computer programs
CTM4DOC: electronic structure analysis from Xray spectroscopy
^{a}Inorganic Chemistry and Catalysis Group, Debye Institute for Nanomaterials Science, Utrecht University, Universiteitsweg 99, Utrecht 3584 CG, The Netherlands, and ^{b}Department of Chemistry, University of Illinois at UrbanaChampaign, Urbana, IL 61801, USA
^{*}Correspondence email: f.m.f.degroot@uu.nl
Two electronic structure descriptions, one based on orbitals and the other based on term symbols, have been implemented in a new Matlabbased program, CTM4DOC. The program includes a graphical user interface that allows the user to explore the dependence of details of electronic structure in transition metal systems, both in the ground and corehole excited states, on intraatomic electron–electron, crystalfield and chargetransfer interactions. The program can also track the evolution of electronic structure features as the crystalfield parameters are systematically varied, generating Tanabe–Suganotype diagrams. Examples on firstrow transition metal systems are presented and the implications on the interpretation of Xray spectra and on the understanding of lowspin, highspin and mixedspin systems are discussed.
Keywords: multiplet simulations; electronic structure; differential orbital covalency; Xray spectroscopy.
1. Introduction
Ledge is a powerful method for the determination of the electronic and magnetic structure of transition metal ions in molecules and solids. The shapes of Ledge Xray absorption spectra are dominated by the excitation of 2p electrons to empty 3d states and can be accurately modelled and interpreted using crystalfield multiplet theory applied to transitions from a 3d^{N} groundstate configuration to a 2p^{5}3d^{N+1} finalstate configuration. This crystalfield multiplet model has been developed by Thole and coworkers (Butler, 1981; Cowan, 1981; de Groot et al., 1990; de Groot & Kotani, 2008; Thole et al., 1988) and forms the basis of the CTM4XAS interface (Stavitski & de Groot, 2010). Systems that have increased covalency cannot be accurately described by crystalfield theory and require the inclusion of chargetransfer channels in their description. In molecular systems, charge transfer can be strongly angulardependent, an observation that led to the development of differential orbital covalency (DOC) simulations, originally applied to inorganic and bioinorganic systems (Hocking et al., 2006, 2007, 2009, 2010; Lundberg et al., 2013; Wasinger et al., 2003). Recently, a number of firstprinciple routes have been developed for solidstatebased methods by the groups of Haverkort (Haverkort et al., 2012, 2014), Hariki (Hariki et al., 2013, 2015) and Ikeno (Ikeno et al., 2011). In the case of molecules, firstprinciple methods are mainly based on restricted active space and they have been developed in the groups of Neese (Roemelt et al., 2013; Maganas et al., 2013), Lundberg (Lundberg et al., 2013) and Odelius (Pinjari et al., 2014).
Multiplet analysis is applied in many experimental studies. It is typically used to determine the valence, the spin state and the crystalfield parameters, by tuning the parameters to optimize the fit with experimental spectra. In a covalent system the DOC is also determined. The electronic states of a transition metal ion in a cubic crystal field can be defined in a variety of descriptions. In an orbital description (strongfield representation), the electronic state is described in terms of occupation numbers of t_{2g} or e_{g} orbitals. Under such a description, the ground state of a highspin 3d^{5} system, for instance, can be described as t_{2g}^{ 3}e_{g}^{ 2}. However, the electronic structure of an openshell system is dominated by strong electron–electron interactions, resulting in nonintegral occupation of oneelectron valence orbitals. In such a system, a more convenient alternative description involves expressing electronic states in terms of their atomic components. This weakfield description shows the relative contributions of atomic term symbols in the ground state. In most cases the term symbol notation in the symmetry of the transition metal ion is most useful as it combines the atomic twoelectron integrals with cubic crystalfield effects. It is important to note here that the term symbol description is only a label and an electronic state usually must be described as a linear combination of term symbols, where the 3d spin–orbit coupling causes a mixture between different spin states.
Fig. 1 gives a systematic description of the main interactions and their consequences on symmetry, including the term symbol notations. If only the crystal field is included, the 3d orbitals are split into t_{2g} and e_{g} manifolds and the state can then be described just as a linear combination of orbital components. Using a 3d^{7} highspin system as an example in Fig. 1, the ground state is then described simply as t^{5}e^{2}. The second interaction is given by twoelectron integrals that give rise to the atomic components (^{4}F). Together, the crystal field and the twoelectron integrals create the cubic term symbols (^{4}T_{1}). The cubic term symbols can be decomposed into their (range of) orbital components or their corresponding atomic components.
The third interaction to consider in Fig. 1 is the 3d spin–orbit coupling, which splits the electronic states in terms of their total angular momentum (J) components. Combined with the twoelectron integrals this yields the atomic term symbols. Combining all three interactions together yields the total term symbol J = Γ_{6} that can be decomposed into its cubic term symbol components, for example α^{4}T_{1}〉 + β^{2}E〉. Alternatively, it can be decomposed into its atomic term symbols. Subsequently, the total term symbol can be developed into its orbital, atomic and J components.
The three interactions described above define one localized 3d^{N} configuration. Inclusion of charge transfer causes this configuration to be mixed with a second configuration 3d^{N+1}L. This second configuration itself can also be decomposed in an analogous manner as the first configuration. After inclusion of charge transfer, a given electronic state can be separated into its base and chargetransfer configuration and each of these configurations can be separated into their three specific components (orbital, atomic and J components), together yielding a picture of the local structure of a transition metal ion. In the general case, this picture must be extended with more ligandtometal chargetransfer configurations, metaltoligand chargetransfer configurations, p–d hybridizations and metaltometal chargetransfer configurations. In solids, translational symmetry will modify this picture further.
To represent the ground and excited states in terms of any of the abovementioned descriptions, a projection method first reported in 2003 is utilized in CTM4DOC (Fig. 2), a new Matlabbased program where three descriptions are implemented to characterize electronic states of transition metal complexes: one that uses a linear combination of atomic term symbols; another one that uses a linear combination of crystalfield term symbols of cubic symmetry; or, for an orbital description, a strongfield representation defined as linear combinations of crystalfield configurations,
To calculate the coefficients α_{ij} in this expansion, this projection method involved a dummy 1sto4p transition (both being spectator orbitals decoupled from the 2p–3d system) between each of the groundstate multiplets (Ψ_{g,i}) and each of the ideal crystalfield configurations, which were constructed by setting to zero all atomic parameters (Slater integrals and spin–orbit coupling parameters). Thus, the resulting calculated intensity obtained for each transition is directly proportional to the expansion coefficients, α_{ij}. By representing the groundstate multiplets according to this expansion, the evaluation of metal3d covalency (of t_{2} and e orbitals) allows the comparison with groundstate density functional theory (DFT) calculations. More recently (Kroll et al., 2015), a similar approach has been used to express the multiplets of the final state (Ψ_{F,j}) of a 2p–3d radiative process in terms of the corresponding excitedstate crystalfield configurations,
From this, the evaluation of the spin state and of the DOC, based on the contributions of chargetransfer configurations with respect to corresponding contributions of crystalfieldbased configurations, has been successfully applied in several studies (Hocking et al., 2006, 2007, 2009, 2010; Wasinger et al., 2003).
CTM4DOC also calculates the metal3d covalency for the ground state and performs a spectral deconvolution in terms of the various orbital contributions, based on the obtained projections for the groundstate and the finalstate multiplets involved in the 2p–3d transition. Additionally, the program calculates Tanabe–Sugano diagrams for the multiplets in the ground and excited states with respect to variations of a given floating parameter (typically related to the crystal field), which is a useful tool for interpretation in optical, and Xray spectroscopies. A quick reference manual for CTM4DOC can be found in the supporting information^{1}.

2. Groundstate projections
The singlepoint calculation of the ground state involves the unoccupied d shell of a transition metal, for which only the F^{2} and F^{4} Slater integrals and the dspin orbit coupling constant are relevant atomic parameters. The F^{2}/F^{4} Slater integrals describe the ddinteractions and they can be rewritten into the Racah B and C parameters (Griffith, 1961).
Fig. 3 shows the three projections implemented in CTM4DOC for Fe^{2+} in three different scenarios under O_{h} symmetry for the ground state. First, the valence spin–orbit coupling and the crystalfield are set to zero. The ground state is a linear combination of t_{2g}^{ 4}e_{g}^{ 2} and t_{2g}^{ 3}e_{g}^{ 3} in 60:40 proportion, corresponding to 100% of the cubic term symbol ^{5}T_{2} and to the atomic term symbol ^{5}D. Then, when the valence (3d) spin–orbit coupling is set to its atomic value, the projection indicates a small amount of the crystalfield configurations t_{2g}^{ 5}e_{g}^{ 1} and t_{2g}^{ 2}e_{g}^{ 4} mixed in, corresponding to a small amount of singlet ^{1}T_{1}. Finally, after turning on the crystal field (with a value of 10Dq = 1.3 eV), the projections indicate almost a pure crystalfield configuration, t_{2g}^{ 4}e_{g}^{ 2}, corresponding to the ^{5}T_{2} crystalfield term symbol and the ^{5}D atomic term symbol. An additional example for Co^{2+} is discussed in the supporting information.
To account for covalency, CTM4DOC can also model the crystalfield projection with chargetransfer configurations. In the current version of CTM4DOC, only ligandtometal chargetransfer (LMCT) parameters can be used, which implies that only the modelling of covalency for donor ligands is possible. In addition, only one chargetransfer state (d^{N+1}L) is currently considered, which is sufficient to account for bonding in most molecular systems with only σ donor ligands. For other complexes, like in some solids, additional chargetransfer states (e.g. d^{N+2}L^{2}) may be required and are not currently implemented. Thus, for chargetransfer calculations the ground state is expressed as an expansion of crystalfield configurations 3d^{N} and of chargetransfer crystalfield configurations 3d^{N+1}L,
Fig. 4 shows an example for the ground state of FeCl_{4}^{−} with parameters given in Table 1, under two different descriptions: (a) crystalfield configurations including LMCT and (b) the corresponding metal3dbased molecular orbitals, revealing their orbital covalency, which could be potentially of great value for structure validation. In this regard, we propose that in combination with fitting of multiplet simulations to experimental data, metal3d covalency can be extracted by performing a followup simulation in CTM4DOC for the ground state, using the fit parameters. We recognize that the fitting (manual or automatic) can lead to multiple solutions. Furthermore, these multiple fits may not only involve the uncertainties related to each multiplet simulation parameter but also include correlations between fit parameters. This implies that the method should involve the evaluation of metal covalencies with uncertainties derived from the fits. Nevertheless, the comparison of these empirically extracted covalencies from experimental data (via the fits) with those calculated by DFT can be used to validate structural models. This combined approach could be an alternative to ab initio calculations applied to spectroscopic techniques subject to multiplet effects where calculation times are prohibitive.
To further illustrate the method, we revisit here examples from a previous study (Wasinger et al., 2003), where the experimental Fe Ledge for a series of well characterized Fe complexes were manually fit with chargetransfer multiplet simulations; the fit parameters are then used to project the ground state into a linear combination of crystalfield configurations and to calculate the DOC; and finally the results are compared with DFT calculations. However, in the original study, the results obtained for the corresponding crystalfield projections are scaled according to the integrated intensity over the Ledge spectra. Here, we omit this step, as we realise now that the projection of the ground state should be independent of any observed intensity. Remarkably, the metal3d covalency obtained for t_{2} and e orbitals using CTM4DOC and the parameters given in Table 1 (from the manual Ledge fits to experiment) are in excellent agreement with the covalency values obtained from their original DFT calculations (see Table 1). To explore the effects of changing the functional, we have performed new DFT calculations for the tetrachloro and hexachloro iron complexes using BP86 and B3LYP. §S2 of the supporting information presents the results from these calculations and their comparison with the values given in Table 1.
3. Finalstate projections
In the current version of CTM4DOC, only the final state related to Ledge in the absence of chargetransfer effects can be calculated. Moreover, only one of the descriptions in terms of crystalfield configurations [as in equation (2)] is available in this version of the program.
In the current version we have implemented a projection for Ledge which shows the total spectrum and a deconvolution in terms of specific transitions to t_{2} and e orbitals in cubic symmetry (or the corresponding orbitals in D_{4h} symmetry). This is useful to interpret the spectrum in terms of a singleparticle model. To accomplish this, each of the multiplets that are allowed in the final state are expanded according to equation (2), and the corresponding groundstate multiplet expanded according to equation (1). Then, the dipole transition integral for each j transition is expressed in terms of such expansions to obtain equation (4),
We note here that any term involving a transition that changes any occupation number by more than 1 is zero. This does not mean that twoelectron processes are not occurring in Ledge Instead, equation (4) explicitly reveals the mechanism from which such transitions are possible, under an orbital description of the involved electronic states. We also note that this description holds under the consideration that any oneelectron transition, which essentially involves putting a 2p electron into a t_{2} orbital [first term of equation (4)] or into an e orbital [second term of equation (4)], has the same 2pto3d oscillator strength (integrated over all x, y and z directions and averaged over all 2p donor orbitals relevant to each component). Then it follows that the intensity projection for each j transition in an Ledge spectrum, which corresponds to transitions to t_{2} or e orbitals, is given by equations (5) and (6), respectively,
Moreover, an interesting observation, which was first apparent in a recent study (Kroll et al., 2015) focusing on demonstrating the projection of the final state in the Ledge of Ti^{4+}, is that in octahedral symmetry, out of the 25 multiplets emerging from the 2p^{5}3d^{1} final state, which collectively carry the expected 6:4 ratio (consistent with the ratio of t_{2}:e holes in a 3d^{0} ground state), only seven are allowed by the electric dipole in Ledge which yields a different t_{2}:e ratio than the one expected based on the number of holes (see Fig. 5b). In other words, the electric dipole operator does not allow transitions to t_{2} and e states in the same proportion under cubic symmetry, and overall there is a preference for e states, likely related to the fact that the relative orientation of the dipole components +1, −1 and 0 of light being parallel to the orientation of orbitals with e symmetry (d_{z 2} and d_{x 2y 2}), and perpendicular to orbitals with t_{2} symmetry. Further, equation (4) provides a nice way to visualize also how multiplet effects promote the mixing of configurations which causes the loss of t_{2} character, again under the assumption that the oscillator strength for individual 2p to 3d transitions (regardless of the dsymmetry) is constant. We note from Fig. 5(a) that, under spherical symmetry (atomic) for d^{0} systems, each multiplet in the final state is composed of exactly 60% and 40% which preserves the expected t_{2}:e ratio. We also observe a recovery of this t_{2}:e intensity ratio as the number of multiplets increase in the final state, like in the case of Fe^{3+} (see Fig. 6) in comparison with Ti^{4+}. The same effect is observed when the symmetry is further lowered, for example from O_{h} to D_{4h}, as shown in Fig. 5(c) where the ratio of e + b_{2} (collectively t_{2} in O_{h}) and a_{1} + b_{1} (collectively e in O_{h}) intensities of ∼55:45 start approaching a 60:40 ratio.
We anticipated also that this ratio should be close to the expected t_{2}:e ratio (based on number of holes) in second and thirdrow transition metals whose Ledge spectra become dominated by 2p spin–orbit coupling and do not affect the mixing of drelated states. However, the calculation of Ledge of second and thirdrow transition metals will only be available in future versions of CTM4DOC. Also, future versions of the program will include the projections in the final state that include LMCT and MLCT states. This will allow for the details on how the t_{2}:e ratio is effectively changed by LMCT and for the decomposition into the additional backbonding orbitals that come into play for MLCT calculations.
4. Tanabe–Sugano and singlepoint energy diagrams
The projections described in the previous sections, as implemented in CTM4DOC, are not only performed for the ground state but also for the rest of the multiplets within an initial or a finalstate configuration. The graphical user interface of CTM4DOC allows the visualization of these additional projections by choosing a different multiplet (other than the ground state) via an energy selector. Conveniently, an energy diagram displaying all multiplets is also calculated. Further, to explore the effect of one of the parameters on the energy and nature, in terms of the abovediscussed electronic structure descriptions, of all multiplets, CTM4DOC extends the same calculations to reproduce Tanabe–Sugano diagrams, from which individual energy diagrams and specific projections to a given state can be extracted and displayed.
This is potentially useful in many different types of spectroscopies. We emphasize here its applicability to soft Xray resonant inelastic Xray scattering (RIXS), where a quick comparison of the RIXS peaks with a groundstate Tanabe–Sugano diagram can reveal crystalfield and chargetransfer parameters (when applicable). In addition, many systems can exist in a state of spin admixture, which cannot be accurately described in terms of a single spinstate configuration. A Tanabe–Sugano diagram helps reveal the conditions under which such situations may arise and assists in providing detailed electronic structure descriptions. For instance, Fig. 7(a) shows a Tanabe–Sugano diagram for a Co^{2+} O_{h} complex in the range of energy for 10Dq of 1–3 eV and with Slater integrals taken at their atomic value. Then, the energy diagram of Fig. 7(b) corresponds to the cut shown in the Tanabe–Sugano near the crossing point (at 2.3 eV) and so are the projections shown for the ground state (highlighted in the energy diagram) in Figs. 7(c) and 7(d), corresponding to a crystalfield configuration description; and a cubic and an atomic term symbol descriptions, respectively.
We note that at a 10Dq of 2.3 eV, with Slater integrals and 3d spin–orbit coupling taking their atomic values, the ground state is a mixture of spin states. It consists, first, of 63% of the cubic doublet ^{2}E, originating from a mixture of mainly the atomic term symbols ^{2}G, ^{2}H, ^{2}D_{1} and ^{2}D_{2}; and, second, of essentially 37% of the quartet ^{4}T_{1}, originating from the 35% of the atomic term symbol ^{4}F and 2% of the atomic term symbol ^{4}P. This mixture of spin states is also apparent from the crystalfield description of Fig. 7(c), which correspondingly reveals 61% of the crystalfield configuration t_{2}^{ 6}e^{1}, 36% of the crystalfield configuration t_{2}^{ 5}e^{2} and 3% of the crystalfield configuration t_{2}^{ 4}e^{3}. An additional Tanabe–Sugano diagram example for Co^{3+} is discussed in the supporting information.
5. Conclusions
In the present manuscript we have introduced CTM4DOC, a Matlabbased program to perform electronic structure calculations for the analysis and interpretation of Xray spectra. Two electronic structure descriptions, one based on oneelectron orbitals and a second one based on multielectron atomic and cubic term symbols, have been implemented. The program is equipped with a graphical user interface that allows the user to explore the evolution of detailed electronic state descriptions in the ground and excited states of transition metal systems subject to changes in the atomic, crystalfield and chargetransfer parameters. Changes in the energies of the electronic states, as crystalfield parameters are changed, can be tracked and summarized as Tanabe–Sugano diagrams.
Several examples in transition metal systems have been presented to illustrate the different features of the program. Furthermore, we demonstrate how electronic systems that are often thought to be simply lowspin or highspin are really of mixedspin nature. This observation could lead to more accurate interpretations of experimental data and better rationale of physical and chemical properties. Moreover, we believe that CTM4DOC is a valuable tool to determine covalency from experiment; first, the set of parameters that best replicates experimental Xray data is determined and then a DOC calculation is performed using these parameters. This combined approach can also be very useful for structure validation when comparing the results directly with DFT calculations, especially for highly correlated systems where ab initio methods aimed to reproduce the experimental data are computationally expensive.
6. Related literature
The following references are mentioned in the supporting information: Becke (1988); DelgadoJaime & DeBeer (2012); Eichkorn et al. (1995); Eichkorn et al. (1997); Grimme (2004); Iikura et al. (2001); Klamt & Schüürmann (1993); Neese (2012); Pantazis et al. (2008); Perdew (1986); Szabo & Ostlund (1989); Weigend (2006); Yanai et al. (2004).
Supporting information
Supporitng text, figures and tables. DOI: https://doi.org/10.1107/S1600577516012443/hf5317sup1.pdf
Footnotes
^{1}A quick reference manual for CTM4DOC is provided. Along with it, the file CTM4DOC_v.1_examples.mat with all examples discussed here is also included. To visualize and explore this file, open via `Load Session' in CTM4DOC. CTM4DOC is directly available through our website. Additionally, §S2 presents the results from new DFT calculations performed for iron chloride complexes of Table 1 using two different functionals, BP86 and B3LYP.
Acknowledgements
MUD and FdG are thankful to the European Research Council (ERC) for their support under advanced grant XRAYonACTIVE (No. 340279), and for valuable discussions with Dr Thomas Kroll, Professor Edward Solomon, Professor Pierre Kennepohl, Professor Erik Wasinger and Professor Serena DeBeer. KZ and JVW are grateful to the Air Force Office of Scientific Research for support under AFOSR Award No. FA95501410314.
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