Valence-Band RIXS: Mechanisms & Applications

• VB-RIXS is a photon-in/photon-out technique that probes valence excitations by creating a core-excited state and monitoring its decay via coherent absorption and emission processes.
• The method distinguishes Raman-like behavior, where energy-loss peaks remain fixed, from fluorescence-like regimes that map closely onto X-ray emission spectra.
• Comprehensive frameworks such as BSE and real-space Green’s functions are crucial for accurately capturing excitonic interactions, matrix elements, and spectral features in diverse materials.

Valence-band resonant inelastic X-ray spectroscopy (VB-RIXS) is a resonant, second-order photon-in/photon-out spectroscopy in which an incident X-ray creates a core-excited intermediate state and the subsequent emission step fills the core hole from the valence manifold, leaving a final state with a valence excitation but no deep core hole. In contrast to core-to-core RIXS, the final state therefore probes excitations within the valence many-body Hamiltonian itself, including bound excitons, crystal-field and multiplet structures, charge-transfer states, and fluorescence-like channels. Because the absorption and emission steps are coupled coherently through the intermediate-state propagator, VB-RIXS is simultaneously sensitive to occupied and unoccupied electronic subspaces, to the light–matter matrix elements, and to the incident-energy detuning (Vorwerk et al., 2020, Roychoudhury et al., 2022, Sundermann et al., 22 Sep 2025).

1. Scattering process and measured quantity

The formal starting point of VB-RIXS is the Kramers–Heisenberg expression for the double-differential cross section. With ground state $|g\rangle$, intermediate states $|n\rangle$, final states $|f\rangle$, incident photon energy $\omega_i$, emitted photon energy $\omega_f$, dipole operator $D$, and intermediate-state lifetime broadening $\Gamma_c$, one writes

$$\sigma(\omega_i,\omega_f)\propto \sum_f \left| \sum_n \frac{\langle f|D^\dagger|n\rangle \langle n|D|g\rangle} {E_g+\omega_i-E_n+i\Gamma_c} \right|^2 \delta(E_g+\omega_i-E_f-\omega_f).$$

This form makes explicit that VB-RIXS is a coherent two-step process: the absorption amplitude into intermediate core-excited states and the emission amplitude into final valence-excited states interfere before the square modulus is taken (Vorwerk et al., 2020, Roychoudhury et al., 2022).

The experimentally relevant energy variable is usually the energy loss, $\omega_i-\omega_f$, which equals the final-state excitation energy. In VB-RIXS this loss window typically spans low-energy valence excitations rather than the large core-to-core losses characteristic of resonant X-ray emission spectroscopy. In iridates, for example, the low-energy window contains spin–orbit excitons and magnons, whereas higher losses contain crystal-field $d$–$|n\rangle$0 transitions; in UO$|n\rangle$1 the low-energy part resolves crystal-field-split multiplets and the higher-loss part contains charge-transfer and fluorescence-like features (Kim et al., 2018, Sundermann et al., 22 Sep 2025).

A central distinction is between Raman-like and fluorescence-like behavior. Below and at the X-ray absorption threshold, the denominators do not pass through resonance, so the energy-loss peaks remain pinned at the final-state excitation energies and do not disperse with incoming photon energy. Above threshold, fluorescence features can dominate, and the spectrum as a function of emitted-photon energy can become closely similar to X-ray emission spectroscopy. This threshold-dependent crossover is explicit in the BSE analysis of $|n\rangle$2-Al$|n\rangle$3O$|n\rangle$4, where Raman-like excitonic features dominate at and below threshold, while above threshold the spectra are well described within an independent-particle approximation and resemble XES (Urquiza et al., 2023).

A frequent oversimplification is to treat VB-RIXS either as a direct map of the loss function or as a fluorescence measurement of occupied states. The published analyses support a more restricted statement. In the Raman-like regime, the loss energies coincide with valence excitations, but not every loss-function peak is visible in RIXS; only those final excitons connected by non-zero absorption and emission amplitudes contribute. In the fluorescence-dominated regime, by contrast, the emitted-energy spectrum can map onto XES and, in the dipole approximation, onto the $|n\rangle$5-projected valence PDOS of the absorbing atom (Urquiza et al., 2023).

2. Excitonic formulation and emission pathways

A fully interacting description of VB-RIXS in solids is obtained by representing both intermediate and final neutral excitations as eigenstates of Bethe–Salpeter Hamiltonians. In the all-electron framework of Vorwerk-Vinson et al., one solves separate BSEs for core-hole excitations and for valence excitations, both within the Tamm–Dancoff approximation. In the compact excitonic basis, the final-state amplitude can be written as

$|n\rangle$6

where $|n\rangle$7 is the core-absorption amplitude, $|n\rangle$8 is the emission “pathway” from intermediate core exciton $|n\rangle$9 to final valence exciton $|f\rangle$0, and the denominator provides resonant enhancement together with core-hole broadening. The cross section then becomes a sum over final excitons weighted by $|f\rangle$1 (Vorwerk et al., 2020).

This representation isolates the microscopic content of a VB-RIXS spectrum. The quantity $|f\rangle$2 determines which core excitons are bright in absorption, $|f\rangle$3 determines how a given core exciton recombines into a particular valence exciton, and coherent interference among nearby intermediate excitons shapes asymmetries, broadening, and multi-peak structures. The same framework yields a two-dimensional “pathway map” as a function of intermediate and final excitation energies. In the reported solids calculations, these maps show a strong band-diagonal structure, reflecting the need for overlap between the excited-electron wavefunctions in the intermediate and final states. Dark core excitons with zero $|f\rangle$4 remain inactive even if $|f\rangle$5 is large (Vorwerk et al., 2020).

Electron–hole interactions enter through full diagonalization of the BSE matrix, whose kernel contains direct screened and unscreened exchange terms. Treating core and valence sectors on the same footing produces consistent transition matrix elements and a common excitonic language for XAS, XRS, and RIXS. In practical VB-RIXS calculations this implies a workflow of ground-state electronic structure, quasiparticle corrections by GW or a scissor shift, solution of the core-level BSE at the selected edge, solution of the valence BSE for optical excitations, evaluation of $|f\rangle$6, $|f\rangle$7, and $|f\rangle$8, and assembly of $|f\rangle$9 over an incident/emitted-energy grid. Because $\omega_i$0 is frequency-independent, repeated evaluation over many $\omega_i$1 points is efficient (Vorwerk et al., 2020, Urquiza et al., 2023).

The necessity of the full excitonic treatment is material-dependent but can be decisive. For the F $\omega_i$2 edge of LiF, the dominant VB-RIXS loss feature at $\omega_i$3–$\omega_i$4 eV is traced to a bound valence exciton at $\omega_i$5 eV above the gap, resonantly coupled to the lowest core exciton at $\omega_i$6 eV. The independent-particle approximation completely misses this dominant feature, showing that full BSE diagonalization is not merely a refinement but can be required for the correct spectral topology (Vorwerk et al., 2020).

3. Alternative theoretical frameworks and controlled approximations

The BSE pathway formalism is not the only established route to VB-RIXS. An earlier ab initio description based on a real-space multiple-scattering Green’s-function formalism and a quasi-boson Hamiltonian reduces the cross section, under the sudden approximation and neglect of exchange between the spectator valence hole and the photoelectron, to a convolution of an effective absorption signal with the X-ray emission signal. Additional many-body corrections are carried by an effective energy-dependent spectral function. In the weak-$\omega_i$7 limit, the effective absorption reduces to ordinary XAS, yielding the “convolution approximation” for VB-RIXS (Kas et al., 2011).

The same real-space Green’s-function scheme provides a computational route that does not require periodicity: a finite cluster is constructed around the absorbing site, screened deep and shallow core-hole potentials are introduced, one-electron Green’s functions are evaluated by real-space multiple scattering, dipole matrix elements are assembled into a nonlocal transition operator, and the final spectrum is obtained by convolution over the emission energy and, if included, over bosonic loss channels. Example calculations for MnO and TiO$\omega_i$8 were reported to give qualitative agreement with experiment, and the method also simulates HERFD line-width suppression (Kas et al., 2011).

A different hybrid framework is CleaRIXS, which combines a constrained-occupation SCF treatment of intermediate core-excited states with a linear-response treatment of final valence excitations. In this construction, core-hole attraction in the intermediate state is described by a non-aufbau determinant, while final valence electron–hole binding is described by a BSE- or TDDFT-type kernel in the space of $\omega_i$9 pairs. The practical consequence is a one-to-one map between each energy-loss feature and the dominant valence–conduction transition encoded in the exciton amplitudes $\omega_f$0. The published implementation states that full VB-RIXS spectra are obtained at roughly the cost of one core-hole DFT calculation plus one BSE run (Roychoudhury et al., 2022).

For some materials classes, additional simplifications become accurate because of broad intermediate-state lifetimes. In iridates, the very short Ir $\omega_f$1 lifetime, with $\omega_f$2 eV, simplifies the incident-energy dependence of $\omega_f$3–$\omega_f$4 excitations and supports the ultra-short core-hole lifetime approximation. In that limit, the Kramers–Heisenberg denominator varies slowly over the valence-excitation scale and the RIXS response approaches a direct probe of the relevant two-particle response function (Kim et al., 2018). In overdoped and optimally doped YBa$\omega_f$5Cu$\omega_f$6O$\omega_f$7, a non-interacting quasi-particle model with a core-hole potential reproduces the weak detuning dependence of the Cu $\omega_f$8-edge RIXS peak, whereas in the strongly underdoped antiferromagnet the same model breaks down, which was interpreted as signaling genuine collective paramagnon excitations (Kanász-Nagy et al., 2015).

Taken together, these approaches delimit the validity of common approximations. Full excitonic formalisms are essential when bound final excitons or strong intermediate-state electron–hole attraction dominate the spectrum. Convolutional and quasi-particle approaches become effective when screening is strong, core-hole lifetimes are short, or the interpretation focuses on broad band-structure effects rather than on threshold excitons (Kas et al., 2011, Kanász-Nagy et al., 2015).

4. Matrix elements, symmetry, momentum transfer, and dichroism

VB-RIXS is not determined by the joint density of states alone. The light–matter matrix elements carry polarization, orbital, and momentum-transfer dependence, and these factors can qualitatively reorganize the observed spectrum. In the beyond-strict-dipole $\omega_f$9 description used for dichroic RIXS calculations, the scattering operator contains matrix elements

$D$0

so the measured intensity depends explicitly on photon helicity and on transferred momentum as well as on the character of the Bloch wavefunctions (Schüler et al., 2022).

For circularly polarized light, the leading dipole selection rule is $D$1. In the dichroic calculations for MoSe$D$2 and 1T$D$3-MoS$D$4, this helicity dependence couples directly to the orbital angular momentum texture of the valence and conduction states and, through it, to the Berry curvature. The reported simulations show circular-dichroism contrast up to $D$5 in monolayer MoSe$D$6 at small $D$7, with sign changes reflecting $D$8 versus $D$9 valley orbital angular momentum, and up to $\Gamma_c$0–$\Gamma_c$1 in inversion-broken 1T$\Gamma_c$2-MoS$\Gamma_c$3 around $\Gamma_c$4 eV and $\Gamma_c$5 a.u., tracking the induced Berry-curvature dipole (Schüler et al., 2022).

The same principle of matrix-element selectivity appears in less explicitly topological settings. In YbB$\Gamma_c$6 and EuB$\Gamma_c$7, RIXS collected at the divalent rare-earth $\Gamma_c$8-edge absorption resonance resembles the unoccupied $\Gamma_c$9 density of states calculated by DFT, which was attributed to transitions between weakly dispersing $$\sigma(\omega_i,\omega_f)\propto \sum_f \left| \sum_n \frac{\langle f|D^\dagger|n\rangle \langle n|D|g\rangle} {E_g+\omega_i-E_n+i\Gamma_c} \right|^2 \delta(E_g+\omega_i-E_f-\omega_f).$$0 and strongly dispersing $$\sigma(\omega_i,\omega_f)\propto \sum_f \left| \sum_n \frac{\langle f|D^\dagger|n\rangle \langle n|D|g\rangle} {E_g+\omega_i-E_n+i\Gamma_c} \right|^2 \delta(E_g+\omega_i-E_f-\omega_f).$$1 states. However, a second resonance appears roughly $$\sigma(\omega_i,\omega_f)\propto \sum_f \left| \sum_n \frac{\langle f|D^\dagger|n\rangle \langle n|D|g\rangle} {E_g+\omega_i-E_n+i\Gamma_c} \right|^2 \delta(E_g+\omega_i-E_f-\omega_f).$$2 eV higher in incident energy despite the absence of a corresponding XAS peak. Its integrated intensity follows $$\sigma(\omega_i,\omega_f)\propto \sum_f \left| \sum_n \frac{\langle f|D^\dagger|n\rangle \langle n|D|g\rangle} {E_g+\omega_i-E_n+i\Gamma_c} \right|^2 \delta(E_g+\omega_i-E_f-\omega_f).$$3, characteristic of Thomson-like indirect scattering, and its strong angular dependence differentiates it from the lower-energy divalent resonance (Sheets et al., 2019).

These examples show that momentum transfer and polarization are not peripheral experimental details. They determine whether VB-RIXS emphasizes direct valence excitons, indirect core-hole–driven channels, orbital textures, or fluorescence-like emission. A plausible implication is that interpretation strategies based only on energy-loss positions will systematically miss information encoded in the matrix elements.

5. Material realizations and experimentally established regimes

Across materials classes, VB-RIXS has been used to access wide-gap excitons, itinerant conduction-band structure, low-energy multiplets in correlated compounds, and transient electronic structure in warm dense matter. Representative cases span first-principles benchmark systems, transition-metal oxides, rare-earth and actinide materials, and femtosecond high-energy-density experiments (Vorwerk et al., 2020, Urquiza et al., 2023, Humphries et al., 2020, Sheets et al., 2019, Kim et al., 2018, Sundermann et al., 22 Sep 2025).

System	Edge or regime	Reported VB-RIXS result
LiF	F $$\sigma(\omega_i,\omega_f)\propto \sum_f \left| \sum_n \frac{\langle f|D^\dagger|n\rangle \langle n|D|g\rangle} {E_g+\omega_i-E_n+i\Gamma_c} \right|^2 \delta(E_g+\omega_i-E_f-\omega_f).$$4 edge	Dominant loss peak at $$\sigma(\omega_i,\omega_f)\propto \sum_f \left| \sum_n \frac{\langle f|D^\dagger|n\rangle \langle n|D|g\rangle} {E_g+\omega_i-E_n+i\Gamma_c} \right|^2 \delta(E_g+\omega_i-E_f-\omega_f).$$5–$$\sigma(\omega_i,\omega_f)\propto \sum_f \left| \sum_n \frac{\langle f|D^\dagger|n\rangle \langle n|D|g\rangle} {E_g+\omega_i-E_n+i\Gamma_c} \right|^2 \delta(E_g+\omega_i-E_f-\omega_f).$$6 eV traced to a bound valence exciton; IPA misses it
$$\sigma(\omega_i,\omega_f)\propto \sum_f \left| \sum_n \frac{\langle f|D^\dagger|n\rangle \langle n|D|g\rangle} {E_g+\omega_i-E_n+i\Gamma_c} \right|^2 \delta(E_g+\omega_i-E_f-\omega_f).$$7-Al$$\sigma(\omega_i,\omega_f)\propto \sum_f \left| \sum_n \frac{\langle f|D^\dagger|n\rangle \langle n|D|g\rangle} {E_g+\omega_i-E_n+i\Gamma_c} \right|^2 \delta(E_g+\omega_i-E_f-\omega_f).$$8O$$\sigma(\omega_i,\omega_f)\propto \sum_f \left| \sum_n \frac{\langle f|D^\dagger|n\rangle \langle n|D|g\rangle} {E_g+\omega_i-E_n+i\Gamma_c} \right|^2 \delta(E_g+\omega_i-E_f-\omega_f).$$9	Al $\omega_i-\omega_f$0 and $\omega_i-\omega_f$1 edges	$\omega_i-\omega_f$2- and $\omega_i-\omega_f$3-edge RIXS spectra fully overlap; Raman-like below threshold, fluorescence-like above
Warm dense Ni	Ni $\omega_i-\omega_f$4 edge at LCLS	Single-shot valence DOS, electron temperature, ionization, and K-edge shift extracted
YbB$\omega_i-\omega_f$5, EuB$\omega_i-\omega_f$6	Rare-earth $\omega_i-\omega_f$7 edges	Divalent resonance maps unoccupied $\omega_i-\omega_f$8 DOS; second resonance appears $\omega_i-\omega_f$9 eV higher without XAS counterpart
Iridates	Ir $d$0 edge	Low-energy $d$1 and higher-energy $d$2 excitations show simple resonance profiles
UO$d$3	U $d$4/$d$5 edges	$d$6 meV/$d$7 meV resolution resolves crystal-field, multiplet, charge-transfer, and fluorescence-like features

In wide-gap insulators, the threshold regime is strongly excitonic. In $d$8-Al$d$9O$|n\rangle$00, the calculated $|n\rangle$01- and $|n\rangle$02-edge RIXS spectra were reported to “strikingly fully overlap also beyond an independent-particle picture,” while the below-threshold part is directly comparable to peaks in the loss function. In LiF, the bound exciton that dominates the VB-RIXS spectrum is entirely lost within IPA, providing a clear counterexample to any claim that independent-particle treatments are generically sufficient (Urquiza et al., 2023, Vorwerk et al., 2020).

In warm dense matter, VB-RIXS has been extended from equilibrium spectroscopy to femtosecond diagnostics. For solid-density nickel heated at the Linac Coherent Light Source, the incident beam was scanned from $|n\rangle$03 to $|n\rangle$04 keV across the Ni $|n\rangle$05 edge, with a $|n\rangle$06 bandwidth, $|n\rangle$07 fs FWHM pulse duration, and a $|n\rangle$08 eV spectrometer resolution over a $|n\rangle$09 eV window around K$|n\rangle$10. Fitting the simplified cross section yielded electron temperatures with $|n\rangle$11–$|n\rangle$12 eV uncertainty, with peak $|n\rangle$13 rising from $|n\rangle$14 eV at $|n\rangle$15 W/cm$|n\rangle$16 to $|n\rangle$17 eV at $|n\rangle$18 W/cm$|n\rangle$19, while the extracted K-edge position shifted downward by a few eV with increasing temperature (Humphries et al., 2020).

For strongly correlated and strongly spin–orbit-coupled materials, the accessible excitation content changes but the basic separation of regimes persists. In iridates, $|n\rangle$20 excitations at $|n\rangle$21–$|n\rangle$22 eV resonate at the white-line peak of the Ir $|n\rangle$23 edge, while low-energy $|n\rangle$24 excitations and magnons resonate about $|n\rangle$25 eV below it. In UO$|n\rangle$26, high-resolution VB-RIXS at the $|n\rangle$27 edge resolves five low-energy lines at $|n\rangle$28 eV and broader charge-transfer structures near $|n\rangle$29 and $|n\rangle$30 eV; at the $|n\rangle$31 edge the corresponding broader structures appear near $|n\rangle$32 and $|n\rangle$33 eV (Kim et al., 2018, Sundermann et al., 22 Sep 2025).

6. Resolution limits, computational cost, and current directions

The information content of VB-RIXS depends strongly on instrumental resolution and on the lifetime scales encoded in the theory. In warm dense Ni, the combined energy resolution was limited to $|n\rangle$34 eV by the FEL bandwidth and spectrometer function, although the temporal resolution remained $|n\rangle$35 fs. In actinide spectroscopy, by contrast, recent U $|n\rangle$36-edge measurements achieved $|n\rangle$37 meV at $|n\rangle$38 and $|n\rangle$39 meV at $|n\rangle$40, sufficient to resolve crystal-field branches that would be merged at coarser resolution (Humphries et al., 2020, Sundermann et al., 22 Sep 2025).

Theoretical cost can be substantial in first-principles excitonic calculations. In the all-electron BSE study of $|n\rangle$41-Al$|n\rangle$42O$|n\rangle$43, convergence of the RIXS spectrum required summing approximately $|n\rangle$44 core excitons and $|n\rangle$45 valence excitons. This is a direct consequence of combining two excitonic manifolds within the Kramers–Heisenberg denominator. At the same time, the published workflow shows where efficiencies enter: the pathway matrix $|n\rangle$46 is frequency-independent, and the BRIXS code combines precomputed core and valence BSE eigenpairs rather than re-solving the many-body problem at each incident energy (Urquiza et al., 2023).

Each formalism introduces a characteristic approximation set. The real-space Green’s-function approach uses static screened core holes, often neglects the full Bethe–Salpeter kernel for the photoelectron–hole interaction, truncates the quasi-boson expansion at single-boson excitations, and may employ muffin-tin or atomic-sphere potentials; the authors explicitly list extension to all orders in the spectral function, full-potential RSGF, and explicit multiplet physics as possible improvements (Kas et al., 2011). The warm-dense-Ni analysis neglects the core-hole potential in the intermediate state, non-dipole transitions, and strong many-body decay channels, which were described as second-order for broad DOS mapping but important for sub-few-eV features (Humphries et al., 2020).

The current frontier is therefore not a single universal approximation but a regime-dependent partition of the problem. Threshold excitons, charge-transfer satellites, and low-energy multiplets demand explicit electron–hole interactions and accurate matrix elements. High-energy fluorescence-like sectors can often be analyzed within independent-particle or convolutional pictures. High-resolution actinide measurements and polarization-resolved calculations in topological materials indicate that VB-RIXS is moving toward simultaneous sensitivity to low-energy many-body structure, high-energy hybridization, and symmetry-resolved matrix-element effects. This suggests an expanding role for VB-RIXS in contexts where XAS, XES, or loss-function measurements alone do not isolate the relevant valence excitations (Urquiza et al., 2023, Schüler et al., 2022, Sundermann et al., 22 Sep 2025).