Multireference Ab Initio Study of Ligand Field d–d Transitions in

Article pubs.acs.org/JPCC

Multireference Ab Initio Study of Ligand Field d−d Transitions in Octahedral Transition-Metal Oxide Clusters Yang Yang, Mark A. Ratner, and George C. Schatz* Argonne−Northwestern Solar Energy Research (ANSER) Center, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States S Supporting Information *

ABSTRACT: We have used multiconfigurational (MC) and multireference (MR) methods (CASSCF, CASPT2, and MRCI) to study d−d transitions and other optical excitations for octahedral [M(H2O)6]n+ clusters (M = Ti, V, Mn, Cr, Fe, Co, Ni, Cu) as models of hematite and other transition-metal oxides of interest in solar fuels. For [Fe(H2O)6]3+, all calculations substantially overestimate the d−d transition energies (∼3.0 versus ∼1.5 eV) compared to what has been experimentally assigned. This problem occurs even though theory accurately describes (1) the lowest d−d transition energy in the atomic ion Fe3+ (∼4.4 eV), (2) the t2g−eg splitting (∼1.4 eV) in [Fe(H2O)6]3+, and (3) the ligand-to-metal charge transfer (LMCT) energy in [Fe(H2O)6]3+. Indeed, the results for Fe3+ and the t2g−eg splitting suggest that the lowest d−d excitation energy in the hexa-aqua complex should be ∼3 eV (or slightly below because of Jahn−Teller stabilization), as we find. Possible origins for the d−d discrepancy are examined, including Fe2+ and low-spin Fe3+ impurities. For the [M(H2O)6]n+ clusters not involving Fe(III), our MR calculations show reasonable correlation (mostly within 0.5 eV) with experiments for the d−d transitions, including consistent trends for the intensities of spin-allowed and spin-forbidden transitions. Our calculations also greatly complement experimental data because (1) experimental results for some species are insufficient or even scarce, (2) some of the experimental peaks were not observed directly but were inferred, and (3) the nature or existence of some shoulder peaks and weak peaks is uncertain. Our MR calculations have also been used to study convergence of the results with choice of active space, including the importance of the “double shell” effect in which there are 10 active d orbitals per transition-metal atom rather than 5. The results show that the larger active space does not significantly change the excitation energy, although it lowers the absolute energies for complexes with high 3d occupations. This indicates that reasonable accuracy can be achieved using MR methods in studies of transition-metal oxide clusters using minimal active spaces. This study establishes fundamental principles for the further modeling of larger cluster models of pure and doped hematite and other metal oxides.

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INTRODUCTION Transition-metal oxides are receiving increased attention for their widespread application in solar energy.1−3 For example, hematite (α-Fe2O3) is a stable, abundant, and cheap semiconductor, and it can absorb solar radiation and generate electron/hole pairs that can be used to split water.4 Unfortunately, its efficiency to convert solar energy to fuels is not satisfactory. One remedy for the low photon-to-current efficiency is to introduce various dopants such as Co, Cu, Mn, and Ni.4−7 A doping cation either resides in an interstitial site or substitutes for iron in an octahedral FeO6 unit. In the latter case, a new octahedral MO6 (M represents a transition metal) fragment is formed. The characterization of optical excitation and subsequent photophysics in transition-metal oxides is essential to understanding their use in the conversion of sunlight to fuels. Two types of fundamental excitations are generally considered, ligand field d−d and ligand-to-metal charge transfer (LMCT) transitions. The latter are both Laporte- and spin-allowed and are thus much more intense than the former. Note that some Laporte-forbidden d−d transitions are made weakly allowed © XXXX American Chemical Society

through vibronic coupling. Such coupling increases both the peak intensity and the excitation energy because the final state is both electronically and vibrationally excited. Ligand field d−d transitions are also spin-forbidden for Fe3+ if only one Fe is involved because all five d orbitals are singly occupied. However, this spin selection rule can be relaxed by spin− orbit interaction or through magnetic coupling between adjacent Fe(III) centers.8,9 An additional type of excitation is the double exciton excitation wherein two irons are excited simultaneously. This paired excitation has been observed in dimeric iron molecules10 and in iron oxide and oxide− hydroxide minerals.8,9 Similar excitations have been discussed in other complexes with more than one metal.11 These paired excitations are more intense and have higher energies compared to single d−d excitations, and in general, their strength and excitation energies are intermediate between single d−d Special Issue: John C. Hemminger Festschrift Received: May 28, 2014 Revised: July 3, 2014

A

dx.doi.org/10.1021/jp5052672 | J. Phys. Chem. C XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry C

Article

experimental data for monomeric [M(H2O)6]n+ clusters suffers from uncertainties as the clusters can readily deprotonate to generate highly absorbing products, and they can dissociate to form penta- and tetracoordinate complexes. For example, a small amount of tetracoordinate [Co(H2O)4]2+ has been proposed to coexist in equilibrium with hexacoordinate [Co(H2O)6]2+.24 The origin of shoulders and weak peaks in the optical spectra are thus uncertain. Our calculations will therefore resolve some of the ambiguities in the interpretation of optical spectra.

excitations and LMCT excitations. To understand the wavelength-dependent efficiency in hematite, it is urgent to determine the nature of states generated at different wavelengths. This can be facilitated by computational studies that provide an atomic-level understanding of the optical absorption. An infinite system such as hematite can be described by a periodic Bloch wave function. However, a computational method that is able to account for the high-level correlation effects in Fe(III) is usually prohibitively expensive for a periodic calculation. Fortunately, optical excitations are localized properties and can be simulated with finite cluster models that resemble fragments of the infinite system.9,12,13 The smallest cluster resembling a MO6 unit is the corresponding hexa-aqua complex [M(H2O)6]n+. This cluster represents the fundamental unit in pure and doped hematite and other related transition-metal oxides, and there are important similarities between the optical properties of the cluster and bulk. In particular, [Fe(H2O)6]3+ has absorption peaks at 1.6, 2.3, 3.0, and 5.2 eV,14,15 while hematite has peaks at 1.5, 2.4, 3.2, 3.9, 4.8, and 5.8 eV.16 Some of the peaks in hematite are not found in [Fe(H2O)6]3+ because they involve excitation of more than one iron atom, but otherwise, the results are similar. Of particular interest is the lowest d−d transition, which is usually assigned to the 1.5 eV peak in hematite and 1.6 eV in the [Fe(H2O)6]3+ cluster. Examining this transition for iron and other transition-metal oxide clusters provides a useful calibration of higher-level theories as the [M(H2O)6]n+ complex is relatively simple, and there are lots of experimental results available for direct comparison between experiment and theory.17 Also, the metal−oxygen bonds are highly ionic; therefore, mixing between metal 3d orbitals and ligand 2p orbitals is small, and differences between aqua, hydroxy, and oxo ligands are not important.18 We further note that the d−d and LMCT bands are well separated in energy for the [M(H2O)6]n+ complexes so that the Laporte-forbidden d−d transitions are not masked by intense LMCT absorption in the optical spectra.18 While six water ligands surrounding the metal cation form the first coordination shell, the second hydration shell is coordinated to the first shell through hydrogen bonds and has been explored by both experiments and theories in past work.19−22 Ligand field d−d transitions are metal-centered and involve only interactions between the metal and the first shell. On the other hand, LMCT transitions involve ligands in the first shell and have a higher possibility to be influenced by the second-shell structure. Accounting for this influence with explicit solvation models is very difficult because of a large number of possible second-shell structures.19−22 The use of an implicit solvation model such as the conductor-like screening model (COSMO) on [Fe(H2O)6]3+, however, does not lead to any significant change in either the d−d or LMCT transition energies.23 In this paper, we present multiconfigurational and multireference calculations, including CASSCF, CASPT2, and MRCI, for the [M(H2O)6]n+ clusters with M = Ti, V, Mn, Cr, Fe, Co, Ni, and Cu. Our purpose is not only to understand d−d transitions in monomeric clusters but also to build the basis for further studies of larger clusters for which experimental data is unavailable or unreliable. The larger clusters are better models of infinite materials, allowing us to investigate double d−d transitions, and they are of interest in their own right as the technology for making size-selected clusters for solar fuels evolves. We also note that even the

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COMPUTATIONAL DETAILS It is a significant challenge to accurately model the electronic absorption spectrum of any transition-metal oxide system.25 While some DFT methods give quantitatively accurate results for some transitions, there can also be important errors that lead to qualitative inaccuracy.23 For example, the calculated lowest d−d transition energy of [Fe(H2O)6]3+ is 1.16 eV for BLYP but 2.14 eV for BHandHLYP.23 Also, the calculated lowest LMCT energy is 2.40 eV for BLYP, 3.27 eV for B3LYP, and 5.48 for BHandHLYP.23 At the very least, a cautious assessment of many different types of density functionals is necessary before they can be used with confidence in further studies. Benchmarks used in the assessment of different functionals can be from experiment or from more advanced computations in the absence of experimental data. Indeed, advanced computational methods are an optimal choice given the typical experimental uncertainties. For transition-metal systems with strong electron correlation effects, an adequate treatment of the electronic structure requires multiconfigurational or multireference methods.26 The complete active space self-consistent field (CASSCF) is a multiconfigurational method to account for the near-degeneracy correlation or static correlation.27,28 In CASSCF, the occupied orbital space is divided into a set of active and inactive orbitals. The electrons and orbitals of interest are included in the active space so that CASSCF(m,n) denotes that m electrons in n active orbitals are included in defining configurations. In state-averaged CASSCF, one set of averaged orbitals is used to calculate states for a specific spin symmetry. To capture dynamic correlation, multireference second-order perturbation theory (MRPT2) such as complete active space second-order perturbation theory (CASPT2) or multireference configuration interaction (MRCI) is necessary. In multistate CASPT2 and MRCI, states of the same spatial and spin symmetries are generated simultaneously. In the present studies, CASSCF, CASPT2, and MRCI with single and double excitations (MRCISD) ab initio calculations were performed using Molpro.29,30 The Dunning correlationconsistent basis set cc-pVDZ (abbreviated as VDZ) was used unless otherwise mentioned.31 The CASSCF wave functions were used as reference states for CASPT2 and MRCISD. To avoid intruder state problems in CASPT2, a level shift of 0.1 hartree was applied.32 Note that some [M(H2O)6]n+ systems (e.g., Fe(II),33,34 V(III),35 and Ni(II)35) have been explored using multireference ab initio calculations in past work. However, systematic results for the d−d transitions in the complete set of period four [M(H2O)6]n+ systems have not been reported. Also, detailed comparison between multireference calculations and experiments is rare. Although different density functionals may give very different excitation energies, they provide similar structural parameters for the ground states.23 Indeed, the structures of [Fe(H2O)6]2+ B



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Table 1. Calculated Metal−Ligand (M−L) Bond Lengths (Å) 3+

[Ti(H2O)6] [V(H2O)6]3+ [V(H2O)6]2+ [Cr(H2O)6]3+ [Mn(H2O)6]3+ [Cr(H2O)6]2+ [Mn(H2O)6]2+ [Fe(H2O)6]3+ [Fe(H2O)6]2+ [Co(H2O)6]3+ [Co(H2O)6]2+ [Ni(H2O)6]2+ [Cu(H2O)6]2+ [FeF6]3− a

1

d d2 d3 d3 d4 d4 d5 d5 d6 d6 d7 d8 d9 d5

M−L1a

M−L2a

M−L3

M−L4

M−L5

M−L6

avg

SD

2.097 2.055 2.168 2.017 1.977 2.105 2.211 2.050 2.158 1.931 2.114 2.077 2.020 2.029

2.099 2.055 2.168 2.017 1.977 2.105 2.211 2.050 2.158 1.935 2.115 2.078 2.020 2.029

2.099 2.055 2.168 2.017 1.982 2.107 2.211 2.050 2.181 1.933 2.120 2.083 2.022 2.029

2.099 2.055 2.168 2.017 1.982 2.107 2.211 2.050 2.182 1.936 2.124 2.083 2.022 2.029

2.101 2.055 2.168 2.017 2.172 2.376 2.211 2.050 2.202 1.931 2.127 2.084 2.303 2.029

2.101 2.055 2.168 2.017 2.172 2.376 2.211 2.050 2.202 1.933 2.128 2.086 2.303 2.029

2.099 2.055 2.168 2.017 2.044 2.196 2.211 2.050 2.181 1.933 2.121 2.082 2.115 2.029

0.001 0.000 0.000 0.000 0.091 0.127 0.000 0.000 0.018 0.002 0.005 0.003 0.133 0.000

L1−M−L2 are along an axis, as are L3−M−L4 and L5−M−L6.

symmetry facilitates the determination of the electron configuration of a state based on the state symbol. Within D2h symmetry, dx2−y2 and dz2 orbitals are labeled by the Ag representation, while dxy, dxz, dyz orbitals belong to the B1g, B2g, and B3g representations, respectively. Note that these three degenerate B1g, B2g, and B3g orbitals under D2h symmetry correspond to T1g or T2g under Oh symmetry. All state and orbital symbols given in this paper are under D2h symmetry unless otherwise noted. In general, multireference methods such as CASPT2 give excitation energies with errors of a few tenths of an eV33 and usually 0.5 eV at the most.37−39 However, an accuracy of 0.2− 0.3 eV does require a saturated active space and large basis set.40 In a previous study of [Fe(H2O)6]2+ using CASSCF and CASPT2 methods, an empirical correction of ∼0.5 eV was suggested for practical choices of basis set and active space.33 This empirical correction was obtained based on larger calculations for a smaller analogous system (such as an atom or ion). We will perform a similar analysis here, but most of our results are based on the smallest basis set and active space that is adequate for later studies that we plan with larger clusters (at least two metal atoms). The minimal active space required to model d−d transitions is five d orbitals and the d electrons occupying these orbitals. A double shell, that is, 10 d orbitals in the active space, has been suggested;41−45 therefore, we will also consider this. In addition, metal−ligand covalent interactions may mix metal 3d and ligand 2p orbitals; therefore, ligand orbitals with significant amounts of metal 3d contributions are also considered in the active space.41,43−45 Unfortunately, 16 active orbitals is at the limit of what is computationally practical;46−48 therefore, we have only included ligand orbitals in the active space when just 5 d-type orbitals are included. Our double shell effect and covalency effect studies will also help us to develop empirical corrections to the minimal active space calculations. Note that all of the transitions in this work are orbitally forbidden (corresponding to g → g transitions). Inclusion of vibronic effects is therefore necessary for these transitions to gain intensity, and this is expected to lead to an increase in the excitation energies compared to what we present. While many of the results presented later lead to predicted lowest excitation energies that are a few tenths of an eV below experiment (such that the inclusion of vibronic effects might improve the agreement), this sort of difference is within the errors of CASPT2 and other

were found to be very similar when calculated using both multireference ab initio and DFT methods,33 and the results were consistent with experimental data. Another study of several Fe(II) complexes including [Fe(H2O)6]2+ reached the same conclusion, and it was further suggested that multireference treatments may be confined to energetic calculations on DFT structures.34 In the present work, the structures of all [M(H2O)6]n+ complexes except [Fe(H2O)6]2+were optimized using the B3LYP hybrid functional and TZP basis set. For [Fe(H2O)6]2+, the BLYP functional and TZP basis set were employed because optimization using the B3LYP method failed to converge. Previous studies have shown that different functionals (including B3LYP and BLYP) give similar structures for [Fe(H2O)6]2+;33,34 therefore, this is probably not an important issue. The nature of the optimized stationary points was checked by a frequency analysis at the same computational level to ensure that the calculation converged to an energy minimum. These DFT calculations were all performed using the ADF code.36 The calculated DFT geometries were used in the ab initio calculations with Molpro. These calculations use Abelian point groups, and the highest molecular symmetry available is D2h. Therefore, D2h symmetry was imposed during the DFT geometry optimization except when this led to structures with imaginary frequencies. To get rid of the imaginary frequency, the calculation was carried out with a lower symmetry. Fortunately deviation from D2h symmetry is mainly due to distortion or rotation of hydrogens in the ligands. As indicated in Table 1, the deviation from D2h is insignificant for the MO6 cores of all [M(H2O)6]n+ complexes. For structures slightly deviating from D2h symmetry, some small modifications were made to recover this symmetry before the geometries were imported into Molpro. Here, the lengths of the two opposite metal−oxygen (M−O) bonds were made equivalent using their average value. It can be seen that the length difference between two colinear M−O bonds is negligible, no more than 0.004 Å. Also, the tilted O−H bonds were rotated so that they were coplanar with the M−O axes. All O−H bond lengths were retained during the rotation. There are many benefits to take advantage of D2h symmetry. First, the computational cost is reduced substantially (by several orders of magnitude in some cases). Second, different irreducible representations correspond to different orbitals, and it is easier to identify and pick out orbitals of interest. Moreover, C



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Table 2. Transition Energies for Fe(H2O)63+ Calculated Using CASSCF(5,5) and Various Basis Sets (eV)

methods that we use; therefore, a more detailed analysis that includes vibronic effects is not warranted.

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RESULTS AND DISCUSSION Structures. Table 1 gives the calculated metal−ligand bond lengths of all hexacoordinate ground-state complexes. Their absorption spectra have been summarized in a book by Jorgensen,17 and we note that the 3d shells range from d1 to d9, with the d0 and d10 systems excluded because they have no d−d transition. The Jorgensen results refer to experiments carried out in solution and in other complex environments, including solids, which are not described by our calculations. Correcting the calculations for environmental effects is not straightforward for the high-level methods that we have used, but fortunately, these effects are not likely to be large for d−d transitions; therefore, we will ignore this issue. The experiments show that all complexes except [Co(H2O)6]3+ have high-spin ground states.24 For [Co(H2O)6]3+, the six d electrons are paired in the three t2g orbitals in the ground state. For comparison, [Fe(H2O)6]2+ has four unpaired electrons and one electron pair in its ground state. In principle, no Jahn−Teller effect is expected for the ground states of the d3, d5, d8, and low-spin d6 complexes. The symmetry of the other complexes can be lowered by Jahn−Teller distortions; indeed, the standard deviation (SD) of the six metal−ligand bond lengths is a direct measure of this geometric distortion. Table 1 shows that the d3, d5, d8, and low-spin d6 systems have negligible SDs. Strictly speaking, distortion should occur for the d1, d2, d7, and high-spin d6 systems; however, their SDs are almost negligible. This is not surprising because the Jahn− Teller effect induced by t2g orbitals is insignificant. The pronounced Jahn−Teller effect for d4 and d9 complexes, due to the odd number of electrons in the eg orbitals, is evidenced by their large SDs. In these complexes, two of the metal−oxygen bonds are considerably longer than the other four, and the differences are as large as 0.2−0.3 Å. The Jahn−Teller effect not only leads to structural distortion but also causes splitting of electronic states and affects the absorption spectra. Further details will be discussed later. The calculated bond lengths in our monomeric clusters are consistent with previous results concerning pure and doped hematite. For example, our calculations indicate that the metal−oxygen bond lengths in [M(H2O)6]n+ are in the order Fe(III)−O ≈ Ti(III)−O < Fe(II)−O. This is also the order found in doped hematite.5 In an attempt to improve the water oxidation efficiency, both Ti doping and oxygen vacancy techniques have been employed.49 Extended X-ray absorption fine structure (EXAFS) results suggest an increased Fe−O bond length as a result of the increase in oxygen vacancies, which improves photoelectrochemical cell (PEC) performance at an optimal concentration.49 The authors of that work proposed that the increased bond length originates from the formation of a lower oxidation state of Fe such as Fe(II). The authors further proposed that too many oxygen vacancies may hinder further improvement in performance because this can result in a strong distortion of the crystal structure. These proposals are supported by our calculations because the increase in Fe−O bond length is considerable in going from [Fe(H2O)6]3+ to [Fe(H2O)6]2+. [Fe(H2O)6]3+ (d5). For the purpose of testing the basis set effect, the d−d transition energies for [Fe(H2O)6]3+ were calculated using CASSCF(5,5) and various basis sets (Figure S1 in the Supporting Information). As indicated in Table 2, the augmented triple-ζ basis set with added diffuse functions (aug-

state

VDZ

AVDZ

VTZ

AVTZ

Ag B1g/B2g/B3g 4 B1g/B2g/B3g 4 Ag 4 Ag 4 Ag

0.00 3.39 4.03 4.26 4.29 4.29

0.00 3.41 4.05 4.28 4.31 4.31

0.00 3.43 4.07 4.31 4.33 4.33

0.00 3.44 4.07 4.32 4.34 4.34

6 4

cc-pVTZ, abbreviated as AVTZ) gives almost the same result as the double-ζ basis set VDZ. Previous studies on Fe(II) and Fe(III) (hydr)oxides also suggested that the double-ζ basis set is as good as the triple-ζ basis set.50 While more extensive basis sets may improve the quality of the results, they are not feasible for the larger clusters of potential interest for future work. Therefore, the VDZ basis set will be used throughout the rest of the study. For [Fe(H2O)6]3+ and other complexes without or with insignificant Jahn−Teller effects, the B1g, B2g, and B3g states are exactly or almost degenerate. In this case, a CASPT2 or MRCI calculation on the B1g state is good enough to represent the B1g, B2g, and B3g states (T1g or T2g states within Oh symmetry). The same strategy has been applied to the two exactly or almost degenerate Ag states corresponding to Eg states within the Oh point group. In Tables 2−4, the B1g, B2g, and B3g states correspond to eg → t2g transitions in which an electron transitions from an eg orbital to a t2g orbital to form an electron pair. The Ag states correspond to the spin−flip of the t2g electrons. It can be seen from Table 3 that the CASSCF(5,5) calculation significantly Table 3. Multireference d−d Transition Energies for Fe(H2O)63+ (eV) state 6

Ag B1g 4 B1g 4 Ag 4 Ag 4 Ag 4

CASPT2(5,5)

CASPT2(5,10)

MRCI(5,5)

exp

0.00 2.89 n/a 3.78 3.82 3.82

0.00 2.85 n/a 3.80 3.84 3.84

0.00 3.02 3.74 4.08 4.13 4.13

0.00 1.56 2.29 3.01 3.05

overestimates the transition energies relative to the assigned experimental results (i.e., ∼3.0 versus ∼1.5 eV). CASPT2(5,5) and MRCI(5,5) lower the results by several tenths of an eV, but the deviations from the experimental results are still at least 1.3 eV. This disagreement is surprisingly large for the methods used even considering that the calculations have ignored solvation effects. We also note that the deviation cannot be explained by spin−orbit coupling. The effect of spin−orbit coupling has been investigated using the Breit−Pauli operator,51 and it was found that splittings calculated using both CASSCF and MRCI wave functions are only several tens of meVs. This is not surprising because small splittings have been reported for many first-row transition-metal complexes.35,52 To explore possible reasons for the overestimation, more molecular orbitals have been added to expand the (5,5) active space (Tables 3 and 4). Five unoccupied d orbitals were added to account for the double shell effect, but neither CASSCF(5,10) nor CASPT2(5,10) leads to satisfactory improvement. D



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Table 4. Transition Energies for Fe(H2O)63+ Calculated Using CASSCF(m,n) with Various Active Spaces (eV) state

(5,5)a

(5,10)b

(13,9)c

(9,7)d

(5,8)e

Ag B1g/B2g/B3g 4 B1g/B2g/B3g 4 Ag 4 Ag 4 Ag

0.00 3.39 4.03 4.26 4.29 4.29

0.00 3.19 3.85 4.12 4.16 4.16

0.00 3.19 4.02 4.32 4.35 4.35

3.25 3.92 4.21 4.25 4.25

3.41 3.99 4.28 4.30 4.36

6 4

due to defects or impurities because it is at least 1 order of magnitude weaker than the other peaks. It is important to note that the 1.5 eV peak looks suspicious even if only experimental data is analyzed. For Fe3+ without any ligands, four sextet → quartet peaks (3.97, 4.35, 4.77, and 6.46 eV) have been found experimentally.59,60 The first three peaks should be related to the formation of an electron pair from two electrons with the same spin, and the last one corresponds to the spin−flip of one electron. The average of the first three, 4.36 eV, is roughly the energy penalty to form an electron pair from the high-spin d5 configuration. The t2g−eg splitting in hematite has been measured to be 1.4 eV.61 This is presumably close to the splitting in [Fe(H2O)6]3+. For the lowest-energy d−d transition in both [Fe(H2O)6]3+ and hematite, one electron drops from an eg orbital to a t2g orbital to form an electron pair. Thus, this transition energy should approximately equal the electron pairing penalty minus the t2g−eg splitting (4.4−1.4 = 3.0 eV). Interestingly, this value is in agreement with our calculation (the average of the lowest two transition energies), but it is much higher than 1.5 eV or the average of 1.5 and 2.3 eV (both may correspond to the eg → t2g transitions and are split by Jahn−Teller interactions involving the eg orbitals). For a further evaluation of our computational results, we calculated the two parameters discussed in the preceding paragraph, the electron pairing penalty characterized by the lowest d−d transition energy in the atomic ion Fe3+ (Table 5)

The smallest active space has five Fe 3d electrons in five 3d orbitals. Five virtual d orbitals were added to the smallest (5,5) active space. c The 3s and 3p orbitals were added to the (5,5) active space. dTwo bonding orbitals were included to account for possible covalent M−L interactions. eThe 4p orbitals were added. a b

Note that multistate CASPT2 calculations within Molpro are only possible when the sum of active orbitals and closed orbitals is no more than 32. Therefore, states in CASPT2(5,10) calculations were solved individually. To study the effect of valence orbitals on the metal atom, 3s and 3p orbitals were added to the (5,5) active space to make a (13,9) active space (Figure S2 in the Supporting Information). The CASSCF(13,9) result nonetheless suffers from the same overestimation compared to experiment. Adding two bonding orbitals involving the ligands does not give much improvement either. This is not surprising as the metal−ligand interactions are only weakly covalent. Actually, the two bonding orbitals in the CASSCF(9,7) calculation are mainly located on the ligands (Figure S3 in the Supporting Information). Our attempt to add the empty 4s orbital did not work because the program automatically selects more favorable orbitals that give the lowest overall energy. Thus, the 4s orbital is eventually replaced by a 4d orbital with the same Ag symmetry even when it is selected at the beginning. A similar issue occurred when we tried to include the 4p orbitals to create a (5,8) active space as they are substantially contaminated by 4f orbital contributions (Figure S4 in the Supporting Information). We conclude from our results that the MR calculations give much higher d−d transition energies compared to the experimental assignments. For example, the lowest-energy sextet → quartet transition is always around 3.0 eV in the calculations regardless of the ab initio method, active space, and basis set. In contrast, this peak was found to be ∼1.5 eV in multiple and independent experiments.14,15,17 Not only has the 1.5 eV peak long been assigned as the lowest-energy d−d transition by experimental chemists,14,15,17 it has previously been used to assess semiempirical,18,53 DFT,23,53 and singlereference ab initio23,53 methods in several theoretical studies. Similar bands have been experimentally reported in studies of iron oxides and oxide−hydroxides with FeO6 units (1.43 eV for hematite,16 1.35 eV for goethite,8 1.33 eV for maghemite,8 1.29 eV for lepidocrocite,8 1.24 eV for Fe3+ doped in MgO,54 1.17 eV for Fe3+ doped in Al2O3,55,56 and 1.11 eV for magnetite57). In all cases, these transitions have been assigned to the lowestenergy sextet → quartet transition.8,9 Our results suggest that either there is a new challenge for multireference methods or the nature of the 1.5 eV peak needs to be reconsidered. Similar disagreement has also been reported in studies of hematite, where based on a CASPT2 study, the authors suggested that the calculated lowest excitation is around 2.5 eV and is comparable to the experimental peak at 1.9−2.3 eV.58 They further suggested that the 1.5 eV peak is

Table 5. Calculated d−d Transition Energies for Fe3+ (eV) state

CASSCF(5,5)

CASPT2(5,5)

exp

Ag 4 B1g/B2g/B3g 4 B1g/B2g/B3g 4 Ag 4 Ag 4 Ag

0.00 4.62 4.62 4.62 4.62 4.62

0.00 4.53 4.53 4.53 4.53 4.53

0.00 4.36

6

Table 6. Calculated LMCT Energies for Fe(H2O)63+ (eV) state

transition

CASSCF(7,6)

CASPT2(7,6)

exp

→ → → → →

0.00 7.75 8.11 6.75 6.94 6.74

0.00 5.69 6.09 4.61 4.81 4.61

0.00 5.16

6

Ag 6 B3u 6 B3u 6 B2u 6 B1u 6 Au

2p 2p 2p 2p 2p

eg eg t2g t2g t2g

and the t2g−eg splitting in [Fe(H2O)6]3+ (Table 6, estimated from the LMCT splitting, as we describe below). Table 5 shows that CASSCF(5,5) and CASPT2(5,5) both closely match experiment for the lowest d−d transition energy in Fe3+. For the LMCT transition, we note that this transition is several orders of magnitude more intense than the spinforbidden d−d transition; therefore, it is a major peak in the absorption spectrum and has a much smaller chance to be misinterpreted. Also, the energy difference between 2p(O) → t2g(Fe) and 2p(O) → eg(Fe) transitions can be used to estimate of the t2g−eg splitting. State-averaging of the six states of interest (i.e., the ground state and five O 2p to Fe 3d chargetransfer states) changes the ground-state energy by ∼3.6 eV E



Article

The pentacoordinate [Fe(H2O)5]3+ and tetracoordinate (tetrahedral) [Fe(H2O)4]3+ also do not offer a plausible explanation for the lower-energy peak despite the fact that pentacoordinate and tetracoordinate defects probably exist in iron oxides and oxide−hydroxides. One possible impurity that could contribute to the peak at 1.5 eV is [Fe(H2O)6]2+ as this has an experimentally known spin-allowed transition at ∼1.3 eV (and our calculations are consistent with this, as described further below). Typically, a spin-allowed d−d transition is 1 or 2 orders of magnitude more intense than a spin-forbidden one. Moreover, in the absorption spectrum of [Fe(H2O)6]3+, the peak at 1.5 eV is only 1/4 as intense as the peak at 3.0 eV. Thus, a tiny amount of [Fe(H2O)6]2+ impurities (e.g., on the order of 1%) is enough to give this peak. It is well-known that iron species exist in a ferrous−ferric equilibrium in natural water,66−69 and Fe(III) can be reduced to Fe(II) at high temperature.68,69 In a previous study of the optical spectra of [Fe(H2O)6]3+, various salts were used,15 and it was found that although the transition energies were independent of the anion, the relative intensity of the lowest-energy peak varied significantly with the anion. This is not inconsistent with our assumption because experimental conditions may affect the Fe(III)/Fe(II) equilibrium and change the relative strength of the peaks from Fe(III) and Fe(II) if Fe(II) impurities exist. There is also negative evidence for Fe(II) being involved. The measured lowest peak for [Fe(H2O)6]2+ is at 1.3 eV, which is slightly to the red of the 1.5 eV peak in [Fe(H2O)6]3+. This is a small difference in energy, but the line shapes are different too. Also, the experimental spectrum of [Fe(H2O)6]3+ has another peak at 2.3 eV, but if the 1.5 eV peak is caused by a spin-allowed transition in [Fe(H2O)6]2+, then the 2.3 eV peak cannot arise from a spin-allowed transition according to calculations that we present later. [Fe(H2O)6]2+ does have an observed peak at 2.5 eV, but its absorption coefficient suggests (and our calculations confirm) that this is spin-forbidden, and we would therefore not have the ratio of 2.5 eV/1.5 eV intensities needed to match experiment. It is possible that the experimental peak at ∼2.3 eV is the CASPT2 peak at ∼2.9 eV. The deviation, 0.5−0.6 eV, is slightly larger than might be expected but is still within the assumed limit of the CASPT2 method. However, if we make this assignment, then the assignment of the experimental peak at ∼3.0 eV becomes an issue given the calculated results in Table 3. Another possibility for the ∼1.5 eV peak is low-spin Fe(III) complexes. Spin crossover of Fe(III) compounds has been known for a long time,70−72 and for hexacoordinate Fe(III) systems, this involves an S = 1/2 and 5/2 equilibrium.70,71 Therefore, there might exist a temperature-dependent highspin/low-spin equilibrium for [Fe(H2O)6]3+ or perhaps a photoinduced spin crossover population leading to low-spin Fe(IIII). The spin-allowed doublet-to-doublet transitions associated with this species were calculated and are given in Table 9. This shows some transition energies well below 3 eV that may be correlated with the observed peaks at ∼1.5 and ∼2.3 eV. Because these transitions are spin-allowed, a small low-spin population is sufficient to produce a peak as intense as that from the dominating high-spin population whose d−d transitions are all spin-forbidden. The negative support for this hypothesis is that the energy difference between the equilibrium structures of the two spin states is quite high (3.8 eV from CASPT2); therefore, an equilibrium population sufficient to explain the results is not possible. Viability of the

(Table S1 and Figure S5 in the Supporting Information). This state-averaging, however, causes only small changes to the LMCT state energies. Therefore, the ground state must be treated independently and the five LMCT states determined together in a state-averaged CASSCF calculation. The LMCT energies obtained using this scheme are very close to those from the scheme in which all states are solved individually (Table S1 in the Supporting Information). For the multistate CASPT2 calculation, this is not an issue because states with different spatial symmetries cannot be mixed. Table 6 gives the calculated LMCT energies for [Fe(H2O)6]3+. The average energy of the two 2p → eg transitions is 5.89 eV, and the average energy of the three 2p → t2g transitions is 4.68 eV. The difference between them (1.2 eV), which is an estimate of the t2g−eg splitting in [Fe(H2O)6]3+, is close to the splitting in hematite (1.4 eV). Therefore, the calculated d−d transition energy in [Fe(H2O)6]3+, the electron pairing penalty in Fe3+, and the t2g−eg splitting are consistent with each other. The latter two are further consistent with experimental values. The average of the two calculated transition energies (5.29 eV) is very close to the experimental LMCT peak at 5.16 eV.62−65 For further understanding of the d−d transition energies, we have studied the octahedral Fe(III) complex, [FeF6]3−. Table 7 Table 7. Calculated d−d Transition Energies for d5 [FeF6]3− (eV) state

CASSCF(5,5)

CASPT2(5,5)

MRCI(5,5)

exp

Ag 4 B1g/4B2g/4B3g 4 B1g/4B2g/4B3g 4 Ag 4 Ag 4 Ag

0.00 3.57 4.14 4.30 4.33 4.33

0.00 3.02 3.51 3.83 3.86 3.86

0.00 3.23 3.84 4.12 4.16 4.16

0.00 1.76 2.44 3.15 3.57 3.74

6

shows results that are quite similar to those in [Fe(H2O)6]3+. All CASSCF, CASPT2, and MRCI calculations significantly overestimate the d−d transition energies compared to experimental values. For example, the lowest transition energy is 1.76 eV from experiments but 3.02/3.23 eV from CASPT2/ MRCI calculations.17 [Fe(H2O)6]3+ is not the only possible hydrated Fe(III) complex in the solution. It may deprotonate or lose water ligands. The deprotonated product, [Fe(H2O)5(OH)]2+, has been examined using CASSCF(5,5), and it is seen in Table 8 that this is unlikely to be the contributor to the 1.5 eV peak. Table 8. Calculated d−d Transition Energies for Some Fe(III) Complexes Using CASSCF(5,5) (eV) state

[Fe(H2O)5(OH)]2+

[Fe(H2O)5]3+

[Fe(H2O)4]3+

1 2 3 4 5 6 7 8 9 10

0.00 3.44 3.52 3.57 3.64 4.04 4.06 4.08 4.14 4.20

0.00 3.27 3.58 3.65 4.12 4.16 4.18 4.24 4.25 4.27

0.00 3.85 3.86 3.87 4.14 4.15 4.16 4.26 4.27 4.28 F



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Table 11. Calculated d−d Transition Energies for d2 [V(H2O)6]3+ (eV)

Table 9. Calculated Doublet-to-Doublet Transition Energies for Low-Spin [Fe(H2O)6]3+ Using CASSCF(5,5) (eV) 2

B1g

0.01 1.18 1.45 2.51 2.59

2

2

B2g

B3g

0.01 1.15 1.42 2.54 2.59

0.00 1.17 1.44 2.54 2.59

2

Ag

state

CASSCF(2,5)

CASPT2(2,5)

MRCI(2,5)

exp

0.00 1.49 3.18 0.00 1.49 3.18 0.00 1.49 3.18 3.17

0.00 1.62 3.09

0.00 1.50 3.06

0.00 2.21 3.19

3.38

3.17

3.19

3

1.17 1.71 1.72 2.22 2.58 2.59

B1g B1g 3 B1g 3 B2g 3 B2g 3 B2g 3 B3g 3 B3g 3 B3g 3 Ag 3

photoinduced mechanism under these circumstances is not known. Note that previous assessments of semiemipirical,18,53 DFT,23,53 and single-reference ab initio23,53 methods with reference to the 1.5 eV peak may have to be reexamined if this peak does not arise from high-spin [Fe(H2O)6]3+. For example, it was previously concluded that hybrid generalized gradient approximation (GGA) methods with 20−25% exact exchange yield excitation energies that are the closest to 1.5 eV.23 However, if the peak at 1.5 eV is reinterpreted, then functionals with higher amounts of exact exchange are required to reproduce the results. This conclusion would match what was found in that work to be necessary to reproduce the LMCT transition energies. [Ti(H2O)6]3+ (d1). In Jorgensen’s book, two absorption peaks are given (2.52 and 2.16 eV) for [Ti(H2O)6]3+.17 The peak at 2.52 eV is very broad and has a shoulder at 2.16 eV. Ti(III) is a d1 system, and there is only one possible d−d transition (t2g → eg transition). Thus, the author interpreted the two peaks as being caused by the Jahn−Teller effect. In another study of the ligand field splitting, the average value (2.33 eV) of such two bands was regarded as the t2g−eg splitting.73 However, our calculations suggest that the Jahn−Teller distortion is too small to cause two bands (Table 10). In addition, assuming a maximal deviation of ∼0.5 eV for calculated excitation energies, it is possible that the peak at 2.16 eV corresponds to our MRCI value of 1.77 eV. Note that the observed absorption spectra of [Ti(H2O)6]3+ show strong variations.74 For example, the above two peaks shift to 2.36 and 1.88 eV in strong hydrochloric acid. It was proposed in that work that hydrolysis at low acid concentrations and formation of chloro complexes in strong hydrochloric acid are responsible for the complicated spectra. It has also been considered that solvation effects contribute to this. [V(H2O)6]3+ (d2). Spin-allowed t2g → eg transitions in the d2 system involve either one or two electrons. Two major peaks, at 2.21 and 3.19 eV, have been observed.17 Our calculations give three spin-allowed transitions (Table 11). In the two B1g excited states, one electron transfers from t2g to eg. In the Ag state, two electrons transfer from t2g to eg. The splitting of the two B1g states is attributable to the Jahn−Teller effect. The B1g state that we calculate (Table 11) at 3.09 eV (CASPT2) lies

only slightly below the Ag state at 3.38 eV (CASPT2). These two transitions combine in the experimental spectrum, and both correspond to the absorption peak at 3.19 eV. Interestingly, the lowest transition energy suffers a relatively large deviation from experiment. Our best calculation gives 1.62 eV (CASPT2), while the experimental value is 2.21. The difference, 0.59 eV, is not outside of the general accuracy range of CASPT2 in view of our small basis set and active space. Similar underestimation was reported in a CASPT2 study of the potential energy surface of [V(H2O)6]3+.35 Those calculations suggested that the energy minimum of the lowest excited state lies 1.22 eV above the ground state. This value is 0.44 eV smaller than the estimated experimental value, which is not available directly. [V(H2O)6]2+ (d3). Four peaks have been identified in the absorption spectrum for [V(H2O)6]2+.17 Three major peaks are at 1.53, 2.29, and 3.46 eV, and a shoulder one is at 1.62 eV. All of our calculated results are in excellent agreement with the experimental ones (Table 12). The calculations show three Table 12. Calculated d−d Transition Energies for d3 [V(H2O)6]2+ (eV) state

CASSCF(3,5)

CASPT2(3,5)

MRCI(3,5)

exp

Ag B1g/B2g/B3g 4 B1g/B2g/B3g 4 B1g/B2g/B3g 2 Ag 2 Ag 2 Ag 2 Ag 2 Ag 2 B1g/B2g/B3g 2 B1g/B2g/B3g 2 B1g/B2g/B3g 2 B1g/B2g/B3g

0.00 1.19 1.97 3.36 2.01 2.01 2.90 3.58 3.58 2.12 2.79 3.27 3.35

0.00 1.26 2.01 n/a 1.89 1.89

0.00 1.28 2.07 3.41 1.98 1.98

0.00 1.53 2.29 3.46 1.62

1.99

2.08

2.29

4 4

Table 10. Calculated d−d Transition Energies for d1 [Ti(H2O)6]3+ (eV) state 2

B1g 2 B2g 2 B3g 2 Ag 2 Ag

CASSCF(1,5)

CASSCF(1,10)

CASPT2(1,5)

MRCI(1,5)

exp

0.00 0.00 0.00 1.69 1.70

0.00 0.00 0.00 1.69 1.70

0.00

0.00

0.00

1.71 1.72

1.76 1.77

2.16 2.52

G



Article

spin-allowed transitions for the d3 V(II) system. The first two arise from the one-electron t2g → eg transition, and the third arises from the two-electron transition. The shoulder peak receives contributions from spin-forbidden transitions. Spinforbidden transitions may also contribute to the experimental band at 2.29 eV. [Cr(H2O)6]3+ (d3). As with the above d3 system, our calculations give three spin-allowed transitions and two spinforbidden ones (Table 13). The calculated spin-allowed

3.09, 3.47, 3.69, and 4.09 eV) and one shoulder peak (3.14 eV) in his book.17 In another work by him, the peak at ∼4 eV is not listed.14 Also in this work, he cited a reference giving only four peaks at 2.43, 3.09, 3.46, and 3.70 eV. The peak at 4.09 eV was reported by Heidt et al., along with a peak at 5.06 eV.75 Our calculations show fewer peaks than the experiments, and the assignment is not unambiguous (Table 14). It appears that CASPT2 gives the best match with experiment, properly assigning four out of six peaks. However, in the CASPT2 results, the calculated transition energy of 2.91 eV may correspond to three of the experimental peaks. Note that the lowest d−d transition is around 3 eV, which is similar to what we found for Fe(III). [Fe(H2O)6]2+ (d6). Fe(II) in this cluster is a high-spin d6 system. It has one spin-allowed transition and some spinforbidden transitions. In his book, Jorgensen assigned the spinallowed peak at 1.29 eV and spin-forbidden peaks at 2.45, 2.62, 2.75, and 3.21 eV.17 He also mentioned a shoulder peak at 1.79 eV. Our calculations suggest a small splitting for the spinallowed t2g → eg transition (Table 15). The calculated spinallowed transition energy is ∼0.45 eV lower than the experimental value. With regard to the spin-forbidden transitions, the calculated and experimental results agree very well. No transition corresponding to the shoulder peak at 1.79 eV can be found in our calculated results. [Co(H2O)6]3+ (d6). Co(III) in this cluster is a low-spin d6 system. Information about its optical spectrum is scarce, and only two spin-allowed peaks have been reported (2.06 and 3.09 eV).17 The CASPT2 calculation failed to converge for the excited states in our calculations. For the lowest spin-allowed excitation energy, the CASSCF and MRCI results are 0.6−0.7 eV smaller than the experimental value (Table 16). This is slightly outside of the general accuracy of multireference methods. The deviation in the second excitation energy is 0.4− 0.5 eV, which is on the large side of the deviation range. Although the scarce spectral data may be an issue, it seems that the computational performance for the low-spin complex is not as satisfactory as that for the other complexes. [Co(H2O)6]2+ (d7). Jorgensen summarized five peaks in his book, 1.02, 1.40, 1.98, 2.41, and 2.67 eV.17 The peak at 2.67 eV is given as a shoulder of the peak at 2.41 eV.76 These two peaks are overlapping, and the shoulder is not evident in the original spectrum covering absorption ranging from 2 to 3 eV. As pointed out by the author, the peak at 2.67 eV was not read directly from the spectrum but was based on an assumption and previous experience. It assumes that the absorption curves (the extinction coefficient as a function of wavenumber) have a simple Gaussian line shape and are consequently symmetric

Table 13. Calculated d−d Transition Energies for d3 [Cr(H2O)6]3+ (eV) state

CASSCF(3,5)

CASPT2(3,5)a

MRCI(3,5)

exp

Ag 4 B1g/B2g/B3g 4 B1g/B2g/B3g 4 B1g/B2g/B3g 2 Ag 2 Ag 2 Ag 2 Ag 2 Ag 2 B1g/B2g/B3g 2 B1g/B2g/B3g 2 B1g/B2g/B3g 2 B1g/B2g/B3g 2 B1g/B2g/B3g

0.00 1.69 2.70 4.36 2.41 2.41 3.71 4.50 4.50 2.54 3.41 4.07 4.21 4.94

0.00 1.77 2.68 4.28 2.23 2.23

0.00 1.79 2.77 4.42 2.35 2.35

0.00 2.16 3.05 4.69 2.60 2.60

2.35

2.47

2.60

4

a

The 4B1g states were determined individually.

transition energies agree well with the experimental energies.17 In addition, two shoulder peaks have been observed in the absorption spectrum.17 These peaks are uncertain because of their low absorption strength. They may correspond to spinforbidden transitions and may also be caused by impurities. Our calculations suggest that the observed peak at 2.60 eV is related to spin-forbidden transitions, but the peak at 1.86 eV fails to match our spin-forbidden results suitably. [Mn(H2O)6]2+ (d5). Mn(II) is isoelectronic with Fe(III) and has no spin-allowed d−d transition. In particular, the bands for [Mn(H2O)6]2+ are so weak that the measurement had to be performed in 10 cm cells.14 The extinction coefficients in [Mn(H2O)6]2+ are 1 order of magnitude smaller than those in [Fe(H2O)6]3+, whose d−d transitions are already spinforbidden. 14 This creates substantial hindrance to the unambiguous determination of the existence of an absorption peak. Jorgensen has summarized six major peaks (2.34, 2.86,

Table 14. Calculated d−d Transition Energies for d5 [Mn(H2O)6]2+ (eV) state

CASSCF(5,5)

CASPT2(5,5)

MRCI(5,5)

exp

Ag 4 B1g/4B2g/4B3g 4 B1g/4B2g/4B3g 4 B1g/4B2g/4B3g 4 B1g/4B2g/4B3g 4 Ag 4 Ag 4 Ag 4 Ag 4 Ag

3.42 3.77 4.67 4.92 3.88 3.88 3.89 4.86 4.86

2.91 3.39 4.22 4.62 3.67 3.67 3.68 4.49 4.49

3.25 3.66 4.45 4.76 3.81 3.81 3.82 4.67 4.67

2.34, 2.86, 3.09 3.47 4.09

6

H

3.69 3.69 3.69



Article

Table 15. Calculated d−d Transition Energies for d6 [Fe(H2O)6]2+ (eV) state 5

B1g B2g 5 B3g 5 Ag 5 Ag 3 B1g 3 B1g 3 B1g 3 B1g 3 B1g 3 B1g 3 B1g 3 B2g 3 B2g 3 B2g 3 B2g 3 B2g 3 B2g 3 B2g 3 B3g 3 B3g 3 B3g 3 B3g 3 B3g 3 B3g 3 B3g 3 Ag 3 Ag 3 Ag 3 Ag 3 Ag 5

CASSCF(6,5) 0.00 0.02 0.00 0.75 0.82 2.29 2.61 2.86 3.22 3.28 3.70 3.75 2.30 2.60 2.85 3.22 3.26 3.70 3.77 2.26 2.62 2.89 3.23 3.31 3.70 3.72 3.16 3.18 3.69 3.86 3.91

CASPT2(6,5) 0.00

MRCI(6,5) 0.00

Table 17. Calculated d−d Transition Energies for d7 [Co(H2O)6]2+ (eV) exp

state

0.00

4

B1g B1g 4 B1g 4 B2g 4 B2g 4 B2g 4 B3g 4 B3g 4 B3g 4 Ag 2 Ag 2 Ag 2 Ag 2 B1g 2 B1g 2 B1g 2 B2g 2 B2g 2 B2g 2 B3g 2 B3g 2 B3g 4

0.80 0.89 2.23 2.51 2.86 3.05 3.13 3.59 3.80

3.06 3.08 3.48 3.75 3.80

0.83 0.91 2.27 2.62 2.89 3.21 3.24 3.70 3.83

3.24 3.25 3.66 3.91 3.96

1.29 1.29 2.45 2.62 2.75 3.21 3.21

1

3.21 3.21

Ag B1g 1 B1g 1 B2g 1 B2g 1 B3g 1 B3g 1

CASSCF(6,5)

CASSCF(6,10)

MRCI(6,5)

exp

0.00 1.39 2.56 1.38 2.56 1.39 2.56

0.00 1.45 2.65 1.45 2.65 1.45 2.65

0.00 1.30 2.60

0.00 2.06 3.09

CASPT2(7,5)

MRCI(7,5)

exp

0.00 0.67 2.83 0.00 0.67 2.85 0.00 0.68 2.82 1.47 1.96 1.98 2.95 2.50 2.52 3.22 2.50 2.53 3.22 2.50 2.51 3.22

0.00 0.81 2.69

0.00 0.65 2.62

0.00 1.02 2.41

1.81 1.95 1.98 3.08 2.29 2.57 3.16

1.43 1.76 1.79 2.97 2.38 2.46 2.99

1.98 1.98 1.98 2.41 2.67

CASPT2 value is 0.82 eV, and the MRCI value is 0.65 eV. For the second spin-allowed transition, the calculated CASPT2 result is at 1.81 eV, while MRCI is at 1.43 eV, and the experiments show two possible peaks, at 1.98 and 1.40 eV. The 1.98 eV peak is stronger, and the other one was assigned as a spin-forbidden transition by Jorgensen. Our calculation does not support the existence of a spin-allowed peak at 2.67 eV. Instead, it is probably a spin-forbidden transition interrupting the symmetric line shape of the spin-allowed band at 2.41 eV. Also, the weakest experimental band at 1.40 eV is missing in our calculation. [Ni(H2O)6]2+ (d8). Three major peaks (1.05, 1.67, and 3.14 eV) and two shoulder peaks (1.91 and 2.73 eV) are listed in the book.17 Our calculation agrees very well with experiment showing three spin-allowed and two spin-forbidden transitions (Table 18). For all five transition energies, the deviation of the multireference methods (CASPT2 and MRCI) is no more than 0.2 eV. It is interesting to note that an accurate description of the neutral nickel atom has been challenging even for multireference methods, and extensive active spaces (at least 10 electrons in 14 orbitals) must be used.42 Our results indicate that the ionic complex is more easily treated. [Mn(H2O)6]3+ (d4), [Cr(H2O)6]2+ (d4), and [Cu(H2O)6]2+ (d9). Unlike the complexes described above where there is no or insignificant Jahn−Teller effect for the ground states, the d4 and d9 complexes show considerable Jahn−Teller effects. This is related to the fact that the t2g orbitals are half or fully occupied and eg orbitals have odd numbers of electrons in the ground states. Jorgensen lists one peak for [Mn(H2O)6]3+ at 2.60 eV, one peak for [Cr(H2O)6]2+ at 1.75 eV, and two peaks for [Cu(H2O)6]2+ at 1.17 and 1.56 eV.17 With respect to [Cu(H2O)6]2+, the original literature indicated that only one broad band was observed.77 The peak at 1.17 eV was inferred by resolving the asymmetric absorption curve into two symmetric ones. For [Mn(H2O)6]3+, Davis et al. reported

Table 16. Calculated d−d Transition Energies for d6 [Co(H2O)6]3+ (eV) state

CASSCF(7,5)

around their maxima if they arise from single transitions.76 The peaks at 1.40 and 1.98 eV were reported in a different work with a spectrum range of 1−2 eV.14 A strong peak at 1.02 eV was also discussed in this study beause its tail could be seen. On the basis of their absorption coefficients, Jorgensen proposed that the peaks at 1.02, 2.41, and 2.67 eV are from spin-allowed transitions.17 He also proposed that the peak at 1.40 eV is from a spin-forbidden transition, and the peak at 1.98 eV may be related to either a spin-allowed or a spin-forbidden transition. Our calculations are generally consistent with his assignment and give three spin-allowed transitions (Table 17). CASPT2 seems to give better results than MRCI. The lowest spinallowed transition is at 1.02 eV from experiments, while the I



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Table 18. Calculated d−d Transition Energies for d8 [Ni(H2O)6]2+ (eV) state

CASSCF(8,5)

3

Ag B1g 3 B1g 3 B1g 3 B2g 3 B2g 3 B2g 3 B3g 3 B3g 3 B3g 1 Ag 1 Ag 1 Ag 1 B1g 1 B1g 1 B2g 1 B2g 1 B3g 1 B3g

0.00 0.76 1.31 3.34 0.76 1.31 3.34 0.75 1.31 3.35 2.30 2.30 3.61 2.95 3.85 2.95 3.84 2.94 3.85

3

a

CASPT2(8,5)a 0.00 0.89 1.48 3.14

2.03 2.03 3.32 2.71 3.66

MRCI(8,5) 0.00 0.85 1.45 3.29

2.23 2.23 3.56 2.86 3.80

Table 20. Calculated d−d Transition Energies for d4 [Cr(H2O)6]2+ (eV) exp

state

0.00 1.05 1.67 3.14

5

CASSCF(4,5)

CASPT2(4,5)

MRCI(4,5)

0.00 0.62 1.34 1.18 1.23

0.00 0.69 1.44 1.27 1.30

0.00 0.64 1.36 1.19 1.23

Ag Ag 5 B1g 5 B2g 5 B3g 5

exp 1.17 1.75

complexes are somewhat sensitive to experimental conditions such as solvent and solid-state perturbations.52,78 In addition, a broad band means that the relative error in the absorption maximum may be large. In particular, the low-energy band for [Cu(H2O)6]2+ was only estimated through resolving an asymmetric peak. This gives a deviation of 0.56 eV (CASPT2 versus experiment), which is slightly beyond the general CASPT2 accuracy. Double Shell Effect. It has been proposed that it may be necessary, especially for complexes with high 3d occupation numbers, to include five extra d orbitals in the active space. To examine the effect of adding the second d shell, CASSCF(m,5) and CASSCF(m,10) results are compared in Table S2 in the Supporting Information. For the d1, d4, d5, d6, and d9 complexes tested, the double shell effect does not lead to any significant change in the excitation energy (or the energy relative to the ground state), but it may substantially lower the absolute energy. The reduction in the absolute energy is more pronounced when there are more 3d electrons. For the d1 complex, the double shell effect does not even lead to any considerable change in the absolute energy. This is consistent with previous conclusions that this effect is more important for complexes with higher 3d occupation numbers. It is very encouraging that the double shell effect only affects the absolute energy rather than the excitation energy because 20 active orbitals are needed to account for it in a transition-metal dimer, and such a large active space would not be affordable.

1.91 1.91 2.73

The 3B1g states were solved individually.

two peaks at 2.53 and 1.11 eV.52 They also noted that the band at 2.53 eV is broad and asymmetric and may contain more than one peak. For [Cr(H2O)6]2+, Fackler et al. studied a number of CrX2·(H2O)6 (X = I, Br, ClO4) complexes in the solid state and in solution.78 They have identified two and sometimes three peaks (∼1.17, ∼1.75, and sometimes ∼2.26 eV). An interesting feature of the Mn(III), Cr(II), and Cu(II) complexes (including the hexa-aqua ones) is a low-energy band (∼1 eV) whose strength is comparable to that of the major band.52 In principle, the major peak should correspond to the t2g → eg transition. The low-energy band has often been attributed to ground-state Jahn−Teller splitting, but other arguments do exist (e.g., excited-state splitting, spin-forbidden transition, charge transfer, and so forth).52 Our calculations suggest a strong splitting of the ground-state eg orbitals (which correspond to the two Ag states within D2h symmetry). Also, the degeneracy of the B1g, B2g, and B3g states is broken, and the range of their energies is ∼0.2 eV (Tables 19−21). This explains why the high-energy band observed is broad. For both the low-energy transition (the Jahn−Teller splitting of eg orbitals) and major transition (t2g → eg), CASPT2 gives better agreement with experiments than MRCI, and both calculations underestimate the results. The CASPT2 splittings of the eg orbitals are 0.77 eV for [Mn(H2O)6]3+, 0.69 eV for [Cr(H2O)6]2+, and 0.61 eV for [Cu(H2O)6]2+. They should correspond to the experimental low-energy bands at 1.11, 1.17, and 1.17 eV respectively. Note that the spectra of these

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CONCLUSIONS Hexa-aqua transition-metal ions, [M(H2O)6]n+, have been investigated using multireference ab initio calculations. These metal−oxygen clusters resemble pure and doped hematite and other corresponding octahedral transition-metal oxides. For [Fe(H2O)6]3+, the calculated d−d transition energies are considerably higher than the assigned experimental ones. In particular, the peak at 1.5 eV is missing in our best calculations even including those for the largest deviation expected based on the method used. Interestingly, this peak at 1.5 eV has been observed in many octahedral iron oxides and oxide− hydroxides. A similar problem was noted in an earlier CASPT2 study of hematite, and our results amplify on the likelihood that this peak in the experimental spectra is from a low population minority iron species or from an impurity.

Table 19. Calculated d−d Transition Energies for d4 [Mn(H2O)6]3+ (eV) state 5

Ag 5 Ag 5 B1g 5 B2g 5 B3g

CASSCF(4,5)

CASSCF(4,10)

CASPT2(4,5)

MRCI(4,5)

0.00 0.68 1.89 1.70 1.75

0.00 0.69 1.91 1.72 1.76

0.00 0.77 2.21 1.96 1.99

0.00 0.72 2.00 1.78 1.82

J

exp 1.11 2.53



Article

Table 21. Calculated d−d Transition Energies for d9 [Cu(H2O)6]2+ (eV) state

CASSCF(9,5)

CASSCF(9,10)

CASPT2(9,5)

MRCI(9,5)

0.00 0.48 0.92 0.79 0.84

0.00 0.51 0.97 0.84 0.89

0.00 0.61 1.23 1.08 1.12

0.00 0.52 0.99 0.85 0.89

2

Ag 2 Ag 2 B1g 2 B2g 2 B3g

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Indeed, an examination of experimental data for the d−d transition in the atomic ion Fe3+ and the t2g−eg splitting in hematite suggests that the lowest d−d transition in [Fe(H2O)6]3+ is much higher than 1.5 eV. Also, our calculations of the d−d transition in Fe3+ and the t2g−eg splitting in [Fe(H2O)6]3+ are consistent with experiments, as are our calculations of the LMCT energy. All of these issues raise an important question concerning the nature of the peak at 1.5 eV. We demonstrated that this is unlikely to be caused by deprotonated or penta-/tetracoordinate Fe(III) species. Small amounts of [Fe(H2O)6]2+ or some other Fe2+ species is one possible explanation for the discrepancy, while a minority lowspin [Fe(H2O)6]3+ or a related species is another possibility. For the other [M(H2O)6]n+ clusters, our calculations correspond to the experimental results reasonably, and they show that many of the transition metals have strong d−d transitions at lower energy than we find for Fe(III). We also note that the inclusion of vibronic effects in the calculations would likely improve the comparison with experiment as the lowest excitation energies that we calculated are mostly below the measured values. Our calculations also clarify some experimental uncertainties in an understandable manner. The double shell effect may substantially reduce the absolute energy of the clusters that we have studied, but it does not cause significant change in the excitation energy relative to the ground-state energy. As such, accurate multireference ab initio results on analogous clusters can be achieved using minimal active spaces and basis sets. This is a promising start and provides a firm basis for further studies of larger clusters that better describe the properties (e.g., excitation involving more than one metal center) in pure and doped hematite and other metal oxides.

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1.17 1.56

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ASSOCIATED CONTENT

S Supporting Information *

Table S1 (LMCT energies using different state-averaging schemes), Table S2 (double shell effect on absolute energies), and Figures S1−S5 (CASSCF active orbitals). This material is available free of charge via the Internet at http://pubs.acs.org.

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exp

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



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Multireference Ab Initio Study of Ligand Field d–d Transitions in

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