DFT methods for the hexaaqua ion to give distances of 2.024, 2.108, and 2.121 Å, respectively [36]. The last two gradient-corrected functional results are much longer than the local density functional result. For iron(III), Remsungnen and Rode found a hexacoordinate structure with 13.4 water molecules on average in the second hydration shell [37] using the Hartree–Fock description for the ion with the first hydration shell and ab initio-generated two- and three-body potentials for the remainder. They found an Fe-O distance of 2.02 Å with the HF/DZP basis set used. The velocity autocorrelation function was used to generate (scaled, 0.89) vibrational frequencies of 443 cm⁻¹ (A_g), 379 cm⁻¹ (E_g), 275 cm⁻¹ (T_2g), and 240 cm⁻¹ (T_1u) [38]. The unscaled results (498, 426, 309, 270 cm⁻¹) are still slightly too low compared with the experiment. A pure two- and three-body potential MD simulation gave a distance of 2.05 Å and 15 water molecules in the second hydration shell [39]. The B3LYP/6−31G* calculation (Fe-VTZ) of hexaaquairon(III) with and without implicit solvation demonstrated that the PCM model of solvation can decrease the Fe-O bond distance in Fe(H₂O)₆³⁺ from 2.039 Å to 1.996 Å (T_h) or 2.005 Å (S₆) [40]. The symmetry lowering was an artifact of the solvation model. The BLYP/6−31G** Pople-style and BLYP/US-PW plane wave basis set with the Vanderbilt pseudopotential calculation gave the same Fe-O distance of 2.064 Å [41]. These results were combined with a Car–Parrinello and classical MD simulation with 32 water molecules and another classical simulation with 512 water molecules to give Fe-O first peak maxima at 2.05, 1.98, and 1.96 Å, respectively. Of note is the shortening of the Fe-O distance by ~0.02 Å. Our static calculations also reproduce the shortening of Fe-O by the second hydration sphere, but it is slightly more pronounced. Another Car–Parrinello 12 ps simulation using 64 water molecules and the PBE96 exchange correlation with a pseudopotential and plane wave basis set gave a first-shell peak at 2.10 Å (average 2.12 Å, σ = 0.088 Å) [42]. The corresponding gas phase calculation (PBE96/pVDZ) gave an Fe-O distance of 2.11 Å. A QMCF-MD simulation with 1000 water molecules at the HF/DZP level gave an inner shell of six water molecules and Fe-O first-shell maxima of 2.03 Å (half-width 0.25 Å), with a second shall average of 13.6 water molecules (range 11–17) [43]. The stretching frequency was found to be 513 cm⁻¹. Their static calculations on the di-, tetra-, and hexahydrate gave slightly longer distances than ours: HF, 1.89, 1.97, 2.06; MP2, 1.87, 1.95, 2.04; and B3LYP, 1.92, 1.95, 2.05 (all in Å). Their CCSD/DZP calculation on the dihydrate gave 1.88 Å, which is between the HF and MP2 result. A MCSCF/(aug)cc-pVTZ and MR-PT2/(aug)-cc-pVTZ calculation on the hexaaqua ion gives an Fe-O distance of 2.059 and 2.032 Å, respectively [44]. Other electronic states were also calculated. For the most part, these results were in good agreement with ours.

5. Conclusions

The HF, MP2, and B3LYP methods were paired with the 6−31G* and 6−31+G* basis sets to study the hydrated high-spin iron(III) ion. The calculated structures of the hexaaqua ion compared favorably with the crystal structures of hexaaquairon(III) salts. The calculated vibrational frequencies of the hexaaquairon(III) ion do not agree with the experimental solution measurements. However, including an explicit second hydration shell in the calculation brings the results to much closer agreement. Symmetry can be used to guide the search for new structures.