Goethite (α-FeOOH) is a common and reactive mineral in the environment 1. α-FeOOH is the most thermodynamically-stable form of the Fe-oxyhydroxides found in soils, groundwater, and acid mine drainage precipitates 2. α-FeOOH is an excellent adsorbent of contaminants (e.g., arsenate) and nutrients (e.g., phosphate) and is an electron receptor for anaerobic bacterial respiration under anoxic conditions (34 and references therein). Consequently, the surface chemistry of α-FeOOH is important for understanding many environmental processes. One of the most stable and well-studied low-index surfaces of α-FeOOH is the (010) surface (Pbnm space group = (100) in the Pnma space group; 56). As a result, this study will focus on the (010) α-FeOOH surface.

Several molecular modeling studies have been published related to α-FeOOH and goethite-H₂O interfaces. For example, Rosso and Rustad 7 published local density approximation (LDA) and generalized gradient approximation (GGA) density functional theory (DFT) calculations on the structures of diaspore, goethite, boehmite, lepidocrocite, akaganeite, guyanaite and grimaldiite that reproduced observed structures of these minerals to within 3%. Rustad et al. 8 used a force field that allowed for the dissociation of H₂O, in order to perform classical molecular dynamics (MD) simulations of the (110) α-FeOOH-H₂O interface. Rakovan et al. 5 combined atomic force microscopy, X-ray photoelectron spectroscopy (XPS), low-energy electron diffraction, and periodic molecular orbital calculations, and proposed that the most stable termination of the (010) α-FeOOH surface was the O2-termination. In the O2-termination, the surface is cleaved between protonated O atoms rather than the O1 non-protonated O atoms of the bulk structure (Fig. 1). Shroll and Stratsmaa 9 performed classical MD simulations with an AMBER-type force field to examine goethite-water interactions and found a mineral influence on the water structure out to 5.5 Å. Kerisit et al. 6 employed a polarizable version of the SPC/E water model, along with a α-FeOOH-H₂O force field 10, to describe the structure of electrolyte solutions near the (010) α-FeOOH surface. Most recently, Aquino et al. 11 performed DFT and Møller-Plesset calculations to investigate the H-bonding of H₂O to clusters of the (110) α-FeOOH surface.

The calculations reported here are different from these previous studies in that they are periodic DFT energy minimizations of bulk goethite and the (010) α-FeOOH-water interface. The use of quantum chemistry to model the surface distinguishes this work from previous classical simulations. Furthermore, the use of periodic boundary conditions separates these results from those of Aquino et al. 11. Energy minimizations performed using periodic DFT can be used to verify the accuracy of the computational approach before expending dramatically more computational resources to perform DFT-based MD simulations of the α-FeOOH-water interface.

The purposes of this study were to test the ability of periodic DFT to:

(1) reproduce the structures and vibrational frequencies of bulk α-FeOOH,

(2) predict the surface bond distances for use in surface complexation models, and

(3) investigate the H-bonding of H₂O at the α-FeOOH-H₂O interface.

Fe-bearing minerals present a significant challenge for DFT calculations because the electronic ground-state of Fe is typically in a high-spin state. Furthermore, α-FeOOH is anti-ferromagnetic and the spin states of individual Fe atoms within a model must be carefully controlled. Once the DFT methodology, as implemented in VASP, can be shown to reliably reproduce static observables, such as crystal structures and vibrational frequencies, then similar computational methods can be employed to perform quantum MD simulations of reactions at the mineral-H₂O interface.

Bulk α-FeOOH and the (010) surface models were built using the Crystal Builder and Surface Builder modules of Cerius² 4.9 12, respectively. The α-FeOOH lattice parameters and initial atomic positions were based upon the previously published experimental X-ray diffraction measurements of Szytula et al. 13. It is imperative to note that the Cerius² 4.9 program uses the Pnma space group for α-FeOOH. In the Pbnm space group the (010) surface, which is commonly used for α-FeOOH 13, is the (100) surface in the Pnma space group (P.J. Heaney, pers. comm.).

The (010) α-FeOOH surface employed in this study was cleaved from the (100) plane of α-FeOOH in the Pnma space group with the O2-termination (Fig. 1). This surface structure is consistent with that of Kendall et al. 14. The 2-D periodic slab generated by this process was stoichiometric, neutral, and symmetric. The bare surface consisted of bridging OH groups (Fe₂OH or μ-hydroxyl), bridging oxo groups (Fe₃O or μ₃-oxo), and 5-fold coordinated Fe atoms. Thus, to hydrate the surface and keep it neutral, H₂O molecules were coordinated to the surface Fe atoms to fulfill octahedral coordination. In this configuration, the hydrated model was neutral and the presence of Fe-OH₂ functional groups did not indicate a positively-charged surface of protonated Fe-OH functional groups, as is commonly assumed for metal oxide surfaces. The question of whether or not H₂O favorably dissociates on this surface was also examined in this study. The O2 atoms are unlikely to accept a second proton (i.e., form an Fe₂OH₂ ⁺ site), so if the (Fe)-OH₂ groups were to donate a proton it would be transferred to the O atoms bonded to three Fe atoms just below the α-FeOOH-water interface (i.e., Fe₃O + H⁺ → Fe₃OH⁺). The relative potential energies of these two configurations were compared. If the O1-termination were used, then dissociation would be more probable because the 5-coordinate surface Fe atoms could adsorb an H₂O molecule and this could dissociate to form Fe-OH and protonated O1 surface atoms (i.e., FeOH₂ + FeO → 2FeOH). However, since Rakovan et al. 5 determined that this termination was less stable, we did not model it in our study.

All calculations were performed with the Vienna Ab Initio Simulation Package (VASP; 15). The projector-augmented wave (PAW) method 1617 was used in combination with the Perdew, Burke, and Ernzerhof (PBE) 18 exchange-correlation functional. The O_h and H_h pseudopotentials, as implemented in VASP, were used for the O and H atoms, respectively. The Fe_pv pseudopotential, which includes 14 electrons in the valence shell and treats the 3-p electrons explicitly, was used for the Fe atoms. The plane-wave kinetic energy cutoff was set to 700 eV. (Note: Softer pseudopotentials and a lower energy cutoff were tested and resulted in much poorer reproduction of the observed goethite crystal structure.) Energy differences of 1 × 10^-4 eV/atom and energy difference gradients of -0.02 eV/Å were used as convergence criteria. The number of unpaired electrons for the Fe atoms were set to five (i.e., high-spin d⁵ electronic configuration), with alternating (010) planes in positive and negative spin directions (Fig. 1; 19). This ensured that the model mimicked the anti-ferromagnetic ground-state of α-FeOOH and that the overall magnetization remained close to zero.

Two bulk α-FeOOH simulation cells were used, namely a 1 × 1 × 1 unit cell model and a 1 × 3 × 2 supercell model. P1 symmetry was applied in both cases. The Monkhorst-Pack 20 scheme was used to generate the k-point sampling grids within the Brillouin zone. For the bulk calculations, a 2 × 6 × 4 grid was used for the 1 × 1 × 1 unit cell model because the unit cell was approximately 9.9 × 3.0 × 4.6 Å. For the 1 × 3 × 2 supercell model, a 1 × 1 × 1 grid (i.e., gamma-point) was used because the simulation cell was nearly cubic with dimensions of 9.9 × 9.0 × 9.2 Å. This combination of direct and reciprocal lattices provided nearly isotropic accuracy with respect to the energy calculations 2122. Any difference in the accuracy of the energy between small and large simulation cells is not significant because we only compared calculated structures and not the energies between different models. The "Accurate" precision level (which uses the energy cut-off input as 700 eV and doubles the number of grid points in the fast Fourier transforms used to describe the charge density), as implemented in VASP, was used in all cases. Each k-point grid for sampling the Brillouin zone utilized a first-order Gaussian smearing function 23 of width sigma = 0.1 eV (i.e., ISMEAR = 0 and SIGMA = 0.1 in the VASP input file) in each calculation.

Variable-cell energy minimizations were performed for the calculations of bulk α-FeOOH, but only the atomic positions were allowed to relax for the energy minimizations of the (010) α-FeOOH surfaces. Vibrational frequencies were calculated for the energy-minimized bulk and surface slab models, using a finite difference method (i.e., each atom is displaced individually around its equilibrium position by approximately 0.1 Å) and numerical solution of the Hessian matrix. This solution assumes a harmonic oscillator, so any anharmonicity that is present in the actual vibrational modes will not be accounted for. This is especially important for O-H stretching modes with stronger H-bonds. Energy minimizations and frequency calculations were performed with both the SP-GGA and SP-GGA+U methods in most cases (only the extraordinarily long frequency calculations for the larger models were not repeated with SP-GGA+U). In both cases, the GGA functional corresponded to the SP-PBE exchange-correlation functional. Dudarev's rotationally invariant approach to the SP-GGA+U method was used here 24. The effective on-site Coulomb and exchange interaction parameters for each Fe atom were set to 4 eV and 1 eV, respectively, as recommended by Rollmann et al. 25. See Rollmann et al. 25 for details on the SP-GGA+U method applied to α-Fe₂O₃.

Periodic slabs with surface areas of 13.86 Ų (1 × 1) and 124.76 Ų (3 × 3) were used for the surface model calculations to test the effect of increasing the surface area on the predicted structures. Slab thickness was 8.88 Å in both cases. Vacuum gaps with z-dimension of 10 Å were used to separate the slabs. Solvated models were built by adding one H₂O molecule per 5-coordinate surface Fe atom with a bond distance of approximately 2.1 Å (2 H₂O molecules in the 1 × 1 and 18 H₂O molecules in the 3 × 3 models). The H₂O molecules were then relaxed using the COMPASS force field 26 while all of the goethite slab atoms were fixed. For the hydrated models, these were filled with H₂O molecules to approximate a density of 1 g/cm³ as closely as possible. This gave a total of 8 and 54 H₂O molecules in the 1 × 1 and 3 × 3 models, respectively. This represents three layers of hydration (L1 – directly bound to the surface Fe atoms, L2 – H-bonding to the L1 layer, and L3 – H-bonding to the L2 layer; see 27 for experimental verification of these layers on TiO₂ and SnO₂ surfaces). The H₂O molecules were initially positioned to optimize their H-bonding with the surface. An orientation that allowed the H₂O to act as an H-bond donor to the O2 surface atoms and H-bond acceptor from the surface H₂O molecules was selected. This initial configuration has not been verified experimentally and will influence the final results of the energy minimizations. Molecular dynamics simulations should be performed to test the reliability of this initial configuration. However, we justify this current decision based on the acidity of the surface groups present. The H⁺ ions in the Fe-OH₂ surface groups should be more acidic than the H⁺ ions on the O2 surface groups 28; hence the Fe-OH₂ sites should be better H-bond donors and the O2 atoms better H-bond acceptors (29 and references therein).

Although the above results represented energy-minimized structures on a neutral surface, and further testing must be performed under various surface charge states with MD simulations, it is instructive to compare the results to previous studies of the goethite-water interface. In our model, we began and ended with a (010) α-FeOOH surface terminated by Fe-OH₂ functional groups. This is similar to the result of Rustad et al. 8 where Fe-OH₂ functional groups were generated from their classical MD simulations. However, the initial structure in 8 was a Fe-OH-terminated surface and the Fe-OH₂ functional groups were created as solvent H₂O molecules transferred protons to the surface and OH^- ions H-bonded to the surface.

A similar discrepancy exists between the current study and the (010) α-FeOOH surface model of Rakovan et al. 5. In 5, a Fe-OH-terminated surface was the final configuration. Rakovan et al. 5 did not use a solvation layer above their (010) α-FeOOH surface and interpreted their XPS data to conclude that 5-fold coordinated surface Fe atoms should be coordinated with OH functional groups. In our model, such a termination was metastable, but the associated H₂O surface was lower in potential energy. This is based on a neutral surface that should represent α-FeOOH (010) at the point of zero-charge (PZC). However, the water layers in this model do not contain H⁺ or OH^-. Consequently, DFT-based MD simulations would be useful to test whether H⁺ transfers occur from the surface to solvent H₂O molecules, resulting in a Fe-OH-terminated surface charge-balanced by H₃O⁺ ions or whether a protonated surface and aqueous OH^- forms as should occur for this surface, which is below its PZC at pH 7. Simply moving H⁺ in these energy minimizations could not test this because the H₂O layers would need to re-arrange significantly to adjust to this H⁺-transfer.

In agreement with a study performed by Kerisit et al. 6, who found ordering of 4 to 5 H₂O layers, the H₂O molecules in our solvated model were ordered to at least 3 layers of H₂O molecules. However, the large degree of ordering in our model structures is partly an artifact of the highly ordered initial state and the fact that energy minimizations at 0 K were performed, not MD simulations at finite temperature. Again, DFT-based MD simulations are necessary to examine the configuration space of the α-FeOOH-H₂O interface.

The studies discussed above did not report Fe-O(H₂) bond distances, nor H-bond distances between solvent H₂O molecules and the α-FeOOH surface, so we cannot compare our DFT-calculated (010) α-FeOOH-H₂O interface structures to these studies quantitatively. However, Aquino et al. 11 did report O-H bond distances and H-bond distances from their DFT calculations, using cluster models of the (110) α-FeOOH surface. The (110) surface in the study of Aquino et al. 11, however, was terminated by OH functional groups, which complicates comparison. Their O-H bond distances ranged from 0.97 to 1.00 with no systematic variation among the terminal Fe-OH (hydroxo), bridging Fe₂OH groups (μ-hydroxo), and Fe₃OH (μ₃-hydroxo). These values were similar to those predicted in our periodic DFT calculations, but H-bonding played a role in lengthening the Fe₂OH hydroxyl OH bonds of the hydrated surface compared to terminal Fe-OH₂ groups in the current work. This effect was not observed for the solvated models.

The range of H-bond distances, 1.6 to 2.3 Å, calculated by Aquino et al. 11 overlaps the range predicted for the solvated 1 × 1 slab of this study. However, as noted above, the H-bond distances were significantly lengthened when the model was expanded to a 3 × 3 surface slab. Consequently, strong DFT-calculated H-bonds were probably an artifact of the system size and the neglect of H-bonding to H₂O layers above the L2 layer. Qualitatively, this was confirmed by the observation that up to three H-bonds were formed between the α-FeOOH surface and one H₂O molecule in the Aquino et al. 11 study. On the other hand, such extensive H-bonding was not observed in this study. The closest result obtained in this work to the H-bonding arrangement in 11 is in Fig. 9 where the FeOH₂ forms both donor and acceptor H-bonds to an H₂O molecule. The third H-bond is to the O atom of the adjacent Fe₂OH group, however. Recent work by Nangia et al. 43 has demonstrated the importance of surface H-bonding networks on influencing the structure of the mineral-water interface, so this factor should be considered in models of mineral surfaces.

The MUlti-SIte Complexation (MUSIC) model developed by Hiemstra et al. 39 is useful for predicting the PZC, as well as macroscopic adsorption behavior of mineral surfaces. Consequently, this type of thermodynamic model has great value in geochemistry and environmental chemistry. MUSIC requires charges, metal-oxygen bond lengths and the number of H-bonds to surface sites in order to calculate the pK_a values of these sites that control surface charging and adsorption. Fitts et al. 38 used second-harmonic generation spectroscopy to show that surface bond lengths and H-bonding arrangements calculated from periodic DFT and classical MD simulations significantly improve the MUSIC-predicted PZC, in contrast to using bulk bond lengths and assumed H-bonding structures. Thus, the structural predictions reported in this study should be useful as input to MUSIC model calculations on charging and adsorption onto α-FeOOH (010) surfaces. The most reliable values are likely to be those determined on the solvated 3 × 3 slab because this model allows for the greatest relaxation and accounts for surface-H₂O H-bond interactions that will be present in bulk potentiometric experiments.

Periodic DFT calculations can reasonably reproduce the structures and vibrational frequencies of bulk α-FeOOH. These methods were also applied to predict the surface structures of the (010) α-FeOOH surface. Model predictions of the surface metal-oxygen bond distances and H-bonding configurations should use a simulation cell comprised of multiple surface unit cells to allow for relaxation beyond what is achievable with single unit cell models. Additional layers of H₂O molecules beyond the L1 layer at metal-oxide surfaces are necessary to reproduce what would be found at the bulk mineral-water interface. The α-FeOOH surface was predicted to have a configuration consisting of FeOH₂, Fe₂OH and Fe₃O groups (i.e., an associated water adsorption model).

JDK performed most of the DFT calculations and was the primary author of the manuscript. KWP performed DFT calculations on the 3 × 3 × 1 supercell of the hydrated surface and determined how to perform the SP-GGA+U calculations. KWP was secondary author of the manuscript, especially with regard to the introduction and methods sections. DLS supervised and reviewed the manuscript throughout its development. All authors read and approved the final manuscript.