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).