Periodic density functional theory calculations of bulk and the (010) surface of goethite
© Kubicki et al; licensee BioMed Central Ltd. 2008
Received: 01 October 2007
Accepted: 13 May 2008
Published: 13 May 2008
Goethite is a common and reactive mineral in the environment. The transport of contaminants and anaerobic respiration of microbes are significantly affected by adsorption and reduction reactions involving goethite. An understanding of the mineral-water interface of goethite is critical for determining the molecular-scale mechanisms of adsorption and reduction reactions. In this study, periodic density functional theory (DFT) calculations were performed on the mineral goethite and its (010) surface, using the Vienna Ab Initio Simulation Package (VASP).
Calculations of the bulk mineral structure accurately reproduced the observed crystal structure and vibrational frequencies, suggesting that this computational methodology was suitable for modeling the goethite-water interface. Energy-minimized structures of bare, hydrated (one H2O layer) and solvated (three H2O layers) (010) surfaces were calculated for 1 × 1 and 3 × 3 unit cell slabs. A good correlation between the calculated and observed vibrational frequencies was found for the 1 × 1 solvated surface. However, differences between the 1 × 1 and 3 × 3 slab calculations indicated that larger models may be necessary to simulate the relaxation of water at the interface. Comparison of two hydrated surfaces with molecularly and dissociatively adsorbed H2O showed a significantly lower potential energy for the former.
Surface Fe-O and (Fe)O-H bond lengths are reported that may be useful in surface complexation models (SCM) of the goethite (010) surface. These bond lengths were found to change significantly as a function of solvation (i.e., addition of two extra H2O layers above the surface), indicating that this parameter should be carefully considered in future SCM studies of metal oxide-water interfaces.
Goethite (α-FeOOH) is a common and reactive mineral in the environment . α-FeOOH is the most thermodynamically-stable form of the Fe-oxyhydroxides found in soils, groundwater, and acid mine drainage precipitates . 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 ([3, 4] 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; [5, 6]). As a result, this study will focus on the (010) 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. . 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 , 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. . 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 , 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. . 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.  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; ). The projector-augmented wave (PAW) method [16, 17] was used in combination with the Perdew, Burke, and Ernzerhof (PBE)  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; ). 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  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 [21, 22]. 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  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 . 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. . See Rollmann et al.  for details on the SP-GGA+U method applied to α-Fe2O3.
Periodic slabs with surface areas of 13.86 Å2 (1 × 1) and 124.76 Å2 (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  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  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 ; hence the Fe-OH2 sites should be better H-bond donors and the O2 atoms better H-bond acceptors ( and references therein).
Bulk structure and vibrational frequencies
Comparison of observed  and calculated lattice parameters and atomic positions for bulk α-FeOOH.
α = β = γ
.05, .75, .20
.05, .75, .20
.05, .75, .20
.06, .75, .20
.20, .75, .71
.20, .75, .71
.20, .75, .70
.20, .75, .68
.30, .25, .22
.30, .25, .21
.30, .25, .20
.30, .25, .19
.45, .25, .70
.45, .25, .70
.45, .25, .70
.44, .25, .70
.55, .75, .30
.55, .75, .30
.55, .75, .30
.56, .75, .30
.70, .75, .78
.70, .75, .79
.70, .75, .80
.70, .75, .82
.80, .25, .28
.80, .25, .29
.80, .25, .30
.80, .25, .32
.95, .25, .80
.95, .25, .80
.95, .25, .80
.94, .25, .80
.15, .25, .95
.15, .25, .96
.15, .25, .96
.15, .25, .94
.35, .75, .45
.35, .75, .46
.35, .75, .46
.35, .75, .44
.65, .25, .55
.65, .25, .54
.65, .25, .54
.65, .25, .56
.85, .75, .05
.85, .75, .04
.85, .75, .05
.85, .75, .06
.10, .75, .40
.08, .75, .41
.08, .75, .40
.09, .75, .40
.40, .25, .90
.42, .25, .91
.42, .25, .90
.41, .25, .90
.60, .75, .10
.58, .75, .09
.58, .75, .09
.59, .75, .10
.90, .25, .60
.92, .25, .59
.92, .25, .59
.91, .25, .60
One significant discrepancy between the observed and calculated structural parameters was the (Fe)O-H bond length. Experimentally, the (Fe)O-H bond length is 0.88 Å, whereas the DFT-calculated (Fe)O-H bond length was approximately 0.99 Å. Given that the IR frequencies and intensities of Fe-OH vibrational modes strongly depend upon O-H bond lengths , a difference of 0.1 Å is expected to be significant. Other calculations, such as molecular orbital theory calculations, also predict O-H bond lengths in the range of 0.96 to 1.00 Å in minerals and Fe-hydroxide molecular clusters [31–34]. The vibrational frequencies predicted by molecular clusters are generally in good agreement with observation for a range of compounds and materials . Furthermore, X-ray diffraction methods are insensitive to the positions of H atoms in crystalline materials and thus their atomic positions must be inferred. Consequently, it is possible that the calculated (Fe)O-H bond length is more accurate than the experimentally estimated (Fe)O-H bond length.
The reproduction of crystal structures can test the ability of a particular method to determine a system's minimum energy position on its potential energy surface. Vibrational frequencies probe the second derivatives of the potential energy surface (i.e., the Hessian matrix) around this minimum and are related to the force constants of bonds. Atomic vibrations can also be used to calculate entropies and other thermodynamic properties of crystals . Consequently, vibrational frequencies are useful observables to calculate and serve to further validate a computational methodology.
Comparison of observed (Expt 1 = ; Expt 2 = ) and calculated IR frequencies for bulk α-FeOOH.
VASP 4.5 does not calculate the IR intensities of the vibrational modes. Thus, it is difficult to assign calculated frequencies unequivocally to observed vibrational modes. Because there are a large number of calculated vibrational frequencies, correlations between model and observed frequencies may be fortuitously accurate. However, given these caveats, the DFT-calculated displacements in Table 2 reasonably correspond to experimentally-measured vibrational modes both from IR spectroscopy and inelastic incoherent neutron scattering . Therefore, we can be confident that the calculations are reasonably modeling the potential energy surface around the minimum energy structure.
Surface structure, H-bonding and vibrational frequencies
The calculated Fe-O(H2) bond length was significantly longer (Fig. 4) than the Fe-O bond it replaced in the bulk structure (2.42 versus 1.87 Å; Fig. 2b, Fig. 3a or 2.38 versus 1.93 for the SP-GGA+U results, Fig. 3c versus 4c). Furthermore, the surface Fe-O bonds shortened from approximately 2.13 to 2.02 Å to compensate for the added H2O molecules, as would be predicted by Pauling's rules. A small difference was predicted between the O-H bond lengths of the adsorbed H2O molecules and the OH functional groups that occupy the (010) α-FeOOH surface. The former were 0.98 Å and the latter were 1.00 Å due to differences in H-bonding. For example, the adsorbed H2O was only weakly H-bonded to the O atom of the Fe-(OH)-Fe linkage (H---O = 2.25 Å and O-H---O = 140°), while the α-FeOOH OH functional groups formed moderately strong H-bonds (H---O = 1.82 Å and O-H---O = 175°). It should be noted that a shorter H---O bond distance and more linear O-H---O angle both correspond to a stronger H-bond . Although this difference may seem insignificant, it has an influence on how solvating H2O molecules interacted with the surface, as discussed below.
Comparison of observed and calculated IR frequencies for the hydrated α-FeOOH surface.
790g, 797h, 805h
In this section, we first compare the solvated (010) α-FeOOH surface to the hydrated (010) α-FeOOH surface for the 1 × 1 slab discussed above. Second, a comparison of the solvated (010) α-FeOOH surfaces generated via energy minimization of the 1 × 1 and 3 × 3 slabs are compared. Third, the vibrational frequencies of the 1 × 1 solvated (010) α-FeOOH slab are examined in comparison to experimental vibrational spectra. Last, a comparison of the potential energies of the solvated surfaces with associated and dissociated H2O is made.
H-bonding between the L1 and L2 layers significantly affected the Fe-O bond lengths in the models used in this study. One H2O molecule acts as a proton donor and acceptor with the surface forming a 1.60 Å H-bond between (Fe)-OH2 functional groups and a 1.55 Å H-bond to the O atom of the Fe-(OH)-Fe surface linkage (Fig. 5). H-bonds can transfer electron density from the O-H bond which allows the O atom in the adsorbed H2O to form stronger bonds with Fe(III) on the surface.
As mentioned above in the Hydrated surface section, the OH functional groups of α-FeOOH were H-bonded to the surface, preventing them from behaving as proton donors to solvent H2O molecules. Because the Fe-OH2 functional groups only acted as proton donors in H-bonds to solvent H2O molecules, a pattern of H2O molecules formed in the L2 layer whereby each H2O molecule formed a donating H-bond to the O of the Fe-(OH)-Fe group and an accepting H-bond to the Fe-OH2 functional group (Fig. 5). The DFT energy minimizations exaggerated this pattern, so MD simulations should be performed to test how stable this structure is at finite temperatures. However, the underlying structure of the mineral should influence the H-bonding pattern of the L2 structure.
Comparison of observed and calculated IR frequencies for the solvated α-FeOOH surface (1 × 1 surface unit cell).
The (010) α-FeOOH surface can form different configurations with respect to H+ positions. The physisorbed H2O surface has been discussed above with an Fe2OH site from the original bulk structure, an FeOH2 terminal group (molecular adsorbed H2O) and no H+ on the Fe3O sites. Dissociated H2O configurations are also possible with H2O adding a H+ to the Fe3O site to form an Fe3OH + Fe2OH + FeOH surface (Fig. 5). Theoretically, a surface comprised of Fe3OH + Fe2O + FeOH2 is another combination. The relative energies of these configurations were investigated using the SP-GGA+U method for the solvated surface because this method is thought to result in more accurate energies . (Note: this was not done for the hydrated surface because H-bonding to the L2 likely plays a significant role in stabilizing the protonation state of the surface.) It is also worth noting here that the interatomic distances predicted using the SP-GGA+U method are similar to those obtained for the SP-GGA method.
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.  where Fe-OH2 functional groups were generated from their classical MD simulations. However, the initial structure in  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. . In , a Fe-OH-terminated surface was the final configuration. Rakovan et al.  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. , 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.  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. , 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.  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.  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  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.  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.  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 pKa values of these sites that control surface charging and adsorption. Fitts et al.  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).
We are grateful to U. Becker for providing the structure of the O2-terminated (010) goethite surface. K. W. Paul appreciates financial support from a University of Delaware fellowship. J. D. Kubicki acknowledges the support of NSF grants CHE-0221934 and CHE-0431328. VASP calculations were performed at the Center for Environmental Kinetics Analysis, a National Science Foundation and Department of Energy Environmental Molecular Sciences Institute at The Pennsylvania State University (Grant No. CHE-0431328). We appreciate advice and guidance from Andrei Bandura and Jorge Sofo regarding VASP utilization, and the constructive criticisms of three anonymous reviewers who greatly increased the quality of this study.
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