A Restricted Open Configuration Interaction Singles Study on M- and L

Article Cite This: J. Chem. Theory Comput. 2018, 14, 4320−4334

pubs.acs.org/JCTC

Restricted Open-Shell Configuration Interaction Singles Study on M- and L‑edge X‑ray Absorption Spectroscopy of Solid Chemical Systems Adam Kubas,†,§ Max Verkamp,∥ Josh Vura-Weis,∥ Frank Neese,*,†,‡ and Dimitrios Maganas*,†,‡ †

Max Planck Institute for Chemical Energy Conversion, Stiftstr. 34−36, 45470 Mülheim an der Ruhr, Germany Max-Planck-Institut für Kohlenforschung, Kaiser-Wilhelm-Platz 1, 45470 Mülheim an der Ruhr, Germany § Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland ∥ Department of Chemistry, University of Illinois, 600 South Matthews Avenue, Urbana, Illinois 61801, United States

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‡

ABSTRACT: In this study the M- and L-edge X-ray absorption spectra of a series of open- and closed-shell solids (TiO2 rutile, α-Fe2O3 hematite, FeS2 pyrite, and the spinel Co3O4) are investigated with the restricted open-shell configuration interaction singles methods (ROCIS/DFT and PNO-ROCIS/DFT) using the embedded cluster approach. ROCIS/DFT type of methods are grounded in wave function-based ab initio electronic structure theory and have shown great performance in the field of X-ray spectroscopy in particular in the field of transition metal L-edge spectroscopy. In this work we show that ROCIS/DFT can be used to calculate and interpret metal M- and L-edge XAS spectra of solids. To this end, clusters with up to 52 metal centers are considered. In all cases good to excellent agreement between theory and experiment is obtained. The experimentally probed local coordination environments are discussed in detail. The physical origin of the observed spectral features is explored through the machinery of natural difference orbitals. This analysis provides valuable information with respect to the core to valence, metal to metal charge transfer, and metal to ligand charge transfer characters of the relativistically corrected many particle states. The influence of the above electronic effects to the spectral shapes and the size of the treated clusters are thoroughly investigated.

1. INTRODUCTION X-ray absorption spectroscopy (XAS) has long been used as a powerful element-specific technique to probe the electronic and geometric structure of molecules and materials.1−4 Since XAS is based on core−electron excitations, the resulting X-ray absorption spectra are classified as K-, L-, and M-edge according to the origin of the excited core electron: 1s, 2s/2p, and 3s/3p/3d, respectively. In contrast to K-edge spectroscopy, the core electron excitation process in 3d transition metal M/ L-edge spectroscopies involves dipole allowed p → d transitions. As a result, intense spectral features are obtained. Furthermore, the method has been proven to be a sensitive probe of the oxidation and spin states as well as the coordination number and identity of the ligands surrounding the metal center.3,5,6 Hence, a host of applications exist including in situ catalyst characterization,7,8 surface science,9 or the biochemistry of metalloproteins.2,10 However, while the L-edge spectra are characterized by sharp and richly structured features, the respective © 2018 American Chemical Society

M-edge spectra usually feature broader line shapes (∼1 eV) due to rapid (200 atoms and more than 3000 basis functions using regular computing nodes or even powerful desktop machines. Hence, it is an attractive method for spectroscopic applications on solid-state systems. In fact the ROCIS/DFT methods have shown excellent performance in treating the metal L-edge problem on classes of molecular systems ranging between mononuclear complexes34,44 up to solid polymetallic clusters such as V2O5,45 CaF2,46 and TiO2.46 Recently, by using the pair natural orbital (PNO) machinery, the PNO-ROCIS and PNOROCIS\DFT variants have been developed.47 These methods have shown spectacular performance in calculating XAS spectra of “real-life” chemical systems, while providing results that are virtually indistinguishable from the original ROCIS predicted spectra. It should be noted that systems featuring an antiferromagnetically coupled ground state cannot be presently treated within the ROCIS and ROCIS/DFT frameworks. In this work, we present a combined experimental and computational study on the transition metal M- and L-edge XAS spectra of open- and closed-shell solids, including the rutile polymorph of TiO2 rutile, the α-Fe2O3 hematite, the FeS2 pyrite, as well as the mixed valence spinel cobalt (II−III) oxide Co3O4.

methods. These calculations were based on a ground state restricted open-shell Kohn−Sham determinant using the B3LYP49−51 functional and def2-TZVP52,53 basis sets of triple-ζ quality54 along with the matching Coulomb fitting basis set in order to accelerate the calculations in the framework of resolution of identity (RI)55 and chain-of-spheres (COSX)56 approximations. For the embedded cluster calculations, all quantum clusters (QCs) were extracted from the crystallographic supercells. In order to account for long-range Coulombic forces, point charge fields (PCs) were constructed including between 1000 to about 8500 charges that were placed on the appropriate crystal lattice nodes. The employed embedding scheme is nonpolarizable in the sense that the positions and magnitudes of the point charges are kept fixed. The number of point charges was adjusted to reflect the stoichiometry of the cluster and to ensure that the inclusion of another layer of point charges does not alter the calculated spectrum in each case. A third boundary region (BR), equipped with capped effective core potentials (cECPs), was introduced between the QC and PC regions in order to avoid spurious electron leakage from the clusters. In particular, a single layer of cECPs, ECP10MDF57 and ECP2MWB58 (included in the def2-SD framework), was used to replace the metal and the oxygen/sulfur atoms, respectively. The charges associated with the cECPs and PCs were obtained by constraining the absolute total charge of the system to a value close to zero (here 10−5), as described previously.34 The embedding procedure is presented graphically for the FeS2 cluster in Figure 1. The number of point charges, cECPs, and the corresponding charges for all clusters are presented in Table 1. Atomic coordinates of all systems were obtained from the American Mineralogist Crystal Structure Database59 and the Crystallography Open Database.60,61 The details of experimental setups as well as structure refinement can be found in the related literature.62−66 In order to ensure convergence of the calculated spectra with respect to the number of nonrelativistic states included in the quasi-degenerate perturbation theory (QDPT) treatment, the ROCIS/DFT type of calculations accounted for 80 states (roots) per multiplicity out of localized ROKS orbitals (the Pipek-Mezey localization scheme67 was used throughout) belonging to a single metal center at a time into a complete canonical acceptor orbital space of the entire cluster. Excited states are classified by their spin quantum number S′ relative to that of the ground state S. SOC selection rules demand the inclusion of states with S′ = S, S′ = S − 1 and S′ = S + 1. The nonrelativistic states of different multiplicity are subsequently used in the QDPT treatment for the SOC operator. The twoelectron SOC operator itself is represented by the spin−orbit mean field operator (SOMF).68−70 The final spectrum of the cluster is obtained as the sum of all individual subspectra obtained for each ion. To account for systematic errors in the calculation of transition energies, the total spectra need to be shifted for comparison to experiment.44,46 It should be emphasized that for a given level of approximation (e.g., ROCIS/ DFT) in conjunction with the basis set incompleteness, the incomplete treatment of scalar relativistic effects and shortcomings in the shapes of the DFT orbitals, these errors are highly systematic, and simple linear regression is sufficient to establish predictive accuracy in the calculated transition energies for any given combination of element and basis set. In the present study however we deal with a limited set of transition metal solids; hence, the element specific shifts need to be

2. COMPUTATIONAL PROTOCOL All calculations were performed with the ORCA suite of programs.48 All of the XAS spectra were computed by employing the ROCIS/DFT33,34 or PNO-ROCIS/DFT47 4321

DOI: 10.1021/acs.jctc.8b00302 J. Chem. Theory Comput. 2018, 14, 4320−4334

Article

Journal of Chemical Theory and Computation

Figure 1. Schematic representation of the embedding approach illustrated for the case of FeS2 solid. The quantum cluster (QC, [Fe6S38]26‑) is surrounded with a number of positively charged cECPs (gray sticks) placed in the positions corresponding to transition metal sites. Subsequently, a large number of point charges (PCs, red dots) are used to describe long-range interactions in a crystal as well as to ensure overall neutrality of the computed system.

Table 1. Composition of the Clustersa solid

cluster

n(cECP)

n(PC)

q(M)

q(X)

TiO2

[TiO6]8− [Ti7O29]30‑ [Ti13O58]64‑ [Ti21O71]58‑ [Ti31O100]76‑ [Ti52O160]116‑ [FeO6]9− [Fe4O20]28‑ [Fe7O27]33‑ [FeS12]10‑ [Fe6S38]26‑ [Fe19S88]48‑ [Co3O12]16‑ [Co9O30]36‑

10 44 78 63 62 108 10 60 118 18 38 78 77 68

8500 8500 8500 8500 8500 8500 2550 2550 2550 5800 5800 5820 1266 1266

2.54 2.54 2.54 2.54 2.54 2.54 1.82 1.82 1.82 1.24 1.24 1.24 1.44 1.44

−1.46 −1.46 −1.46 −1.46 −1.46 −1.46 −1.34 −1.34 −1.34 −0.83 −0.83 −0.83 −1.34 −1.34

Fe2O3

FeS2

Co3O4

applying an energy dependent Gaussian broadening, in which the half-width of the Gaussian curves was chosen as a square root of the transition energy (in eV) multiplied by a constant factor (0.03 and 0.1 for L-edge and M-edge, respectively). The resulted broadening parameters range between 0.6 and 0.9 eV, respectively. As a comparison measure of the shape of two different spectra the mean absolute error of the calculated areas MAEarea1,2 is used, which reads according to the relation: MAEarea1,2 = %

|area1 − area 2| area1

As has been described recently47,74 the individual contributions to a particular SOC state are obtained by (1) determining the leading nonrelativistic states that contribute to the SOC state of interest with a large oscillator strength and (2) analyzing the nature of these nonrelativistic states in terms of leading one electron excitations via evaluation of the natural difference orbitals (NDOs)75,76 for these states (alternatively, the natural transition orbitals NTOs can be used as well).77

a n(cECP): number of capped ECPs; n(PC): number of point charges; q(M) and q(X): charge of the metal M and counterion X in the cECP and PC layers, respectively. The charges were chosen in a way that the total charge of the quantum region, cECP layer, and the PC field is zero.

3. GEOMETRIC STRUCTURES The structures of the clusters used in this study are presented in Figure 2. In each case, we present the top and side views along with the ligand field of the basic unit. TiO2 rutile is the thermodynamically stable polymorph of titanium(IV) oxide. It crystallizes in the P42/mnm space group,62 and the basic building blocks are the TiO6 octahedrons. Each unit has an elongated axial Ti−O bond (1.986 Å) as compared to equatorial Ti−O bonds (1.945 Å). The O− Ti−O angles for equatorial ligands are equal to 81° and 99°. As a consequence, the local crystal field around the Ti atoms is lowered from Oh to D2h. As it is shown in Figure 2 cluster models containing up to 52 titania atoms ([TiO6]8−, [Ti7O29]30‑, [Ti13O58]64‑, [(Ti21O71)]58‑, [(Ti31O100)]76‑, and [(Ti52O160)]116‑) were chosen to represent the TiO2 rutile coordination environment. Iron(III) oxide in the hematite (α-Fe2O3) phase is a promising candidate for photochemical water splitting.78 It forms trigonal crystals of R3̅c symmetry.63 Each iron atom is ligated by six oxygen atoms of which three fac-located have Fe−O bond distance of 1.946 Å and the other three Fe−O bond length of 2.116 Å. The local coordination environment of the FeO6 units is C3v symmetric. As shown in Figure 2, cluster models containing one [FeO6]9−, four [Fe4O20]28‑, and seven [Fe7O27]33‑ iron centers were chosen to represent the

refined by systematic benchmark studies of well-defined mononuclear complexes. The preliminary shifts for Ti, Fe, and Co metal centers amount to 9.4, 16.8, and 16.9 eV for calculated L-edge as well as 4.2, 2.3, and 4.8 eV for the M-edge spectra, respectively. We note that in the case of solids containing Ti and Fe centers these shift values agree very well with those evaluated previously for a limited set of mononuclear titanium and iron complexes.34 The transition intensities were obtained by a zeroth order multipole expansion to include electric dipole contributions to the absorption transition rate. In metal M- and Ledge X-ray absorption spectroscopy, the higher energy M2 and L2 signals are typically more broad relative to their lower energy M3 and L3 counter parts as they are subject to distortions and broadening due to the Coster−Kronig Auger decay71−73 as well as Fano-type resonances.11,73 Such decay processes cannot be isolated and estimated accurately experimentally and are not included in the ROCIS/DFT calculations. It is a common practice to broaden the calculated spectra by applying a Gaussian broadening to account for the experimental spectrometer resolution and a Lorentzian one to account for the core-hole lifetime. In this work we have approximated the convolution of these two effects using a single Gaussian for each calculated transition. We should note that slightly better visual agreement between experimental and theoretical L2 regions was observed by 4322


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Co−O bond lengths are remarkably similar, 1.923 Å for Co3+ and 1.928 Å for Co2+. The octahedral site undergoes a distortion that lowers the symmetry from Oh to C2v, while the tetrahedral sites preserve essentially a Td symmetry. As shown in Figure 2, two cluster models containing three and nine cobalt atoms [Co3O12]16‑ and [Co9O30]36‑ were considered to represent the coordination environment of Co3O4. Both chosen clusters contain tetrahedrally and octahedrally coordinated Co(II)O4 and Co(III)O6 centers. It should be emphasized that this choice of clusters allows for a compact and uniform treatment of the solid structure rather than treating the clusters with isolated Co(II)O4 and Co(III)O6 centers and completely neglecting their natural inter electronic communication.8,81

4. ELECTRONIC STRUCTURE OF GROUND- AND CORE-EXCITED STATES IN A SIMPLE PARTICLE−HOLE PICTURE (IV) Ti in a distorted octahedral (D2h: TiO6) network has ground state with a 2p63p63d0 electron configuration (1Ag). Upon np core electron excitation, the final state manifold consists of closed-shell and open-shell np53d1 electron configurations with total spin S′ = 0 and S′ = 1. As has been discussed previously46 and also shown in Figure 3a, in this symmetry five excitations per polarization and spin state arise: 1A1 → 3 × 1,3(B2 + A1 + A2 + B1) corresponding to the 2px,y,z → 3dxy,xz,yz,x2−y2,z2 single electron excitations that construct the final state multiplet. High spin (HS) Fe(III) in distorted octahedral (C3v: FeO6) networks possess nondegenerate ground state 6A1 with the open-shell 2p63p63d5 electron configuration. Upon np core electron excitation final states np53d6 electron configurations are formed, thus leading to states with total spin S′ = 3/2 and S′ = 5/2, respectively. The three components of the angular momentum operator transform as Lxy, E and Lz, A1. As it is seen in Figure 3b the multiplet structure of the final state is constructed by at least three single electron excitations per multiplicity involving the npxy → 3dxz,yz, npxy → 3dx2−y2,z2 and npz → 3dz2, respectively. Low spin (LS) Fe(II) (or isoelectronic Co(III)) in distorted octahedral (C2v: FeS6, CoO6) networks possess nondegenerate ground state 1A1 with a closed-shell 2p63p63d6 electron configuration. Upon np core electron excitation, final states with np53d7 electron configurations are formed, thus leading to states with total spin S′ = 0 and S′ = 1. Under this symmetry the three components of the angular momentum operator transform as Lx, B2; Ly, B1; and Lz, A1. Hence, the multiplet structure of the final state is correspondingly constructed by at least two single electron excitations per multiplicity involving the npx,y,z → 3dx2−y2, z2 single electron excitations (Figure 3c). Finally, Co(II) in a cubic ligand field (Td: CoO4) has a highspin ground state 4A1 with the 2p63p63d7 electron configuration. Upon np core electron excitations, the resulting final state has the np53d8 configuration that leads to a complex final state multiplet structure that involves states with total spin S′ = 1/2 and S′ = 3/2. The multiplet structure is then constructed by single electron excitations of npx,y,z → 3dxy,xz,yz and npx,y,z → 3dx2−y2,z2 character (Figure 3d).

Figure 2. Graphical presentation of the transition-metal cluster models used in this study to model core-level absorption spectra of (a−e) TiO2, (f−h) α-Fe2O3, (i−k) FeS2, and (l and m) Co3O4. In each case, the symmetries of the basic building blocks are provided schematically along with the symmetry label. Color code: Ti, pink; Fe, cyan; Co, violet; O, red; and S, yellow.

coordination environment of the α-Fe2O3. It should be noted that, in α-Fe2O3, the high-spin (S = 5/2) Fe sites are ferromagnetically coupled within a single layer and antiferromagnetically coupled between layers. This suggests that a proper treatment of such systems might require taking into account at least to some extent the magnetic coupling between adjacent sites.79 It has been shown recently12 that the experimental Fe M-edge XAS spectrum of α-Fe2O3 indicates the presence of high-spin Fe(III) centers. Hence given the primarily local nature of corelevel excitations, as well as for computational convenience, we have chosen to treat these clusters if all Fe sites are ferromagnetically coupled (e.g., each Fe(III) center carries five unpaired electrons). Iron(II) sulfite (pyrite) crystallizes in the cubic Pa3̅ space group.66 The crystal lattice is composed from iron ions and the disulfite S22− moieties. All Fe−S bonds in the FeS6 octahedron are of the same length (2.263 Å); however, variation on the magnitude of the bond angles lowers the local symmetry to C2v and lifts the orbital degeneracy. As shown in Figure 2 and Table 1, cluster models containing up to 19 iron centers ([FeS12]10‑, [Fe6S38]26‑, and [Fe19S88]48‑) were chosen to represent the FeS2 coordination environment. Cobalt(II, III) oxide is a mixed valence compound (spinel) in which both Co2+ and Co3+ ions are present in 1:2 stoichiometry; thus, it is sometimes referred to as Co[Co2O4]. The crystal structure possesses a cubic Fd3̅m space group.64 The structure of this oxide is especially interesting due to the fact that the cobalt centers exist in oxidation states II and III forming corner sharing Co(II)O4 and Co(III)O6 A and B sites. In fact, high spin Co2+ ions occupy the tetrahedral A sites (magnetic moment of 3.02 μB), while in the octahedral B sites low spin Co3+ ions are present.80 The spin-pairing difference is mainly due to small ligand-field splitting at the tetrahedral site as well as lower oxidation state of cobalt present in this unit. Despite the differences in the oxidation and spin states, the

5. EXPERIMENTAL SPECTRA The experimental L- and M-edge XAS spectra for TiO2 rutile, FeS2 pyrite, α-Fe2O3 hematite, and the spinel Co3O4 have been previously reported.8,12,82−86 The respective XAS spectra are visualized collectively in Figure 4. 4323


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Figure 3. MO splitting diagram of the local (a) Ti(IV)O6 rutile (D2h), (b) Fe(III)O6 (C3v), (c) Fe(II)S6 and Co(III)O6 (C2v), and (d) Co(II)O4 (Td) cores. In addition, the adapted 2p/3p-3d transitions in the one electron particle/hole approximation are visualized assuming ideal D2h, C3v, C2v, and Td ligand field splitting, respectively.

The Ti M-edge XAS spectrum of rutile (Figure 4a) consists essentially of two broad bands located at 46 eV and 38 eV. The respective experimental L-edge XAS spectrum consists of seven distinguishable peaks. In particular, the L3 region contains two satellite signals located at 456−457 eV, a high intensity signal at 458 eV, and two overlapping signals at 460 and 461 eV. On the other hand, the L2 region is less informative containing only two well-separated signals at 463.5 and 466.5 eV. The intensity ratio of these overlapping signals is characteristic to the polymorph being investigated.87−89 The experimental Fe M-edge XAS spectrum of the iron(III) oxide α-Fe2O3 (Figure 4b) has an intense band located at 57.5 eV and two lower energy signals located at 53.5 and 55.5 eV respectively. Two spectral features dominate the L3-edge located at 707.5 and 709 eV. In addition a higher-energy tail is observed between 710 and 715 eV. Similarly, the L2 peak is split into two relatively broad signals at 721 and 722.5 eV. The higher energy tail is not well resolved. The recorded spectra of the pyrite are shown in (Figure 4c). In comparison to iron(III) oxide both Fe M- and Fe L-edge XAS spectra of the iron(II) sulphite are observed at lower energies (∼1 eV) due to the smaller effective charge on Fe. M-edge as well as the L2-edge XAS spectra consist of only one broad signal located at 57 and 720 eV, respectively. The L3-edge XAS spectrum contains two overlapping signals located

Figure 4. Experimental M- and L-edge XAS spectra of all examined solids. 4324


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distribution R(r)np the 3p orbitals are much more diffuse than the 2p orbitals. An example is provided in Figure 5 for the Ti4+ ion. In addition, the SOC integrals roughly scale as ⟨r−3⟩np, where ⟨r−3⟩np is the area under the curve of the radial

at 707 and 708 eV as well as a high-energy tail in the range between 710 and 715 eV. Similarly, the Co M-edge XAS spectrum of the Co3O4 spinel is quite broad with the maximum located at 64 eV (Figure 4d) along with a low intensity prepeak around 58.5 eV. On the other hand, the experimental L3 edge spectrum shows two distinct signals located at 778 and 781 eV in 2/3 intensity ratios. There is also a less well-defined shoulder observed at about 785 eV. The L2 edge region is not well resolved, although it seems that it consist of two peaks, a major feature at 795 eV and a less intense feature about 1 eV higher in energy.

distribution function

Figure 5. Free-ion 2p and 3p radial distribution function for Ti4+ ions.

Table 2. B3LYP/def2-TZVP Calculated ⟨r−3⟩2/3p Values for the Ti, Fe, and Co Atoms in a.u.−3 Units 4+

Ti Fe2+/3+ Co2+/3+

⟨r−3⟩2p

⟨r−3⟩3p

64 112 128

8 14 16

r3

. By evaluating the ⟨r−3⟩2p and

⟨r−3⟩3p integrals for the Ti4+, Fe2+/3+, and Co2+/3+, it can be shown (Table 2) that the M-edge SOC effects are expected to be at least 1 order of magnitude smaller in comparison to the L-edge SOC effects. Thus, SOC induced splittings on the order of 1−1.5 eV are expected. A more quantitative picture is provided in Figures 6 and 7 which present a comparison of calculated L- and M-edge spectra with and without SOC state mixing. As it is seen in Figure 6 in the case of the Ti and Fe L-edge spectra of clusters [TiO6]8− and [FeO6]9− the correct space of the final state multiplets as well as the correct number of the various members of each spin multiplet, which results in the L3 and L2 components of the respective L-edge spectra, are only observed when SOC is included in the calculations. Likewise in the case of the calculated M-edge spectra presented in Figure 7 the influence of SOC on the shape of the spectral envelope is strongly pronounced. Qualitatively, in the case of [FeS12]10‑ and [Co3O12]16‑ clusters inclusion of SOC results in a shift of the nonrelativistic spectrum to higher and lower energies respectively. On the other hand in the case of [TiO6]8− and [FeO6]9− clusters inclusion of SOC results in a redistribution of intensity throughout the spectral envelope. 6.2. Quantitative Analysis of the Calculated Features in M and L-Edge Spectra. 6.2.1. Closed Shell Clusters (TiO2 Rutile and FeS2). The ROCIS/DFT (or the PNO-ROCIS/ DFT) calculated M-edge and L-edge XAS spectra for the cluster models [TiO 6 ] 8− , [Ti 7 O 29 ] 30‑ , [Ti 13 O 58 ] 64‑ , [(Ti21O71)]58‑, [(Ti31O100)]76 [(Ti52O160)]116‑, [FeS12]10‑, [Fe6S38]26‑, and [Fe19S88]48‑ are presented in Figures 8 and 9. As it is seen in the case of TiO2, satisfactory agreement with the respective experimental spectra is obtained only when clusters with more than six Ti centers are considered while in the case of FeS2 the employed models provide similar spectra that are all in good agreement with the experiment. This implies that in the case of FeS2 the calculated spectra converge fast with respect to the size of the employed cluster. In fact, the mean absolute errors (MAE) in the areas between the

6. THEORETICAL CALCULATIONS 6.1. Spin−Orbit Coupling Effects. Prior to the analysis of the calculated spectra we emphasize the importance of the inclusion of SOC in the calculations. As it is shown in Figure 4 in the case of the metal L-edge XAS spectra, the explicit treatment of SOC is evidently necessary due to the experimental observation of the L3 and L2 components. However, its impact in the M-edge region is less clear. Based on their radial

atom

|R(r )np |2 r 2

Figure 6. Comparison of Ti and Fe L-edge spectra calculated without (red) and with (blue) spin−orbit of clusters [TiO6]8− and [FeO6]9−. 4325


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Figure 7. Comparison of Ti, Fe, and Co M-edge spectra calculated without (red) and with (blue) spin−orbit coupling for clusters [TiO6]8− and [FeO6]9−, [FeS12]10‑, and [Co3O12]16‑.

Figure 8. Experimental (black) versus calculated ROCIS/DFT or PNO-ROCIS/DFT M- and L-edge spectra for clusters [TiO6]8− (blue), [Ti7O29]30‑ (red), [Ti13O58]64‑ (green), [(Ti21O71)]58‑ (purple), [(Ti31O100)]76‑ (dark red), and [(Ti52O160)]116‑ (orange). The percentages in parentheses are the MAE in the areas between the calculated M- and L-edges spectra of the various cluster models when compared to the respective spectra of the larger [(Ti52O160)]116‑ cluster.

Figure 9. Experimental (black) versus calculated DFT/ROCIS M- and L-edge spectra for clusters [FeS12]10‑ (blue), [Fe6S38]26‑ (red), and [Fe19S88]48‑ (purple). The percentages in parentheses are the MAE in the areas between the calculated M- and L-edges spectra of the various cluster models when compared to the respective spectra of the larger [Fe19S88]48‑ cluster.

calculated M- and L-edges spectra of the cluster models [FeS12]10‑and [Fe6S38]26‑ when compared to the respective spectra of the larger [Fe19S88]48‑ cluster range between 2 and 3%. On the other hand in the case of TiO2 the calculated M- and L-edges spectra converge quite slowly with the size of the employed clusters. In fact the MAE errors of both M- and L-edge XAS spectra of the [TiO6]8−, [Ti7O29]30‑, [Ti13O58]64‑, [(Ti21O71)]58‑, and [(Ti31O100)]76‑ clusters range between ∼46% and ∼5% when compared to the respective M- and L-edges XAS spectra of the largest [(Ti52O160)]116‑ cluster and drop below 4% on clusters composed by 31 Ti atoms. In particular, as it is seen in Figures 8 and 10 the intensity of the low energy band (1) in the M-edge region (∼43 eV), the

position of the overlapping bands (2 and 3) in the L3 (460 and 461.5 eV) and L2 (464.5 and 466 eV) regions, and intensity ratios in general can only be successfully predicted by considering contributions from all Ti centers in clusters [Ti7O29]30‑, [Ti13O58]64‑, [(Ti21O71)]58‑, [(Ti31O100)]76‑, and [(Ti 52O 160 )] 116‑ . This is in accordance with previous studies.22,46,87,88,90,91 In order to get further insight, we chose to analyze in detail the clusters [Ti7O29]30‑ and [Fe6S38]26‑. As it is seen in Figures 10 and 11 in both clusters, the subspectra arising from individual Ti and Fe centers resemble the spectra shape of the monometallic clusters [TiO6]8− and [FeS12]10‑. In [Ti7O29]30‑ the SOC state mixing is more pronounced in the L-edge than in the M-edge 4326


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Figure 10. ROCIS/DFT calculated M-edge (left) and L-edge (right) XAS spectra of the [Ti7O29]30‑ cluster model with excitations that originate either only from one titanium atom (bottom blue line) or from all transition metal atoms (middle red line) compared to the experimental spectrum of the bulk TiO2 (rutile, black). Single titanium atom spectrum is further decomposed into contributions from S′ = 0 (dark violet) and S′ = 1 (orange) spin states. Band assignment is performed on the basis of natural difference orbitals (NDOs), drawn with 0.03 au isosurface value. Only the acceptor NDOs are visualized. The spectra are convoluted with Gaussian widths between 0.6 and 0.9 eV.

generated by the [TiO6]8− cluster or from individual Ti centers deviate strongly from experiment while the agreement improves rapidly with the number of metal centers in the studied clusters. It can be concluded that in the case of TiO2 saturation of the cluster size is difficult to achieve and requires very large cluster models with more than 30 Ti centers. On the other hand in the case of FeS2 saturation of the cluster size is much faster and any cluster that considers the local octahedral coordination FeS6 environment will provide nearly identical spectra. This is directly related to the nature of the involved transitions in the two clusters, MMCT and MLCT, besides the 2/3p → 3d core to valence excitations. 6.2.2. Open Shell Clusters (α-Fe2O3 Hematite). The ROCIS/DFT (or the PNO-ROCIS/DFT) calculated M- and L-edge XAS spectra for the cluster models [FeO6]9−, [Fe4O20]28‑, and [Fe7O27]33‑ are presented in Figures 12 and 13. As it is shown in Figure 11 likewise to the TiO2 case for both M- and L-edge XAS spectra better agreement between theory and experiment is observed by considering excitations from all four iron atoms of the [Fe4O20]28‑ cluster (red spectra) or all seven

spectral features. In fact the individual Ti center L-edge spectral features are almost equally composed of singlet and triplet nonrelativistic states (47% and 53%, respectively) while the respective M-edge spectral features are primarily dominated by singlet nonrelativistic states. On the other hand, in cluster [Fe6S38]26‑ the individual Fe center M- and L-edge subspectra involve significant contributions from nonrelativistic states arising from electron configurations with total spin S′ = 0 and S′ = 1, respectively (M edge: 53%/47% and L-edge: 38%/62%). NDO analysis shows that bands 1−3 and 1−2 in clusters [Ti7O29]30‑ and [Fe6S38]26‑ are dominated by states involving local core to valence excitations (2p/3p → 3dxy, 2/3p → 3dxz,yz 2/3p → 3dz2,x2−y2 and 2/3p → 3dx2−y2, 2/3p → 3dz2 respectively). On the other hand, band 4 in cluster [Ti7O29]30‑ involves a metal to metal charge transfer (MMCT) type of excitation, while bands 3−4 in cluster [Fe6S38]26‑ involve a metal to ligand charge transfer (MLCT) type of excitation. In particular MMCT states involve donor NDOs between two different absorber metal centers to the same acceptor NDO. These interactions are not taken into account when only one metallic center is considered. In fact, the spectral features 4327


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Figure 11. ROCIS/DFT calculated M-edge (left) and L-edge (right) XAS spectra of the [Fe6S38]26‑ cluster model with excitations that originate either only from one iron atom (bottom blue line) or from all transition metal atoms (middle red line) compared to the experimental spectrum of the bulk FeS2. The differential spectrum plotted with the dashed olive line was calculated by subtracting the normalized spectrum of a single atom origin from the spectrum obtained for the full cluster. The single iron atom spectrum is further decomposed into contributions from S′ = 0 (dark violet) and S′ = 1 (orange) spin states. Band assignment is performed on the basis of natural difference orbitals (NDOs), drawn with 0.03 au isosurface value. Only the acceptor NDOs are visualized. The spectra are convoluted with Gaussian widths between 0.6 and 0.9 eV.

iron atoms of the [Fe7O27]33‑ cluster (purple spectra). Once again the calculated M- and L-edge XAS spectra arising from only one and from seven iron centers deviate substantially (MAE7−1 = 41.8% and 54.8%, respectively). This is mainly due to the spectral features located at 54.2 and 712 eV of the M- and L-edge XAS spectral regions, respectively. The MAE errors of both M- and L-edge XAS spectra of the [Fe4O20]28‑ cluster drop below 7% and 1% when compared to the respective M- and L-edges XAS spectra of the largest [Fe7O27]33‑ cluster. In order to get further insight we choose to analyze in detail cluster [Fe4O20]28‑. As it is seen in Figure 13 bands 1−6 in the M- and L-edge XAS spectra regions are dominated by states with high spin electron configurations (S′ = 10). According to the NDO analysis (provided also in Figure 13) bands 1, 2, 4, 5

and 3, 6 in the Fe M- and L-edge spectra are dominated by states involving core to valence single electron excitations (2p/3p → 3dxy,xz,yz and 2p/3p → 3dz2,x2−y2 respectively) in accordance with expectations from particle-hole theory presented in Figure 3. However, as indicated by the metal composition of the acceptor NDOs, the M-edge 3p → d excitations are quite delocalized containing significantly enhanced MMCT character relative to the respective L-edge excitations. Once again analysis of the donor NDOs show that the MMCT states involve electron excitations from different absorber metal centers to a common acceptor space involving those absorber centers. 6.2.3. Mixed Valence Clusters (Spinel Co3O4). As discussed in the Geometric Structures section, the minimum cluster models to represent the spinel Co3O4 coordination environment 4328


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excitation vs MLCT or LMCT) and (2) the multiplet structure of the final states. For the clusters [Ti7O29]30‑, [Fe6S38]26‑, [Fe4O20]28‑, and [Co3O12]16‑, the relevant results are summarized in Tables 3 and 4. As it is seen in both M- and L-edge regions, ∼75−85% of the calculated spectral features are associated with states involving single electron excitations into the valence 3d, 4s, and 4p molecular orbitals, while an additional 15−25% involves states with MLCT or MMCT characters. In the cases where the calculated spectra are dominated by combinations of core to valence and MMCT states, both types of states contribute intensity to both the low- and high-energy bands of the calculated spectra. In all of the studied cases, MMCT type states involve electron excitations from two or more different absorber metal centers into a common acceptor range of orbitals involving all absorber centers. As a result, saturation of the spectral features requires larger cluster models in order to properly cover the nonlocal effects. By contrast, in the cases where the calculated spectra are dominated by combinations of core to valence and MLCT states, it was shown that the core to valence states primarily contribute to the low energy bands of the spectra while the MLCT ones dominate the high energy spectral regions. MLCT states involve only one absorber metal center; hence, saturation of the spectral features depends on the locally probed coordination environment around the metal centers and much less on the size of the employed cluster or the number of the treated metal centers. As it is collectively seen in Tables 3 and 4, state combinations with core to valence and MMCT characters lead to M-edge XAS spectra with relative simple multiplet structure in which the relativistically corrected states are dominated by nonrelativistic states with total spin S′ = S. This in accordance with the fact that in general the SOC effects are more pronounced in the L-edge spectra in comparison to the respective M-edge XAS spectra. As it is shown in Tables 3 and 4, this is for example the case of the M-edge XAS spectra of the clusters representing the TiO2 rutile, α-Fe2O3 hematite, and Co3O4 spinel solid environments. However, the respective L-edge XAS spectra or the M- and L-edge XAS spectra involving combinations of core to valence and MLCT states have complicated multiplet structures in which all parent nonrelativistic states with all possible total spins S′ = S and S′ = S ± 1 may similarly contribute. As it is seen in Table 4, this is particularly the case for cluster [Fe6S38]26‑ in which the Fe M-edge XAS spectrum contains significant contributions from nonrelativistic states with total spin S = S′ (38%) and S = S′ ± 1 (62%), respectively. Finally there are several experimental XAS spectral features in the M-edge region like for example those located at around 38 eV in the case of TiO2 and at around 53.5 eV in the case of Fe2O3 the relative intensity and energy position of which are not correctly reproduced by the ROCIS/DFT method. This is due to the fact that in the current ROCIS/DFT implementation the configuration interaction space is saturated with single configuration state functions. Hence it is expected that the ROCIS/DFT will not be able to describe the correct energy position and relative intensities of spectral features for which treatment of dynamical correlation to include the effect of doubles excitations is essential. This implies the need to improve the physical content of the ROCIS methods by concentrating on a more detailed treatment of dynamic electron correlation effects.

Figure 12. Experimental (black lines) versus ROCIS/DFT and PNOROCIS/DFT M- and L-edge calculated XAS spectra generated by considering one Fe center in clusters [FeO6]9−, [Fe4O20]28‑, or [Fe7O27]33‑ (blue line) and by considering all Fe atoms in clusters [Fe4O20]28‑ (red line) and [Fe7O27]33‑ (purple line). The percentages in parentheses are the MAE in the areas between the calculated M- and L-edges spectra of the various cluster models when compared to the respective spectra of the larger [Fe7O27]33‑ cluster.

are the [Co3O12]16‑ and [Co9O30]30‑ which contain simultaneously tetrahedrally coordinated Co(II)O4 as well as octahedrally coordinated Co(III)O6 centers. Figures 13 and 14 show the ROCIS/DFT calculated M-edge and L-edge XAS spectra for the cluster models [Co3O12]16‑ and [Co9O30]30‑. As it is shown in Figure 14 the calculated M- and L-edge spectra arising from three or nine cobalt centers deviate substantially (MAE9−3 = 16.7% and 56.8%). The calculated M- and L-edge XAS spectra are further deconvoluted in terms of individual contributions arising from the tetrahedrally coordinated Co(II)O4 and the octahedrally coordinated Co(III)O6 centers. In Figure 14 this analysis is visualized for the [Co3O12]16‑ cluster. As it is seen although the M-edge XAS spectrum is mainly dominated by the Co(II)O4 centers, the corresponding L-edge XAS spectra contain contributions from both the Co(II)O4 and Co(III)O6 centers. In particular, by focusing on the L3 region, the signal located at 778 eV is dominated solely by Co(II)O4 centers whereas the one at 781 eV is dominated by Co(III)O6 ones. Moreover according to the NDO analysis, performed at the bottom of the Figure 15, the calculated M- and L-edge signals (bands 1−4) are mainly dominated by core to valence single electron excitations (2/3p(Co(II)) → 3dxy,xz,yz and 2/3p(Co(III)) → 3dz2,x2−y2). On the other hand the M-edge signal located at 64 eV contain contributions from both 3p(Co(II)) → 3dxy,xz,yz and 3p(Co(III)) → 3dz2,x2−y2 excitations. It should be noted that in both edges bands 2 and 4 show significant MMCT characters. As it is shown in Figures 13 and 14, the [Co3O12]16‑ cluster contains only one Co(II) center, and hence better agreement between theory and experiment is obtained when the [Co9O30]30‑ cluster is employed. This cluster contains multiple Co(II) and Co(III) centers, and thus it can better describe the contributing states of MMCT character. 6.3. Insights into the Intensity Mechanism of M- and L-Edge Spectra of Solids. In the above section it was shown that the energy position and the relative intensity of the calculated spectral features of the M- and L-edge XAS spectra are greatly influenced by (1) the general character (local 4329


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Figure 13. ROCIS/DFT calculated M-edge (left) and L-edge (right) XAS spectra of the [Fe4O20]28‑ cluster model with excitations that originate either only from one iron atom (bottom blue line) or from all transition metal atoms (middle red line) compared to the experimental spectrum of the bulk α-Fe2O3. The differential spectrum plotted with the solid olive line was calculated by subtracting normalized spectrum of a single atom origin from the spectrum obtained for the full cluster. The single iron atom spectrum is further decomposed into contributions from the same (S′ = 10, dark violet) and lower spin states (S′ = 9, orange). Band assignment is performed on the basis of natural difference orbitals (NDOs), drawn with 0.03 au isosurface value. Only the acceptor NDOs are visualized. The spectra are convoluted with Gaussian widths between 0.6 and 0.9 eV.

7. CONCLUSIONS In this work we have presented a systematic study of the metal M- and L-edge X-ray absorption spectra of a number of closedand open-shell transition metal solids, namely, TiO2 rutile, α-Fe2O3 hematite, FeS2 pyrite, and the Co3O4 spinel solids by employing the embedded cluster approach together with ROCIS/DFT or the PNO-ROCIS/DFT methods. In all studied cases satisfactory agreement between theory and experiment was observed. It was shown that in all cases the multiplet effects are more pronounced in the L-edge than in the respective M-edge XAS spectral features. Both M- and L-edge XAS spectra proved to be sensitive probes of the local coordination environment and the metal oxidation states of the cobalt centers in the Co3O4 spinel solid. It was shown that the natural difference orbitals (NDOs) can be used within the ROCIS/DFT framework to qualitatively and quantitatively analyze the states dominating the calculated spectral features. In both M- and L-edge spectra the dominating states involve metal core to valence single electron excitations while certain calculated features contain significant MLCT and MMCT character. In particular it was

Figure 14. ROCIS/DFT M- and L-edge differential XAS spectra (left and right, respectively; green lines) calculated between small ([Co3O12]16‑, dark cyan line) and large ([Co9O30]30‑, red line) Co3O4 models. Experimental spectra are shown for reference in black. The spectra are convoluted with constant Gaussian broadening 1.2 eV. 4330


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Figure 15. ROCIS/DFT calculated M-edge (left) and L-edge (right) XAS spectra of the [Co3O12]16‑ cluster model with contributions from Co(II)O4 (dashed blue line) and Co(III)O6 centers (dashed green and dotted violet lines) compared to the experimental spectrum of the bulk Co3O4 (black). Band assignment is performed on the basis of natural difference orbitals (NDOs), drawn with 0.03 au isosurface value. Only the acceptor NDOs are visualized. The total spectrum presented either as a red or cyan solid line is a sum of subspectra from all transition metal ions with 0.8 and 1.2 eV constant Gaussian broadening.92

Table 3. Percentage Contribution of Various Excitation Classes to the L- and M-Edge XAS Spectra L-edge

M-edge

cluster

2p-3d

2p-4s/4p

MLCT

MMCT

3p-3d

3p-4s/4p

MLCT

MMCT

[Ti7O29]30‑ [Fe6S38]26‑ [Fe4O20]28‑ [Co3O12]16‑

69.2 78.2 68.3 75.3

4.0 1.1 17.7 7.1

2.4 16.7 1.9 1.1

20.5 0.5 12.2 16.3

62.6 71.4 62.2 60.5

5.7 1.0 11.7 16.2

3.3 23.5 2.0 2.3

22.0 1.0 23.2 19.5

Table 4. Percentage Contribution of Various Spin States S′ to the L- and M-Edge Spectra L-edge

a given cluster contribute similar spectra. In fact in the case of FeS2 solid, in which only core to valence and MLCT excitations contribute to the spectral features, the overall spectra resembled the subspectra contributed by each individual atom. In all other studied cases MMCT excitations were found to prominently contribute to the spectral envelope in both the M-edge, L3, and L2 XAS spectral regions. The presented calculation protocol shows that a simultaneous analysis of the M- and L-edge XAS spectra is instrumental to provide information regarding the local coordination environment oxidation and spin states of these solid chemical systems.

M-edge

cluster

S′ = S

S′ = S ± 1

S = S′

S′ = S ± 1

[Ti7O29]30‑ [Fe6S38]26‑ [Fe4O20]28‑ [Co3O12]16‑

47 53 89 68

53 47 11 32

89 38 98 97

11 62 2 3

shown that, unless the calculated spectral features contain significant MMCT contributions, the different metal centers in 4331


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Taken together, these results demonstrate that combined experimental and theoretical metal M- and L-edge X-ray spectroscopy provides a strongly predictive analytical tool for applications in the field of solid-state catalysis. Further developments in our laboratory are currently on going in order to explore the capabilities and the applicability of such analytical tools.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Josh Vura-Weis: 0000-0001-7734-3130 Dimitrios Maganas: 0000-0002-1550-5162 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.K., D.M., and F.N. acknowledge financial support from the Max-Planck Society. A.K. acknowledges partial support from the National Science Centre, Poland Grant No. 2015/17/D/ ST4/00112. This material is based upon work supported by the National Science Foundation under Grant No. 1555245 (to J.V.-W.). M.V. acknowledges support from the Springborn Graduate Fellowship. The reviewers of the manuscript are acknowledged for their constructive comments.

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REFERENCES

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