Theoretical X-Ray Spectroscopy of Transition Metal Compounds
TTheoretical X-Ray Spectroscopy of Transition Metal Compounds
Sergey I. Bokarev ∗ , Oliver K¨uhn † Article Type:
Advanced Review
Abstract
X-ray spectroscopy is one of the most powerful tools to access structure and properties of matterin different states of aggregation as it allows to trace atomic and molecular energy levels in courseof various physical and chemical processes. X-ray spectroscopic techniques probe the localelectronic structure of a particular atom in its environment, in contrast to UV/Vis spectroscopy,where transitions generally occur between delocalized molecular orbitals. Complementaryinformation is provided by using a combination of different absorption, emission, scattering aswell as photo- and autoionization X-ray methods. However, interpretation of the complexexperimental spectra and verification of experimental hypotheses is a non-trivial task andpowerful first principles theoretical approaches that allow for a systematic investigation of a broadclass of systems are needed. Focussing on transition metal compounds, L -edge spectra are ofparticular relevance as they probe the frontier d -orbitals involved in metal-ligand bonding. Here,near-degeneracy effects in combination with spin-orbit coupling lead to a complicated multipletenergy level structure, which poses a serious challenge to quantum chemical methods.Multi-Configurational Self-Consistent Field (MCSCF) theory has been shown to be capable ofproviding a rather detailed understanding of experimental X-ray spectroscopy. However, it cannotbe considered as a ’blackbox’ tool and its application requires not only a command of formaltheoretical aspects, but also a broad knowledge of already existing applications. Both aspects arecovered in this overview. ∗ Institut f¨ur Physik, Universit¨at Rostock, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany † Institut f¨ur Physik, Universit¨at Rostock, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany a r X i v : . [ phy s i c s . c h e m - ph ] J un INTRODUCTION
Spectroscopy is one of the most powerful tools in physics and chemistry and, in particular, X-rayspectroscopy attracts special attention not at least due to the ongoing developments at X-ray FreeElectron Laser (XFEL) facilities and High Harmonic Generation (HHG) sources.
1, 2
As a conse-quence of the localized nature of the core orbitals, the transition operator acts locally and that is whyX-ray excitation probes the local electronic structure of a particular atom embedded in its chemicalenvironment. This is in contrast to, e.g., UltraViolet/Visible photon energy range (UV/Vis) spec-troscopy, where transitions usually occur between delocalized valence Molecular Orbitals (MOs).The combination of different absorption, emission, scattering, photoionization, and diffraction X-ray techniques allows addressing various aspects of static properties as well as photoinduced andchemical dynamics in steady-state and time-resolved X-ray spectroscopy.
1, 3–6
Sharpening the ex-perimental probe does not guarantee, however, an increase of the acquired knowledge as the com-plexity of the detected signal increases as well when addressing more and more intricate effects.Thus, for interpretation of the experimental data, theoretical modeling is mandatory.This review presents an overview on theoretical approaches with a focus on first principleselectronic structure methods and in particular on multi-reference wave function techniques. Appli-cations to the interpretation of experimental data will be shown to provide a means for dissectingdifferent structural and dynamical problems. We will restrict ourselves to isolated molecules andcomplexes as they appear in the gas or solution phases. Extended periodic systems like crystalsare explicitly excluded from consideration as respective theoretical models represent a huge self-standing body of methods which are reviewed elsewhere.
5, 7, 8
Here, we address only discrete thepre-edge structure leaving XANES and EXAFS parts of the absorption edge aside since especiallythe latter provides more information on the geometric rather than electronic structure. Although computational X-ray methods have been developing apace with quantum chemistrysince its infancy, the 21st century heralds an explosive growth of theoretical studies of X-ray spectra.These studies are mainly devoted to the K -edge (1 s ) spectra of second period and heavier elementsusing single reference methods. It is barely possible and not aimed in the current review to coverthem all, including the numerous applications (for reviews see References 5, 10–12). Therefore,mostly the soft X-ray photon energy range (0.1-1.0 keV) and 4th period Transition Metal (TM)complexes will be considered. Thus, mainly metal L -edges will be discussed corresponding toexcitation or ionization from 2 p core orbitals. 2 X-RAY SPECTROSCOPY
Spectroscopy in the UV/Vis spectral range is a standard tool for probing electronic transitions andrelated processes, taking place within the valence state manifold. Here, participating electronicstates often exhibit rather delocalized electron densities. This makes an interpretation in terms oflocal changes difficult, e.g., with respect to specific bonds. In contrast, X-ray transitions involveat least one orbital, which is localized on an atom, thus providing a local probe of the electronicstructure. Moreover, core level energies vary vastly between different atoms what provides elementspecificity. Further, core orbitals with non-zero angular momentum lead to pronounced relativisticeffects, e.g., caused by Spin-Orbit Coupling (SOC). Conventional X-ray spectroscopy is plaguedby the rather short lifetime of the core-hole (4-10 fs), which limits the resolution (a possible wayto overcome this problem is High Energy Resolution Fluorescence Detection X-ray AbsorptionSpectrum (HERFD-XAS) ). However, this can be turned into an advantage by using the core-hole lifetime to clock ultrafast dynamics (“core-hole clock”). Finally, X-ray wavelengths are inthe range of relevant molecular length scales, enabling better spatial and time resolution.The list of problems, which X-ray spectroscopy typically addresses includes probing the chemi-cal environment and bond distances,
9, 15–17 the nature of chemical bonds as well as oxidation,spin, and solvation states.
Further, it is indispensable for the investigation of solids and sur-faces.
3, 4, 25, 26
While steady-state X-ray spectroscopy is well-established, emerging time-resolved ex-periments are pushing the limits to enter the regime of even the fastest nuclear and electron dynam-ics. This becomes possible due to the fact that the optical cycle of X-rays is in the few attosecondrange. Among others,
1, 6 recent examples include, e.g., the study of the (photo)reaction dynam-ics of acetylene isomerization upon core-hole formation, photodissociation of simple flourides, ligand exchange and transient photodynamics in metal complexes, photocatalytic processeson metal surfaces,
32, 33 nanosecond dynamics of dissociation and reassociation of insulin, and ul-trafast electron and nuclear wave packet dynamics using time-resolved and transient photoelectronspectroscopy.
35, 36
The advantages of X-ray as compared to UV/Vis spectroscopy come at the price of requiringlarge scale facilities like synchrotrons and XFELs. However, recent progress in HHG sources scalesdown the size of the setups to the table-top. Moreover, prominent recent developments includelow-power X-ray (bremsstrahlung) tubes and plasma-based X-ray pulse generation. It is3nticipated that within a few years a table-top intense and stable source of isolated ultrafast XUVand X-ray pulses will appear what will herald the “golden age” of frequency- and time-resolvedX-ray spectroscopy giving rise to a plethora of new techniques and perspective directions. E c o n t i n u u m occ.unocc.core ε kin IPAbsorption PES NXES RIXS(RXES) AES CK(Auger) s i V / V U SAX
E IP SX I R AES
CKUV/Vis
SAX
P E S L L SEX N Neutral Ionized L L e-e-e- e c ne l a v e r o c (a) (b) N e- ( N -1) e- Figure 1: (a) Processes relevant for X-ray spectroscopy viewed from the MO picture standpoint.For absorption, the respective process in UV/Vis range is also shown. (b) Processes relevant forX-ray spectroscopy in the many-body state picture. (Color code for arrows: photon absorption– black, radiative decay of a core hole – red, non-radiative decay – blue). Abbreviations: Pho-toelectron Spectrum (PES), Non-resonant X-ray Emission Spectrum (NXES), Resonant InelasticX-ray Scattering (RIXS), Resonant X-ray Emission Spectroscopy (RXES), Auger Electron Spec-trum (AES), Coster-Kronig (CK).For TM compounds, L -edge spectra enjoy great popularity.
4, 5
They are due to excitation orionization from the 2 s and 2 p orbitals ( L -shell) of the target atom to bound or continuum states,respectively. Being interested in frontier d -orbitals containing information on the nature of metal-ligand or metal ion-solvent interaction, dipole-allowed excitation from 2 p orbitals is addressed morefrequently. Traditionally, solid state samples have been investigated. However, recent developmentsin the vacuum liquid microjet technique have broadened the applicability of soft X-ray spectroscopyto highly volatile liquid solutions.
39, 40
Excitation of a system from its electronic ground state to a high-energy and highly non-equilibrium core-excited state triggers a number of relaxation processes which either representlosses or can be used as a spectroscopic probe of material properties. Let us consider them on theexample of 2 p L -edge spectroscopy as illustrated in Figure 1. In general there are two pictures with4espect to the electronic states that can be used for the discussion. First, the intuitive and there-fore widely used “single electron” MO picture (panel (a)). Second, the more appropriate “statepicture”, which is based in the many-particle eigenstates of the electronic Hamiltonian (panel (b)).Upon absorption of a soft X-ray photon (0.1-1.0 keV), a 2 p core hole is created at the metalcenter accompanied by K -edge ionization of the ligands. This process is depicted on the very leftof Figure 1 and is subject of X-ray absorption spectroscopy. In this case, an electron is excitedfrom a localized core MO to delocalized bound one. Within the state picture, the X-ray AbsorptionSpectrum (XAS) is due to transitions within the N -electron manifold, i.e. between the ground andcore-excited states of the (“neutral”) system.The core-electron can also be excited into the continuum and the photocurrent of the outgoingelectrons can be measured yielding a PES. In the state picture this corresponds to a transitionbetween the initial N -electron (“neutral”) system and the N − N − d → p and 3 s → p pathways, which represent about 1% of the total decay probabilityfor the first-row TMs. Depending on whether the initial excitation has been into a bound or con-tinuum state one speaks about resonant (RXES) or RIXS or the NXES, respectively. The majordecay channel is due to Auger, CK decay, and Interatomic Coulombic Decay (ICD) autoionizationprocesses (Figure 1a). In the state picture these processes involve a transition between the N and N −
5, 44
Absorption spectroscopy provides the most simple type of spectra considered in the context ofX-ray excitation as long as the discrete pre-edge structure is considered only. Viewed in the MOpicture, XAS probes the unoccupied valence orbitals. Conventional X-ray experiments operate inthe weak-field regime and the respective absorption amplitude, proportional to the absorptioncross section at frequency Ω, can be expressed in terms of first-order time-dependent perturbation5heory (Fermi’s Golden Rule): X (Ω) = (cid:88) g f ( E g , T ) (cid:88) i (cid:12)(cid:12)(cid:12) (cid:104) i | ˆ d · e in | g (cid:105) (cid:12)(cid:12)(cid:12) Λ( E g + Ω − E i ) . (1)Here, the | g (cid:105) and | i (cid:105) states are ground and core-excited final states with respective energies E g and E i . Due to finite temperature T , there can be several initial states populated according tothe Boltzmann distribution f ( E g , T ) = exp ( − E g /kT ) / (cid:80) j exp ( − E j /kT ). This especially appliesto TM complexes, where the initial high degeneracy of the electronic states can be lifted due toJahn-Teller effect and SOC, leading to several levels accessible by thermal excitation.Further, ˆ d and e in in Eq. 1 are electric dipole operator and polarization vector of the incominglight. The lineshape function, Λ( E g + Ω − E i ), characterizes the density of the final states and,thus, the form of the absorption bands. Effects to be considered here include the width of theincoming excitation pulse, homogeneous (lifetime) broadening due to Auger and radiative decay,and inhomogeneous broadening due to influence of environment, e.g., solvent or phonon broadeningin solid state. These broadening parameters can be often fitted to best reproduce the experiment.To address XAS theoretically in dipole approximation, one thus needs to calculate energies ofthe core-excited electronic states with respect to the ground one and the ground-to-core-excitedtransition dipole moments. Note that for hard X-ray radiation the dipole approximation breaksdown and higher order multipole moments have to be considered. For L -edge spectra, additional complexity enters due to strong SOC, triggered upon core-holeformation and within the 3 d states themselves. This leads to a characteristic shape of the spectrumcontaining two groups of bands, so-called L and L edges (see Figure 4 below). Staying in the weak-field perturbational regime, the radiative decay of the core-excited state | i (cid:105) prepared upon absorption of X-ray light can be described within second-order perturbation theory.The respective Kramers-Heisenberg expression for the emission amplitude reads R (Ω , ω ) = (cid:88) g f ( E g , T ) (cid:88) f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) i (cid:114) Γ i π (cid:104) f | e ∗ out · ˆ d | i (cid:105) (cid:104) i | ˆ d · e in | g (cid:105) E g + Ω − E i − i Γ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × Λ( E g + Ω − E f − ω ) . (2)It contains the same ingredients as the XAS expression, Equation (1), where in addition ω and e out denote the energy and polarization of the emitted photon, Γ i = 1 /τ i is the inverse lifetime of core-6xcited state | i (cid:105) and state | f (cid:105) represents the final valence state. Notice that due to the fact thatthe sum over i appears under the square, the radiative channels ending up in the same final state f interfere with each other. This photon-in/photon-out process represents an electronic analogueof the well-known vibrational Raman spectroscopy. The only difference is that one addressestransitions not between vibrational levels but between different electronic states. Formally, RIXScorresponds to a non-linear χ (3) process and therefore is analogous to four-wave mixing but withthe last two interactions being of spontaneous rather than stimulated nature. Thus, it is onlylinearly proportional to the intensity of the incoming light. Drawing on this analogy the overallspontaneous light emission signal can be separated into two contributions: sequential incoherentfluorescence where a population in the excited state is created followed by radiative decay and adirect “coherent” Raman process. While in the former case resonant excitation is required, RIXSalso works for off-resonant excitation, i.e. involving only coherences between the participatingstates.In the following we will focus on RIXS only. The RIXS intensity can be recorded as a functionof incoming, Ω, and outgoing, ω , photon energy. Thus, it provides a two-dimensional spectrum,resolving both absorption and emission processes. Often one discusses one dimensional cuts atfixed excitation energy of this spectrum only. Alternatively, the so-called Partial FluorescenceYield (PFY) spectrum at a given excitation energy Ω can be obtained by integrating with respectto the emission energies in some energy window ω ∈ [ ω , ω ] F (Ω) = (cid:90) ω ω R (Ω , ω ) dω , (3)providing a spectrum analogous to XAS. Viewed in the orbital picture (Figure 1a), RIXS givesaccess to the unoccupied MOs as well as to the occupied valence MOs. In state picture (Figure 1b),absorption and emission processes become coupled because of orbital relaxation due to core-holeand valence excitation relaxation. Thus, RIXS approximately maps the valence excited states.Calculating RIXS is a more elaborate task as compared to XAS since these core-hole and valenceexcitation states may have different requirements to the computational protocol. Photon-in/electron-out spectroscopy represents a powerful suit of methods.
25, 49, 50
Depending onwhether the X-ray photon energy exceeds the binding energy of the core-electron or is below thecore ionization threshold one distinguishes between non-resonant PES and Resonant Photoelectron7pectrum (RPES), cf. Figure 1a. PES provides direct information on the energies of core orbitals,e.g., the value of their SOC splitting, if viewed from the orbital picture perspective. In the statepicture, it corresponds to a transition from e.g. ground state of non-ionized system to the core-excited states of ionized one. Moreover, the shake-up excitations, where ionization is accompaniedby valence excitation, can provide valuable insights into the electronic and geometric structure. The resonant counterpart, RPES, contains two processes – direct ionization from the valencelevel and autoionization of the core-excited state. The former one represents a usual PES whereelectrons from valence MOs are ejected having kinetic energies comparable to that of the absorbedX-ray photon. The latter one is an Auger process, i.e. a transition between core-excited state ofthe non-ionized manifold and the valence state of the ionized manifold accompanied by emission ofan electron as depicted in Figure 1b. In terms of orbital picture, it corresponds to core-hole refill bya valence electron and simultaneous emission of another valence electron with high kinetic energy.The total RPES cross section assuming integration over all directions of the outgoing photo-electron reads P (Ω , ε kin ) = (cid:88) i f ( E g , T ) (cid:88) f + Λ( E f + + ε kin − E g − Ω) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:104) f + ψ el ( ε kin ) | e in · ˆ d | g (cid:105) + (cid:88) i (cid:104) f + ψ el ( ε kin ) | ˆ H − E i | i (cid:105) (cid:104) i | ˆ d · e in | g (cid:105) E g + Ω − E i − i Γ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (4)The structure of this expression is a hybrid of first-order Fermi’s Golden rule for direct photoioniza-tion (first term under square) and second order Kramers-Heisenberg-like term for autoionization.The system starts from thermally populated initial states | g (cid:105) . The incoming X-ray light either ion-izes the system by transition to a final state | f + ψ el ( ε kin ) (cid:105) , which is the properly antisymmetrizedproduct of the bound N − | f + (cid:105) and the state of the free electron | ψ el ( ε kin ) (cid:105) , having kinetic energy ε kin . Alternatively, the same state can be obtained by excitationto the intermediate bound core-excited state | i (cid:105) , which then decays non-radiatively, mediated bythe Hamiltonian ˆ H of the system. This decay is essentially due to the Coulomb coupling which isdominating in the respective matrix element.There are some more comments needed at this point. First of all, due to the coinciding finalstate of both processes there is an interference between the direct and Auger parts of the signal.This interference is present also for decay channels from different intermediate states in the Augerpart of the signal. Second, Equation (4) sheds light on the nature of the parameters Γ i entering alsothe RIXS expression, Equation (2). In fact, Γ i = 2 π (cid:80) f + | (cid:104) f + ψ el ( ε kin ) | ˆ H − E i | i (cid:105) | , representing8he total Auger decay rate of core-excited state | i (cid:105) . In principle, radiative decay would add tothe total rate, but for TMs autoionization is the dominating process. Next, we note that herewe assumed the simple product form | f + ψ el ( ε kin ) (cid:105) of the final state and thus correlations betweenthe ionic remainder and the outgoing electron are neglected. A more elaborate scheme takinginteraction of ionization channels requires a scattering theory formulation. Finally, RPES can bealso viewed as a 2D type of spectrum as it depends on the incoming photon energy Ω and thephotoelectron kinetic energy ε kin .As compared to XAS and RIXS, PES is analogous to the former and AES to the latter. The dif-ference is that PES contains information on the core-excited states of the ionic remainder, whereasAES contains that of the core states of the non-ionized system and valence manifold of ionized sys-tem (see Figure 1b). RPES calculations are more elaborate due to the need to compute integralsbetween bound and continuum states to obtain matrix elements of the of the dipole moment andmolecular Hamiltonian ˆ H − E i .In the weak field regime, PES cross sections can be calculated in the framework of the DysonOrbital (DO) formalism. For illustration, we leave the second term in Equation (4) asideand consider only the direct photoionization represented by the first term. Assuming the strongorthogonality between the state | ψ el ( ε kin ) (cid:105) and MOs of the initial system, the expression can berewritten in terms of quasi-one-electron states | Φ DO f + g (cid:105) which are called DOs. It represents an N − | Φ DO f + g (cid:105) = √ N (cid:104) f + | g (cid:105) N − . Using the DO, thetransition matrix element (neglecting polarization dependence) can be expressed as (cid:104) f + ψ el ( ε kin ) | ˆ d | g (cid:105) N ≈ (cid:104) ψ el ( ε kin ) | ˆ d | Φ DO f + g (cid:105) . (5)Here, the subscripts N and 1 denote N -electron and one-electron integration, respectively. Thisformal reorganization allows to separate system and electronic structure method dependence ofthe intensity from the actual representation of the photoelectron. Once the DO is computedfor a particular pair of bound states, | g (cid:105) and | f + (cid:105) , it can be further used to estimate transitiondipoles for different kinetic energies ε kin and even different forms of | ψ el (cid:105) . Commonly used freeelectron functions include the plane wave expansion in terms of partial waves or Coulomb waves. The former is appropriate when photodetachment from a negatively charged species is considered,whereas the latter accounts for spherical Coulomb potential of the ionic remainder. In case of TMcomplexes, one needs many terms in the expansion to realistically approximate the free electronfunction. This approach has facilitated interpretation of experimental XUV and X-ray PESs of9ransition metal complexes in solution.
44, 56–58
Finally, we would like to mention the so-called Sudden Approximation (SA). It assumes thatthe cross section for a pair of states, | i (cid:105) and | f (cid:105) , is proportional to | Φ DO fi | . Thereby the kineticenergy dependence of the cross section is neglected such that no bound-to-continuum integrals inEquation (5) need to be calculated. This approximation is usually considered to be justified if thenature of the transitions and thus of the DOs is similar and kinetic energies are large. MCTDHF
TD-MCSCFnon-SCF SCF f r equen cy t i m e orbital relaxation do m a i n MR-CIMR-CCMR-PT
RASPT2NEVPT2
ΔSCF
HF, KS-DFT Δε TP-DFT, FOA LR TDDFT, ADCEOM/LR-CC CI DFT-CI, LFM
RT-TDDFTTD-CI MCSCF
CASSCFRASSCF ---
Figure 2: Overview on different methods discussed in this review, grouped according to threecriteria: (i) frequency (energy) vs. time domain, (ii) explicit orbital optimization, i.e. SCF vs.non-SCF, and (iii) reference for construction of excited state basis is based on a single or multipleelectronic configurations. (MR-CI, MR-CC, and MR-PT are multi-reference CI, CC, and PT,respectively. Other abbreviations are defined in the text.)10 .1 General remarks
Theoretical studies of X-ray spectra can be divided into two classes: 1) investigations of closed-shellmolecules containing mainly the second-period atoms addressed with single-reference methods and2) works reporting on open-shell systems, primarily TM complexes. In general, first principlesquantum chemical predictions for excited electronic states of TM compounds are significantly moredifficult than for compounds containing main group elements. TM chemistry is closely connectedto near degeneracy effects, also called static correlation, making methods based on a single SCF-like reference configurations unreliable. Thus, computationally more demanding multi-referenceapproaches are often required to obtain quantitative accuracy. In addition, the treatment of d -electrons requires the inclusion of dynamic electron correlation. Finally, especially for core-excitedstates, relativistic effects, e.g., SOC, have to be taken into account.An overview of available methods is provided in Figure 2. In general, corresponding meth-ods can be formulated in time (E,F,G,H) and energy (A,B,C,D) domain, which amounts to solvingeither the time-dependent or stationary Schr¨odinger equation. The spectroscopic observable (Equa-tions (1), (2), and (4)) are formulated in terms of the eigenvalues and eigenstates of the stationarySchr¨odinger equation, ˆ H | Ψ (cid:105) = E | Ψ (cid:105) . However, the spectra could be equivalently expressed in asFourier transforms of multi-time dipole correlation functions. For instance for XAS, these aretwo-point and for RIXS four-point dipole correlation functions.
Beyond this perturbationalregime, i.e. for strong field excitation, or in cases of (shaped) ultrashort laser pulses one has toresort to explicit solution of the time-dependent Schr¨odinger equation including the field-matterinteraction. Compared to the Quantum Chemistry (QC) methods, Real-Time (RT) propagationapproaches have received less attention so far.Another, “dimension” of the classification in Figure 2 is with respect to the type of MOsthat are used to express the many-electron states. In fact, every calculation starts from an MOoptimization; the question is whether it is done within the actual excited state method as well(B,D,F,H) or the MOs are fixed during excited state calculations (A,C,E). Finally, QC methods canbe distinguished according to the used reference space into single (A,B,E,F) and multi-configuration(C,D,H) approaches. Note that we are not aware of the methods which fall into category G.In the following, we will review ab initio QC methods, focussing on MCSCF-based methods infrequency domain (Figure 2D and C) as applied to X-ray spectroscopy of TM complexes. However,other methods mentioned in Figure 2 will be briefly discussed as well.11 .2 Basic concepts: SCF
The HF method is the most basic MO approach discussed here. Although it is not directly ap-plied to core-level spectroscopies it is central for understanding other methods. It can be char-acterized by the following points: (i) The N -electron wave function Ψ N ( r , ..., r N ), describingthe ground state of the system, is represented as a single Slater determinant Φ N ( r , ..., r N ) =det { φ a ( r ) φ b ( r ) ...φ c ( r N ) } of one-electron MOs { φ i ( r j ) } (for inclusion of spin, see Section 3.6).(ii) The optimal set of { φ i ( r j ) } is searched variationally to obtain | Φ N (cid:105) = | Φ g (cid:105) giving the bestpossible estimate E HF of the exact ground state energy ¯ E under the restriction of (i), see Eq. (6).(iii) The electronic Hamiltonian contains one- and two-electron terms ˆ H = (cid:80) i ˆ h ( i ) + (cid:80) ij ˆ g ( i, j ).HF theory is a mean-field approach where the latter are approximated by a one-electron potential (cid:80) ij ˆ g ( i, j ) ≈ (cid:80) i ˆ V eff ( i ). Matrix elements of this one-electron operator are expressed via Coulomb, J kl , and exchange, K kl , two-electron integrals as (cid:104) φ k | ˆ V eff (1) | φ k (cid:105) = (cid:80) l ( J kl − K kl ). (iv) The prob-lem is reformulated in terms of a one-electron eigenvalue equation ˆ f i | φ i (cid:105) = ε i | φ i (cid:105) , yielding orbitals φ i ( r j ) and their energies ε i . Since the Fock operator, ˆ f i , depends on the orbitals { φ i ( r j ) } theequation is solved iteratively (SCF scheme). Essentially, this is done via a search for unitarytransformations ˆ U (ˆ κ ) = exp(ˆ κ ) of orbitals which give a minimal ground state energy E HF ( κκκ ) = min κκκ (cid:104) Φ g | exp( − ˆ κ ) ˆ H exp(ˆ κ ) | Φ g (cid:105) (6)where ˆ κ is an anti-hermitian one-electron operator. Thus, the orbital variations are parametrizedin terms of the elements of the skew-hermitan matrix κκκ , which represents ˆ κ in the basis of MOs;one may think of the elements of κκκ as corresponding to pairwise orbital rotations.In practice, one uses the Linear Combinations of Atomic Orbitals (LCAO) form, where the MOsare represented in an atomic basis, { χ k ( r ) } , φ i ( r ) = (cid:88) k c ik χ k ( r ) . (7)Thus the coefficients { c ik } are variationally optimized.According to its mean-field nature, HF theory does not take into account correlations due tothe electron-electron interaction. Electron correlation is at least a two-body effect and shouldbe distinguished from the one-electron orbital relaxation accounted for via the SCF procedure.Different types of correlations can be identified if we adopt for the moment a classical point of12iew. Two electrons in an atom will try to “occupy” orbits with different radii (radial correlation)or reside on different sides of the nucleus (angular correlation). This type of correlation is calleddynamic (or weak) and can be accounted for by allowing | Ψ N (cid:105) to have in addition to the leading HFterm | Φ N (cid:105) a small admixture of other determinants built on the same set { φ i } but with differentinteger occupation numbers (excited determinants). This approach will be considered later inthe context of the CI method in Section 3.3.1. There are cases, when a single leading term for | Ψ N (cid:105) cannot be found since several of determinants have comparable weights. This more extremecase leads to complete break-down of HF approximation and is referred to as static (or strong)correlation. It will be considered in the framework of MCSCF theory in Section 3.3.2. Finally,one may distinguish differential correlation, when the correlation energy is different in differentelectronic states. The KS-DFT method belongs to the same class as HF, see Figure 2B. The approach is ratherdifferent though. Backed by the Hohenberg-Kohn theorems it formulates the energy as a func-tional of electron density ρ ( r ) subject to the variational principle. Since ρ ( r ) is a three-dimensionalquantity and thus simpler than the 3 N -dimensional wave function, it offers distinct benefits fromthe viewpoint of computational efficiency. In practice, one considers the Kohn-Sham scheme,where a single determinant is postulated for the ground state. The density is determined as ρ KS ( r ) = (cid:80) i | φ KS i ( r ) | , with { φ KS i ( r ) } being the occupied Kohn-Sham MOs. This approachaims at the exact description of a system, adopting a formally non-interacting electron ansatzfor the wave function. The “non-classical” terms which constitute the difference between thisHF-like solution and the exact one are thus transferred to the one-electron potential ˆ V eff ( i ).The energy functional of the electron density which is variationally minimized reads E [ ρ KS ( r )] = T KS [ ρ KS ( r )]+ J [ ρ KS ( r )]+ E el − nuc [ ρ KS ( r )]+ E XC [ ρ KS ( r )], containing kinetic energy of non-interactingsystem, “classical” Coulomb repulsion energy, electron-nuclear attraction energy, and the eXchange-Correlation (XC) energy E XC [ ρ KS ( r )] comprising the difference between exact and approximate( T KS ) kinetic energies, the effect of electron correlation, and exchange interaction. Various ap-proximate E XC exist and numerous techniques for the construction of this functional have beendevised.All shortcomings of Density Functional Theory (DFT) result from the approximate nature of E XC . Most notable is the so-called self-interaction error. In HF, the interaction of an electron13ith itself is canceled by the exact relation J ii − K ii = 0. This is, however, not true anymore ifapproximate XC functionals are considered. This appears to be critical in many respects andespecially for the core-levels of atoms (see Section 3.5 below). The KS-DFT method has replaced HF in modern applications. Although being also of one-electronnature, it describes independent quasi-particles, implicitly accounting for (dynamic) electron cor-relation. Both methods can be used to generate a suitable basis of orbitals for subsequent electroncorrelation treatment by single-reference methods (Figure 2A), such as CI, PT, or CC approachesas discussed below. The choice of a proper orbital basis is more important for core-excited statesthan for valence-excited ones. The reason is that removing an electron from the core region wherethe density of electrons is very high significantly influences electrons occupying other shells duesubstantial changes in the screening of the nuclear charge, which can be associated with an ad-ditional polarization potential. Thus, most of the orbitals will strongly relax to become moretightly bound in energy and more confined in space. Naturally, this type of orbital relaxationis very important for core ionization or the description of the subsequent NXES; although theremight be exception, see Reference 67. This may require to perform an SCF orbital optimization toaccount for the core-hole excited state to accurately predict X-ray spectra. The simplest way to doso is to obtain relaxed orbitals where the core-excited atom is replaced by the next element fromthe periodic table. Within the so-called Z + 1 or equivalent core approach these orbitals are usedfor calculations of the parent molecule. Important applications include the interpretation of PESsand thermodynamic data. Although both HF and DFT techniques were described in the previous section as purely groundstate methods, in principle, they can be applied to excited states as well. In this case, the excitedstate is also represented as a single Slater/Kohn-Sham determinant analogously to the ground state;this is called ∆
SCF method since the transition energy is be obtained as the difference betweenrespective SCF energies. Its application to core-excited states has been first suggested in Reference70. Because some of the features of the ∆SCF method will be important for the later discussion ofMCSCF calculations of core-excited states, more details will be provided in the following.A major issue for all methods which involve orbital optimization (cf. Figure 2B,D) is the so-called variational collapse . Let us assume that we have setup a wave function with a hole in somecore orbital and an additional electron in a virtual orbital and naively perform a variational SCF14ptimization (note that no spatial symmetry is assumed and the spin coincides with that of theground state). For a randomly chosen initial orbital guess, the virtual orbital hosting an excitedelectron will be transformed to the core one in a series of unitary transformations at every SCFiteration. In other words, without additional tricks the wave function is likely to collapse to theground state and core-excited states cannot be accessed. The only reason why such calculationsof deep core-hole excitations might converge to the desired local minimum solution is that usuallythese deep-lying core orbitals have a negligible overlap and couplings with other MOs. For largermolecules and shallow holes this, however, might not work leading to oscillations, divergence, or tocollapse to the ground state.To circumvent this problem, several schemes have been suggested (Figure 2B). In approach I ,one searches consecutively for higher-lying variational solutions, e.g., under the constraint thateach next solution should be orthogonal to all previously found, although this additional conditionis not always rigorously applied, see below. This might be a tedious procedure if more than afew core-excited states are needed. Alternatively, in approach II , a one-electron orbital basis isconstructed in such a way, that the differences of orbital energies give good estimates of the X-raytransition energies (∆ ε approach ).Several different schemes along the lines of approach I have been suggested to perform a con-strained search. They differ in the construction of occupied and virtual MOs involved in the excita-tion and in setting their occupation numbers, see Reference 71 and references therein. Exemplarily,one should mention the application of a multiple hole/particle algorithm within orthogonality con-strained DFT to the calculation of K -edge spectra. Moreover, state-following algorithms can beapplied to avoid the variational collapse without explicit setting the orthogonality constraint. Here,one specifies the desired occupation number pattern for the initial guess and attempts to main-tain it during SCF iterations. This can be accomplished by applying different kinds of projectorsto separate core orbitals from others at every iteration, see Reference 73 and references therein.Alternatively, one can utilize a maximum overlap criterion
74, 75 to identify the core-hole orbital ateach SCF step. In principle, one can also apply conventional SCF subject to a special constructionof the initial guess as discussed in References 76, 77. Here, in the first step the singly-occupied coreorbital is frozen and thus it cannot mix with other MOs via unitary transformations ˆ U (ˆ κ ), i.e. onegets a constrained SCF minimum. Next, one uses this partially relaxed wave function to convergeto the local energy minimum, corresponding to a fully relaxed core, making use of quadraticallyconverging algorithms. 15n approach II, one aims at obtaining orbitals partially relaxed due to presence of the corehole such that differences of orbital energies, ε i , are a good approximation of transition energies E i − E g = (cid:104) Φ i | ˆ H | Φ i (cid:105) − (cid:104) Φ g | ˆ H | Φ g (cid:105) ≈ ε i − ε g . Several versions have been suggested, differing inthe included fraction of the core hole and excited electron causing orbital relaxation, e.g., full orhalf of the core hole and full neglect or inclusion of the excited electron.
78, 79
One of the prominentrepresentatives of approach II is the
Transition Potential DFT (TP-DFT) method reviewed inReference 81. Here one chooses a half-occupied core orbital and neglects the effect of the excitedelectron. Such a choice of a partial occupation can be viewed from a more general viewpoint ofensemble DFT,
82, 83 where the number of electrons in the system varies from N − N . For theexact XC potential, the energy between points with different integer particle numbers E ( N − E ( N ) changes linearly with fractional occupation n and the derivative of the energy exactlyamounts to the orbital energy ∂E/∂n i = ε i at any value of n i ∈ ( N − , N ). This is howevernot the case for approximate XC functionals as the energy behaves nonlinearly. According toLagrange’s mean value theorem the exact relation between the energy derivative and ε i is restoredat some point between N − N electron numbers. In a sense, TP-DFT is pretending to searchfor this optimal point approximating it by N − /
2. More generally, fractional occupation can beconsidered as a “fitting” parameter to generate optimal orbitals for X-ray calculations.
86, 87
Note that in a number of works an even more simplified protocol belonging to approach II, theso-called
Frozen Orbital Approximation (FOA) , has been used. Here, only ground state SCF iscarried out and intensities and energies are fully based on transition matrix elements and energydifferences between ground state Kohn-Sham orbitals. FOA is thus completely neglecting excitedstate orbital relaxation and in this respect it is similar to Koopmans’ theorem approach for HF orits analogues for DFT
84, 89, 90 as applied to X-ray PES. However, it gives reasonably good resultsfor K -edge XAS and NXES of TM compounds.
91, 92
For approach I, the calculation of transition matrix elements is quite complicated in case of non-orthogonal wave functions and can lead to spurious results. Within approach II they are usuallyestimated on a quite approximate basis as X (Ω) ∝ | (cid:104) Φ i | ˆ d · e in | Φ g (cid:105) | ≈ | (cid:104) φ i | ˆ d · e in | φ g (cid:105) × S | .Here, | Φ g (cid:105) and | Φ i (cid:105) are SCF solutions for ground and i th core-excited states and | φ g (cid:105) and | φ i (cid:105) are orbitals corresponding to the respective one-electron excitation. Thus, the transition matrixelement between N -electron states is replaced by a one-electron integral. The factor S is the N − The main advantage of the ∆SCF method is that one can use the well-developed machinery ofground state SCF theory for obtaining excited state properties without much of new development.One should keep in mind, however, that in case of DFT-based techniques the accuracy of the under-lying density functional is of vital importance. ∆SCF-like methods have been widely applied to the K -edges of the second period elements from small molecules
75, 94, 95 to macromolecules
96, 97 and con-densed phases.
79, 81, 93, 98
It has been even attempted to calculate energies of L -edge transitions ofSi, P, S, Cl, and Cu containing systems, although without proper treatment of relativistic effectslike SOC. Finally, ∆SCF can also be used to optimize electronic states with multiple holes residingon the core or valence MOs as relevant for Auger spectroscopy and shake-up XAS transitions.
99, 100
Apart from ∆SCF-like approaches adapting the HF or KS-DFT ground state to compute higher-lying states, there is a plethora of single-reference energy-domain excited state methods; cf. Fig-ure 2A,B. The respective excited states can be represented in the general form | Ψ i (cid:105) = ˆ E i | Ψ g (cid:105) , (8)where | Ψ g (cid:105) is the ground state obtained with any suitable method and ˆ E i is an excitation operator.In single-reference methods, | Ψ g (cid:105) is usually taken to be a single ground state Slater (or Kohn-Sham)determinant, | Φ g (cid:105) , and orbitals are not further optimized. We start the discussion of this class ofmethods with the CI technique, being conceptually the simplest although not computationally mostefficient representative. The CI framework facilitates understanding of other (even conceptuallydifferent) methods as it provides a very pictorial tool of wave function composition.CI is one of the most logical ways to improve on the SCF description. First it provides asystematic way for inclusion of electron correlation and second it can be naturally extended to thecalculation of excited states. Since ground state and excited Slater (or Kohn-Sham) determinantsform a complete basis, a CI state i can be written as | Ψ CI i (cid:105) = (cid:88) j C ij | Φ j ( { φ fixed k } ) (cid:105) . (9)Here, the determinants | Φ j (cid:105) are built from the fixed SCF orbitals { φ fixed k } , obtained prior to the CIprocedure, e.g., by ground state HF or KS-DFT. In practice, the coefficient matrix C is obtained17y writing the electronic Hamiltonian in the basis of determinants which is iteratively diagonalizedusing the Davidson algorithm. Using all possible determinants in the expansion Equation (9) corresponds to the exact FullCI (FCI) solution for a given basis set. Its computational complexity grows exponentially withthe number of basis functions and truncation schemes are usually applied. Starting point is theclassification of the terms in Equation (9) as being unexcited reference, singly, doubly, triply, andso on excited electronic configurations: | Ψ CI (cid:105) = | Φ g (cid:105) (cid:124)(cid:123)(cid:122)(cid:125) ref. + (cid:88) ia C ia | Φ ai (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) singles + 14 (cid:88) ijab C ijab | Φ abij (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) doubles + ... (10)In this notation, indices i, j, ... and a, b, ... denote occupied orbitals from where an electron is excitedand virtual orbitals to which electrons are excited in the determinant | Φ ab...ij... (cid:105) , respectively. To makethe problem computationally tractable, one truncates the expansion Equation (10) including, e.g.,all determinants up to a given excitation class. This gives rise to CI Singles (CIS), CI SinglesDoubles (CISD) methods, and so on. Since electron correlation is at least a two-body effect, doublesare necessary to correct for it on top of a HF reference. Singles are important for description ofcharge density and thus for properties like transition dipole moments. If only singles are included,it might provide a good tool for spectroscopy with near-HF quality for the excited states.From the viewpoint of quantum chemistry the core-excited electronic states are not any differentfrom the valence-excited ones. But, the problem is that one would need to calculate thousandsor even millions of states to reach the highly lying levels relevant for X-ray spectroscopy. Thisproblem stems from the fact that, if one is interested in the n th eigenvalue, the efficient Davidsonalgorithm standardly used for this purpose, diagonalizes at least an n × n matrix and thusobtains non-interesting eigenvalues below n . Since matrix diagonalization is the main computationalbottleneck such a brute force approach wastes computational resources if this is possible at all. Letus consider, for instance, two systems H O ( K -edge) and [Fe(H O) ] ( L -edge). If one takes aquite moderate triple-exponent TZP basis and accounts for singly and doubly excited configurationswhich is the minimal reasonable choice to include electron correlation within the CI approach, thetotal amount of valence states is 46 872 and 136 370 360, respectively. This means that to reach thefirst 1 s - or 2 p -core-excited state one would need to explicitly calculate a huge number of states,where in case of XAS often only few low-lying ones are of relevance for the spectrum (due to thefactor f ( E g , T ) in Equation (1)). 18here are several ways to circumvent this difficulty. First of all, one can stay within theconventional Davidson scheme and project onto a smaller subspace of trial functions excludingmost (or even all apart from the ground state) valence states (Figure 3). In practice one makesuse of the concept of an Active Space (AS) of MOs, where certain orbitals (i.e. the core orbitalsfor XAS calculations) are allowed to change their occupations, while for the others electronicexcitations are forbidden (also called Core-Valence Separation (CVS) ). Equivalently, thesubspace of electronic configurations can be limited according to simple Koopmans’ estimates ofthe state energy based on the orbital energies as realized within Restricted Excitation WindowTime-Dependent Density Functional Theory (REW-TDDFT). Val. 1h-core 2h-core V a l . - c o r e2h - c o r e a) RAS3RAS2RAS1SecondaryInactive/FrozenInactive/Frozen Fe 2pσ3dσ 3d * π* ( )CO /Fe 3dn3d Other core MOsOther occupied MOsOther unoccupied MOs
HF CI // F u ll C I ho l e e – b) Figure 3: (a) Illustration of the projection of the full Hamiltonian to the subspace of limited valence-and core-excited configurations (blue blocks), neglecting the respective off-diagonal coupling blocks(red crosses). (b) Principal MO subspaces used in the RASSCF method. The typical AS isexemplified for the Fe(CO) complex. The table on the right demonstrates the level of orbitaloptimization (HF) and CI for each subspace. (Valence electronic excitations are depicted by redand core excitations by blue arrows, respectively.)This “reduced subspace” concept is illustrated in Figure 3a. It shows a pictorial view of the CI-like Hamiltonian matrix expressed in the basis of electronic configurations resulting from valence,single core-hole (1h-core), and double core-hole (2h-core) excitations. The overall dimensions ofthe Val–Val block are those thousands or millions mentioned before. In fact, in XAS one is usuallyinterested in relatively small sub-block (marked with blue color) of the 1h-core–1h-core block in-cluding only few tens, hundreds or even thousands of states if only the discrete pre-edge region is19f interest. In case of RIXS, one needs in addition a number of low-lying valence-excited states alsodenoted as a blue sub-block of the Val–Val block. The 2h-core–2h-core block is usually completelyneglected. The same holds for the coupling Val–1h-core and 1h-core–2h-core blocks. The motiva-tion for neglecting off-diagonal couplings between two configurations from v (valence) and c (core)is provided by PT. Here, the error is estimated to be of the order of V vc / ( E c − E v ) in the wavefunction and | V vc | / ( E c − E v ) in energy, where V vc is the off-diagonal Hamiltonian matrix elementand E c − E v is the usually large difference between energies of these configurations standing onthe matrix diagonal. In fact, the restricted subspace approach is equivalent to neglecting specifictype of two-electron integrals of the form (cid:104) ab | cd (cid:105) , where three of the orbitals a, b, c, d are valenceand one is core orbital or vice versa and where orbitals in the bra are both core and in the ketboth valence. Hence the diagonalization of the enormously large CI matrix in Figure 3acan be replaced by a Davidson diagonalization of the small (blue) sub-blocks. The uniform errorintroduced by CVS was estimated to be of the order of 0.5-1.0 eV for K -edges of the second periodelements. Another way to access core-states is to modify the Davidson algorithm itself. For instance, inReference 111, a modification of the standard algorithm employing the so-called “root-homing”technique has been suggested. It utilizes the maximum overlap criterion (similar to ∆SCF) toidentify the desired state during each Davidson iteration step. A more recent modification builds on the idea of Reference 113 and introduces an intermediate projection of the matrix to bediagonalized onto the subspace of the trial vectors having energies in a predefined window. Afterdiagonalization the eigenvectors are projected back onto the full space. The approximate energy ofthe trial vector is estimated as the orbital energy difference. This approach is superior to the simpleindex fixing used in REW-TDDFT as it systematically includes contributions out of the initiallypredefined reduced space in a consistent way. This gives a good approximation for the high-lyingstates in the full space of trial vectors without explicit calculation of the lower-lying ones. Forthe high density of states typical for X-ray spectra, a more advanced adaptive hybrid algorithmcan be applied to improve the convergence.
Apart from CI, so far such approaches wererealized for Linear-Response TDDFT (LR-TDDFT) and Equation-Of-Motion CoupledClusters (EOM-CC) methods discussed below.In the truncated CI ansatz, Equations (9) and (10), the underlying one-electron orbitals deter-mine accuracy and convergence of the CI expansion. That is why the choice of the orbital basis is ofconsiderable importance for the overall efficiency. It is known that virtual MOs from ground state20F calculations are too diffuse especially for large basis sets and thus represent a poor choice. Thesuggested remedies are to use improved virtual orbitals, natural orbitals, or even KS-DFTorbitals.
In Reference 87, different types of the relaxed initial orbitals (restricted, unrestricted,and restricted open-shell HF and KS-DFT) have been considered and their performance for core-excitations within CIS is deduced by comparison to experiment. It was found that a HF referencewhere orbitals are relaxed to the core-hole is most suitable. Ground state frozen orbitals are notthat good and produce errors of order of 10 eV in core excitation energies.CI has been traditionally used for predicting X-ray spectra due to K -edge transitions in smallmolecules. For L -edge spectra of TM complexes, the Configuration Interaction onKS-DFT orbitals (DFT-CI) approach enjoys great popularity. In this method, the standard CI ˆ E ( i ) operator is applied to the ground state KS-DFT determinant ( | Φ KS (cid:105) ). The idea is to take thedynamical correlation effects at the level of DFT into account and to incorporate stronger (static)correlation effects by CI. As a result, the number of important determinants and thus the sizeof the Hamiltonian matrix can be substantially reduced. This method largely relies on the bettersuitability of Kohn-Sham orbital energies as estimates of the electronic transition energies. Sincethese orbital energy differences are dominating the diagonal of the CI Hamiltonian matrix in thebasis of determinants, it makes the major contribution to the final CI energies.
However, sincethe CI is built on top of quasi-particle orbitals which already implicitly include dynamic electroncorrelation a number of points have to be considered. First, double counting of correlation effectsshould be excluded which can be done by energy-dependent exponential scaling of the CI couplingsbetween determinants.
Moreover, the two-electron integrals calculated on KS-DFT MOs areoverestimated and need to be scaled down even in case of CIS, where no double counting occurs dueto its “uncorrelated” character.
The need to parametrize this scaling makes the method semi-empirical in nature although the number of parameters is kept as small and universal/transferableas possible. The parameters determined, e.g., by least-squares fitting to experiment, depend onmany aspects: XC functional, multiplicity and character of the target excited states, and so on.Thus, the variants usually applied to valence TM spectroscopy need to be adjusted to addresscore spectra, e.g., by changes of the virtual orbital energies.
The hybrid nature of DFT-CI provides another distinct advantage making use of the correlatedorigin of the Kohn-Sham MOs. For TM complexes with covalently bound ligands it reproduces theeffect of bonding better than HF which predicts bonds to be in general too ionic. There are twovariants of DFT-CI which were applied to X-ray spectra of TMs. In their approach, Ikeno et al.21tudy crystalline metal oxide phases within a cluster approach.
No corrections for double-counting or scaling of two-electron integrals is performed. As an XC functional the simplest localdensity approximation is applied, thus containing no exact exchange. Note that the CI problemis solved only in small subspace of MOs containing few electrons (essentially 2 p − n d m + n , where n = 0 ,
1, and also 2 p ligand orbitals to account for charge-transfer effects) and the other electronsand their interaction with active ones are taken in an averaged approximate way. This makesthis approach being strongly related to the Ligand Field Multiplet (LFM) semi-empirical methoddescribed below, although no fitting of parameters beyond usual DFT level is performed. Despiteof the absence of fitting parameters and the approximations introduced, this method demonstratesa remarkable level of agreement with experimental XAS and other types of spectra. A similarCI approach based on state-averaged relativistic Dirac-Hartree-Fock orbitals (no DFT) has beenapplied to X-ray 4 f and 2 p PESs of UO and MnO, respectively.The second DFT-CI method specifically designed to calculate TM L -edge spectra is DFT-basedRestricted Open-shell Configuration Interaction Singles (ROCIS) . The restricted open-shell SCFcalculation ensures the proper reference ground state configuration with unpaired electrons whichis typical for TM complexes. Being a DFT-CI technique, it needs rescaling of two-electron integralsto additionally reduce electron-electron interaction similar to LFM. In practice, three parameterswhich are universal across the periodic table are introduced: two scaling Coulomb and exchangeterms on the diagonal of the CI matrix and one common scaling parameter for all off-diagonalelements. With this it puts emphasis on the diagonal of the CI matrix containing basically theexcitation energy for a given orbital pair. These scaling coefficients are optimized only for twodensity functionals: B3LYP and BHLYP in a specific def2-TZVP(-f) atomic basis. Thus, thismethod is semi-empirical making use of a training set containing experimental spectra. From theviewpoint of CI it accounts only for single electron excitations thus being intrinsically of SCFquality for the excited electronic states and not including any additional electron correlation ontop of KS-DFT MOs. Although being a relatively crude approximation, it allows to keep theconfiguration space very compact at the price of neglecting shake up excitations. Therefore, itcan significantly underestimate the density of states in energy regions where such satellite bandsare occurring. On the other hand, this approach is computationally very efficient, especially inconnection with a local correlation treatment.
It has been applied to models including up toseveral tens of TM atoms.
This method is one-component (see Section 3.6) but scalar relativisticeffects and SOC are included what is prerequisite to reproduce L -edge X-ray spectra. Overall,22his approach provides fairly good agreement with XAS experiments at least for highly-symmetricsystems. Finally, we briefly mention
LFM theory as another semi-empirical CI-like method. It has playedan important role in understanding the complex structure of X-ray spectra during the last threedecades. Within the MO paradigm, the covalent bonding situation is usually described as a one-electron effect resulting in mixing of orbitals of bound atoms. LFM follows more the valence bondtheory route, where these effects are taken on the level of many-electron CI between configurationswith localized electrons.LFM theory has its origin in ligand field theory developed back in 1950s, but there are severalmodifications to accommodate the finer spectroscopic features of TM complexes. It is an atomictheory focussing on an isolated metal ion with, e.g., a 3 d n configuration (corresponding to anelectronic state | d n (cid:105) ). The in general complicated chemical environment is taken into account asa perturbation via an effective electric field that disturbs the spherical symmetry of an atom. Since the effect of ligands goes beyond effective electrostatic interaction especially in case of highlycovalent metal-ligand chemical bond, this model needs to be significantly extended. The majorextension is to incorporate sophisticated bonding contributions in terms of charge-transfer betweenmetal and ligand.
For this purpose in the framework of X-ray spectroscopy, one needs to considernot only | p d n L (cid:105) and | p d n +1 L (cid:105) electronic configurations with “neutral” ligand ( L ) andeither fully filled 2 p -shell (2 p ) or having one core hole (2 p ), but also states where an electron istransferred, e.g., from the ligand to the 3 d -shell – | p d n +1 L + (cid:105) and | p d n +2 L + (cid:105) .If one extends the number of basis functions in which the LFM Hamiltonian is written to thesize of a few or even tens of thousands electronic configurations, upon diagonalization one obtainsa complex structure of eigenvalues called multiplets. These multiplets originate from the manifoldof atomic levels, which are additionally split in energy due to the ligand field and charge-transfereffect.Often, LFM theory serves as a model for semi-empirical fitting of a number of parameters toreproduce experiments (for an overview see Reference 4). The Coulomb and exchange interactionswithin core and valence manifold as well as between them are parametrized via Slater-Condonintegrals precalculated at the HF level and tabulated for each element. Additionally, their numericalvalues are often reduced to 80% or even smaller, to roughly account for electron correlation. The3 d valence and the 2 p core SOC is also introduced semi-empirically as a parameter within LS or jj coupling limits (see below) depending on the TM ion considered. Another parametrized23nput for the Hamiltonian is the ligand field. In case of high symmetry TM complexes, e.g.,octahedral or tetrahedral, only one effective parameter is used. However, the number of ligand-field parameters grows when symmetry is lowered. Finally, the charge-transfer parameters areadjusted to fit experimental results. The overall number of adjustable parameters can becomequite large, thus complicating the fitting procedure. Due to its semi-empirical nature this theoryrepresents a valuable tool to postanalysis of experiments but has only limited predictive power.The transition energies are usually calculated with an accuracy not better than 1.0 eV but in somecases an impressive agreement with experiment has been achieved. To parametrize more complicated ligand fields the necessary information can be taken fromab initio calculations. The tandem of the simplicity of LFM theory with the versatility of firstprinciples approaches gave a rebirth to the well-established concept.
Moreover, band structurecodes can be utilized for such a parametrization in case of solids.
Multi-configurational methods have been introduced to describe bond dissociation, distorted ge-ometries, near-degenerate ground states, and electronically excited states. During decades of ap-plication, these methods have acquired a reputation of being the “golden standard” for this type ofproblems.
From the 1980’s on, when the first efficient and rapidly converging algorithms forMCSCF have been developed, also K -edge core-level spectra for diatomics and small polyatomicmolecules have been calculated.
76, 154, 155
During the last years investigations of L -edge X-ray spec-tra of TM compounds emerged, where it has been shown to be able to cope with highly excitedstates in multiconfigurational situations. The MCSCF method is a hybrid of the HF and CI approaches. It employs the CI multi-determinantal ansatz, Equation (10), for the many-body state, | Ψ MCSCF i (cid:105) = (cid:88) j C ij | Φ j ( { φ ( i ) k } ) (cid:105) , (11)with the important difference that not only the CI coefficients, C ij , are variationally minimized,but also the set of coefficients c ( i ) kl , within the LCAO ansatz φ ( i ) k ( r ) = (cid:88) l c ( i ) kl χ l ( r ) . (12)24he MCSCF wave function can be parametrized as | Ψ MCSCF i (cid:105) = exp(ˆ κ ( i ) ) (cid:88) j C ij | Φ j ( { φ ( i ) k } ) (cid:105) , (13)where operator exp(ˆ κ ( i ) ) describes an orthogonal transformation of a set of MOs { φ ( i ) k } which arespecific to state | i (cid:105) . The energy functional can be expressed as E MCSCF i ( C , κκκ ) = min C ,κκκ (cid:80) jk C ij C ik (cid:104) Φ j | exp( − ˆ κ ( i ) ) ˆ H exp(ˆ κ ( i ) ) | Φ k (cid:105) (cid:80) k C ik . (14)Upon joint optimization of CI and MO coefficients, the one-electron MO basis adapts to the de-scription of electron correlation effects. This allows to quite reliably account for static correlation(multiconfigurational character) and (near-)degeneracies of the electronic states, which are typicalfor TM compounds.However, this simultaneous treatment of orbital relaxation and electron correlation leads to aconsiderable increase in complexity of the actual simulations. The number of CI configurations andMO coefficients quickly increases with the number of atoms and the number of basis functions peratom. The straightforward application of Equation (11) considering all orbitals on equal footing israther demanding if not impossible already for medium-sized systems. There have been attemptsto individually select the most important configurations, e.g., via perturbation theory. Amore robust and popular solution is provided by the concept of the AS. At the most generallevel it corresponds to the disjoint subsets of MOs for which different levels of CI expansions areperformed. Historically, the most popular variant is the Complete Active Space SCF (CASSCF)approach, cf. Figure 3b. Here, the orbitals are classified in three groups: (i) those which staydoubly occupied are called inactive, (ii) those which are unoccupied, virtual or secondary, and (iii)active ones, forming a subspace where all possible distributions of some number of active electronsare allowed, thus corresponding to a subspace FCI calculation. It is important to note that inactiveorbitals are treated at the HF level and that is why the scaling with the physical size of the systemis not prohibitive. The exponential growth of the computational demands can be thus mitigated oreven avoided by treating only the local correlated subsystem within the AS. For fine tuning withrespect to important configurations, one can introduce additional subspaces restricting the numberof electrons which can be excited from or to them. This leads to the RASSCF method ifthe number of active subspaces is equal to three or, in general, to the Generalized Active SpaceSCF (GASSCF) method or similar approaches.
25o sum up, MCSCF unites the benefits of the SCF and CI allowing not only for orbital relaxationupon core-hole formation but also for strong correlation effects. Therefore, it optimally suits thetreatment of TM systems which are well-known for static correlations. Moreover, as we haveseen for the case of CI for core-level calculations, the most pragmatic and robust approach to core-excited states employs restricted subspaces of orbitals and excitations between this subspaces. Thisnaturally leads to the idea of using CASSCF or RASSCF for X-ray spectroscopy. It should also suitto probe with X-rays the excited state dynamics involving bond cleavage or conical intersections,being a native habitat of MCSCF-based methods.The choice of included orbitals in the AS fixes electron configurations considered within CItreatment. This has to be done in a balanced way as in general correlation effects do not naturallysplit into strong and weak ones but continuously fill the range of importance. Thus, the selectionof the AS has been for a good while considered to be an art. As of now there is a consensuson how to select a more or less optimal AS in general non-pathological cases.
The ASshould comprise: (i) orbitals which are involved in the excitation to a particular state i , (ii) allorbitals hosting unpaired electrons, (iii) strongly correlating orbitals – usually respective bondingand antibonding MO pairs, (iv) MOs which are responsible for near-degeneracy, e.g., σ and σ ∗ ifthe chemical bond is dissociating or degenerate d -orbitals for TMs or f -obritals for lanthanidesand actinides, (v) more specifically in case of 3 d -TMs, 3 s and 3 p orbitals might be included forthe elements in the beginning of a row to include semicore correlation. For the later elements of arow, additional set of 4 d orbitals is required to recover radial dynamic correlation of the electronsin the non-bonding d -orbitals already in the AS (double-shell effect). Note that some of the rulesmight be redundant since, for instance, 3 d electrons are strongly correlating and at the same timerespective orbitals can be singly occupied and degenerate. Naturally, to perform a computationallyfeasible calculation, the number of active MOs as well as active electrons needs to be kept as smallas possible.On the one hand, core-excited states can be considered as quite difficult from the viewpointof selection of AS since the number of d -electrons changes in course of excitation/deexcitation or(auto)ionization. On the other hand, in practice this does not represent a problem. Exemplarily,let us consider, how to choose the AS for a core state RASSCF calculation of the Fe(CO) modelsystem, see Figure 3b. Although Fe(CO) has a high symmetry, it is not obeying atomic selectionrules. However, X-ray excitation involves a very localized part of the electronic system, havingpredominantly atomic nature. Thus, we can still use approximate angular momentum (∆ l = ± L -edge transitions one needs to include at least2 p and 3 d orbitals into the AS as illustrated in Figure 3b. The AS is further subdivided into RAS1,RAS2, and RAS3 subspaces. Usually, one is interested in singly core-excited states whereas doublycore-hole states are not relevant (Figure 3a). To account for this and simplify the calculation, 2 p orbitals are put in a separate subspace RAS1. Further, only one electron is allowed to be excitedfrom RAS1 to other active orbitals, meaning that RAS1 contains 6 (for valence states) or 5 (for corestates) electrons distributed in the three 2 p orbitals. The 3 d electrons are usually highly correlatedwhat especially applies to Fe(CO) being a covalent complex. Therefore, one would prefer toinclude them in a separate subspace, RAS2, and construct FCI-type configurations. Apart fromthe doubly-occupied non-bonding n d orbitals which reside almost exclusively on the Fe atom,there is a pair of bonding/antibonding σ d and σ ∗ d orbitals which are essential for correlationtreatment. Moreover, one of the two main transitions in XAS is due to the 2 p → σ ∗ d excitationand in RIXS the σ d → p relaxation plays an important role. Additionally, one can includeligand orbitals, e.g., in the RAS3 subspace and limit the number of electrons which are allowed tobe excited to them from RAS1 and RAS2, e.g., to one. It is reasonable to include π ∗ (CO) / Fe3 d orbitals which have a mixed π ∗ (CO) and non-bonding 3 d character. These orbitals are involvedin the 2 p → π ∗ (CO) / Fe3 d transitions giving rise to an intense absorption feature. These emptyligand-dominated orbitals can be included to describe effects of backdonation and explicit charge-transfer excitations as analyzed for the Fe(CO) and [Fe(CN) ] − / − .
19, 158
Remarkably, theradial nodal structure of these MOs resembles that of the “double-shell” 4 d orbitals. Thus, beingcorrelated, or in other words allowing for higher than single excitations to RAS3, they mighthelp to describe radial 3 d correlation. However, if one focuses on reproducing transition strengthssomewhat sacrificing accuracy in energy, single excitations should suffice. If one is interested inother charge-transfer effects, one might add additional orbitals to RAS1 or RAS3 spaces. All otheroccupied orbitals are set inactive and stay always doubly occupied.The complexity of the selection of the AS for TM compounds strongly varies with the cova-lence of the metal–ligand bond. One can state that covalent complexes correspond to particularlystrong static correlation, whereas ionic complexes can be essentially singly-configurational.
In case of ionic complexes [M(H O) ] n + , the AS can be kept even more compact including just three2 p and five 3 d orbitals ( t g non-bonding and two e g anti-bonding orbitals in octahedral symme-try).
21, 44, 53, 157, 171
Other filled orbitals with 3 d character, for instance, the bonding e g counterpart,can be added to describe further shake-up and charge-transfer transitions as exemplified for, e.g.27FeCl ] − and [Mn(H O) ] / complexes. Inclusion of 4 s orbitals into the AS accounts forallowed 2 p → s excitation, having, however, lower intensity without notable influence on L -edgespectra. It might be still important for accurate reproduction of experimental spectra.
As a byproduct of having both filled and unfilled (or half-filled) valence MOs in the AS oneobtains a fraction of valence electronic states. In fact, states with valence 3 d → d character canbe used to calculate RIXS and RPES. That is why, one should always take care that not onlycore but also bright valence states (in a sense of core → valence radiative relaxation) are representedappropriately. This condition is automatically fulfilled for the AS shown in Figure 3b. Finally,by choosing the AS one can switch on and off various effects, e.g., in addition to 3 d → p backrelaxation one can account for dipole-allowed 3 s → p decay by simply adding 3 s orbitals to theAS. As a note in caution: Imposing constraints on orbital subspaces does not in general case solvethe problem of variational collapse. The matrix κκκ in Equations (13) and (14) contains terms whichare interchanging active MOs with inactive and virtual ones. Effectively, those orbitals, whichcorrespond to the largest coupling elements in the Hamiltonian, will stay in or enter the AS. Thus,there is always a probability that the calculation will not converge to the desired local minimumof the energy (with core orbitals in the AS) but to the global one, where core MOs are rotatedout of AS. As mentioned already for the case of single-reference SCF, if the initial orbital guessis reasonable then, due to small overlap of 2 p orbitals with the valence ones and using at least asecond order algorithm, the initial orbitals will tend to stay active and the system will fall into theclosest local minimum of energy. In our experience, this procedure works quite well for high-spinstates if the orbitals optimized for the lowest valence excited states are used as a starting guess.Nevertheless, a systematic convergence to the desired local minimum cannot be ensured and oftenfails for medium- and low-spin electronic states of TM complexes. A possible solution might be touse a core-hole relaxed initial guess as suggested by ˚Agren et al.
76, 77
Non-desired rotations can also be suppressed by restricting the elements of κκκ mixing 2 p orbitalswith others to zero, i.e. to perform a constrained SCF search. This approach is justified bythe findings in References 66, 173, i.e. that a hole in some deep-core shell does not influencemuch other orbitals in the same shell and thus relaxation of 2 p orbitals can be neglected (see alsoReferences 76, 77). In addition, one can utilize symmetry arguments in case of centrally symmetricsystems. Another “brute force” solution is to freeze all occupied orbitals in the inactive subspace, while28till allowing for rotations between RAS n subspaces and virtual orbitals. This approach stillincludes some orbital relaxation due to possibility to “borrow” some orbital contributions from thevirtual space and redistribute them among active MOs. Although it sounds quite restrictive, infact, it produces results which agree with experiments fairly well as illustrated in Section 4. Thisapproach allows to efficiently compute a few thousands of the electronic states which are needed tocover all relevant valence and core levels of the neutral and ionized systems as depicted in Figure 1b.In fact, doing an MCSCF calculation produces non-orthogonal electronic states since MOs fordifferent states i are not orthogonal with all the negative implications for the calculation of thetransition properties. It is connected to notable numerical effort as the ( C , c ) set needs to bedetermined for each electronic state separately, whose number for core-excited states can easilyreach few thousands. To circumvent this and improve convergence properties, the state-averagingprocedure is routinely employed. Within this approach the averaged ensemble density and energyover several electronic states are obtained via variational minimization of the functional E av = (cid:88) i w i E MCSCF i , ( w ≥ w ≥ ... ≥ w N ) , (15)with respect to a common set of orbital rotations. According to the most general formulation ofthe respective variational theorem by Gross-Oliveira-Kohn, the weighting coefficients should notincrease with the state number. In practice, the weights { w i } are usually taken to be equal. Thus,the index i for the set of orbitals in Equation (13) can be omitted as they are the same for all states | Ψ MCSCF i (cid:105) in the ensemble. Having a common set of MO coefficients, a more balanced descriptionof several states is achieved and the difference in energies is resembled by the CI. Such a procedureallows to obtain a set of orthogonal non-interacting electronic states. However, via averaging,the orbitals loose their optimality for the CI-expansion. Nevertheless, averaging allows to attainquite good agreement with experiment. The state averaging for core states has been suggested byMcWeeny and used to interpret results of X-ray Photoelectron Spectrum (XPS) and Augerspectroscopies.On the one hand, due to the large number of core-excited states which usually need to beaccounted for, state averaging is a highly advantageous procedure as it reduces the computationaleffort. On the other hand, if the optimal orbitals for different states are substantially different, thisstate-averaging procedure becomes not eligible. One might expect such a situation if a number ofvalence and core states are optimized simultaneously since the former require no relaxation uponthe influence of the core hole and the latter do. (A separate averaging of valence and core states29as been also suggested. ) The unexpected practical solution to this problem is that even ifone does democratic averaging over all the states obtained within a given AS, the results agreewell with experiments. The democratic average can be viewed as accounting for non-integer core-hole occupation somewhat similar to the TP-DFT approach. Thus, the orbitals represent somecompromise between needs of valence and core states. In passing we note that this reminds onthe procedure where all the irrelevant lower-lying states are averaged with the interesting higher-lying ones to improve convergence and mitigate variational collapse.
The description of electron correlation within MCSCF can be considered as unbalanced since itaccounts for near-degeneracy effects (static correlation) but in addition also partially for the dy-namic correlation within the AS. The MOs outside the AS are treated at the HF level with therespective moderate costs, see Figure 3b. A more balanced description requires a more completeaccount for the dynamic correlation contribution and can be achieved via PT applied on top ofMCSCF. Here, one considers the correlation in an approximate way due to, for instance, singleand double excitations between inactive, active, and virtual orbital subspaces in case of PT up tosecond order. The second order energy correction reads E MR − PT i = − (cid:88) j | (cid:104) Φ j | ˆ V | Ψ MCSCF i (cid:105) | E j − E MCSCF i + δ , (16)where | Ψ MCSCF i (cid:105) is the MCSCF state, Equation (13), to be corrected by PT, and configurations | Φ j (cid:105) with respective energies E j correspond to the single and double excitations mentioned above,ˆ V is the difference between the true Hamiltonian and its zero-order approximation, and δ is thelevel shift. Among different flavors of Multi-Reference (MR)-PT differing in the definitionof the zero-order approximation, Complete Active Space second order PT (CASPT2) andits restricted variant RASPT2 have enjoyed popularity for excited states of TM complexes ingeneral as well as for TM L -edge X-ray spectra. A concise introduction to this methodlisting main peculiarities and inherent problems can be found in Reference 189; for a more detaileddiscussion see also References 167, 188.At this point we would like to highlight two issues: (i) CASPT2 or RASPT2 cannot correctfor the effect of a too small AS. In fact, usually a larger AS than the minimal one is requiredto get accurate PT results. This is reflected in the AS selection rules listed above, for instance,the necessity to include “double shell” d -orbitals. In principle, this effect can be mitigated by30sing the multi-state version of RASPT2. Here, the effective Hamiltonian is constructed in thebasis of single-state RASPT2 wave functions and subsequently diagonalized giving a more precisedescription. However, it might be impractical for core states due to their large number.(ii) The energy expression in Equation (16) can be subject to singularities due to the vanishingdenominators, giving rise to the so-called intruder state problem. This is especially severe forcore-excited states which can easily get degenerate with some | Φ j (cid:105) configuration. The naturalsolution is to increase the AS if possible. Another solution is to apply a real-valued level shift δ (Equation (16)), which artificially lifts the degeneracy with the unwanted weakly-interactingintruder states. An even more elegant solution is the introduction of purely imaginary level shifts, which is in fact utilized for the core-level computations. Nevertheless, choosing a universal levelshift which is good for a large number of states at the same time might be problematic and onehas to stay at the RASSCF level. Perhaps the best solution is known as N -electron Valence secondorder PT (NEVPT2). It explicitly includes two-electron terms in the zero-order Hamiltonianand is manifested to be intruder state-free. However, it has not yet been widely applied to X-rayspectra calculations, except the work in Reference 172.Finally, we note that the effect of RASPT2 depends on the covalence of the complex, withrelatively small effect for aqueous metal ions
21, 156 but large impact for systems, like metal hex-acyanides, where the affordable AS is not big enough to include all important ligand-dominatedMOs.
Recently, the RASSCF/RASPT2-based multi-reference approach to X-ray spectroscopy of metalcomplexes was shown to be quite efficient in unraveling the nature of different transitions in staticXAS,
24, 44, 157, 158, 172, 174, 180, 194
RIXS,
18, 19, 21, 156, 171, 196, 197 and core PES
44, 51, 53, 198, 199 as well asdynamics studies.
22, 31, 56, 200
Selected examples will be presented in the Section 4.
A conceptually different approach is the direct solution of the Time-Dependent Sch¨odinger Equation(TDSE) (for a review, see Reference 201) employing a time-dependent Hamiltonian, ˆ H ( t ) = ˆ H mol +ˆ V int ( t ), with the field-free molecular Hamiltonian ˆ H mol and the molecule-external field interactionˆ V int ( t ). The state | Ψ( t ) (cid:105) can be written in some time-independent basis, e.g., | Ψ( t ) (cid:105) = (cid:80) i a i ( t ) | Φ i (cid:105) ,with the temporal evolution recast in the form of time-dependent coefficients. This expansion canbe done on the level of orbitals, leading to the time-dependence in MO coefficients. It results inthe TD-SCF method, e.g., Real-Time Time-Dependent Density Functional Theory RT-TDDFT31n case of the KS-DFT ansatz for the wave function. If one assumes time-independent electronicconfigurations (e.g., Slater determinants) built on time-independent MOs allowing CI coefficientsevolving in time one gets the Time-Dependent CI (TD-CI) method. Uniting both approaches resultsin the Time-Dependent MCSCF (TD-MCSCF) technique, which in the most general form canbe also viewed as an Multi-Configurational Time-Dependent HF (MCTDHF) method. The advantage of solving the TDSE is that core-excited states can be accessed right awayby choosing the proper frequency of the oscillating external field. Further, for pulsed excitationone can tune the width of the frequency window to be investigated.
Besides focusing on theabsorption of light, shaping the envelope of the pulse opens a way to discuss the excitation ofintricate superpositions of eigenstates and quantum control of the system’s dynamics in real time.Various real time methods have been applied for simulations of the core spectra and dynamicsin the core-excited states such as RT-TDDFT,
TD-CI, non-Dyson Algebraic Diagram-matic Construction (ADC), real-time time-dependent Equation Of Motion (EOM)-CC, and ρ -TD-RASCI. While real time-domain propagation methods are emerging, perturbative response function ap-proaches enjoyed considerable popularity over the last decades. Since it falls outside the mainscope of the review, we will briefly discuss non-SCF single-reference methods only (for an ex-tensive review, see Reference 12). Response approaches are based on a perturbation expan-sion of the time-dependent observables in frequency domain, (cid:104) Ψ( t ) | ˆ O | Ψ( t ) (cid:105) = (cid:104) Ψ | ˆ O | Ψ (cid:105) + (cid:80) ω R (1) ( ω ) E ( ω ) e − i ω t+ γ t + . . . , where an external monochromatic field E ( ω , t ) = E ( ω ) e − i ω t +c . c . is assumed. In lowest order only the term linear in the field is retained and the limit γ → R (1) ( ω ) wherethe operator ˆ O is the dipole moment operator. Moreover, by using correlation functions involvingdifferent operators one can generate the full realm of various propagator methods. The mainadvantage of the so-called polarization propagator methods and closely related Green’s functionapproaches is that the spectrum is determined directly from poles and residues of the propaga-tor. This allows to avoid explicit calculation of the initial and final states and thus to minimize theerror in differential correlation.Among the propagator methods the ADC scheme and in particular its second-order variant32DC(2) has been actively developed recently. As far as core level calculations are concerned,including CVS, it goes back to the 1980’s with early applications to X-ray spectra of smallmolecules reported in References 109,110,218. More complex recent applications have been reportedfor XAS,
RIXS,
PES and in combination with the Stieltjes method or a grid-basedrepresentation of the ionized electron to AES and ICD.
The most widely used response function approach is LR-TDDFT whose versatility in predicting K -edge spectra of molecules containing main group elements is well documented, for a review seeReference 10. In the K -edge case, the relevant electronic states are essentially single-configurationaland thus can be safely treated by a single-reference method. Being related to the CIS method, LR-TDDFT needs to incorporate the same tricks as discussed for the CI method. These includeconfiguration space or energy window restriction or modification of the diagonalisation pro-cedure itself leading to an energy specific solver.
LR-TDDFT is prone to problems due to the so-called self-interaction error. It correlates withthe overlap between the occupied and unoccupied orbitals, corresponding to an electronic tran-sition.
This overlap is very small for core-to-valence and especially for core-to-Rydberg tran-sitions due to different radial sizes of very confined core and notably more delocalized virtualvalence and Rydberg orbitals. Consequently, the energies of core excitations are usually substan-tially underestimated by LR-TDDFT with standard XC-functionals.
Similar to charge-transfervalence-valence transitions, energies of core transitions were found to be sensitive to the amountof exact exchange in the XC-functional.
Here, exact exchange refers to orbital-dependentexchange integrals, analogous to the HF K kl term, but calculated using Kohn-Sham orbitals.
65, 232
As a remedy, functionals with an adjusted amount of exact exchange have been suggested, such asBH . LYP for carbon K -edge transitions and the CV-B3LYP for K -edges of second period ele-ments, where core regions are described by BHHLYP (50% HF exchange) and valence with B3LYP(20% HF exchange). Another, more gentle way is to split the Coulomb operator into short-rangeand long-range parts depending on the interelectron distance and switch exact exchange betweendifferent values with a smooth function. Most important for core spectroscopy is the short-rangecorrection as introduced in References 229, 234, leading to a remarkable improvement of K -edgetransition energies. For other correction methods designed for prediction of X-ray spectra, see alsoReferences 235–237.LR-TDDFT has been successfully applied to main group element K -edges
10, 112, 116, 238 demon-strating good accuracy and predictive power. Attempts have also been made to calculate L -edges33f TMs. In particular, the REW-TDDFT method has been used to demonstrate dif-ferent kinds of non-linear X-ray spectroscopies such as stimulated RIXS to probe the dynamics ofelectronic and nuclear wave packets in the excited states.
Another way to treat correlation on top of SCF is provided by the Coupled Cluster (CC)technique. In contrast to the linear form of the CI expansion, it employs an exponential ansatz forthe excitation operator such that the ground state wave function can be obtained, e.g., from the HFsolution as | Ψ g (cid:105) = exp( ˆ T ) | Ψ HF (cid:105) . The operator ˆ T = ˆ T + ˆ T + ... generates singly, doubly, and so onexcited configurations similar to the CI method. One also does a hierarchical truncation accordingto excitation level as in the truncated CI discussed above. Its immense success for ground statestudies suggests an extension to the excited electronic states.The CC approach can be generalized to treat excited states within the EOM or LR for-malisms, giving the same energies but different transition moments. The advantage over conven-tional CI is that the excited state wave function is not a linear combination of the determinants buthas an exponential counterpart inherited from the reference state. Essentially, it can be obtainedfrom Equation (8), where | Ψ g (cid:105) = exp( ˆ T ) | Ψ HF (cid:105) and E i has the same structure as in the CI case.The operator exp( ˆ T ) effectively accounts for higher-order excitations due to its exponential formand E i treats differential correlation with respect to the ground state.Both EOM and LR-CC have been applied to K -edge core spectra recently. For example, EOM-CC with singles and doubles excitations has been applied on top of the HF reference with a core-holeobtained with maximum overlap method for emission spectroscopy of water. Alternatively, theenergy specific algorithm, as discussed in Section 3.3.1, was used together with EOM-CC for aset of molecules containing main group atoms.
Different approaches have also been used to getcore-ionized potential energy surfaces in case of the ClF molecule.
In Reference 248, a two-stepapproach is utilized, describing separately the effect of core-hole relaxation and an excited boundelectron by means of electron-attachment variant of EOM-CC. A simpler reduction of the excitationspace is introduced for the LR-CC method.
The EOM-CC and LR-CC methods fail if the ground state cannot be described by a single-reference CC wave function. In contrast to CI, the extension of the CC idea to the multi-referencecase is non-trivial due to the problem with definition of a proper vacuum state leading to a numberof technical complications, for review see Reference 251. However, various flavors of multi-referenceCC have been applied to core-spectra of small main group molecules recently. .6 Relativistic effects Relativistic effects are traditionally considered for the valence chemistry of heavy elements.
However, they also play an important role for core electrons as they experience the (almost) un-screened Coulomb potential of the nucleus. For instance, these effects make s and p core electronseven more tightly bound to nuclei resulting in an additional stabilization of the s and p / levels anddestabilization of the p / , d , and f ones. This adds different energetic shifts, e.g., to L -edge(2 p / → d ) and L -edge (2 p / → d ) transitions. Further, SOC is determining the characteristicshape of L , -edge spectra.A rigorous treatment of relativistic effects is provided by the Dirac equation, for reviews seeReferences 258–263. Here, the wave functions are four-component vectors, corresponding to posi-tive (electronic) and negative (positronic) energy states. Four-component theory, equipped with aproper electron correlation treatment, is without doubt the “golden standard” of accuracy in rel-ativistic calculations. But, the considerable computational costs limit their applicability to atomsor small (highly-symmetric) molecules. Examples of four-component core calculations include (i)DFT-CI based calculations of MO n − (M=Ti, Fe, Ni, etc) and UO − clusters, (ii) LR-TDDFT calculations of K - and L -edges of H S and Si, Ge and S halides, and (iii) ADC-basedinvestigations of XeF n N -edge spectra as well as of photoionization of (HI) and (LiI) andof MnO. Accounting for negative energy states contributes little to the physical and chemical processesmost relevant for molecules containing no very heavy elements. This situation is reflected by thefact that in case of low and medium nuclear charge Z there are solutions of the Dirac equationhaving dominant contributions from positive energy states. Various techniques have been designedto decouple both components such that the small component is implicitly included in calculationof the large component. This can be done in principle within any predetermined level of accuracyyielding so-called exact two-component methods. Examples include the Zero-Order Regular Ap-proximation (ZORA) and Infinite-Order Two-Component (IOTC) methods, which are oftenused for core-level calculations. The transition from four- to two-component theory can also bedone via the Douglas-Kroll-Hess protocol which applies a sequence of unitary transforma-tions eliminating the coupling order-by-order and being exact in the limit of an infinite number oftransformations. This yields a block-diagonal Hamiltonian represented as a perturbational sum,where in addition to block diagonalization the spin-dependent and spin-free parts can be separated,35hus building a basis for distinguishing scalar and, e.g., SOC relativistic effects. In practice, how-ever, one transforms only one-electron terms as the transformation of two-electron integrals is veryinvolved and is often sacrificed for the sake of efficiency. This might lead to inaccuracies in L /L splittings as discussed in Reference 206.Such a block-diagonalization of the Dirac Hamiltonian with subsequent neglect of the negativeenergy block leads to two-component schemes. Here, SOC is usually taken into account at thestage of the variational SCF orbital optimization leading to the so-called jj -coupling limit. Thejustification behind is that for high- Z elements SOC becomes comparable to the largest two-electronintegrals and hence significantly influences the one-particle MO basis. Examples of 2-componentcalculations include (i) ZORA DFT-CI based calculations of oxides, (ii) ZORA LR-TDDFTcalculations of K - and L -edges of Ti complexes, of chemical shifts in K α , emission lines ofYbF n and Nb oxide, XAS of gas phase TM oxochlorides, short-chain oligothiophenes XASand XPS , (iii) real-time DFT calculations of XAS for TM chlorides and oxochlorides, (iv)RASSCF IOTC calculations of XPS of noble gases.
For compounds containing low and medium Z elements the most pragmatic approach is toneglect the spin-dependent part of the Hamiltonian in the two-component scheme, resulting in aone-component approach. It corresponds to the usual solution of the Schr¨odinger equation with thedifference that scalar relativistic effects are taken into account. The latter have been shown to becrucial for predicting the correct energetics of X-ray spectra. Typically, scalar relativisticeffects are treated at the second-order Douglas-Kroll-Hess level.For L -edge spectra the magnitude of SOC is larger than typical energy level spacings and thusit must be taken into account. Within the one-component protocol SOC is included as a pertur-bation of the spin-free problem in the framework of the so-called quasi-degenerate perturbationtheory. Here, the SOC Hamiltonian, ˆ H SOC , is represented in the basis of spin-free spatial wavefunctions multiplied by an appropriate spin part. The SOC-coupled eigenenergies and eigenstatesare obtained via matrix diagonalization and have the form of linear combination of spin-free states, | Ψ SF , ( S,M S ) n (cid:105) , with complex coefficients | Ψ SOC a (cid:105) = (cid:88) n ξ ( S,M S ) an | Ψ SF , ( S,M S ) n (cid:105) . (17)In practice for core-level calculations, it is advisable to use as many states for the zero-orderstate basis set {| Ψ SF , ( S,M S ) n (cid:105)} as are available for a given AS. Note that since in this approachnot the total momenta of the individual electrons are coupled as in the jj -limit, but rather the total36ngular momenta L and spins S of many-body electronic states, it is referred to as LS -couplinglimit.The expectation values and transition matrix elements are then calculated for the SOC-coupledeigenstates {| Ψ SOC a (cid:105)} . Therefore, formally spin-forbidden transitions may gain notable intensityin X-ray spectra which is borrowed from the allowed ones via SOC-mixing.
21, 44, 157
In passing wenote that this mixed nature of SOC eigenstates is also reflected in the structure of the DOs, whichbecome complex orbitals having spin-up and spin-down components due to mixing of differentmultiplicities. For such an a posteriori treatment of SOC the operator ˆ H SOC in the Breit-Pauli form iscommonly used. It can be represented in terms of one-, ˆ h SOC ( i ), and two-electron, ˆ g SOC ( i, j ),operators As usual, multi-center two-electron terms require the most effort in the calculation ofˆ H SOC matrix elements in the basis of spin-free electronic states. However, they cannot be neglected(especially for light elements) without notable decrease in accuracy.Let us consider a matrix element of ˆ H SOC between two Slater determinants differing in theoccupation of the i th and j th orbitals, whereas occupations of other orbitals, { n k } , are the same,i.e. H ij = (cid:104) i | ˆ h SOC (1) | j (cid:105) + 12 (cid:88) k n k [ (cid:104) ik | ˆ g SOC (1 , | jk (cid:105) − (cid:104) ik | ˆ g SOC (1 , | kj (cid:105) − (cid:104) ki | ˆ g SOC (1 , | jk (cid:105) ] . (18)An efficient mean-field type approximation is to fix the occupation numbers { n k } to, e.g., groundstate HF values. Further, a speedup is achieved when the molecular mean-field is approximatedby the superposition of the mean-fields of the constituent atoms. Accounting for the locality of SOC,decaying much faster with distance than the Coulomb interaction, and the fact that the largestcouplings are expected for core orbitals which are close to the respective nucleus, one can staywith one-center two-electron terms only. This completely separates molecular SOC into additiveatomic contributions. The { n k } are standardly calculated by the HF method for neutral atoms.This approximation is called Atomic Mean-Field Integrals (AMFI) and appeared to be quiteaccurate. Numerous applications in the field of X-ray spectroscopy confirmed that this techniqueprovides remarkable accuracy not only for valence but also for core-electrons as is evidenced bythe prediction of L /L energy separations and general shape of XAS, XPS, and RIXS. It is thebasis of the SOC treatment in the ROCIS-DFT method and of all multi-reference calculationsdiscussed in Section 4.Finally, we comment on the atomic basis sets which can be employed for core-level calculations.37he usual way of implicit inclusion of relativistic effects via core-potentials is naturally notsuitable for this purpose. In this approach the core electrons are removed from the respectiveatoms, essentially simplifying the nodal structure of the MOs, and their effect is accounted by aneffective pseudopotential and only transitions between valence levels can be rigorously calculated.However, there have been reports of two-component calculations using pseudopotentials with ana posteriori restoration of the core nodal structure. However, in general utilization of all-electron basis sets is prerequisite for explicit calculations of X-ray spectra. Moreover, basis sets needto contain very tight functions to describe the core region; to increase the accuracy one might evende-contract otherwise contracted core functions.
For this purpose, basis sets specifically designedfor correlation and at the same time relativistic treatment from the families cc-pwCVXZ-DK andANO-RCC are usually utilized. For calculations of core spectra of compounds containing atomsfrom the first three periods the IGLO bases have also been developed.
Solvated transition metal ions have been shown to provide a model system for metal-ligand in-teractions and in particular for the electronic structure at the metal-solvent interface.
44, 157, 171
InFigure 4 XAS and RIXS spectra for the prototypical ferrous [Fe(H O) ] complex are shown. Thisweak-field high-spin (quintet) complex has d configuration and a Jahn-Teller distorted tetragonalsymmetry. The d z and d x − y orbitals of the iron ion form bonding σ d and antibonding σ d ∗ orbitals of e g symmetry with the a orbitals of the water molecules. The other 3 d orbitals of t g symmetry form π -type MOs with the water 1 b and 1 b orbitals, where the mixing is notably weakeras compared to the σ orbitals. That is why they can be denoted as non-bonding n d MOs. Notethat symmetry labels assume octahedral symmetry for simplicity.Calculated and measured absorption spectra are in rather good agreement as shown in Fig-ure 4a.
44, 171
Focusing on the L edge, analysis of the core-excitations reveals that almost all intensetransitions are of 2 p → σ d ∗ type, but with substantial admixture of 2 p → n d character. Noticethat this mixing of σ d ( e g ) and n d ( t g ) orbitals strongly depends on the considered ion. For Fe ,for instance, it has been found that the low-energy part of the L edge is dominated by t g -typetransitions. The multiplicity of the core-excited states strongly depends on energy.
Forenergies up to about the L maximum spin-conserving quintet-quintet transition dominate. The38egion up to 715 eV is characterized by strongly mixed quintet-triplet transitions. The L -edge iscomprised essentially of spin-forbidden quintet-triplet transitions, which borrow intensity by mixingwith spin-allowed ones. (a)Exp. PvFYExp. PcFY Calc. XASCalc. PcFYCalc. PvFY (b) Calc. RIXSExp. RIXS 0-5-10 Energy loss, eV I n c o m i ng pho t on ene r g y , e V R I XS i n t en s i t y , a r b . un . PFY intensity, arb. un. 2p3s3d3p
X-ray2p3d3p3s X-rayRIXSPvFYPcFY(c)S→mixed→SS→mixed→S-1
Figure 4: Experimental and calculated (RASSCF/RASPT2) spectra of the aqueous Fe ion (an[Fe(H O) ] cluster is used in simulations, see inset). (a) Absorption spectra obtained in differentmodes: true XAS (calculated – black), valence PFY (Equation (3)) with ω ∈ [660 , V FY (experiment – blue, calculation – magenta) and core PFY ω ∈ [605 , C FY (experiment – red; calculation – green). (b) Respective RIXS spectra in energy loss( ω − Ω in Equation (2)) representation. The green and gray rectangles denote the energy range ofthe spin-allowed and formally spin-forbidden transitions, respectively. (c) Orbital scheme showingdifferent radiative relaxation channels. (adapted from References 44, 171)RIXS spectra of [Fe(H O) ] shown in Figure 4b can be used to characterize the electronicstructure in more detail. For excitation across the L -edge up to its maximum, most prominentis the elastic Rayleigh feature at the energy loss of 0 eV. Inelastic peaks at about -1.2 eV energyloss are due to emission from the doubly occupied t g orbital and the respective shake-off satellites.With increasing excitation energy the intensity of inelastic peaks also increases as compared tothe elastic ones. This evidences the fact that below -1.2 eV valence excited states are of tripletcharacter. Essentially, the excitation occurs from the ground quintet state with spin S = 2 to aspin-mixed core-excited state (see also Section 4.3 and Figure 6b). The features with energy losses39rom 0 to -1.2 eV correspond to S → mixed → S transitions. Due to the mixing in the intermediatestate, the new formally spin-forbidden radiative channels S → mixed → S − L -edge is of minorimportance, evidencing the predominant ionic character of the metal-ligand bond. Respectiveresults obtained for probing the oxygen K -edge, yield rather similar spectra for [Fe(H O) ] andfor pure water, thus supporting the weak mixing (ionic) picture. Recording a true XAS in transmission mode is a difficult task for optically-thick condensedphase samples, such as solutions, even if microjets are utilized. Therefore, experimentally oneusually obtains it in the PFY detection mode, Equation (3). The natural question arising inthis respect is how large are the distortions of the PFY relative to the true XAS and how theycan be reduced.
Figure 4a shows the two experimental PFY spectra resulting from 3 d → p (P V FY) and 3 s → p (P C FY) dipole-allowed radiative relaxation of the initially created core hole, see panel (c). The comparison of the calculated XAS and PFY spectra stemming from these twochannels evidences that the core variant P C FY should be a more reliable probe of XAS in contrastto P V FY. The reason for the distortions of the latter has been identified as being the openingof new radiative relaxation channels via strong SOC in the core-excited state, cf. discussion inprevious paragraph for RIXS.The RIXS spectra record photon-in/photon-out events and thus are determined by dipole se-lection rules as is evident from the Kramers-Heisenberg Equation (2). In Reference 44, it has beendemonstrated for [Fe(H O) ] that a back-to-back analysis of RIXS and photon-in/electron-outRPES, which is free from such selection rules, provides complementary information on the elec-tronic states. In particular, comparing intensities, one can obtain information on the competitionbetween radiative and non-radiative decay channels of the core-excitation. XAS and RIXS spectroscopy applied to metal-ligand complexes can provide an atom-specific, chemi-cal state selective, crystal field symmetry and orbital symmetry resolved description of the electronic40tructure.
In References 18, 19, 176, it has been shown that this allows for scrutinizingtraditional chemical concepts of metal ligand bonding. Within the valence bond structure theory,metal-ligand bonding is described by the so-called σ -donor/ π -acceptor mechanism. For example,the formation of the covalent Fe–CO bonds in Fe(CO) is accompanied by charge donation from the5 σ orbital of CO to the σ d orbital of the Fe and back donation to the 2 π ∗ orbital of CO. In termsof the MOs of the metal-ligand complex, the donation/back donation is expressed by the mixingof the respective Fe and CO orbitals. The extent of this mixing can be addressed by identificationof charge-transfer transitions in RIXS spectra.
510 520 530 540 690 695 700 705 710
Oxygen K α Iron L E x pe r i m en t T heo r y I n t en s i t y ( a r b . un i t s ) Emitted photon energy (eV)LMCTMLCT a ’ a ’ a ’ e’,e”e’,e”e’,e”e’,e”a ’ a ’ π CO π CO σ CO σ π CO CO + C ha r ge - t r an s f e r Lo c a l O K-edgeFe L-edge2p n3d(e’,e’’) 1s n3d(e’,e’’)2p 3d(a ’)
1s CO(b) (a)
Figure 5: (a) Experimental and theoretical RIXS spectra of the Fe(CO) complex at iron L -edgeand oxygen K -edge. The colored areas show the range of metal-to-ligand (MLCT) and ligand-to-metal (LMCT) charge-transfer transitions which carry information about the strength of covalentbonding as explained in the respective insets. (b) Transition density difference plots which showthe localization of the excited electron (red) and the electron which is refilling the core hole (blue)within the complex. (adapted from Reference 18)For illustration, let us consider the RIXS amplitude for the Fe L -edge in the simplified pictureconsidering only MOs. The dipole operator can be written as ˆ d = d p → d | d (cid:105) (cid:104) p | + h . c . Supposethat the 3 d character of the absorbing state is given by the MO coefficient C (abs)3 d of a virtual orbital,to which the core electron is excited, i.e. | Ψ (abs) (cid:105) = ˆ d | p (cid:105) = d p → d C (abs)3 d | d (cid:105) . Thus, core excitationprojects out all MO contributions apart from the 3 d due to locality and relatively strict dipole selec-41ion rules. The 3 d character of the respective state for emission shall be given by the coefficient C (em)3 d of the occupied orbital from which the core hole is refilled, i.e. | Ψ (em) (cid:105) = ˆ d | d (cid:105) = d d → p C (em)3 d | p (cid:105) .Hence, the RIXS amplitude in Equation (2) becomes R ∝ | d p → d | | C (abs)3 d | | C (em)3 d | . This suggeststhat comparing the intensities of different RIXS channels provides access to the character of theinvolved MOs. This has been demonstrated in References 18 and 19 for Fe(CO) and [Fe(CN) ] − ,respectively. Needless to say that this reasoning applies to other absorption edges as well.In Figure 5a, exemplary RIXS spectra for the Fe L -edge and the O K α -edge of Fe(CO) areshown. In case of the Fe L -edge, there are, on the one hand side, local d − d type radiativerelaxation transitions of n d ( e (cid:48) , e (cid:48)(cid:48) ) to 2 p character (see density difference plots in panel (b)). Onthe other hand side, inelastic peaks of σ d ( a (cid:48) ) to 2 p character are clearly discernible for differentexcitation energies. From the density difference plots (note especially the negative part shown inblue) in Figure 5b the ligand to metal charge-transfer character of these core-hole refill transitionsbecomes apparent. Based on the simple model outlined before the intensity ratio of these twotypes of transitions can be used to characterize the degree of orbital mixing. The same argumentapplied to the O K -edge leads to the identification of the n d ( e (cid:48) , e (cid:48)(cid:48) ) to 1s transition as markerbands for the metal-to-ligand charge-transfer character. Taking both results together gives a meansfor quantifying the MO composition in terms of atomic orbitals or equivalently σ -donor/ π -acceptorcharacter of the metal-ligand bonding. In order to establish the general nature of this argument, the[Fe(CN) ] − complex has been investigated using RIXS spectroscopy. Comparing band intensitiesbetween [Fe(CN) ] − and Fe(CO) it was verified that CN − is a stronger σ -donor but weaker π acceptor than CO, which is in accord with chemical intuition. Following the building principles of natural water-oxidation systems, manganese-based nanostruc-tured catalysts have attracted considerable attention.
X-ray spectroscopy is sensitive to theoxidation state of metal atoms. In Reference 21, this has been shown in a spectroscopic investiga-tion of the redox evolution of manganese systems relevant for water photo-electro-oxidation.
In the experiment, Mn precursors with different oxidation states have been doped into a nafion ma-trix, where all converted into Mn II . These species can be oxidized upon application of a bias andreduced under visible light illumination. To follow this reaction, XAS as well as RIXS spectra havebeen calculated for the model compounds [Mn II (H O) ] , [Mn III (H O) ] , and [Mn IV L(OMe) ] + (cf. Figure 6). Although the actual experimental systems are much more complex, the local42
00 400 600 800 1000 1200 14000.000.250.500.751.00 S p i n c on t r i bu t i on State number
200 400 600 800 1000 12000.000.250.500.751.00 S p i n c on t r i bu t i on State number val . d b cval . db ceval . a b c
100 200 300 400 500 600 7000.250.500.751.00 S p i n c on t r i bu t i on State number
SS-1S+1 a b cd ce
640 645 650 655
Theory: y t i s ne t n . nu . b r a , Incoming photon energy, eV II Mn TEY doped II Mn TEY biased
III
Mn TEY biased
Experiment: IIIIIIV b [Mn (H O) ] II/III 2+/3+2 6 [Mn L(MeO) ]
IV +3
XAS i IIIIIIV (a) (b)
Figure 6: (a) Experimental and calculated (RASSCF/RASPT2) XAS spectra for Mn species indifferent oxidation states. (b) Contributions of states with ∆ S = 0 , ± L and L bands which is about 11eV, almost independent on the oxidation state. It follows that the SOC constant is about 7.3 eV.This large SOC constant leads to considerable coupling of core excited states of different spin, i.e.if S denotes the spin of the electronic ground state, core-excited states are mixtures of S − S ,43nd S + 1 spin states. Any interpretation of the XAS spectra has to account for this fact. Thedecomposition of the spin-orbit coupled wave functions into ∆ S = 0 , ± S = 0 or ∆ S = − L band only, the L band has a similar assignment, although the spin mixing is even stronger). (i) Mn II : Here, thelow-energy shoulder (a) is of dominantly 2 p → d ( t g ) character, comprising ∆ S = 0 transitions.The main band (b) is shaped by intense 2 p → d ( e g ) transitions with ∆ S = 0 , −
1. The peaksdenoted by (c) are due to mostly spin-forbidden ∆ S = − t g /e g character.(ii) Mn III : The spectrum is dominated by 2 p → d ( e g ) transitions, but also contains shake-up3 d ( e g ) → d ( t g ) transitions in region (b). Starting from the main peak (b) ∆ S = 0 , − S = − IV : thespectrum is similar to Mn III , but also contains notable contribution from ∆ S = +1 transitions inregions (b,d,e). The influence of the PT2 correction to the RASSCF is also illustrated in Figure 6.Apart from an overstabilization of the ground state which needs to be compensated by a constantshift of the spectrum, its shape hardly changes as can be seen in Figure 6 what is an evidence ofmostly ionic metal-ligand bond character.Analysis of inelastic RIXS spectra revealed that for all species most features are due to spin-forbidden transitions, mediated by core-excited states similar to what has been discussed foriron complex in Section 4.1. Note that intricate spin mixing of the core-excited state gives riseto the exciting ultrafast spin-flip dynamics which can be triggered by XFEL and HHG X-raypulses. Straightforward extension of the RASSCF-based approach to the calculation of core-excited states ofmulti-center metal compounds would require active spaces that go beyond the capabilities of currenthard- and software. Even if one would be able to perform such a calculation, the spectra wouldconsist of hundreds of thousands of transitions, which would render any detailed interpretation ofsuch spectra a challenging task. In order to address this problem, we have recently developed anapproach which follows the logic of the so-called Frenkel exciton model.
The latter is used44o describe the coupling of low-lying valence excitations in molecular aggregates and crystals, i.e.in situations where monomers assemble as a consequence of the van der Waals interaction. Sinceall what is needed is a localized excitation, the Frenkel exciton picture might apply to the presentcase of core-holes even if multiple metal centers in covalently bound complexes are considered.The principal idea is to introduce metal center specific excitations which are in turn coupled bythe Coulomb interaction. Thus, the Frenkel exciton Hamiltonian readsˆ H FE = (cid:88) M (cid:88) A E A M | A M (cid:105)(cid:104) A M | + 12 (cid:88) MN (cid:88) A,B,C,D J MN ( A M B N , C N D M ) × | A M (cid:105)(cid:104) D M | ⊗ | B N (cid:105)(cid:104) C N | . (19)where the states A M − D M are eigenfunctions of the monomeric Hamiltonian, i.e. H M | A M (cid:105) = E A M | A M (cid:105) , and the Coulomb coupling can be expressed, e.g., in dipole approximation J MN ( A M B N , C N D M ) ≈ d A M D M · d B N C N | X MN | − X MN · d A M D M )( X MN · d B N C N ) | X MN | , (20)where d A M D M = (cid:104) A M | ˆ d | D M (cid:105) is the transition dipole matrix element and the vector X MN connectsmonomers M and N .The approach is illustrated in Figure 7c for the example of a dimer. The different monomericstate manifolds will be denoted as ground | g M (cid:105) , valence-excited, | v M (cid:105) , and core-excited, | c M (cid:105) ,states. Equation (20) contains couplings between all possible transitions in the dimer system. Inthe following, we will make use of the fact that the actual observables of interest, namely XAS andRIXS spectral amplitudes, are of first and second order with respect to the interaction with theexternal field. This suggests to employ either a one- or two-particle basis as shown in Figure 7c. Inthe former, states of the type | a g (cid:105) and | g a (cid:105) ( a = v, c ) are incorporated, while the latter includesin addition states of type | a , b (cid:105) ( a, b = v, c ) and is in principle exact for the dimer. Note thatthis effectively corresponds to a CISD-like treatment of the interaction between monomers in thecomposite system with X-ray specific preselection of configurations, while monomers are treatedwithin the usual RASSCF procedure. In order to make this approach computationally feasible, anumber of approximations can be introduced as detailed in Reference 196. Most important in thisrespect is the preselection of core-excited states to construct the monomeric basis functions, whichis based on an energy window criterion for the excitation light pulse.45 ne-particle basisTwo-particle basis Monomer1 Monomer2 I n c o m i ng pho t on ene r g y , e V -10 -8 -6 -4 -2 0 2Emitted photon energy, eV710.4712.0715.4722.2725.271 .0 9712.3715.1(a) N o r m a li z ed R I XS , na r b . u . -10 -8 -6 -4 -2 0710.4712.0715.4 N o r m a li z ed R I XS , na r b . u . Emitted photon energy, eV2(b) (c)
Figure 7: (a) Left: calculated normalized XAS for [heme B-Cl] (black) and [heme B-H O] + (redfilled curves). Right: normalized RIXS spectra for selected excitation energies. (b) CalculatedRIXS spectra for the hemin dimer (red filled curves) with the -COOH groups pointing in thesame direction. The monomer spectrum is also shown ( ×
2, black lines). Note that monomerand dimer XAS are indistinguishable on this scale. (c) Different choices of excitonic bases and thecorresponding transitions included in the dimer coupling. For the one-particle basis, only transitionsfrom or to the ground state have been included. For the two-particle basis, de-excitations from anarbitrary core-excited state to any other state are allowed. (adapted from Reference 196)In a proof-of-concept study, this approach has been applied to investigate the signatures ofaggregation of hemin in solution.
Hemin shows a solvent dependent aggregation behavior, stay-ing monomeric in polar solvents like ethanol or DMSO, while it forms dimers in water solution.Experimentally, the effect of aggregation was addressed by means of soft X-ray Fe L -edge XAS aswell as by NXES and RIXS on the example of DMSO and aqueous solutions. The generalshape of the spectra for both cases was quite similar and the pronounced difference in broadeningsfor RIXS as well as a 1.3 eV energy shift in off-resonant NXES was attributed to aggregation.However, judging the effect of aggregation one should account for the labile equilibrium betweendifferent species in solution that also could be a source of spectral changes.In Figure 7a, XAS as well as selected RIXS spectra are shown for the bare hemin molecule46[Heme B-Cl] ) as well as for the case where the axial Cl − ligand has been exchanged by a watermolecule ([Heme B-H O] + ). The XAS spectrum shows a distinct sensitivity of the L band to thetype of ligand. In the present case, changing the ligand from Cl − to H O causes a splitting of the L peak in two components at 710.0 eV and 710.9 eV. While the main peak is due to 2 p → d x − y and d z excitations, the splitting results from the energetic lowering of 2 p → d z with respect to2 p → d x − y transitions.The RIXS spectra of the monomer show a trend similar to the one discussed in Section 4.1. Forexcitation energies above 720 eV, the core-excited states are mostly of quartet type. Here, the mostintense emission goes to valence-excited states with high quartet contributions. The RIXS spectrafor both species differ mostly in the inelastic peaks, whereas the elastic peak has a comparableintensity in both spectra.The comparison of [Heme B-Cl] dimer and monomer spectra is shown in Figure 7b. The XASspectra are essentially identical, i.e. there is almost no effect due to dimerization. This can beattributed to the intensities of the metal 2 p → d and 3 d → d transitions relevant for L -edgeX-ray spectra that are lower (due to smaller radial overlap and dipole selection rules) than thoseof the π → π ∗ and n → π ∗ transitions usually discussed in the case of aggregates of organic dyes.However, distinct fingerprints of dimerization can be seen in RIXS, in both the inelastic and elasticfeatures. Overall, the magnitude of the effect of dimerization is comparable to the differencesobserved upon ligand exchange in the monomer. Hence, the unequivocal identification of spectralfeatures due to solvent dependent dimerization is not a straightforward task. There is a plethora of methods which have been developed to calculate core-excited states forprediction of various X-ray spectroscopic signals. For TM compounds beyond the K -edge, multi-reference approaches provide the methods of choice as they can deal with the intricate multipletstructure shaped by static and dynamic correlations and substantial SOC. However, limitations aredue to the methods’ complexity and associated computational effort. Here, active space approachesand in particular RASSCF are playing a prominent role as they are, in principle, exact and system-atically improvable by increasing the size of the active space. In practice, however, methods likeRASSCF cannot be considered as a “black box” due to the strong influence of the choice of the AS.Recent progress in the field of Density Matrix Renormalization Group (DMRG) and automated47S selection certainly mitigates this problem. For polymetallic complexes, merging establishedideas of electronic structure theory with models developed in other areas such as the Frenkel excitonHamiltonian could broaden the range of systems for theoretical X-ray spectroscopic studies.To date, there is a considerable record of theoretical and experimental applications of frequency-domain X-ray techniques to study TM complexes in various environments. With the emergingexplicit time-domain techniques, either using XFEL or HHG facilities, nonlinear spectroscopy inthe X-ray domain will play a role for unraveling electron dynamics, comparable to that of UV/Visin case of (non-)Born-Oppenheimer electron-nuclear dynamics.
ACKNOWLEDGEMENTS
We would like to acknowledge the financial support from the Deutsche Forschungsgemeinschaft viagrant No. BO 4915/1-1 (S.I.B.) and Ku952/10-1 (O.K.) and the Deanship of Scientific Research(DSR), King Abdulaziz University, Jeddah, (Grant No. D-003-435).
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