Strengths of covalent bonds in LnO2 determined from O K-edge XANES spectra using a Hubbard model

In LnO2 (Ln = Ce, Pr, and Tb), the amount of Ln 4f mixing with O 2p orbitals was determined by O K-edge X-ray absorption near edge (XANES) spectroscopy and was similar to the amount of mixing between the Ln 5d and O 2p orbitals. This similarity was unexpected since the 4f orbitals are generally perceived to be “core-like” and can only weakly stabilize ligand orbitals through covalent interactions. While the degree of orbital mixing seems incompatible with this view, orbital mixing alone does not determine the degree of stabilization provided by a covalent interaction. We used a Hubbard model to determine this stabilization from the energies of the O 2p to 4f, 5d(eg), and 5d(t2g) excited charge-transfer states and the amount of excited state character mixed into the ground state, which was determined using Ln L3-edge and O K-edge XANES spectroscopy. The largest amount of stabilization due to mixing between the Ln 4f and O 2p orbitals was 1.6(1) eV in CeO2. While this energy is substantial, the stabilization provided by mixing between the Ln 5d and O 2p orbitals was an order of magnitude greater consistent with the perception that covalent bonding in the lanthanides is largely driven by the 5d orbitals rather than the 4f orbitals.

For CeO2, the unperturbed ground state is |L 1 4f 1 ⟩, which consists of four wavefunctions y1-y4.This state corresponds to a "hole" in the O 2p A2u 1 orbital (L 1 ) and one electron in the Ce 4f A2u orbital (4f 1 ), which is the main component of the G7 state if spin orbit coupling is included.The energy of y1-y4 is assigned as 0. The excited state that interacts with |L 1 4f 1 ⟩ is |L 2 4f 0 ⟩ and consists of a single wavefunction y5.The energy of |L 1 4f 1 ⟩ is set to zero, and the energy of |L 2 4f 0 ⟩ is U'.The individual states are shown in Table S1.Only states in which the electron can hop from the occupied O 2p A2u orbitals into the Ce 4f orbital without changing sign may interact with y5; these states are y1 and y4.In other words, only y1 and y4 have non-zero, offdiagonal matrix elements.The other states cannot interact with y5 and are excluded from the model.
Table S1: A2u symmetry, L 1 4f 1 basis states for the Hubbard molecule model of LnO2.Only the states in red are used in the model.
The Hubbard molecule model Hamiltonian for the interaction between the |L 1 4f 1 ⟩ and |L 2 G7 0 ⟩ in CeO2 can be described using the following matrix, t is the electron hopping term (interaction integral), which is analogous to the off-diagonal matrix element, Hij, (orbital interaction integral) in MO theory, and U is the energy of the unperturbed excited state with respect to the ground state.

Hubbard molecule model for TbO2: O 2p and Ln 4f, A2u mixing
Unlike CeO2 and PrO2, the unperturbed ground state of TbO2 in the HMM involves a tetravalent lanthanide and has two electrons in the OA2u (L 2 ) orbital.In the unperturbed ground state, |L 2 4f 7 ⟩, the oxygen A2u orbital is doubly occupied and each of the f-orbitals is singly occupied.While the actual ground state is an octet, in the HMM, the ground state is simplified to a doublet: y1 is spin up and y2 is spin down.Likewise, in the unperturbed charge transfer state, the oxygen A2u orbital is singly occupied, and the corresponding orbital on Tb is double occupied: y3 is spin up and y4 is spin down.Only y3 can interact with y1, and only y4 can interact with y2.The energy of |L 2 4f 7 ⟩ is set to zero, and the energy of |L 1 4f 8 ⟩ is U'.

Figure S1
. HMM for the 4f interaction in TbO2.
The Hubbard molecule model Hamiltonian for the 5d Eg orbitals in LnO2 can be described using the following matrix for each of the two spatially independent wavefunctions y1 and y2.
and t is the electron hopping term (interaction integral), which is analogous to off-diagonal matrix element, Hij, (orbital interaction integral) in MO theory.
The Hubbard molecule model Hamiltonian for the 5d T2g orbitals in LnO2 can be described using the following matrix for each of the three spatially independent components of y1.

HMM to second order for CeO2 4f interaction
The energies can be determined to second order by solving |A-ES| = 0, where A is Hamiltonian for the HMM and S are the overlap integrals.Sii is equal to 1 and there is only one unique value of Sij, which can be shortened to S. For the CeO2 4f interaction, the determinant is given below From the first 2 rows of A, a1 = a4 and E±a1=(-t-E±S)a5 or a5 = -E±/(t+ E±S) a1, where ai is the coefficient of yi.Using a1 = 1, the unnormalized wavefunction is y± = y1 + y4 -E±/(t+ E±S)y5.The normalization constant, N, is 1 42 + ( ± /(t +  ± S)) # ⁄ .The y± can be simplified by dividing by √2 to give ⁄ can be more conveniently represented as |L 1 4f 1 ⟩ and y5 by |L 2 4f 0 〉, to give the following:

S10
HMM to second order for PrO2 4f interactions.
The model for PrO2 is the same as for CeO2.

HMM to second order for TbO2 4f interactions.
The HMM for TbO2 can be expressed as doubly degenerate (spin degenerate) with

HMM to second order for LnO2: O 2p and Ln 5d interactions
The HMMs for the T2g and Eg interactions are essentially the same apart from their spatial degeneracy.In both cases, the interaction can be broken into non-degenerate cases (e.g., the double degenerate Eg interaction can be divided into the dz 2 and dx 2 -y 2 interactions).For each of the non-degenerate cases, the HMM can be expressed as Solving |A-ES| = 0 gives the same solution as for CeO2 4f interactions.

Experimental details
Magnetic susceptibility measurements.In an argon filled glovebox, samples were loaded into 3 mm OD quartz tubes by sandwiching them between two plugs of quartz wool.The samples were compressed into a pellet by squeezing them between two quartz rods.The quartz rods were removed, and the ends of the tube were capped by inserting them into septa for 7 mm tubing.The capped tube was removed from the glovebox.The center of the tube was wrapped with a Kimwipe, saturated with liquid nitrogen, and sealed with a propane/oxygen torch.Variable temperature magnetization data were recorded at 1 T, 2 T, and 4 T using a Quantum Designs MPMS SQUID magnetometer.Variable temperature magnetization was corrected for the diamagnetism of the quartz wool using Pascal's constants for covalent compounds, χQW = 3.7 × 10 -7 emu g -1 (no correction for the diamagnetism of the quartz tube is needed as it never leaves the SQUID coils).Molar susceptibility was calculated using the following equation: Where χmol is the molar susceptibility, Mmeas is the measured magnetization, Mferro is the magnetization of the ferromagnetic impurity, which is temperature and field-independent; χQW is the contribution to the susceptibility due to the quartz wool, χdia is the diamagnetic correction determined using Pascal's constants, and H is the applied field.Two ferromagnetic impurities are commonly encountered in laboratory samples, ferrous metals and magnetite or other ferrites from the oxide coating on stainless steel lab equipment.Of these, magnetite is far more likely to be encountered.In general, the magnetization of ferromagnets is temperature independent below the Curie temperature, which is 860 K for magnetite, so magnetization of the impurity is temperature independent for this experiment.The magnetization of magnetite reaches saturation at approximately 0.2 T, above which the magnetization is ~90 emu/g.Below this field, the magnetization of magnetite is roughly linear with applied field.
Based on the assumption that the impurity is magnetite or a related ferrite resulting from the abrasion of stainless steel lab equipment, the data were corrected for a temperature and field independent ferromagnetic impurity.Mferro was allowed to vary to minimize the least squares difference between χmol at different fields.Variable temperature magnetic susceptibility data including before and after the correction for ferromagnetic impurities are included in the SI.
Diffuse reflectance (DR) measurements.DR spectra were obtained with an Ocean Optics T3000 spectrometer equipped with a diffuse reflectance probe.Samples were smeared onto a glass microscope slide covered with several layers of poly-tetrafluoroethylene (PTFE) tape.The blank spectrum was obtained from the PTFE-covered microscope slide prior to taking the data on the compound.Reflectance data were converted to F(R∞) using the Kubelka-Munk transform. 1he DR spectra were normalized by setting the lowest absorbance of the spectrum to zero (Figures S1 and S2).
Charge-transfer band gap energy (EBG) determination.2][3] The band gap was determined using the approach described by Makula et al. 1 Briefly, the data are plotted as [F(R∞)•hu] 2 vs hu (DR spectra) or (a•hu) 2 vs hu (XANES spectra), where a is the absorbance.The regions below and above the transition are fit to straight lines.The intercept of these lines is the band gap.In the case of the O 2p to 4f transitions in the DR spectra of LnO2, the intercept with the x-axis was used instead of fitting the region before the transition.To determine the CeO2 O 2p to Ce 5d band gaps, the CeO2 absorption spectrum reported by Niwano et al. was digitized using the program UN-SCAN-IT, and the band gaps were determined using the approach described by Makula et al. 1,[4][5] The uncertainty in Tauc plot determinations of band gaps is typically reported as 0.03 eV.Based on work on CeO2 thin films, we assume the uncertainty is slightly larger, 0.05 eV, for the band gap determined from the UV-Visible DR data (the numerical error from fitting the Tauc plots is smaller). 6 Least Squares fitting and uncertainty analysis.Least squares fitting of the XANES pre-edge energies to the charge transfer band gap energies with errors in both dimensions was performed as described in "Numerical Recipes." 7 Uncertainties in the modeled parameters, sf, were determined using  9 # = ∑ ^:9 , where sxi is the uncertainty of measured property xi, and the derivatives were determined numerically.

Figure S3 .
Figure S3.Tauc plots for CeO2 from the data published by Niwano, et al.5 Band gap given in the upper left corner of each plot.

Figure S4 .
Figure S4.Tauc plots for CeO2 O K-edge XANES pre-edge peaks.Band gap given in the upper left corner of each plot.

Figure S5 .
Figure S5.Tauc plots for PrO2 O K-edge XANES pre-edge peaks.Band gap given in the upper left corner of each plot.

Figure S6 .
Figure S6.Tauc plots for TbO2 O K-edge XANES pre-edge peaks.Band gap given in the upper left corner of each plot.

Table S1 :
4he ground state is |L4Eg 0 ⟩, which corresponds to O 2p Eg4and Ln 5d Eg 0 .This state is spatially degenerate, and the model may be more conveniently applied by considering each component of the ground state separately since they interact with different excited states.The spatially degenerate wavefunctions of ground state are given by y1 and y2, which correspond to the empty dz 2 and empty dx 2 -y 2 orbitals and the matching, filled O 2p SALCs, respectively.The energy of y1 and y2 is 0. The excited state that interacts with y1 is |L 3 Eg 1 ⟩ , which 16-fold degenerate and given by y3 -y18.The individual states are shown in TableS1. Oly states in which the electron can hop from the occupied O 2p Eg orbitals into the Ln 5d Eg orbitals without changing sign can interact with the ground state: y1a can interact with y2 and y7, and y1b can interact with y12 and y17. Te other states cannot interact with y1 and are excluded from the model. E 3 5d Eg 1 basis states for the Hubbard molecule model for LnO2. Te spatially degenerate orbitals are indicated by the curved brackets.Only the states in red are used in the model.O 2p Eg 3 Ce 5d Eg 1 states states

Table S1 :
6he ground state is |L 6 T2g 0 ⟩, which corresponds to O 2p T2g6and Ln 5d T2g 0 .This state is triply spatially degenerate and consists of wavefunctions y1, y2, and y3.The energy of y1, y2, and y3 is 0. The excited state that interacts with y1, y2, and y3 is |L 5 T2g 1 ⟩ , which 36-fold degenerate and given by y4 -y39.The individual states are shown in TableS1.Only states in which the electron can hop from the occupied O 2p Eg orbitals into the Ln 5d Eg orbitals without changing sign can interact with the ground state: y1a can interact with y2 and y7, and y1b can interact with y12 and y17, and y1c can interact with y30 and y37.The other states cannot interact with y1 and are excluded from the model. T2g 55d T2g 1 basis states for the Hubbard molecule model of LnO2. Te spatially degenerate orbitals are indicated by the curved brackets.Only the states in red are used in the model.O 2p T2g 5 Ln 5d T2g 1 states for (dxz)(dyz)(dxy) states