Quantifying the influence of 3d–4s mixing on linearly coordinated metal-ions by L2,3-edge XAS and XMCD

The mixing valence d and s orbitals are predicted to strongly influence the electronic structure of linearly coordinated molecules, including transition metals, lanthanides and actinides. In specific cases, novel magnetic properties, such as single-ion magnetic coercivity or long spin decoherence times, ensue. Inspired by how the local coordination symmetry can engender such novel phenomena, in this study, we focus our attention on dopants (Mn, Fe, Co, Ni, Cu) in lithium nitride to accept innovation from molecular magnetism in a high symmetry P6/mmm solid-state crystal. The linear coordination environment results in strong 3d–4s mixing, proving to be an ideal series to investigate the role of d–s mixing and bonding on electronic structure and magnetism. It is shown that L2,3-edge XAS can be applied to experimentally identify the presence of 3d–4s mixing and the influence this has on the ligand-field splitting. XMCD specifies how spin–orbit coupling is affected. The combined spectroscopies are analysed to determine the effect of 4s mixing with support from ab initio calculations. The results provide new insight of relevance to future applications, including quantum information processing and the sustainable replacement of rare earths in magnets.


S.2.1 Computational Details
Periodic DFT calculations on Li 2 (Li 1−x Cu x )N were performed using the plane-wave pseudopotential DFT method available within the code CASTEP 1 .The generalised gradient approximation for the exchange-correlation energy was selected in the form of a PBE functional revised for solids 2 .
Self-consistently generated ultrasoft pseudopotentials were used for both PBE and PBE+U calculations.A kinetic energy cutoff of 750 eV for the wave function, together with a (6x6x6) Monkhorst-Pack k-point grid, were determined as parameters for convergence calculations.A (10x10x10) k-point grid was used instead for calculating the density of states (DOS).Self-consistent calculations were performed to a convergence value of 1×10 −7 eV.Due to the isolated nature of Cu atoms in Li 2 (Li 1−x Cu x )N, we operated with a supercell constructed from the hexagonal cell of Li 3 N (space group P6/mmm) with a single dopant atom.The structure was generated using the sample method as previously reported for Li 2 (Li 1−x Fe x )N 3 .A smearing of 0.1 eV was applied to the computed eigenvalues to improve the k-point convergence.The angular dependence of Cu L 3 -edge was calculated, including the effects of a core-hole 4 and using the same k-point grid as previously used for the DOS.The ground state DFT was also expanded by expressing the exchange-correlation potential in terms of local-density band theory via the PBE+U method 5 .The electronic properties were calculated with the simplified, rotational-invariant formulation developed within the linear response approach 6 .An effective U value of 3 eV was included in the calculations.Based on the ground-state energy evaluation and the spectroscopic results, angular-momentum dependent orbital occupation was determined with Löewdin charge analysis on top of ground-state, converged DFT wavefunctions.X-ray absorption spectra were computed by extracting the matrix elements for electronic interband transitions from the ground state DFT, including the local effects of the 2p 5 core-hole as implemented in the CASTEP code.An energy shift of 933.38 eV was applied to match the experimental data and normalised through trapezoidal integration of the simulated spectrum.
Transition broadening as a consequence of instrumental resolution (Gaussian) and core-lifetime effects (Lorentzian) was set as 0.2 and 0.7 eV FWHM, respectively.

S.2.2 Cu L 2,3 -edge analysis
Periodic DFT calculations were performed within CASTEP 1 utilising a 3x3x3 supercell Li 3 N matrix doped with a single Cu atom at the Wyckoff 1b position.Partial density of states (pDOS) were mapped upon the converged system to compare the interpretation to molecular DFT approaches within the main body of the text.pDOS calculations permit an extension to Mulliken population analysis to isolate individual bands and orbitals of a selected atom.Unoccupied Cu-3d states according to molecular bonding symmetries (dσ, dδ and dπ) are in agreement with both angular dependent spectroscopic labelling and molecular DFT (Figure 3) approaches to the pronounced transitions observed within the Cu L 3 -edge spectra (I-III), Figure S2.Related Löewdin population analysis are tabulated with respect to molecular orbital number within Table S1.(exchange) as given by:

S.4 Ni(I) absorption cross-section selection rules
where where a ground-state wavefunction, ψ gs is determined through diagonalisation of the hybridisation Hamiltonian matrix: Where required, differential orbital symmetry adapted metal to ligand charge-transfer 12 is introduced to reproduce back-bonding contributions of the π and δ-bonds.Ligand back-donation is treated with parameters, V π and V δ and individual ∆ values, ∆ π and ∆ δ .Contributions of 3d-4s hybridisation is symmetry restricted to the dσ-orbital (3d z 2 ) and introduced with an energy separation, ∆ 4s and orbital overlap parameter, V 4s .Additionally, further exchange Slater-Condon-Shortley integrals are included in the ground, G 2 ds and excited states, G 1 ps , G 2 ds .The ground state wavefunction is expressed as multiconfiguration linear combination of |3d n ⟩ , 3d n−1 4s and 3d n−1 L − and was systematically explored by fitting calculated spectra to experiment, to comprehensively deduce the independent bonding contributions for the series of linear transition metal complexes studied.The charge-transfer ligand-field multiplet parameters were informed by the Cu TD-DFT calculations (discussed within the main body of text) including the relative energy ordering of characteristic satellite features of dσ < dδ < dπ.
Ab initio ligand-field theory calculations provide a first measure of Slater-Condon-Shortley integrals weightings from first principles, Table S3, from which optimised parameters were itteratively deduced, Table S2.Both ground-state (1-shell) and excited-state (2-shell) calculations were performed where the latter was computationally feasible.The active space of a 1-shell calculation is as described within the main body of the text, where the 2-shell enables the calculation of 2p3d Coulomb and exchange integrals.Rotation of the three 2p x,y,z orbitals within the active space expands the calculation to N electrons in eight orbitals, where N = 13 and 14 for TM = Fe (56 quartets and 168 doublets) and Co (28 triplets and 36 singlets) for the respective monovalent calculation and TM = Co (56 quartets and 168 doublets) and Ni (28 triplets and 36 singlets) for divalent; tabulated values collated within Table S3.An 80% reduction is applied to Hartree-Fock deduced parameters resulting from the over-estimation of electron-electron repulsion found for the free ion where 2-shell calculations were impractical.Broadenings of all calculated transitions were convolved with a full-width half-maximum Gaussian of 0.25 eV representative of the experimental instrument resolution, and a varying Lorentzian broadening over the L 3 and L 2 -edges to account for the core-hole lifetimes, Table S5.

S.5.2 Ab initio ligand-field multiplet calculations
Ab initio ligand-field theory calculations were performed on Li   S3).Excited state (2-shell) calculations were attempted where feasible (See S.5.1.for further details).to spectral positions and intensities.Complete tabulated values for the series as stated in Table S2.

Figure S1 presents experimental Li 2 (
Figure S1 presents experimental Li 2 (Li 1−x Cu x )N L 2,3 -edge XMCD spectra.The absence of any dichroism is indicative of a system exhibiting no magnetic phenomena upon application of a 14 T field.This strongly suggests a near-closed shell Cu(I), d 10 valence with spectral features corresponding to orbital and ligand hybridisation; full discussion within the main body of text.

Figure
Figure S2: a) Single crystal angular dependent Cu L 3 -edge spectra.(Top) Experimental spectra with background subtraction.(Bottom) Periodic DFT calculated spectra.0 • corresponds with E ⊥ c and 90 • with E ∥ c. b) Periodic DFT calculated partial density of states (pDOS), visualising predominant contributions of Cu-d states above the Fermi energy level.

Figure S3 :
Figure S3: DFT deduced molecular orbital energy level diagram with related isosurface plots.

Figure
Figure S4 presents angular dependent Ni(I) (d 9 ) L 2,3 -edge XAS and XMCD calculations of various ground-state ligand-field configurations.The main L 3 -and L 2 -edge dipole transitions at 852 and 869.2 eV respectively exhibit pronounced angular dependencies which are contingent to the orbitallocation of the lone electron hole.The presence of Ni(I) L 2 -edge absorption intensity within the experimental spectra (Figure8h) precludes a Ni(I) ground term 2 D 3/2 (dσ 2 dπ 4 dδ 3 ) (FigureS4e); due to a lack of L 2 -edge peak within the calculation.Further isolation of the orbital-location of lone electron hole can be deduced through identification of the opposing angular dependencies within the experimental measurements of the XAS L 2 -and L 3 -edges (Figure6).This compels an electronic ordering of dσ < dδ < dπ or dδ < dσ < dπ; as an unoccupied dσ orbital (L = 0) predicts equivalent XAS angular dependencies (FigureS4c) in addition to an incorrect negative-positive XMCD L 2,3 -edge signal, FigureS4h.This technique facilitates an understanding of the general electronic occupation of Ni(I); but it is limited to primitive labelling of the lone electron hole to an E 1g doublet.To further quantify the electronic ordering of the complete d-orbital manifolds ab initio and charge-transfer ligand-field multiplet calculations are applied within the main-body of the text.

A
k and g k represent the angular coefficients, and F k and G k are the radial integrals of the direct and exchange interactions, respectively.Spin-orbit coupling parameters, ĤSO = N i=1 ξ (r i ) l i • s i of the 3d n manifolds for Mn and Ni, are left consistent with atomic values; while Fe and Co are scaled to experimentally deduced values previously reported by temperature-dependent L 2,3 -edge XAS 3 and EPR 10 .2p 5 core-hole spin-orbit coupling strengths of all transition metal ions remain consistent with the atomic values.The crystal-field potential, ĤCF =V CF (r i , θ i , φ i ) = k,m r k C k m (θ, φ)within normalised spherical harmonics is solved for the local symmetry of the transition metal complex as previously defined within the D ∞h point group, which is equivalent to D 6h when Dq = 0.This results in a ligand-field d-orbital energy splitting of an A 1g (d z 2 ) singlet, and two E doublets, E 1g (d xy , d yz ) and E 2g (d x 2 −y 2 ,d xy ) with energies that are adjusted through the ligand-field parameters, Ds and Dt.Metal-ligand covalency is σ symmetry permitted via A 1g , π symmetry permitted via E 1g and δ symmetry permitted via E 1g .Orbital covalency is includedwithin the model via a metal-to-ligand charge-transfer (MLCT) interaction acting between 3d n and 3d n−1 L − configurations, where L is a supplementary set of (3d) ligand orbitals symmetry permitted for metal-ligand mixing.MLCT or ligand-to-metal charge transfer (LMCT) can be used to simulate the effect of metal-ligand covalency on the measured spectra.The energy separation between configurations, ∆ L , and valence bond configuration interaction mixing is given by V L 11

2 (
Li 1−x TM x )N where TM = Mn, Fe, Co and Ni to gain a parameter-free insight of the series.Both mono ([Li 14 TMN 2 ] 9+ ) and divalent ([Li 14 TMN 2 ] 10+ ) TM fragment calculations were performed with Slater integrals, spin-orbit coupling strengths and ligand-field splitting results (Table S2) used as input values for a representative angular dependent L 2,3 -edge XAS multiplet calculation, Figure S5.The SA-CASSCF-NEVPT2 AILFT calculations are most representative of the monovalent oxidation state reproducing the primary multiplet excitations (0 -∼4 eV) of the series.The higher energy satellite intensities are absent within these calculations resulting from the limited active space (5 d-orbitals) as described within the main body of the text, thus indicating the requirement for an extended multi-configurational multiplet Hamiltonian to comprehensively replicate the observed spectral features.Attempts at excited-state (2-shell) AILFT calculations were unsuccessful in accurately replicating the observed XAS spectra and require further exploration beyond this study's scope.

Figure
Figure S7 illustrates a systematic exploration of each individual multi-configurational ground-state of Li 2 (Li 1−x Ni x )N through charge-transfer ligand-field multiplet theory.The high-symmetry (D 6h ) pocket TM ions occupy within lithium nitride results in orbital degeneracies which can be considered as completely orthogonal to each other.This results in symmetry restricted orbital hybridisation and charge-transfer configurations which are virtually uncoupled to one another permitting individual optimisation of the observed high-energy satellite spectral features through ∆ 4s,δ,π and V 4s,δ,π

Figure S6 :
Figure S6: Energy level diagram series trend of divalent Li 2 (Li 1−x TM x )N (where TM = Mn, Fe, Co, and Ni) calculated as one electron eigenfunctions employing a SA-CASSCF-NEVPT2 AILFT calculation.

Figure S7 :
Figure S7: Systematic exploration of the multi-configurational interaction energy difference, ∆ dependencies upon normal incidence (

Table S3 :
Mono and divalent SA-CASSCF-NEVPT2 AILFT deduced Li a Experimentally deduced from EXAFS measurements, Co 13 and Fe 3 .b SCF energy minimised within the B3LYP functional.

Table S4 :
Ground-state expectation values of the electronic occupation of 3d, 4s and ligand orbitals for Li 2 (Li 1−x TM x )N of the optimised charge-transfer ligandfield multiplet parameters (TableS2).

Table S5 :
Full-width half maximum, (FWHM) Lorentzian core-hole life time broadenings applied to calculated L 2,3 -edge spectra.Gaussian FWHM consistent throughout all calculations of 0.25 eV.
S.7 Grazing incidence XAS and XMCD