Computational Modeling of Octahedral Iron Oxide Clusters

Article pubs.acs.org/JPCC

Computational Modeling of Octahedral Iron Oxide Clusters: Hexaaquairon(III) and Its Dimers Yang Yang, Mark A. Ratner, and George C. Schatz* Argonne-Northwestern Solar Energy Research (ANSER) Center and Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States S Supporting Information *

ABSTRACT: Octahedral monomeric and dimeric iron oxide clusters represent the basic units in many iron oxide and oxide-hydroxide minerals. In this paper, we provide a detailed theoretical analysis of the structural and optical properties of the most important of these clusters in a vacuum and in an aqueous environment. An evaluation of various computational methods was performed on the experimentally well-known monomer [Fe(H2O)6]3+, and it is found that all methods provide similar and reliable structures. Most density functional theory (DFT) methods reasonably reproduce the spin-forbidden sextet−quartet d−d transition energy, which also resembles the lowest transition energies in many infinite octahedral iron oxide systems. On the other hand, Hartree−Fock (HF) and MP2 methods significantly overestimate this energy. The ligand-to-metal charge transfer (LMCT) energy is highly sensitive to the method employed, with the closest agreement with experiment provided by the BHandHLYP functional. Thermodynamic property calculations suggest that dimerization reactions starting from [Fe(H2O)6]3+ are highly exothermic in a vacuum. In contrast, these reactions have insignificant energy changes in solution, though the singly μoxo bridged dimer is slightly favored. The electrostatic repulsion between two charged monomers hinders their close contact. The singly μ-oxo bridged dimer suffers less from this because of its maximal Fe−Fe distance, which is consistent with the existence of stable crystal structures for this dimer. A comparison between the calculated structures and experimental results suggests that several dimer species coexist in solution. The calculated ferromagnetic and antiferromagnetic states of the dimers are found to have comparable energies and structures. While the singly μ-oxo and doubly μ-hydroxo bridged dimers have spin states that are well separated in energy, the spin states in the triply μ-hydroxo bridged dimer pack closely. The single excitation d−d transition in the dimer structure is comparable in energy to the d−d transition in the monomer, while the double excitation d−d transition, i.e., simultaneous excitation of two iron centers, has a higher excitation energy that is 1.6−2.6 times the single excitation energy but below the LMCT energy. This means that doubly excited states can be populated during the non-radiative relaxation of iron oxide clusters following initial photoexcitation of the LMCT state.

1. INTRODUCTION

Many iron(III) oxide and oxide-hydroxide minerals contain octahedral networks in which each iron atom is surrounded by six oxygen atoms. In hematite, for example, adjacent iron atoms are ferromagnetically coupled and share edges along the (001) direction. In the [001] direction, adjacent iron atoms are antiferromagnetically coupled and share their face (bridged by three oxygen atoms). Accordingly, monomer and dimer cluster models can be developed that are related to these structures. The simplest model is the octahedral monomer [Fe(H2O)6]3+ which represents the smallest fragment in many iron oxide minerals. An advantage to the use of this model is the existence of abundant experimental data that can be used to calibrate computational methods. For example, the cation can be stabilized by suitable counteranions in the solid state and then the structure can be determined by X-ray single crystal diffraction11−13 and low-temperature neutron diffraction techniques.14 Moreover, the cation structure can also be determined by X-ray absorption fine structure (XAFS)15 and X-

Iron(III) oxide and oxide-hydroxide minerals are earthabundant materials.1 They are widely utilized in many physical, chemical, geological, and biological processes. For example, hematite (α-Fe2O3) is a promising candidate for the photocatalytic splitting of water.2 A complete description of the electronic structure of a mineral system can be achieved from electronic structure calculations using periodic Bloch wave functions. Unfortunately, self-interaction errors for DFT methods are large for Fe(III) because of its five highly localized 3d electrons.3 Much effort has been devoted in remedying this problem, with the most common solution involving use of the DFT+U method.4−6 However, the U term is not uniquely defined and has been varied depending on the specific system studied. An alternative to explore the iron oxide system is to use finite cluster models resembling the basic units in the infinite system.7−10 This is especially useful for localized optical properties such as excitation involving a single Fe as translational symmetry is broken and nonperiodic density functional theory is more accurate.9 © 2013 American Chemical Society

Received: August 12, 2013 Published: September 24, 2013 21706

dx.doi.org/10.1021/jp408066h | J. Phys. Chem. C 2013, 117, 21706−21717

The Journal of Physical Chemistry C

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Figure 1. Chemical structures of the dimers. Color codes for the oxygen atoms: red, bridging oxygen (Ob); magenta, apical oxygen (Oa); blue, terminal oxygen (Ot); green, equatorial oxygen (Oe).

ray scattering techniques16 in solution. In addition, its ligand field d−d transition17−19 and ligand-to-metal charge transfer (LMCT) transition20−23 absorption peaks have been well identified and assigned in highly acidic solutions. This establishes a standard for an assessment of electronic structure methods for handling excited state properties. As mentioned above, it is necessary to characterize [Fe(H2O)6]3+ in highly acidic solutions. This is because it can participate in many deprotonation and oligomerization reactions at higher pH, and the generated products are also highly absorbing. The oligomerization process can be regarded as the preliminary step to building an infinite network. The dimerization process is the simplest and most fundamental oligomerization process. Experimentally known bridged diiron molecules indicate that the dimers can be singly, doubly, or triply bridged by μ-oxo or μ-hydroxo bridges (Figure 1).24 These dimers can be viewed as the precursors for various iron oxide minerals.25−28 They are more explicit models of the solid state than the monomer for several reasons. First, different iron oxide minerals can have different networks. Even in one mineral, different dimer constructions may exist in different directions. In hematite, double and triple bridges coexist whose dimer properties are substantially different. Therefore, dimer models can better simulate the diversity of structures in iron oxide minerals. In addition, adjacent iron atoms can be magnetically coupled and the magnetic coupling can only be modeled when there is more than one iron atom. Third, besides the ligand field d−d and LMCT transitions, an additional type of excitation in dimers is the double exciton excitation wherein two Fe3+ are excited simultaneously. This paired excitation has been observed in diiron molecules29 and probably in iron oxide and oxide-hydroxide minerals.30 A key property of these excitations for energy-related applications is that iron-localized excitons are produced via a spin-allowed transition. The two iron atoms in a dimer can either be ferromagnetically or antiferromagnetically coupled. The antiferromagnetic coupling leads to an open-shell singlet state in which two high

spin iron atoms have opposite spins. This singlet state is different from the closed-shell state whose spin density is paired everywhere. In the conventional DFT method, a spinunrestricted calculation of a singlet state tends to behave like a spin-restricted one and converges to a closed-shell state. To model the antiferromagnetic state, one approach is to assume that the interaction between two iron atoms is small.31,32 Thus, the antiferromagnetic state can be approximated by the ferromagnetic state in which the two high spin iron centers are preserved. Previous studies have suggested that the ferromagnetic and antiferromagnetic solutions for diiron molecules have small energy differences, but they are both markedly different from the spin-paired (restricted) solution.32 The second approach is to directly model the antiferromagnetic state with the broken symmetry method applied to the dimer, through which the physically realistic spin density is retained. In this paper, we present DFT results for [Fe(H2O)6]3+ as well as several possible dimer structures, for the purpose of both calibrating a variety of DFT methods for predicting structures and excited states and providing insights on the influence of exchange-repulsion effects on the monomer and dimer properties. Comparisons with experimental data will be included, and ultimately, we hope to use this work to determine which variant of DFT can be used to describe larger clusters that better approximate hematite.

2. COMPUTATIONAL DETAILS Calculations on the [Fe(H2O)6]3+ cluster were carried out using various density functional and wave function methods. Among these methods, the BLYP33,34 is a generalized gradient approximation (GGA) method. B3LYP*,35 B3LYP,36 B1LYP,37 and BHandHLYP38 are hybrid functionals that include Hartree−Fock (HF) exchange. They together form a sequence with increasing percentage of HF exchange. B3LYP* is a variant of the more well-known B3LYP hybrid functional in which HF exchange is 15% rather than 20%. It has been suggested that this modified version yields reliable energetics for transition 21707



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metals such as iron.35 PBE39 and BP8633,40 are also GGA methods similar to BLYP. PBE0 is a hybrid form of PBE with 25% HF exchange.41 M06-L42 and TPSS43 are called metaGGA’s, as these include the kinetic energy density. M0644 and TPSSH45 are hybrid forms of the meta-GGA’s. Besides these density functionals, two wave-function-based methods, HF and MP2, were also employed. Note that the structures and d−d transitions of [Fe(H2O)6]3+ were explored using GGA, HF, and MP2 in an earlier study.46 However, no hybrid GGA or metaGGA was used and the frequency analysis was not performed at the MP2 level. Also, LMCT transitions were not examined in that study. The water solvation effect for [Fe(H2O)6]3+ and some other [M(H2O)6]n+ complexes has been examined using various implicit and explicit models.47−51 For example, [Fe(H2O)6]3+ was treated with the polarizable continuum model (PCM) and the isodensity polarizable continuum model (IPCM).48 With PCM, the authors obtained a Th structure with imaginary frequencies and a distorted S6 energy minimum with tilted and rotated water molecules. However, their IPCM calculation suggested that the Th structure was more stable. The authors further employed PCM on [Al(H2O)6]3+ and also obtained the S6 distortion. In view of the closed shell Al3+ cation, the authors attributed the S6 distortion to an artifact of the solvation model. Moreover, the solvation models only lower the energy of [Fe(H2O)6]3+ by less than 2 kcal/mol compared to the vacuum structure. The vacuum structure and Th PCM structure are also similar. In explicit solvation investigations, [M(H2O)6]n+[H2O]12 (M = Mg, Al, Ti, Zn, and so forth) were considered, each with 12 second-shell water molecules.47,49−51 Here, it was found that the geometry change of the core caused by the second shell water is small. Therefore, we deem the vacuum geometries of [Fe(H2O)6]3+ and its dimers to be a reasonable approximation for their solution geometries in water. In our calculations, the conductor-like screening model (COSMO) was used to model aqueous solvation.52 The dielectric constant of water was set to be 78.39, and the radius of the rigid water sphere was chosen to be 1.93 Å. On the basis of the verified computational methodology of [Fe(H2O)6]3+, the dimer structure was studied using the BLYP functional unless otherwise mentioned. The various dimer structures were defined by a combination of the number of bridges and the nature of the μ-bridge (see Figure 1 for details). For example, 2OH means that it is bridged by two hydroxo groups. One of the dimers (2OH) was also calculated using the B3LYP hybrid functional for an evaluation of the influence of exact exchange. Unfortunately, our attempts to optimize the geometries of 1OH and 3O led to their dissociation. For the same reason, the geometries of only the (5/2, ±5/2) states of 2O were obtained. Some SCF calculations failed to converge to the designated states, but fortunately most succeeded. Thus, the big picture is not affected by our convergence problems and sensible conclusions can be reached. In all calculations, the geometry was first optimized using the specified method. The nature of the optimized stationary point was checked by a frequency analysis to ensure that the calculation converged to an energy minimum. The vertical d−d transition energy was calculated from the energy difference between the ground state and a higher spin state at the ground state geometry. All DFT calculations were performed within the ADF package,53 and the TZP (valence triple-ζ plus one polarization function) basis set was employed. All wave function calculations were performed within the Q-Chem

package,54 and the Hay−Wadt LANL2DZ basis set was used for Fe with its 1s, 2s, and 2p electrons represented by the LANL2DZ effective core potential.55 The 6-311G(d,p) basis set was used for H and O.

3. MONOMERIC CLUSTER [FE(H2O)6]3+ Structures. The central Fe(III) has five d electrons, so sextet, quartet, and doublet states are possible ground states. Detailed electron configuration information from our calculations is given in the Supporting Information. The Mulliken charge and spin at the central iron verify the spin state that a calculation converges to (Tables 1 and 2). For [Fe(H2O)6]3+, Table 1. Ground State (Sextet State) of the Monomer [Fe(H2O)6]3+ Calculated Using Various Methods BLYP B3LYP* B3LYP B1LYP BHandHLYP BP86 PBE PBE0 M06L M06 TPSS TPSSH HF MP2

Xa

dFe−O (Å)

qb

spinc

0 15 20 25 50 0 0 25 0 27 0 10 100 100

2.094 2.056 2.054 2.051 2.032 2.071 2.073 2.036 2.048 2.027 2.061 2.048 2.055 2.047

1.37 1.50 1.55 1.60 2.14 1.32 1.32 1.54 1.49 1.81 1.42 1.50 2.04 1.73

4.14 4.29 4.34 4.38 4.53 4.20 4.19 4.42 4.28 4.35 4.22 4.32 4.72 4.64

a

X: percentage of exact exchange. bq: Mulliken charge at the central iron atom. cspin: Mulliken spin density at the central iron atom.

all methods suggest the sextet state as the ground state with an electron configuration (t2gα)3(egα)2. In the sextet state, the central iron atom and six oxygen atoms form a regular octahedron. In other words, the six Fe−O bond lengths are equal and the three O−Fe−O axes are perpendicular to each other. All methods give similar Fe−O bond lengths (Table 1), which are all in very good agreement with the experimental values.11−16 For density functional methods, incorporation of exact exchange reduces the bond length. This decrease is evident in the sequence of BLYP, B3LYP*, B3LYP, B1LYP, and BHandHLYP. The same trend can be found by comparing PBE, M06-L, and TPSS with their respective hybrid forms. In addition, the meta-GGA’s give shorter bonds compared to those given by GGA’s. With regard to the wave function methods, the MP2 bond length is slightly shorter than the HF one. This is probably related to correlation effects that are included in MP2. All methods suggest that the quartet state lies above the sextet state but below the doublet state. The electron configuration of the quartet state is (t2gα)3(t2gβ)1(egα)1, and the configuration of the doublet state is (t2gα)3(t2gβ)2 (detailed information is given in Table S1 in the Supporting Information). The structures for these states both deviate from a regular octahedron, though the Fe−O bond lengths at opposite positions are equal. In general, the functional dependence for the sextet state can also be found in the quartet state; i.e., (1) a higher amount of exact exchange leads to a shorter bond, (2) the meta-GGA’s give smaller bond lengths than the GGA’s, and (3) the correlation effect in the 21708



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Table 2. Quartet and Doublet States of [Fe(H2O)6]3+ Calculated Using Various Methods θ (deg)

dFe−O (Å)

BLYP B3LYP* B3LYP B1LYP BHandHLYP BP86 PBE PBE0 M06L M06 TPSS TPSSH HF MP2 BLYP B3LYP* B3LYP B1LYP

Xa

Fe−O1

Fe−O2

Fe−O3

0 15 20 25 50 0 0 25 0 27 0 10 100 100

2.142 2.099 2.098 2.095 2.073 2.118 2.121 2.080 2.084 2.076 2.103 2.092 2.099 2.088

2.048 2.010 2.007 2.005 1.978 2.026 2.029 1.989 2.008 1.978 2.018 2.004 1.987 1.984

1.958 1.928 1.926 1.925 1.908 1.934 1.936 1.909 1.932 1.909 1.929 1.919 1.936 1.914

0 15 20 25

2.006 1.968 1.962 1.962

1.986 1.955 1.957 1.955

1.969 1.943 1.946 1.947

avg quartet 2.049 2.012 2.010 2.008 1.986 2.026 2.029 1.993 2.008 1.988 2.017 2.005 2.007 1.995 doublet 1.987 1.955 1.955 1.955

θ1b

θ2

θ3

qc

spind

Δd (Å)e

ΔE (eV)f

5 4 3 3 0 5 5 3 4 3 5 4 0 0

1 0 0 0 0 1 0 0 0 0 0 0 0 0

1 0 0 0 0 1 1 0 1 0 0 0 0 0

1.30 1.48 1.53 1.58 2.00 1.24 1.25 1.52 1.43 1.40 1.33 1.47 1.88 1.51

2.68 2.76 2.78 2.81 2.85 2.72 2.72 2.84 2.79 2.80 2.73 2.78 2.95 2.92

0.045 0.044 0.044 0.043 0.046 0.045 0.044 0.043 0.040 0.039 0.044 0.043 0.048 0.052

0.88 1.14 1.24 1.34 1.79 0.92 0.95 1.46 1.65 1.72 0.84 1.07 3.30 2.49

3 2 2 2

1 0 0 0

5 3 3 3

1.25 1.53 1.59 1.64

0.95 0.98 0.99 1.00

0.107 0.101 0.099 0.096

0.95 1.34 1.49 1.63

a X: percentage of exact exchange. bθ: the average of the four absolute differences between the O−Fe−O bond angle and 90° in a FeO4 plane. cq: Mulliken charge at the central iron atom. dspin: Mulliken spin density at the central iron atom. eΔd = dsextet − dquartet/doublet. fΔE = Equartet/doublet − Esextet.

MP2 method reduces the bond length. In spite of the slightly different absolute values provided by the different methods, the trend from the sextet state to the quartet state is the same regardless of the method. The Fe−O bond lengths in the quartet state split into three groups. The long one is even longer than the one in the sextet state, while the intermediate and short ones are shorter than the one in the sextet state. In the end, the average bond length in the quartet state is smaller by about 0.04 Å compared to that in the sextet state. Moreover, unlike the mutually perpendicular O−Fe−O axes in the sextet state, the O−Fe−O bond angle in a quartet state may slightly deviate from a normal angle. The deviation angle in a plane, θ, is defined as the average of the four absolute differences between the O−Fe−O bond angles and 90°. Most density functionals except BHandHLYP give at least one nonzero deviation angle in each case. For these density functionals, inclusion of exact exchange tends to reduce this angle. The deviation angles given by the two wave function methods, HF and MP2, are virtually zero. This explains the zero deviation angle in the BHandHLYP method. For those structures with nonzero deviation angles, the largest deviation angle is formed by the two shorter Fe−O bonds. The doublet state results show similar functional trends to the sextet and quartet states, though this trend is less evident, since B3LYP*, B3LYP, and B1LYP calculations give comparable results. Nonetheless, the decrease of the bond length and deviation angle from a GGA to a hybrid GGA is apparent in the results. Unlike the quartet state whose largest deviation angle comes from the plane perpendicular to the longest Fe−O bond, the largest deviation angles in the doublet state come from both planes perpendicular to the longest and shortest Fe− O bonds. Another difference between doublet and quartet states is that the doublet state has a smaller range in Fe−O bond lengths. The longest Fe−O bond is shorter than the corresponding one in the quartet state, while its shortest bond

is longer than the corresponding one in the quartet state. The average value in the doublet state is smaller by 0.06 Å. All the Fe−O bond lengths in the doublet state are smaller than that in the sextet state, and the average value is smaller by about 0.10 Å. Spin State Energies. As mentioned above, all calculations give the same order of spin state energies: Esextet < Equartet < Edoublet. The energies of the quartet and doublet states relative to their ground states are given in Table 2. Unlike the structure which is rather method-insensitive, the spin state energy is moderately sensitive to the computational method. A higher amount of exact exchange is associated with a larger energy gap. For example, the sextet−quartet gap increases from 0.88 eV in BLYP to 1.79 eV in BHandHLYP and the sextet−doublet gap increases from 0.95 eV in BLYP to 1.63 eV in B1LYP. Though the sextet and quartet states are well separated, the quartet− doublet gap is relatively small, around a couple of tenths of eV. For the sextet−quartet gap, the two meta-GGA’s give substantially different values, which are 1.65 eV for M06-L and 0.84 eV for TPSS, respectively. The former one is close to the value from the hybrid functional BHandHLYP with a high percentage of exact exchange, while the latter one is close to the value from a GGA (0.88 for BLYP, 0.92 for BP86, and 0.95 for PBE). This is in contrast with their performance in structure calculations, in which M06-L and TPSS provide similar results. This is also in contrast with the performance of GGA’s, which give similar results in both structures and spin state energies. While the spin state energy is only moderately dependent on density functional, the wave function methods considerably overestimate the sextet−quartet energy gap. The HF method gives an exceptionally high value, 3.30 eV, which is approximately 4 times that from a GGA. Accordingly, incorporation of more exact exchange leads to a larger energy gap. The reason for the HF overestimation is overstabilization of the higher spin state due to correlation of electrons with 21709



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close to that from GGA. In addition, the HF calculation gives a much higher sextet−quartet transition energy than is given by DFT calculations. The MP2 calculation decreases the HF value by about 0.8 eV, but the result is still pretty high compared with DFT calculations. The sextet−quartet and sextet−doublet transitions differ by roughly 0.3 eV. In other words, the Franck−Condon quartet and doublet states do not lie far away from each other. The sextet−quartet d−d transition energy was determined for [Fe(H2O)6]3+ to be around 1.53 eV.17−19 In particular, this energy is reminiscent of the lowest-lying absorption peaks for Fe3+ in many periodic iron oxide systems with octahedral constructions: 1.43 eV for hematite,56 1.35 eV for goethite,30 1.33 eV for maghemite,30 1.29 eV for lepidocrocite,30 1.24 eV for Fe3+ doped in MgO,57 1.17 eV for Fe3+ doped in Al2O3,58,59 and 1.11 eV for magnetite.60 They have all been assigned as 6A → 4T1 transitions. With the experimental values as the reference, most DFT calculations give excellent results, with deviations within several tenths of an eV. To test the influence of the solvation model, the conductor-like screening model (COSMO) was applied. The solvated sextet−quartet d−d transition energy using the BLYP functional is 1.15 eV, which is virtually the same as the vacuum value. In general, some of the hybrid GGA’s perform better than the corresponding GGA. Thus, the energies for hybrids with 20−25% exact exchange are 1.54 eV for B3LYP, 1.63 eV for B1LYP, and 1.64 eV for PBE0, while the GGA results are 1.16 eV for BLYP and 1.25 eV for PBE. However, in view of the extra computational burden of such calculations, a GGA is often regarded as a reasonable choice, especially for large-size systems such as periodic hematite. Interestingly, the meta-GGA’s and hybrid meta-GGA’s yield acceptable results, but they do not outperform GGA’s and hybrid GGA’s. As with the spin state energy, the HF calculation significantly overestimates the sextet−quartet transition energy. The MP2 calculation somewhat improves the HF result, but the improvement is not satisfactory. Although HF and MP2 results significantly overestimate the d−d transition energy, they give relaxation energies comparable to those from DFT methods. The relaxation energy from the Franck−Condon quartet state to the energy minimum of this quartet state is around 0.2−0.3 eV regardless of the method. LMCT Transitions. While d−d transitions in [Fe(H2O)6]3+ are spin forbidden, LMCT transitions are spin allowed and thus more intense. The major observed absorption peak of [Fe(H2O)6]3+, occurring at 5.16 eV, corresponds to the water-to-iron charge transfer band. The LMCT transition energy was calculated using both time dependent density functional theory (TDDFT) and wave function treatments (Table 4). As with the d−d transition energy, inclusion of the COSMO has an insignificant effect. With the BLYP functional, the excitation energy including the solvation model is 2.42 eV and the one without the solvation model is 2.40 eV. Different from previous properties discussed, the LMCT transition energy is highly sensitive to the functional employed. For example, the result is 2.40 eV for BLYP but 5.48 eV for BHandHLYP; thus, a higher amount of exact exchange leads to a considerably higher energy. Even in the sequence of B3LYP*, B3LYP, and B1LYP with a sequential increase of only 5% in the percentage of exact exchange, the excitation energy is increased by more than 0.2 eV each step. Therefore, choosing the appropriate functional is essential for generating accurate or at least qualitatively correct LMCT energies. For [Fe(H2O)6]3+,

parallel spin in the Fermi hole. On the other hand, this effect has little influence on structure, for which DFT and HF yield similar results. By adding more electron correlation effects to the HF method, MP2 reduces the energy gap to 2.49 eV, which is nonetheless still considerably higher than those from DFT methods. Note that the subtle difference between DFT structures and wave-function-based structures in Tables 1 and 2 could partly be due to differences in their basis sets and computation packages, but the significant energy gap difference should be mostly from the method itself. Ligand Field d−d Transitions. The vertical d−d transition energy, the energy difference between the ground state and a Franck−Condon excited spin state, provides an opportunity to refer to experimentally observed transitions. The sextet ground state of [Fe(H2O)6]3+ has all of its five 3d orbitals singly occupied. Therefore, all possible d−d transitions involve spin flip and are spin-forbidden. In iron oxide minerals and complexes with more than one iron center, this spin selection can be released through magnetic coupling between adjacent iron centers.30 Spin−orbit coupling makes the transition weakly allowed, even for the isolated molecule, but including this in the calculations would be very difficult. The vertical d−d transition energies were calculated using various DFT and wave function methods, and the results are given in Table 3. Also given in Table 3 are the Mulliken charge and spin confirming the Franck−Condon state that the calculation converges to. Table 3. Vertical d−d Transition Energies of [Fe(H2O)6]3+ Calculated Using Various Methods (eV) X BLYP B3LYP* B3LYP B1LYP BHandHLYP BP86 PBE PBE0 M06L M06 TPSS TPSSH HF MP2 BLYP B3LYP* B3LYP B1LYP

0 15 20 25 50 0 0 25 0 27 0 10 100 100 0 15 20 25

E

Erelaxa

sextet → quartet 1.16 0.28 1.32 0.18 1.54 0.30 1.63 0.29 2.14 0.35 1.18 0.26 1.25 0.30 1.64 0.18 1.86 0.20 2.01 0.30 1.14 0.30 1.30 0.23 3.52 0.21 2.75 0.26 sextet → doublet 1.41 0.46 1.68 0.34 1.80 0.31 1.88 0.26

q

spin

1.28 1.40 1.44 1.48 2.00 1.23 1.22 1.42 1.39 1.67 1.32 1.39 1.91 1.92

2.63 2.75 2.78 2.80 2.86 2.67 2.67 2.84 2.73 2.80 2.69 2.77 2.94 2.94

1.24 1.33 1.38 1.44

0.87 1.00 1.02 1.01

a

Erelax: The relaxation energy from the Franck−Condon quartet/ doublet state to the energy minimum of this quartet/doublet state.

As with the spin state energy, the d−d transition energy is moderately sensitive to the functional. It is as expected that they share the same functional dependence because they are both energetics-based properties. Thus, the following trends are familiar from the spin state energy section. A higher amount of exact exchange leads to a larger transition energy. All GGA’s give similar sextet−quartet transition energies, while the two meta-GGA’s behave substantially differently. M06-L gives a value close to that from BHandHLYP, while TPSS gives a value 21710



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Table 4. LMCT Transition Energies of [Fe(H2O)6]3+ Calculated Using Various Methods (eV) BLYP B3LYP* B3LYP B1LYP BHandHLYP BP86 PBE PBE0 M06L M06 TPSS TPSSH CIS CIS(D)

X

E1

f1

E2

f2

0 15 20 25 50 0 0 25 0 27 0 10

2.40 3.04 3.27 3.56 5.48 2.54 2.54 3.76 3.14 3.67 2.71 3.11 9.20 5.02

0.061 0.059 0.058 0.056 0.053 0.061 0.061 0.057 0.065 0.062 0.063 0.061 0.063

6.56 7.36 7.56 7.79 9.42 6.81 6.81 8.06 7.80 8.14 7.12 7.55 12.16

0.140 0.152 0.156 0.152 0.161 0.139 0.137 0.140 0.142 0.149 0.142 0.150 0.105

2[Fe(H 2O)6 ]3 + → 2OH + 2H3O+

(1B)

2[Fe(H 2O)6 ]3 + → 3OH + 3H3O+

(1C)

2[Fe(H 2O)6 ]3 + + 2H 2O → 2O + 4H3O+

(1D)

2[Fe(H 2O)5 (OH)]2 + → 2OH + 2H 2O

(2B)

→ 3OH + 3H 2O

(2C)

2[Fe(H 2O)5 (OH)]2 + → 3OH + 2H 2O + H3O+

(2D)

2[Fe(H 2O)5 (OH)]2 + → 2O + 2H3O+

(2E)

The dimerization energetics were investigated both in a vacuum and in water with the latter using the COSMO model (Table 5). In a vacuum, all processes have significant reaction energies. Table 5. Thermodynamic Properties of Several Reactionsa reaction

ΔEelb

ΔEnucc

ΔSvac

ΔHvacd

ΔGvace

ΔEsolf

ΔGsolg

1A 1B 1C 1D 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F

−84 −35 −181 −301 207 256 143 110 −10 −146 49 −146 −97 217 266 120

2 −1 −2 1 1 −2 −4 −4 0 1 −3 −1 −4 1 −2 −3

6 31 64 42 −15 10 44 44 22 10 25 34 58 −37 −12 22

−83 −36 −182 −300 207 254 140 107 −10 −145 46 −146 −100 217 264 117

−84 −46 −203 −313 212 250 126 93 −17 −148 38 −157 −118 229 267 110

72 42 196 318 −213 −243 −133 −89 33 143 −30 154 124 −245 −276 −122

−12 −4 −7 5 −1 7 −8 4 16 −5 8 −2 6 −17 −9 −11

All units are in kcal/mol except that of ΔSvac, whose unit is cal/(mol· K). bEel: electronic internal energy. cEnuc: nuclear internal energy. d Hvac = Eel + Enuc. eGibbs free energy in a vacuum, Gvac = Hvac − TSvac. f Esol: solvation energy. gGibbs free energy in water solution, Gsol = Gvac + Esol; presumably the reaction occurs at constant volume and no work is done. a

The dimerization reactions from [Fe(H2O)6]3+ are highly exothermic, while the reactions starting from [Fe(H2O)5(OH)]2+ or [Fe(H2O)4(OH)2]+ are highly endothermic (except eq 2E). The difference is attributable to the large energy release associated with the deprotonation reaction of [Fe(H2O)6]3+ (eq 3).

4. DIMERIC CLUSTERS Dimerization Thermodynamics. The dimerization process can be regarded as the initial stage to form huge aggregates. Since [Fe(H2O)6]3+ readily deprotonates, different monomeric species may dominate the dimerization in solution at different pH values. Moreover, in view of the structures of the dimers, [Fe(H2O)6]3+, [Fe(H2O)5(OH)]2+ ,and [Fe(H2O)4(OH)2]+ are all plausible reactant candidates. The possible dimerization reactions are depicted in eqs 1A−1D and 2A−2E (using the structures and abbreviations in Figure 1). (1A)

(2A)

[Fe(H 2O)5 (OH)]2 + + [Fe(H 2O)4 (OH)2 ]+

only the BHandHLYP hybrid functional yields a reasonable excitation energy. Errors arising from an inappropriate functional could be 2−3 eV. All GGA’s severely underestimate the energy. The meta-GGA’s yield somewhat larger values, which nonetheless still suffer from significant underestimation. Incorporation of exact exchange reduces the degree of underestimation. This is understandable in view of the significant overestimation of the LMCT energy using the configuration interaction singles (CIS) method. The experimental energy is roughly midway between the CIS energy and a typical GGA energy. Therefore, BHandHLYP strikes a balance with the amount of exact exchange that is included. CIS(D) is a correlated excited state treatment, and can be regarded as the excited state analogue of the ground state MP2 method.61,62 It lowers the CIS value by approximately 4 eV (to 5.02 eV), leading to good agreement with the experimental result. In spite of the diverse excitation energies given by various methods, they all suggest that the lowest excitation peak is from 3-fold degenerate transitions. In addition, they predict another more intense peak approximately 4 eV blue-shifted. We are unaware of any experimental observation of this intense peak, but we note that it would correspond to a VUV wavelength and would be technically difficult to study.

2[Fe(H 2O)6 ]3 + + H 2O → 1O + 2H3O+

2[Fe(H 2O)5 (OH)]2 + → 1O + H 2O

[Fe(H 2O)6 ]3 + + H 2O → [Fe(H 2O)5 (OH)]2 + + H3O+ (3)

For the dimerization process to occur spontaneously in a vacuum, it should occur starting from [Fe(H2O)6]3+ directly (such as in eqs 1A−1D) rather than go through [Fe(H2O)5(OH)]2+ (such as in eqs 2A−2E). Moreover, the 3OH dimerization reaction is more exothermic than the 2OH one. Likewise, the 2O dimerization reaction is more exothermic than the 1O one. This is as expected because more energy is released when more bridges are formed. In addition, forming a μ-oxo bridge is more energetically favorable than forming a μhydroxo one. This is also understandable because μ-oxo is a better μ-bridge connecting two metal atoms. The reaction energies become significantly different when solvation (using COSMO) is included. Interestingly, a highly 21711



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Table 6. Selected Distances (Å), Angles (deg), and Relative Energy Levels (eV) of the Low-Lying States of the Dimers 1O 1Og 1O 1O 1O 1O 2OH 2OH 2OH 2OH 2OH 2OH 2O 2O 3OH 3OH 3OH 3OH 3OH 3OH

S1a

S2

Fe1−Ob

Fe2−O

Fe1−Fe2

Fe1−Ob−Fe2c

Ed

q1e

spin1f

q2

spin2

5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2 5/2

5/2 3/2 1/2 −5/2 −3/2 −1/2 5/2 3/2 1/2 −5/2 −3/2 −1/2 5/2 −5/2 5/2 3/2 1/2 −5/2 −3/2 −1/2

2.121 2.084 2.128 2.115 2.111 2.122 2.108 2.125 2.125 2.105 2.106 2.111 2.152 2.148 2.100 2.119 2.127 2.092 2.099 2.100

2.121 2.084 1.994 2.114 2.077 1.995 2.108 2.051 1.987 2.106 2.052 1.989 2.144 2.133 2.100 2.038 1.964 2.097 2.030 1.966

3.836 3.679 3.736 3.691 3.623 3.650 3.343 3.323 3.279 3.358 3.229 3.227 2.733 2.585 2.866 2.855 2.796 2.874 2.819 2.746

179 179 179 180 180 179 108 110 109 110 108 107 92 88 88 88 88 89 88 87

0.50 0.74 1.12 0.00 0.87 1.06 0.12 0.54 0.76 0.00 0.57 0.80 0.36 0.00 0.04 0.23 −0.01 0.00 0.08 0.07

1.32 1.28 1.29 1.34 1.33 1.32 1.39 1.37 1.37 1.39 1.39 1.38 1.26 1.24 1.33 1.32 1.29 1.35 1.35 1.33

4.11 3.46 3.93 4.01 4.00 3.94 4.14 4.07 4.07 4.10 4.13 4.10 3.99 3.85 4.10 4.06 4.03 4.05 4.05 4.06

1.32 1.28 1.20 1.34 1.25 1.20 1.39 1.35 1.29 1.39 1.33 1.28 1.26 1.24 1.33 1.32 1.30 1.33 1.31 1.27

4.11 3.46 1.05 −4.01 −2.43 −0.56 4.14 2.88 1.06 −4.10 −2.61 −0.82 3.99 −3.86 4.10 2.83 1.19 −4.05 −2.62 −0.85

S1: spin at the first iron atom. bFe1−O: the average of the six Fe−O bond lengths around the first iron atom. cOb: bridging oxygen atom. dThe energies of the (5/2, −5/2) states were scaled to zero. eMulliken charge at the first iron atom. fMulliken spin density at the first iron atom. gThe nominal (5/2, 3/2) state is actually a symmetrical linear combination of the (5/2, 3/2) and (3/2, 5/2) states. a

kinetic properties such as the diffusion process. Two cationic monomers must diffuse close to each other before the dimerization process can take place. It has been noted that the electrostatic repulsion between charged iron oxide clusters can affect their oligomerization, since bringing two charged species close results in an energy penalty.64 While it is hard to quantitatively characterize kinetic stability, the structural features of the dimer backbone give qualitative information which is important to stability. The Fe−Fe distances are about 3.7 Å for 1O, 3.3 Å for 2OH, 2.9 Å for 3OH, and 2.6 Å for 2O. A shorter distance is related to larger repulsion and consequently a greater energy penalty. In this sense, the former two are more favored than the latter two. This means that 1O is likely favored by kinetic factors in addition to thermodynamics. Many previous studies assumed 2OH as the primary dimer in solution,21−23,65−67 while other investigations claimed 1O is the leading species.68,69 In addition, 1O has been found in crystals generated from solution,70−72 although molecular stability might differ in the solid state. In spite of evidence from solution studies, no 2OH has been isolated as far as we know. A number of experimental studies performed in solution provide further valuable data.25−28,73,74 Only monomers were detected at very low pH, but dimers form at higher pH. At least two and up to four Fe−Fe distances were reported in these studies.25−28,73,74 The distances found were 2.9, 3.1, 3.5, and 3.9 Å. Evidently, these distances involve a mixture of the various dimers rather than a single species. By comparing with the calculated Fe−Fe distances, only 2O has too short a Fe−Fe distance to fit in any experimental value. This could be due both to the significant electrostatic repulsion hindering the approach of two monomeric species to make 2O and to the fact that 2O is less stable than the other dimers. In our further discussion, we will focus on 1O, 2OH, and 3OH. As with the monomer, each iron center in a dimer may adopt three different spin states (S = 5/2, S = 3/2, and S = 1/2), where S refers to the projection of the spin along the Fe−Fe

negative reaction energy in a vacuum is mostly balanced out by its positive solvation energy. Likewise, a highly positive vacuum reaction energy is mostly balanced out by a negative solvation energy. As a result, all reactions have small reaction energies in solution. The significantly negative vacuum reaction energy of the deprotonation reaction (eq 3) becomes slightly negative in the solution. This is consistent with a previous study employing a different solvation model, the united atom Hartree−Fock/ polarizable continuum model (UAHF/PCM).63 Thus, reactions starting from different monomers have limited energetic differences in solution, and more than one type of monomer may participate in the dimerization process simultaneously. With all other factors being ignored, the interconversion reactions between dimers (eqs 4A−4F) suggest that 1O is slightly more favored than the others (i.e., the free energy is negative for all processes that convert other dimers into 1O). However, these reactions have only small reaction energies in the solution. Therefore, it is unlikely that 1O will dominate over other dimers under all conditions. Thermodynamically, it is more likely that several dimer types as well monomer types will coexist in the solution, with the composition dictated by pH and other factors.

1O → 2OH + H 2O

(4A)

2OH → 3OH + H3O+

(4B)

1O → 3OH + H3O+ + H 2O

(4C)

2O + 2H3O+ → 1O + H 2O

(4D)

2O + 2H3O+ → 2OH + 2H 2O

(4E)

2O + H3O+ → 3OH + 2H 2O

(4F)

Structures. The thermodynamic calculations presented above determine the relative thermodynamic stability of various dimers. However, the dimerization process is also controlled by 21712



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comparison, a Fe−Fe bond length is usually around 2.5 Å.75 With regard to detailed Fe−O bonds, the Fe−Ob bonds are shorter than the apical bonds, which are further shorter than the equatorial bonds. Adding protons to the μ-oxo bridges in 2O generates 2OH. The additional protons lead to tremendous changes in the molecular structure. As previously mentioned, the (5/2, ±5/2) states in 2O have relatively large structural differences. In contrast, the differences between the (5/2, ±5/2) states in 2OH are small. Each iron center in 2O has two equivalent Fe− Ob bonds, which become quite different in 2OH. The additional proton in 2OH points toward the shorter Fe−Ob side, and its H−Ob···Ob angle is 180°. The Fe−Fe distance and Fe−O−Fe angle in 2OH are considerably larger than those in 2O. The differences between 2O and 2OH can be attributed to their different bridges. The μ-oxo bridge is divalent, but the μhydroxo bridge is monovalent. When bridging two iron atoms, the μ-oxo bridge is a stronger bridge and is able to force the two iron atoms closer. The Fe−Fe distance and Fe−O−Fe angle in 2OH are considerably larger than those in 3OH. This is not surprising because 3OH has one more μ-hydroxo bridge. On the other hand, the Fe−O−Fe angle in 3OH is similar to that in 2O, whose Fe−Fe distance is even smaller. This further supports that the μ-oxo bridge is stronger than the μ-hydroxo one. In spite of the many differences, 1O, 2OH, and 3OH share a structural feature which has already been noted for the monomer. A higher spin iron center (not necessarily in a higher spin molecule because an antiferromagnetic molecule may have no net spin) is correlated with a larger Fe−O bond length on average. Consequently, the (5/2, ±5/2) states usually have a larger framework than the corresponding (5/2, ±3/2) and (5/2, ±1/2) states. Thus, it is expected that vibrational relaxation from a Franck−Condon-produced spin state to the energy minimum of this spin state undergoes a structural compression. Spin State Energies. As discussed previously, the ferromagnetic and antiferromagnetic solutions have similar structures. This is also true of their spin state energies. In particular, the largest ferromagnetic−antiferromagnetic energy differences come from the (5/2, ±5/2) states of 1O and 2O, which also exhibit the largest structural differences. For all the dimers, the (5/2, −5/2) state is more stable than the (5/2, 5/ 2) state. With regard to the other states, the antiferromagnetic state may not be as stable as its corresponding ferromagnetic state in some cases. The coupling strength of a dimer is represented by the energy difference between the (5/2, −5/2) and (5/2, 5/2) states. Thus, the coupling strengths of 1O and 2O are stronger than those of 2OH and 3OH. This is probably because the introduction of a proton at the μ-bridge interrupts the coupling. For the two μ-oxo bridged dimers, 1O has a stronger coupling even though 2O has one more μ-oxo bridge. This is probably related to the linear Fe−O−Fe backbone in 1O compared to the angular backbone in 2O (Table 6). Likewise, the more angular 3OH has a smaller coupling strength than 2OH despite the additional μ-hydroxo bridge in 3OH. In 1O and 2OH, the low-lying spin states are well separated. Their energies are ordered (5/2, ±5/2) < (5/2, ±3/2) < (5/2, ±1/2). This is as expected in view of the order in the monomer. In 1O, the (5/2, ±1/2) states lie ∼1.1 eV higher than the (5/2, −5/2) ground state. The energy levels of the antiferromagnetic states in 1O are reminiscent of the energy

axis. The combination of two such spin centers yields many different spin states. For convenience, the spin state of a dimer will be denoted (S1, S2), in which S1 is the spin of one iron and S2 is the spin of the other iron. The low-lying states of a dimer include both the ferromagnetically coupled (5/2, 5/2), (5/2, 3/ 2), (5/2, 1/2), and (3/2, 3/2) states and the antiferromagnetically coupled (5/2, −5/2), (5/2, −3/2), (5/2, −1/2), and (3/ 2, −3/2) states. Among them, the (5/2, 1/2) and (3/2, 3/2) states have the same total spin, and so do the (5/2, −5/2) and (3/2, −3/2) states. As such, we find that geometry optimization of the (3/2, 3/2) state undesirably converges to the (5/2, 1/2) state, and optimization of the (3/2, −3/2) state undesirably converges to the (5/2, −5/2) state. Geometric data for the lowlying spin states are given in Table 6 and Table S4 in the Supporting Information. Note that only distinct converged results are given in the table. It has been suggested that the antiferromagnetic solution can be used to approximate the ferromagnetic one, as they yield comparable results, especially when the adjacent iron centers interact weakly. This seems a reasonable approximation for the iron oxide dimers, whose ferromagnetic and antiferromagnetic solutions yield comparable geometric features. Among these solutions, the (5/2, ±5/2) states of 1O and 2O present relatively large differences between their ferromagnetic and antiferromagnetic structures. The (5/2, −5/2) state of 1O is its ground state, and its calculated structure agrees with the experimental crystal structure. The calculated Fe−O−Fe backbone is linear, and the reported Fe−O−Fe angles in the crystal structures are either 180 or 170°.70−72 The Fe−Ob (bridging oxygen atom) bond with a calculated length of 1.85 Å is the shortest. This is comparable to the experimental value, 1.78 Å.70−72 The longest Fe−O bond is the Fe−Ot (terminal oxygen atom) one at the trans position of the Fe−Ob bond. Its calculated length is 2.29 Å, while the experimental value is 2.12 Å. The axial Fe−O bonds have intermediate lengths, whose ranges are 2.11−2.16 Å (calculated) and 2.04−2.07 Å (experiment). In general, the calculation reproduces the primary features of the crystal structure. The systematic slight overestimation of bond lengths in the calculation may originate from the functional employed. On the basis of the monomer results, incorporation of exact exchange is expected to reduce bond length. It is also possible that the squeezed crystal structure is mainly caused by crystal packing. All the spin states including the (5/2, −5/2) ground state in 1O have Fe−Fe distances of around 3.7 Å. This maximal Fe− Fe distance benefits from the linear Fe−O−Fe backbone, which is probably related to the μ-oxo bridge and the steric hindrance between two aqua iron centers. Besides the (5/2, ±5/2) states, the (5/2, 3/2) state also has structurally equivalent halves. This suggests that the (5/2, 3/2) calculation actually converged to a state with two identical iron atoms. The Mulliken spin of the iron atom in 1O is bigger than that of an iron atom whose spin is 3/2 but smaller than that of an iron atom whose spin is 5/2 (Table 6). Thus, the state is probably a symmetrical linear combination of the (5/2, 3/2) and (3/2, 5/2) states. The geometries of the (5/2, ±5/2) states of 2O have been optimized. These are the two intuitively lowest-lying spin states. Compared with 1O, they have much shorter Fe−Fe distances in spite of comparable Fe−Ob bond lengths. To accommodate two μ-oxo bridges, 2O has to adopt an angular Fe−O−Fe backbone. The significant angularity decreases the Fe−Fe distance significantly. Its Fe−Fe distance is only 2.73 Å for the (5/2, 5/2) state and 2.59 Å for the (5/2, −5/2) state. For 21713



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Table 7. Ferromagnetic and Antiferromagnetic d−d Transition Energies of the Dimers Using the BLYP Functional (eV) ground state 1O (5/2, 5/2) 1O (5/2, 5/2) 1O (5/2, 5/2) 1O (5/2, −5/2) 1O (5/2, −5/2) 1O (5/2, −5/2) 2OH (5/2, 5/2) 2OH (5/2, 5/2) 2OH (5/2, 5/2) 2OH (5/2, −5/2) 2OH (5/2, −5/2) 2OH (5/2, −5/2) 3OH (5/2, 5/2) 3OH (5/2, 5/2) 3OH (5/2, 5/2) 3OH (5/2, −5/2) 3OH (5/2, −5/2) 3OH (5/2, −5/2)

S1

S2 f

Stotala

f

5/2 5/2 3/2 5/2 5/2 3/2 5/2 5/2 3/2 5/2 5/2 3/2i 5/2 5/2 3/2 5/2 5/2 3/2i

3/2 1/2 3/2 −3/2 −1/2 −3/2 3/2 1/2 3/2 −3/2 −1/2 −3/2i 3/2 1/2 3/2 −3/2 −1/2 −3/2i

4 3 3 1 2 0 4 3 3 1 2 0 4 3 3 1 2 0

excitation g

single single doubleh single single double single single double single single double single single double single single double

Ed−db

Erelaxc

Eadiad

q1e

spin1e

q2

spin2

0.41 1.15 1.46 1.19 1.53 1.91 0.68 1.15 1.79 0.89 1.30

0.17 0.53

0.24 0.62

0.32 0.47

0.87 1.06

0.26 0.51

0.42 0.64

0.32 0.51

0.57 0.80

1.26 1.29 1.23 1.33 1.32 1.26 1.37 1.38 1.27 1.39 1.38

3.47 3.88 2.97 3.98 3.93 2.60 3.98 4.07 2.78 4.11 4.09

1.26 1.18 1.22 1.23 1.21 1.24 1.30 1.24 1.27 1.29 1.24

3.47 1.10 2.95 −2.35 −0.53 −3.12 3.05 1.10 2.78 −2.66 −0.82

0.74 1.28

0.18 −0.05

0.84 1.15

0.08 0.07

1.32 1.33 1.24 1.34 1.34

4.07 4.09 2.86 4.06 4.08

1.25 1.18 1.24 1.24 1.18

2.87 1.14 2.85 −2.74 −1.02

0.92 1.23 1.80 0.91 1.22

a Stotal = S1 + S2. bEd−d: the d−d transition energy. Note that the d−d transition occurs between states with the same magnetic coupling. cErelax: the relaxation energy from the Franck−Condon d−d state to the energy minimum of this state. dEadia: the adiabatic energy difference between a spin state energy minimum and the lowest-lying spin state with the same magnetic coupling. eMulliken charge and spin density of the Franck−Condon state. fThe nominal Franck−Condon (5/2, 3/2) state is actually a symmetrical linear combination of the (5/2, 3/2) and (3/2, 5/2) states. gsingle: single excitation involving only one iron center. hdouble: double excitation involving the simultaneous excitation of two iron centers. iThese calculations failed to converge to the desired states.

Table 8. TDDFT Calculation (BHandHLYP) of LMCT Excitation Energies of the Dimersa (eV) 1O (5/2, 5/2) 1O (5/2, −5/2) 2OH (5/2, 5/2) 2OH (5/2, −5/2) 3OH (5/2, −5/2) 3OH (5/2, −5/2) a

E1

f1

E2

f2

E3

f3

E4

f4

2.41 3.38 3.84 4.13 4.30 4.35

0.029 0.063 0.032 0.043 0.022 0.020

2.62 3.84 4.11 4.96 4.87 4.74

0.010 0.171 0.016 0.018 0.011 0.010

3.17 6.06 4.69 5.45 5.39 5.36

0.189 0.040 0.029 0.120 0.013 0.034

5.72 6.21 5.09 5.51 5.54 5.89

0.029 0.089 0.022 0.045 0.019 0.031

Only transitions whose oscillation strengths are larger than 0.01 au are given.

levels of the monomer [Fe(H2O)6]3+ (0.00/0.87/1.06 eV in the former versus 0.00/0.88/0.95 eV in the latter). This may be related to the fact that the two monomeric units in 1O are bound by a single bridge and each unit retains some properties of an isolated monomer. For 2OH, the (5/2, ±1/2) states are ∼0.8 eV higher than the (5/2, −5/2) ground state. This is moderately smaller than that in 1O but still significant. The energy levels of the antiferromagnetic states in 2OH are 0.00/ 0.57/0.80 eV. They are moderately more densely spaced than in 1O and [Fe(H2O)6]3+. In significant contrast with these, the low-lying states in 3OH are very closely spaced. Except for the (5/2, 3/2) state, which lies ∼0.2 eV above, the other states cluster in a range of less than 0.1 eV. In particular, the (5/2, 1/ 2) state is almost isoenergetic with the (5/2, −5/2) state, which is the ground state for 1O and 2OH. In view of the energy levels, the singly bridged 1O and the doubly bridged 2OH are substantially different from the triply bridged 3OH. This may be related to the three-dimensional constraint in 3OH. Single d−d Excitations. In a dimer, d−d excitation is less likely to occur from a ferromagnetic spin state to an antiferromagnetic spin state and vise versa, since this requires the spin flip of more d electrons. Therefore, both the (5/2, 5/ 2) and (5/2, −5/2) states can be viewed as the ground states. In other words, the ferromagnetically coupled Franck−Condon states have the same geometry as the (5/2, 5/2) state, while

antiferromagnetically coupled Franck−Condon states have the same geometry as the (5/2, −5/2) state. The transitions will therefore be called ferromagnetic and antiferromagnetic transitions, respectively. For convenience, a d−d excitation involving only one iron center will be called a “single” excitation and the single d−d excitation from the (5/2, 5/2) state to the (5/2, 1/2) state will be called the single sextet−doublet excitation. The single d−d transitions present features that are similar to the spin state energetics (Table 7). Again, it is worth noting that the nominal Franck−Condon (5/2, 3/2) state of 1O is actually a symmetrical linear combination of the (5/2, 3/2) and (3/2, 5/2) states because it has two indistinguishable iron atoms. For 1O and 2OH, the antiferromagnetic transition energy is slightly greater than the corresponding ferromagnetic one. On the other hand, such two energies are almost identical in 3OH. As with the spin state energies, the antiferromagnetic transition energies in 1O highly resemble transition energies in the monomer. In spite of somewhat comparable transition energies, 1O and 2OH have very different relaxation energies from 3OH. In 1O and 2OH, the relaxation energy after a single sextet−quartet transition is about 0.3 eV and that after a single sextet−doublet transition is about 0.5 eV. Both are still close to those in the monomer. In contrast, the corresponding values in 3OH are considerably larger, around 0.8 and 1.1 eV, 21714



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explanation also rationalizes the systematically higher d−d transition energies from B3LYP. In spite of the moderately higher absolute transition energies from B3LYP, the relative magnitude remains similar.

respectively. This may be because the more constrained structure in the triply bridged dimer undergoes a higher energy change due to geometry distortion. LMCT Excitations. As justified in the Supporting Information, the geometries of the dimers were first optimized using the BLYP functional and then the spin-allowed LMCT transition energies were calculated using the BHandHLYP hybrid functional. The results are given in Table 8. Because of the existence of several types of oxygen atoms, the LMCT transitions of the dimers are more complicated than those in the monomer. Interestingly, the ferromagnetic and antiferromagnetic states have somewhat distinguishable transitions. As discussed in the manuscript, the ground states for 1O, 2OH, and 3OH are all the antiferromagnetic (5/2, −5/2) states. Therefore, our analysis can be limited to the (5/2, −5/2) states. The dominant excitation for 1O is at 3.84 eV, which is in excellent agreement with the experimental dimer peak at 3.70 eV.21−23 Also, the diffuse reflectance UV/vis spectroscopy of 1O gives a very broad peak ranging from 250 to 400 nm (3−5 eV).70,71 2OH and 3OH also have peaks close to these experimental values, with 2OH at 4.13 eV and 3OH at 4.35 eV. However, note that they are not the highest oscillator strength transitions for 2OH and 3OH. Other stronger transitions occur at higher energy (above 5 eV), so it appears that the match between theory and experiment is best for the 1O structure. However, the possible contribution from 2OH and 3OH cannot be excluded. Double d−d Excitations. Besides the single d−d excitation and LMCT transition, the simultaneous excitation of two iron centers is possible, wherein the magnetic coupling of the iron atoms is released, and there is no spin selection rule constraint on each iron center. For convenience, such transitions are called double excitations to be distinguished from the single excitations. The double sextet−quartet excitation energies were calculated using the BLYP functional and are given in Table 7. Note that such a calculation sometimes undesirably converges to a more stable state with the same total spin. This is especially the case for the (3/2, −3/2) singlet state which is destabilized by several eV compared to the (5/2, −5/2) singlet state. Due to this, the two double antiferromagnetic excitation energies in 2OH and 3OH are missing. In all three dimers, the Franck− Condon (3/2, 3/2) state lies above the (5/2, 1/2) state which has the same total spin. This is not surprising because the single sextet−doublet transition energy is only slightly higher than the single sextet−quartet transition energy. For the same reason, the (3/2, −3/2) singlet state of 1O lies above its (5/2, −1/2) state which also involves spin flip of two electrons from the (5/ 2, −5/2) state. The double excitation energies we have calculated are 1.6−2.6 times their corresponding single excitation energies. Note that the double sextet−quartet excitation of 1O does not have a corresponding single excitation because the nominal (5/2, 3/2) state has two identical iron centers. Influence of Exact Exchange. To check the influence of the amount of exact exchange, the B3LYP hybrid functional was chosen to study 2OH. The results given in Tables S5 and S6 in the Supporting Information can be compared with those from the BLYP functional. The Fe−O bond length produced by the B3LYP method is ∼0.04 Å smaller than the BLYP result. This is consistent with the trend seen in the monomer. In addition, the B3LYP method yields a slightly smaller ferromagnetic− antiferromagnetic gap. This is explainable because exact exchange tends to stabilize the high spin state. The same

5. CONCLUSIONS The experimentally well-known molecule [Fe(H2O)6]3+ has been used to verify; computational methods are then used to study larger iron oxide clusters. All methods are able to generate reliable structural features. Most DFT methods can give reasonable d−d transition energies, which however are significantly overestimated by HF and MP2 methods. The calculated result of the LMCT transition energy is highly sensitive to the method employed, and a hybrid method is necessary even for a qualitative description. Dimerization reactions starting from [Fe(H2O)6]3+ are thermodynamically highly favorable in a vacuum. These reactions have very small reaction energies in solution, with the formation of the singly μoxo bridged dimer being slightly favored over other dimer structures. The electrostatic repulsion between two charged monomers hinders their close contact. The singly μ-oxo bridged dimer suffers less from this because of its maximal Fe−Fe distance. This is supported by the existence of stable crystal structures of this dimer. Further comparison between the calculated structures and experimental results in solution indicates that mixtures of dimer species will be common. The ferromagnetic and antiferromagnetic states of the dimers have comparable structures and energies. Vibrational relaxation from a Franck−Condon excited spin state to the energy minimum of this spin state leads to a structural compression. While the singly μ-oxo bridged and doubly μ-hydroxo bridged dimers have well separated spin states, the spin states in the triply μhydroxo bridged dimer are closely spaced. The triply bridged dimer also differs from the other two because it has a much higher relaxation energy from the Franck−Condon excited d−d state. The double d−d excition in a dimer has a higher excitation energy than the single d−d excitations, and also the monomer d−d excitations, but is below the LMCT energy. This means that doubly excited states can be populated during the non-radiative relaxation of iron clusters following initial photoexcitation of the LMCT state.

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ASSOCIATED CONTENT

S Supporting Information *

Table S1 (electron configuration of monomer in different spin states), Tables S2 and S3 (assessment of functionals for calculation of LMCT transition energies), Table S4 (dimer structure properties), Tables S5 and S6 (B3LYP results for the 2OH dimer). Also provided is a detailed discussion of the choice of the functional for LMCT transitions. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: 847-4915657. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Argonne-Northwestern Solar Energy Research (ANSER) Center, which is an Energy Frontier 21715



Article

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Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001059.

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Computational Modeling of Octahedral Iron Oxide Clusters

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