Density matrix based time-dependent configuration interaction approach to ultrafast spin-flip dynamics
FFULL PAPER
Density matrix based time-dependent configuration interactionapproach to ultrafast spin-flip dynamics
Huihui Wang a , Sergey I. Bokarev a , Saadullah G. Aziz b , and Oliver K¨uhn a a Institut f¨ur Physik, Universit¨at Rostock, Albert-Einstein-Str. 23-24, 18059 Rostock,Germany; b Chemistry Department, Faculty of Science, King Abdulaziz University, 21589Jeddah, Saudi Arabia
ARTICLE HISTORY
Compiled October 3, 2018
Abstract
Recent developments in attosecond spectroscopy yield access to the correlated mo-tion of electrons on their intrinsic time scales. Spin-flip dynamics is usually consid-ered in the context of valence electronic states, where spin-orbit coupling is weakand processes related to the electron spin are usually driven by nuclear motion.However, for core-excited states, where the core hole has a nonzero angular momen-tum, spin-orbit coupling is strong enough to drive spin-flips on a much shorter timescale. Using density matrix based time-dependent restricted active space configura-tion interaction including spin-orbit coupling, we address an unprecedentedly shortspin-crossover for the example of L-edge (2p → ∼ KEYWORDS
Spin-orbit coupling, configuration interaction, density matrix, electron dynamics
1. Introduction
The rapid development of high harmonic generation techniques [1–3] has recently en-abled studies of processes occurring in atoms, molecules, and nanoparticles, which aretriggered by high-energy radiation including soft X-ray light [4–6] on ultrashort timescales approaching few tens of attoseconds [7]. The new light sources allow to triggerand control ultrafast electronic processes via preparation of complex superpositions ofquantum states and to analyze their subsequent evolution. Attosecond spectroscopyhas a huge potential to study atomic and molecular responses to incident light [8, 9].It provides access to, e.g., electron correlation manifesting itself in the entanglementof bound- and photo-electrons (shake-ups), Auger and interatomic Coulomb decay, aswell as to the coupling of electrons in plasmonic systems [1–3, 10].Among the most prominent examples of such processes is the oscillatory charge(hole) migration following the ionization from a localized moiety in molecules [10–15]. Remarkably, such electron dynamics is driven solely by the electron correlation,
CONTACT S. I. Bokarev Email: [email protected] a r X i v : . [ phy s i c s . a t m - c l u s ] N ov INTRODUCTION ? ]. Thespin-orbit couplings (SOC) between valence excited states, however, are small andspin-crossover is essentially driven by nuclear motion since it requires the nuclearwavepacket to pass through a region of near-degeneracy of two states of different mul-tiplicity (for review see Ref. [ ? ]). Thus the time scale is determined by the relatedvibrational periods (see also Ref. [22]). In passing we note that this is also a typical timescale for spin transfer between magnetic centers in polynuclear metal complexes [23].For 2p core-excited electronic states of transition metal complexes, however, themagnitude of SOC increases dramatically. Therefore, one expects the spin dynamicsto change from a nuclear to an electronically driven process. It was recently demon-strated [24] that electronically driven spin-crossover after core excitation in transitionmetals indeed takes place on a few femtosecond time scale, thus, being faster than thecore-hole lifetime. This process can be considered as an elementary step of the conven-tional nuclear dynamics driven spin-crossover [19], analogously to the above mentionedcharge migration. In both cases, electronic wave packet dynamics is ultimately coupledto nuclear motions, eventually leading to charge or spin localization.Here, we further elaborate on this finding, by reformulating the time-dependentrestricted active space configuration interaction method with account for SOC withinthe open system density matrix formalism. Inclusion of strong correlation effects intothe model is essential at this point, since transition metal complexes are known to havea multi-configurational nature of the wave function, sometimes even in the groundstate. The importance of strong SOC and multi-reference effects especially appliesto the 2p core-excited electronic states. In the present work, we study the influenceof different excitation regimes and pulse characteristics as well as phenomenologicalinclusion of Auger decay. The article is organized as follows: First, we present thetheoretical model and computational details in Sections 2 and 3. Subsequently, wediscuss the results of the application of the developed methodology to the prototypicalFe(II) compound [Fe(H O) ] representing a model for the solvated Fe ion in Section4. Conclusions and outlook are given in Section 5 THEORY
2. Theory
Ultrafast spin-flip is investigated using the Time-Dependent Restricted Active SpaceConfiguration Interaction method in its density matrix formulation ( ρ -TD-RASCI),which is similar in spirit to the techniques proposed in Refs. [23, 25, 26]. As comparedto TD-RASCI, the density matrix formulation offers a convenient way of treatingdissipative dynamics of open quantum systems. Working within Born-Oppenheimerapproximation and provided that processes under study occur much faster than theperiod of nuclear motion we assume the clamped nuclei approximation. Here, the nucleiare fixed at the ground state equilibrium positions, and thus we solve the electronicSchr¨odinger equation only. The question whether the nuclear motion can be neglectedfor the early-time dynamics has triggered an ongoing debate [27–30]. Here, we assumethat the system is excited far from conical intersections and that the considered timeinterval is shorter than the relevant vibrational periods.In the present case of the interaction of a molecular system with X-ray light, whereelectronic transitions possess a very local character, the focus is put on a fairly smallsubsystem containing the absorbing atom with its first coordination shell. To accountfor dissipation due to the more extended environment or electronic relaxation pro-cesses, which are not treated explicitly, it is natural to represent the system in termsof the reduced density operator ( ˆ ρ ) evolving in time according to ∂∂t ˆ ρ = − i [ ˆ H, ˆ ρ ] + D ˆ ρ . (1)Here, ˆ H is the Hamiltonian operator of the subsystem of interest and D is the dissi-pation superoperator, which accounts for different dissipation processes.The reduced density operator is represented in the basis of Configuration StateFunctions (CSFs), | Φ ( S,M S ) j (cid:105) , with the total spin S and its projection M S : ρρρ ( t ) = (cid:88) j,j (cid:48) ρ ( S,M S ,S (cid:48) ,M (cid:48) S ) j,j (cid:48) ( t ) | Φ ( S,M S ) j (cid:105) (cid:104) Φ ( S (cid:48) ,M (cid:48) S ) j (cid:48) | . (2)The CSFs are constructed using a time-independent molecular orbital (MO) basis,optimized at the restricted active space self-consistent field [31] level, prior to propa-gation. The Hamiltonian in the CSF basis reads H ( t ) = H CI + V SOC + U ext ( t )= (cid:18) H h H l (cid:19) + (cid:18) V hh V hl V lh V ll (cid:19) + (cid:18) U h ( t ) 00 U l ( t ) (cid:19) , (3)where we separated blocks of low ( l ) and high ( h ) spin states. In Eq. (3), H CI is theconfiguration interaction (CI) Hamiltonian containing the effect of electron correlation.The eigenstates of this spin-free Hamiltonian H CI will be called spin-free (SF) states.The structure of the SF state can be viewed as linear combination | Ψ SF (cid:105) = C | Ψ (cid:105) + (cid:88) ia C ai | Ψ ai (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) + (cid:88) i 3. Computational details The outlined approach is applied to the spin-dynamics in the [Fe(H O) ] complex(see Fig. 1a)) representing a model of the solvated Fe ion, whose X-ray absorptionand resonant inelastic X-ray scattering spectra were discussed in Refs. [34, 36, 43].The dynamics, first described in Ref. [24] and discussed below, is driven solely bythe electronic coupling (SOC and electron correlation), while nuclei are fixed at theground state equilibrium positions taken from Ref. [46]. To justify this approxima-tion, we note that the considered time interval of 15 fs is shorter than the relevantvibrational periods. For [Fe(H O) ] , the Fe–O stretching and O–Fe–O deformation COMPUTATIONAL DETAILS UnoccupiedRAS2 ! Full CIRAS1 ! b)a) xz y . . . Figure 1. a) General view of the [Fe(H O) ] complex; the lengths of three pairs of Fe–O bonds are alsogiven (in ˚A). b) MO active space used for the TD-RASCI calculation. modes, potentially influencing the 2p → 3d core excited electronic states, have periodsabove 100 fs.The active space used in the ρ -TD-RASCI calculations contained 12 electrons dis-tributed over the three 2p (RAS1 subspace) and five 3d (RAS2 subspace) orbitals (cf.Fig. 1b)) to describe the core excited electronic states corresponding to the dipoleallowed 2p → 3d transitions [34, 36, 41, 43]. The number of holes in RAS1 was limitedto one, whereas full CI was done within RAS2. This active space included up to 4h4pconfigurations and resulted in 35 quintet ( S = 2) and 195 triplet ( S = 1) electronicstates, directly interacting via SOC according to the ∆ S = 0 , ± S = 3) are not possible with this active space. Account-ing for the different M S components, the total amount of the SF and SOC states was760, where 160 are valence and 600 core ones. The respective calculations are denotedas RASCI(1,2) below. Notice that both, the account for 4h4p excitations and SOCare essential to recover the dynamics of the highly correlated core-excited states. Inaddition singlet states were also included to test the influence of second-order SOCeffects; denoted as RASCI(0,1,2). Their number increased the dimensionality of thebasis by 170 states in total (50 valence and 120 core). To account for scalar relativisticeffects, the Douglas-Kroll-Hess transformation [47] up to second order was utilized. Tocorrect for weak correlation effects the single-state second-order perturbation theorycorrection (RASPT2) [48] was added to the diagonal of H CI matrix written in the SFbasis, which was then back-transformed to the CSF basis. Thus, due to their diagonalnature this corrections influenced SOC only implicitly via the relative energetics ofthe interacting states. To avoid intruder states in RASPT2 calculations, an imaginarylevel shift [49] of 0.4 E h was introduced. 1s,2s, 3s, and 3p orbitals of iron as well as 1sorbitals of oxygen were kept frozen in RASPT2.Evaluation of the H i , V ij , and (cid:126) d ii matrix elements in Eq. (3) was performed witha locally modified version of the MOLCAS 8.0 [50] quantum chemistry suite, applyingthe relativistic ANO-RCC-TZVP basis set [51, 52] for all atoms. PhenomenologicalAuger decay rates Γ k were set to 0.4 and 1.04 eV for the L and L edges, respec- RESULTS 4. Results In Fig. 2a) the L-edge absorption spectrum of [Fe(H O) ] as calculated with differentmethods is compared to experiment [43]. It has a shape characteristic for transitionmetals, featuring the L ( J = 3 / 2) and L ( J = 1 / 2) bands split due to the SOC. Thissplitting is 12.7 eV (SOC constant is 8.5 eV) what corresponds to a timescale of about0.33 fs. Here RASCI(1,2) and RASPT2(1,2) denote the results of respective methodstaking triplets ( S = 1) and quintets ( S = 2) states into account. They are comparedto the results including singlet states ( S = 0). One observes a fairly good agreementof the shape of the spectrum, with different theoretical methods giving very similarresults. The influence of the perturbation theory correction and inclusion of singletson the spin dynamics will be discussed below.Panel b) of Fig. 2 illustrates the degree of spin-mixing for the SOC states (Eq. (5))in the case of RASCI(1,2), which is the main setup discussed in the following. It canbe seen that the valence excited states are mostly pure quintets or triplets. In contrastthe core excited states are dominantly spin mixtures, with the degree of spin-mixingvarying with state energy. The low-energy flank of the L -band contains pure quintets,whereas moving to higher energies the contributions of triplets start to dominate. Inthe present study, we assume ultrafast preparation of the superposition of stronglyspin-mixed 2p core hole states in different regimes. Thus, Fig. 2b) will provide areference for the spin-nature of the involved eigenstates.Below we discuss the three different excitation regimes, illustrating the dynamics ofultrafast spin-crossover. In regime I , (cid:126)E ( t ) = 0 and it is assumed that a particular SFstate, which is a superposition of strongly spin-mixed 2p core hole states [34, 39], hasbeen instantaneously prepared. This somewhat artificial initial condition will serve asa reference, which highlights the spin dynamics driven solely by SOC.In regime II , the system is initially in the ground state with the M S -components ofthe lowest closely lying electronic states being populated according to the Boltzmanndistribution at 300 K: ρρρ = diag { Z − exp( E i /kT ) } , (9)where Z = (cid:80) i exp( E i /kT ). In fact, the three lowest quintet states have non-negligiblepopulation, which yields 15 M S microstates. The core hole is created and thus spindynamics is driven by an ultrashort X-ray pulse (cid:126)E ( t ) linearly polarized along differentdirections.Finally, in regime III an instantaneous population of the particular CSF is ad-dressed, being a strong superposition of the SF eigenstates, while the SOC and ex-ternal field are switched off ( V SOC = and (cid:126)E ( t ) = 0). This regime corresponds toa sudden excitation and allows to explore the effect of electron correlation separatelyfrom SOC. It is similar to those regarded before for hole migration, for a review seeRef. [13]. RESULTS 705 710 715 720 725 L L J=3/2 J=1/2 a) I n t en s i t y Energy / eV ExperimentRASCI(1,2)RASCI(0,1,2)RASPT2(1,2)RASPT2(0,1,2)SOC splitting1 2 3 4 0 0.2 0.4 0.6 0.8 1 0 100 200 300 400 500 600 700 Valence L L b) W e i gh t StateTripletQuintetSF val SF core Figure 2. a) X-ray absorption spectrum of [Fe(H O) ] as calculated with different methods (see text)and compared with experiment (partial electron yield from 2p3d3d channel) [43]. Arrows denote the ex-citation energies considered in regime II, see text. b) Decomposition of the SOC eigenstates into quintet( (cid:80) n,M S | a ( S =2 ,M S ) mn | , red bars) and triplet ( (cid:80) n,M S | a ( S =1 ,M S ) mn | , grey bars) SF states (cf. Eq. (5)). The par-ticular contributions of valence SF (SF val , green bars) and core SF (SF core blue bars) states used in regime Ito the different SOC states are also shown. Numbered ranges reflect the bandwidths of 0.5 eV (1-4) and 5.0 eV(1’-4’) pulses with carrier frequencies denoted in panel a) in terms of the involved SOC states. Regime I We have chosen two representative quintet SF states, i.e. number 7 and 111, as initialstates for investigating the SOC-driven spin dynamics which are denoted as SF val and SF core below. For the contributions of SF val and SF core to the stationary SOCeigenstates see Fig. 2b). Other states demonstrated very similar dynamics and are notconsidered further.Preparation of SF core , which corresponds to M S = +2 (four spin-up electrons) andhas contributions of SOC states from essentially the whole core hole excited L andL bands (see blue bars in Fig. 2b), demonstrates intricate dynamics. It is illustratedin Fig. 3, where the total populations of all quintet and triplet states with (solid lines)and without (dashed lines) Auger decay (panel a) as well as the detailed evolutionof different M S components (panel b) are shown. As a consequence of strong SOC,the population spreads over both quintet and triplet states such that the total tripletpopulation becomes even larger than the corresponding quintet one within about 1 fs(Fig. 3a)). The population transfer occurs according to the ∆ M S = 0 , ± S = 2 , M S = +2) → ( S =1 , M S = +1) transitions (cf. Fig. 3b)). Quintets with M S = − − k , in Eq. (8) common for all L and L states. Interestingly, the spin-flip occurs faster than the fastest Auger decay of4 fs.Panel c) of Fig. 3 shows snapshots of the spin-density difference, ρ ↑ − ρ ↓ , evolution.Because of this quintet-triplet population transfer, ρ ↑ notably decreases during thefirst 3 fs. After about 4 fs the system almost equilibrates, i.e. the 760 electronic statesact like an “electronic bath”, and the corresponding populations of M S microstatesoscillate around their mean value (Fig. 3b)). The spin density changes relatively slowlyfrom the dominating ρ ↑ to the dominating ρ ↓ and back due to the partial revivals of thequintet’s positive and negative spin projections. The fast modulation in Figs. 3a) andb) with a period of ≈ and RESULTS P opu l a t i on Time / fsQuintets/no AugerTriplets/no AugerQuintets/with AugerTriplets/with Auger 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14b)S Ms P opu l a t i on Time / fs2 +22 +12 02 -12 -21 +11 01 -1 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 P opu l a t i on Time / fs Figure 3. a) Evolution of the total population of the quintet ( S = 2) and triplet ( S = 1) electronic states afterinstantaneous excitation to the SF core state (regime I). Results with and without Auger decay are presented.b) Same as a) without Auger decay for the total population of the M S components of quintet and tripletelectronic states; initial state SF core has M S = +2. The inset gives the analogous evolution after instantaneousexcitation to the valence-excited quintet SF val state ( M S = +1), demonstrating much slower dynamics. c)Evolution of spin-density difference, ρ ↑ − ρ ↓ (red - positive, blue - negative, contour value 0.001) for the caseshown in panels a) and b). L bands. It is roughly the same for all interacting states and is an intrinsic propertyof the 2p core-hole. Thus, core-excited states demonstrate an unprecedentedly fastpurely electronic spin-flip dynamics, which is two orders of magnitude faster than thatdriven by nuclear motion in conventional spin-crossover [21].SF val is a superposition of valence excited SOC states (see green bars in Fig. 2b)).It turns out that it features a rather weak SOC, such that there is almost no dynamicshappening within the considered time window of 15 fs. The inset in Fig. 3 showspopulation dynamics of different M S components of both quintet and triplet stateswithout Auger decay. In fact one can see relatively slow redistribution of populationbetween different quintet states and their M S microstates. Remarkably, the populationof triplet states stays almost exactly zero during the considered time period. Therefore,this initial condition for valence excitation will not be discussed further. Regime II The spin dynamics upon excitation with ultrashort soft X-ray laser pulses polarizedalong X -direction (Fig. 1) with different carrier frequency and bandwidth/durationis shown in Fig. 4. The respective energies and bandwidths in terms of the involvedeigenstates are marked as numbered arrows and numbered ranges in Fig. 2a) and b),respectively. Ranges with 0.5 eV bandwith (8.3 fs pulse duration) and 5.0 eV bandwidth(0.8 fs pulse duration) are denoted as intervals with and without primes, respectively.This notation is consistent with the names of panels in Fig. 4. Here, the excitationfrequencies correspond to spectral regions with small and notable SOC mixing.For regime II, one can see similar ultrafast spin-flip dynamics as in regime I. How-ever, the population of all triplet states stays below 40% within the time period of 15 fsin most cases. As compared with regime I, most notable is the absence of the rapidoscillations. This is due to the fact that the temporal width of the pulse is longer than RESULTS P opu l a t i on Time / fsField envelopeGroundQuintetsTriplets00.20.40.60.81 -6 -4 -2 0 2 4 6 81) P opu l a t i on Time / fs 0 2 4 6 8 10 122’)Time / fs -6 -4 -2 0 2 4 6 82)Time / fs 0 2 4 6 8 10 123’)Time / fs -6 -4 -2 0 2 4 6 83)Time / fs 0 2 4 6 8 10 124’)Time / fs -6 -4 -2 0 2 4 6 84)Time / fs Figure 4. Regime II: Spin dynamics initiated by the explicit field excitation with different carrier frequenciesand bandwidths corresponding to: 1) (cid:126) Ω =706.9 eV, 2) 708.4 eV, 3) 711.5 eV, and 4) 719.8 eV. Panels withprimes correspond to bandwidth (cid:126) /σ = 5 . (cid:126) /σ = 0 . E =2.5 E h e − bohr − for pulses with 5.0 eV bandwidth and E =1.5 E h e − bohr − for 0.5 eV. Auger decay is accounted for. Totalpopulations of core-excited quintet and triplet states are shown by black and red lines, respectively. Theenvelope of the excitation pulse is shown as filled grey curve. The population of M S -components of the groundand first two excited states is shown by the blue line. the 0.3 fs oscillation period dictated by SOC, i.e. the effect is smeared out. Further,compared to regime I, there are more slowly oscillating components in Fig. 4. This canbe traced to the fact that the initial state before excitation is an incoherent thermalmixture of different M S components (Eq. (9)). Hence, the pattern of ∆ M S = 0 , ± W/cm and are barely accessible by current laser setups in the soft X-rayregime. However, these field strengths, despite of their large magnitudes, correspondto the weak field regime for soft X-ray wavelengths, with Keldysh parameters γ > d max E = 2.7 eV and 1.6 eV for the broad and for thenarrow pulses, respectively. Therefore, we do not include strong field effects such as(multiphoton) ionization. In fact, E has been chosen merely to have an appreciabledepletion of the ground state for illustration purposes. Indeed, the dynamics triggeredby much weaker pulses (10 W/cm ) qualitatively agrees with the strong ones [24].The variation of polarization directions leads to very similar dynamics even if (cid:126)e isaligned along Z and Y axes which correspond to the largest differences in the Fe–Obond lengths (Fig. 1). This means that for complexes with only slight distortions fromoctahedral symmetry the effect discussed here is of almost isotropic nature. However,for larger distortions we expect a stronger influence of polarization and thus, for thesolution phase, averaging over polarization directions needs to be performed accounting RESULTS W e i gh t s State CSF 3CSF 2CSF 1SF val SF core a) 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14b) P opu l a t i on Time / fsInitial CSF 2CSF 3CSF 1CSF 4CSF 5 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14c) P opu l a t i on Time / fs Initial CSF 9CSF 10CSF 8CSF 7CSF 21 Figure 5. Regime III: a) three largest CSF contributions to the CI wave function for the valence and corequintet SF states. The state number accounts for spin-degeneracy. b) and c) correspond to the propagation ofthe impulsively excited CSFs 2 and 9, respectively, representing the largest contributions to the spin-free statesSF val and SF core marked with arrows in panel a). for the free tumbling of the solute. For ordered phases (crystals) the variation ofpolarization might be an additional parameter to control the actual spin-state mixtureand respective dynamics.In order to validate the chosen setup we have also considered the inclusion of sin-glet states and of the RASPT2 correction. The singlet states (RASCI(0,1,2)) are notdirectly interacting with the initially prepared incoherent quintet spin-state mixture.The respective absorption spectra are shown in Fig. 2a) and one can conclude thatinclusion of singlets has a rather minor effect on the spectra. The same holds truefor the respective dynamics where the populations of singlet states stay in most casesbelow 5%, see Supplement. Thus, despite of the strong SOC, indirect coupling doesnot play a role in the considered time interval of 15 fs.The same conclusion holds true for the effect of dynamic correlations introducedvia RASPT2 corrections to the energies. This is already apparent from the absorptionspectra in Fig. 2a). Although the inclusion of this correction changes the effective cou-plings between the individual SF states due to changes of energy level separations, asa net effect the overall dynamics stays qualitatively almost the same, see Supplement.This underlines the good accuracy of the RASCI level of approximation at least forthis system. Regime III Electron correlation can be quite substantial for core excited states. To study its effectseparate from the SOC, we have considered the dynamics of initially prepared elec- CONCLUSIONS AND OUTLOOK val and SF core discussed in Section 4.1,respectively. Recall, that the latter have been chosen as representatives for the valenceand core excited electronic states.The subsequent dynamics is shown in Fig. 5b) and c). In case of the CSF 2 (panelb)), some redistribution of the population occurs within 15 fs slightly modulated bythe coherences with other electronic configurations. In contrast, the strong electroncorrelation in the core-excited state leads to an intricate dynamics with the ultrafastdecay of the initial CSF 9 population with multiple revivals within the 15 fs timeperiod (panel c)). The timescale of such dynamics is in general faster than that ofthe core-hole migration phenomena observed for ionized systems [13]. Thus, not onlystrong SOC itself but also electron correlation are important for the electron spindynamics in the core-excited transition metal complexes. 5. Conclusions and Outlook In this article, we have studied ultrafast spin-flip dynamics driven solely by SOC, whichshould be typical for states having core-holes with a nonzero orbital momentum. Forthis purpose we have formulated a density matrix based time-dependent restrictedactive space configuration interaction method suitable for the description of openquantum systems. This enabled to include phenomenological Auger decay. On theexample of a prototypical third-period transition metal complex it was shown thatsoft X-ray light can trigger spin-flips, which are faster than the lifetime of the 2p corehole ( ≈ ≈ 10 fs for Fe L and L , respectively) [53]. Modifications of the pulsecharacteristics such as carrier frequency and pulse duration were shown to be effectivein controlling the actual spin mixture to quite some extent with modest effort.The notable dependence of spin state yields on the pulse parameters calls for anexperimental verification. A possible direct way to address such spin dynamics wouldbe the upcoming time-resolved non-linear X-ray spectroscopy, e.g., stimulated reso-nant inelastic X-ray scattering (SRIXS) [54–56]. In this technique the mixed-spin corestates can be projected onto the manifold of pure-spin valence states, which are usu-ally energetically well separated in transition metal complexes, see, e.g., discussionin Refs. [39, 43]. Thus, the relative SRIXS intensities in the respective energy ranges(0-1.5 eV for quintets and 1.5-8.2 eV for triplets in the case of [Fe(H O) ] system)would provide information on the time-evolution of the contribution of pure spin statesto a mixed one. Given the rapid progress in high harmonic generation [4–7] and free-electron lasers [8] and the expected establishment of time-resolved techniques suchas SRIXS, [54–56] the experimental proof of the effect discussed here and its use formanipulating spin dynamics appears to be within reach. EFERENCES Disclosure statement Authors declare no competing financial interests. Funding This work was supported by the Deanship of Scientific Research (DSR), King Abdu-laziz University, Jeddah under Grant No. D-003-435. References [1] M. F. Kling and M. J. J. Vrakking, “Attosecond electron dynamics,” Annu. Rev. Phys.Chem. , vol. 59, pp. 463–492, 2008.[2] F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. 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