An Ab Initio Correction Vector Restricted Active Space Approach to the L-Edge XAS and 2p3d RIXS Spectra of Transition Metal Complexes

We describe an ab initio approach to simulate L-edge X-ray absorption (XAS) and 2p3d resonant inelastic X-ray scattering (RIXS) spectroscopies. We model the strongly correlated electronic structure within a restricted active space and employ a correction vector formulation instead of sum-over-state expressions for the spectra, thus eliminating the need to calculate a large number of intermediate and final electronic states. We present benchmark simulations of the XAS and RIXS spectra of the iron complexes [FeCl4]1–/2– and [Fe(SCH3)4]1–/2– and interpret the spectra by deconvolving the correction vectors. Our approach represents a step toward simulating the X-ray spectroscopies of larger metal cluster systems that play a pivotal role in biology.

They involve one-and two-step transition processes, respectively, between the 2p-core and 3dvalence orbitals, is depicted in Fig. 1.The dipole-allowed nature of these spectra ensures high intensity and energy resolution.However, the effects of ligand-fields, presence of different multiplet spins, and spin-orbit coupling, complicate the interpretation of the spectra.Theoretical models are thus essential.Resonant inelastic X-ray scattering (RIXS) Figure 1: Schematic of the one-and two-step processes of L 2,3 -edge XAS and 2p3d RIXS spectroscopies.
Recently, several ab initio methods have appeared to compute L-edge XAS [1,4,6] and RIXS [6] spectra.These methods use sum-over-state expressions [1,4,6].However, such approaches become impractical when there are a large number of intermediate and final states, as is expected to be the case when simulating larger bioinorganic clusters [2,3].
A route to computing spectra is the correction vector (CV) formulation [7], where frequencydependent response equations are solved to obtain the CVs, which determine the spectrum at each frequency.Here, we describe an ab initio implementation of the CV approach for L-edge XAS and 2p3d RIXS spectra, within a restricted active space model of the correlated transition metal electronic structure [8].The outline of the paper is as follows.In section 2, we introduce the ab initio relativistic Hamiltonian and the CV approach for L-edge XAS and 2p3d RIXS spectra.We also describe how to deconvolve the spectra to separate different electronic effects.
In section 3, we describe the geometries, molecular orbitals, active space models, wave function ansatz, and methods to optimize the wave function ansatz [9,10,11].In section 4, we compute the XAS and RIXS spectra for monomeric ferrous and ferric tetrahedral iron complexes and compare them to available experimental spectra.By deconvolving the theoretical spectra, we interpret the contribution of different electronic effects and states to the peaks in the XAS and RIXS spectra, highlighting the role of certain electron correlations.In section 5 we provide some perspective on future developments of this approach.

Spin-orbit Hamiltonian
We start from an ab initio Hamiltonian containing spin-orbit coupling described within the mean-field Breit-Pauli (BP) approximation.In second quantization, this is where h ij and (ij|kl) are the ab initio one-and two-electron integrals, h BP ij are the mean-field Breit-Pauli matrix elements, and Tij ( T x,y,z ij ) are the Cartesian triplet operators, Further discussion of the Breit-Pauli mean-field Hamiltonian can be found in Ref. [12].

XAS/RIXS spectra from correction vectors
The XAS spectral function (S) and RIXS cross-section (σ) (averaged over all orientations, directions, and polarizations of the scattered radiation) can be written as with and where |Ψ 0 ⟩ and E 0 are the ground-state wavefunction and energy, μ is the dipole operator, ω ex and ω em are the energies of the incident and scattered radiation, and η and η ′ are Lorentzian broadening factors.We note that we only consider resonant terms in this work.
Following this, the XAS/RIXS quantities can be computed in three steps: (1) solve for |Ψ 0 ⟩, (2) solve for the response (from the correction vector equations), and (3) compute S/σ from the correction vectors.Specifically, after obtaining |Ψ 0 ⟩, we compute the CVs ({A λ (ω ex )}) by solving and obtain the XAS spectral function For the RIXS cross section, we solve for an additional set of CVs ({B ρλ (ω ex , ω em )}), and compute the cross section using

Interpretation of correction vectors
The CVs can be formally expanded in a sum-over-states: |B ρλ (ω ex , ω em )⟩ = where Ψ I , E I are eigenstates, eigenvalues of H respectively.On resonance (i.e. a divergent denominator in Eq. ( 16), Eq. ( 17)), |Ψ I ⟩ is the final state.This has core-excited character in XAS and valence-excited character in RIXS.Note that final states in XAS are the intermediate states in RIXS.

Deconvolution
To interpret the L-edge XAS and RIXS spectra, we deconvolve the intermediates into particlehole and spin contributions.
We define the particle-hole components for XAS (S ia (ω ex )) and for RIXS (σ as where the correction vector On resonance, we interpret S (ph) ia as the amplitude of the core (i) to valence (a) excitation in the XAS final state, and σ (ph) ia as the amplitude of the valence (i) to valence (a) excitation in the RIXS final state.In XAS, we further define the total valence (particle) contribution by summing over the core (hole) contributions To deconvolve the spectra into different spin contributions for XAS (S S (ω ex , ω em )), we apply spin projection operators (P S ), We use here Löwdin's spin projector, [13] For all deconvolution schemes, the sum of the deconvolved spectra is the total spectrum.
For the spectra calculations, we used Lorentzian broadening factors of η = 0.3 eV and η ′ = 0.1 eV in Eqs. ( 9) and (11), respectively.We simulated RIXS spectra using the incident radiation energy (ω ex ) that results in the maximum intensity of the L 3 band in the XAS spectra.All calculations were performed using the PySCF [15,16,17] and Block2 [18] packages.

Active space models
A RAS ansatz can be written in occupation number form as where m i and n j are the occupations (0, 1) in the two active subspaces, RAS1 and RAS2, respectively, and RAS1 consists of M occupied orbitals with a maximum number of holes (M hole ), i.e.M i=1 m i ≥ M − M hole .RAS2 consists of N orbitals with no restrictions on the electron occupancy, except for the total number of electrons in the RAS spaces, i.e., M i=1 m i + N i=1 n i = N RAS elec .
Additional RAS partitions can be introduced.

Matrix product state (MPS) implementation
We implement the RAS ansatz within the matrix product state formalism.We rewrite the configuration coefficients in Eq. 27 as where D × D matrices, and A m 1 and A n N are 1 × D and D × 1 vectors, respectively.All the matrices (and vectors) contain complex elements.Here we choose the bond dimensions D so that the RAS ansatz is exactly represented: there is no MPS compression, and the MPS formalism is only used to simplify the implementation.
To compute ground-states we used the density matrix renormalization group (DMRG) algorithm [9,19,20], and we used the dynamical DMRG algorithm [10] to solve the correction vector equations in Eqs. 12 and 14.

Active space construction
To construct the active space for the RAS ansatz, we used a technique introduced in a previous study of iron-sulfur clusters.[21] We first performed unrestricted DFT calculations for the highspin state without spin-orbit coupling using the BP86 functional [22,23] in the ANO-RCC-VDZP basis [24].Then, we computed unrestricted natural orbitals as the eigenvectors of the sum of alpha and beta DFT density matrices.We then identified active space orbitals from the unrestricted natural orbital occupation numbers, and localized the orbitals within the active space to improve the convergence of DMRG and dynamic DMRG algorithms.

Results and Discussion
Using the above formalism, we simulated the L-edge XAS and 2p3d RIXS spectra of the We compare our results to experimental spectra, normalizing the maximum intensity of the L 3 -edge band of XAS to 1 and that of the highest intensity band of RIXS to 0.2 (for the ferrous complexes), 0.8 (for the tetrachloride ferric complex), and 0.6 (for the tetrathiolate ferric complex).We first discuss the simulations of the L-edge XAS spectra for the ferrous complexes and [Fe II (SCH 3 ) 4 ] 2− using the RAS1(2p 6 Fe )RAS2(3d Fe ) active space model (see Sec. In the bottom panels of Fig. 3, we present the deconvolved spectra for the different spin components.For both [Fe III Cl 4 ] 1 -and [Fe III (SCH 3 ) 4 ] 1 -, the largest contribution is for the same spin-component as the (high-spin) ground state (S = 2), with contributions of 88% and 83%, respectively.The contributions of the ∆S = 1 transitions are 12% and 17%, while those of the ∆S = 2 transitions are negligible.Interestingly, the percentage of the ∆S = 1 contribution is similar for both complexes, possibly due to similar spin-orbit coupling strengths resulting from the same oxidation state.two clear bands, as opposed to only one in the experiment.These differences can be better understood using the deconvolved spectra in Fig. 5, which we now discuss.
The top panels of Fig. 5 represent the valence orbitals of the natural orbitals for the groundstate RAS wave functions.In contrast to the ferrous complexes, the natural orbitals of the ferric complexes have significant mixing with 3p orbitals of the ligands, due to the higher oxidation state of Fe.An exception to this is the 3d   However, in [Fe III (SCH 3 ) 4 ] 1 -, the band positions of 3d x 2 −y 2 are shifted by −2 eV relative to those of 3d z 2 .The shift in the 3d x 2 −y 2 bands is the main reason for the disagreement between the theoretical and experimental spectra in the right panels of Fig. 4. The small mixing between the 3d x 2 −y 2 orbital and the 3p ligand orbitals in Kohn-Sham density functional theory, used to construct the active space, contributes to this shift.
In the bottom panels, we present the deconvolved spectra for the different spin components.
For both complexes, the largest contribution (60%) comes from states with the same spin as the (high-spin) ground states (S = 2.5).Contributions of 40% are observed for the ∆S = 1 transition for both ferric complexes, while the contribution of the ∆S = 2 transition is negligible.The higher contributions for the ∆S = 1 transition in the ferric complexes, as compared to the ferrous complexes, can be attributed to the stronger spin-orbit coupling due to the higher oxidation state of Fe. Figure 6 shows the experimental spectra (left panels) and theoretical spectra (center and right panels) with different active space models (see Sec. Fe ) active space (that includes the the four occupied sigma-bonding orbitals between the Fe and Cl/S atoms) two bands correctly appear, at 0.00 and 0.36 eV for [Fe II Cl 4 ] 2− and 0.00 and 0.54 eV for [Fe II (SCH 3 ) 4 ] 2− .This emphasizes the importance of electron correlation between the 3d orbitals and the sigma-bonding orbitals to reproduce the energy splitting in this energy range.
In the second energy region (1-4 eV), the experimental spectra contains broad bands, with half maximum intensity (HM) at 2.10 and 3.31 eV for the chloride complex, and 1.49 and 3.45 eV for the thiolate complex, and with full widths at half maximum (FWHM) of 1.21 and 1.96 eV, respectively.The minimal active space model has narrower bands with HM at 2.12 and 3.07 eV (chloride complex) and 1.86 and 3.42 eV (thiolate complex), and FWHM of 0.95 and 1.56 eV.Including the sigma-bonding orbitals broadens the bands with HM at 2.24 and 3.50 eV (chloride complex) and 1.88 and 3.84 eV (thiolate complex) and FWHM of 1.26 and 1.96 eV.
These latter results are closer to what is seen in experiment, but are slightly shifted to positive energy.
In the third region (4-6 eV) of the experimental spectra, there is no representative band for the chloride complex, while there is a broad band in a range of 3.5-6 eV for the thiolate complex.The corresponding band in the theoretical spectra for the thiolate complex is much narrower than the experimental band.We ascribe the difference to missing certain important states, such as the ligand-to-metal charge transfer (LMCT) states, in the active space models.
H→P in Eq. 29) for three particle sets (P) (scale is in arbitrary units).In each spectrum, the different contributions of the three hole sets (H) are represented by dashed lines of different colors, and the corresponding transitions are depicted in the orbital diagram next to the x-axis by dashed arrows (using the same color scheme) next to the x-axis.The inset depicts the approximate ground state electronic configuration and the three sets of 3d orbitals used to compute the average particle-hole contributions.The remaining graph at the rear represents the spin-state contribution to the final states (σ (s) S ) defined in Eq. 23.
We further analyze the RIXS spectra of the ferrous complexes (in the larger active space model) by deconvolving them, as shown in Fig. 7.In each panel, the spectrum in the rearmost position shows the spin-state deconvolution defined in Eq. 23.The ground state for both complexes is a spin quintet (S = 2).Thus, the red curve, which depicts quintet contributions, shows the spin-allowed transitions, while the green and blue curves depict spin-forbidden transitions with ∆S = 1, 2, respectively.The remaining three spectra from front to back represent the  Similarly, we divide the RIXS spectra of the ferric complexes into three regions as shown in Fig. 8.In the first region (0-1 eV), both the experimental (left panels) and theoretical spectra with different active space models (center and right panels) show a single band at 0 eV, similar to experiment.In the second region (1-4 eV), the experimental spectrum of the chloride complex shows three representative bands at 1.83, 2.34, and 2.80 eV, while the thiolate complex exhibits a broader band with indistinct peaks in the range of 0.5-3.5 eV.The minimal active space model yields two main bands at 2.58 and 3.42 eV for the chloride complex, and a slightly broader band in the range of 2-3.5 eV for the thiolate complex.In the chloride complex, the larger active space model splits the two bands into three main bands at 2.46, 3.44, and 3.88 eV and two minor bands at 2.78 and 3.00 eV.In the thiolate complex, in the larger active space the bands away from 0 eV are broadened, but still do not match the widths of the experimental bands.
Experimental measurements using magnetic circular dichroism (MCD) spectroscopy [26] show d-d transition bands between 0.9 ∼ 1.39 eV, while the ligand-to-metal charge transfer (LMCT) bands begin at ∼ 1.Finally, we deconvolve the theoretical spectra of the larger active space model, using the same schemes as used in Fig. 7. Figure 9 shows the deconvolved spectra for [Fe III Cl 4 ] 1 -and [Fe III (SCH 3 ) 4 ] 1 -in the left and right panels, respectively.Like in the ferrous complex, the first region (0-1 eV) is dominated by spin-allowed transitions, while the second and third regions (1-6 eV) are characterized by spin-forbidden transitions of ∆S = 1, 2. We observe that in the ferric complexes, there is a larger contribution from the ∆S = 2 transitions as compared to the ferrous complexes.Furthermore, the other spectra in the front and middle panels show the average particle-hole contributions from three sets of particles and holes, namely, {d z 2 },

Conclusion
In this work, we presented a new ab initio technique to compute L 2,3 -edge XAS and 2p3d RIXS spectra, based on the correction vector approach and a restricted active space ansatz.We obtain good general agreement between our theoretical simulations and experimental spectra in a set of mononuclear tetrahedral ferrous and ferric iron complexes.Our results highlight the importance of selecting an appropriate active space for the spectroscopy, and the role of electron correlation between the metal and ligand electrons in determining certain spectral features.
Improved simulations should incorporate additional orbitals into the active space treatment, for example, to treat LMCT/MLCT states; improve the active space through orbital optimization; and include dynamic correlation effects in the spectra.
The elimination of a sum-over-states computation in the correction vector formulation of XAS and RIXS spectra removes a major limitation in the simulation of spectra for multi-nuclear transition metal complexes.Simulations of larger iron-sulfur cluster X-ray spectra are currently underway in our group and will be presented elsewhere.

Figure 3 :
Figure 3: Natural orbitals of the ground state RAS wave function (top panels) and deconvolved XAS spectra for particle (middle panels) and spin (bottom panels) contributions.[Fe II Cl 4 ] 2− (left panel) and [Fe II (SCH 3 ) 4 ] 2− (right panel).The particle contributions are measured in the natural orbital basis shown in the top panels.

4. 2 L 1 -Figure 4 :
Figure 4: L-edge XAS spectra of ferric tetrachloride and tetrathiolate complexes, left and right panels, respectively, with the same format as Fig. 2.

Figure 5 :
Figure 5: Natural orbitals of the ground state RAS wave function (top panels) and deconvolved XAS spectra for particle (middle panels) and spin contributions (bottom panels).[Fe III Cl 4 ] 1− (left panel) and [Fe III (SCH 3 ) 4 ] 1− (right panel).The particle contributions are measured in the natural orbital basis shown in the top panels.

4. 3 Figure 6 :
Figure 6: 2p3d RIXS spectra of [Fe II Cl 4 ] 2− and [Fe II (SCH 3 ) 4 ] 2− in upper and lower panels, respectively.The experimental spectra in the left panels are from Ref. 3. Theoretical spectra with two different active space models are in shown in the center and right panels, respectively.

Figure 7 :
Figure 7: Deconvolved theoretical RIXS spectra for [Fe II Cl 4 ] 2 -(left panels) and [Fe II (SCH 3 ) 4 ] 2 -complexes (right panels) using the larger active space model.Each panel shows four deconvolved spectra.From front to back: The first three show the average particle-hole (valence-to-valence excitation) contributions (σ (ph) average particle-hole contributions defined as follows.(To simplify the analysis, we consid-broad band in the 1-4 eV range, with a minor contribution from ∆S = 2 transitions in the 3-4 eV range.The band can be further divided into three parts based on the dominant spin-flip transitions: transitions of d yz,xz → d xy,x 2 −y 2 , d yz,zx and d xy,x 2 −y 2 → d xy,x 2 −y 2 , d yz,zx , transitions of d z 2 → d xy,x 2 −y 2 , d yz,zx , and transitions of d x 2 −y 2 ,xy , d yz,xz , d z 2 → d yz,xz and d z 2 → d xy,x 2 −y 2 .In the energy range of 4-6 eV, the d − d contributions are similar to those in the higher part of the broad band in the 3-4 eV range.

{d x 2
−y 2 }, and {d xy , d yz , d xz }.In contrast to the ferrous complexes, all transitions to d z 2 , d x 2 −y 2 , and d xy,yz,xz contribute to the total spectra across the entire energy range of 0-6 eV in the ferric complexes.The low energy part of the band in the 2-3 eV region is dominated by spin-flip transitions of d xy,yz,xz → d z 2 , d x 2 −y 2 , d xy,yz,xz character, while above this window there is a mixture of various particle-hole contributions.