# Nonlinear spectral dispersion in resonant Auger scattering from SF 6 for studying nuclear potentials and dynamics

V. Kimberg, O. Travnikova, P. Krasnov, N. Boudjemia, T. Marchenko, Renaud Guillemin, I. Ismail, D. Koulentianos, A. Santos, D. Céolin, et al.

► To cite this version:

V. Kimberg, O. Travnikova, P. Krasnov, N. Boudjemia, T. Marchenko, et al., Nonlinear spectral dispersion in resonant Auger scattering from SF 6 for studying nuclear potentials and dynamics. Physical Review A, 2024, 110 (6), pp.062814. 10.1103/PhysRevA.110.062814. hal-05350517

HAL Id: hal-05350517

https://hal.science/hal-05350517v1

Submitted on 23 Nov 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Nonlinear spectral dispersion in resonant Auger scattering from SF$_{6}$ for studying nuclear potentials and dynamics

V. Kimberg,¹,*, O. Travnikova,²,³,† P. Krasnov,⁴ N. Boudjemia,² T. Marchenko,²,³ R. Guillemin,²,³ I. Ismail,²,³
D. Koulentianos,²,⁵ A. C. F. Santos,⁶ D. Céolin,³ M. N. Piancastelli,²,
R. Püttner,⁷ F. Gel'mukhanov,¹ and M. Simon²,³

¹Theoretical Chemistry and Biology, KTH Royal Institute of Technology, S-106 91 Stockholm, Sweden

²Sorbonne Université, CNRS, UMR 7614, Laboratoire de Chimie Physique-Matière et Rayonnement, F-75005 Paris, France

³Synchrotron SOLEIL, l'Orme des Merisiers, Saint-Aubin, BP 48, F-91192 Gif-sur-Yvette Cedex, France

⁴International Research Center of Spectroscopy and Quantum Chemistry – IRC SQC, Siberian Federal University, 660041 Krasnoyarsk, Russia

⁵Department of Physics, University of Gothenburg, Origovägen 6B, SE-412 96 Gothenburg, Sweden

⁶Instituto de Física, Universidade Federal do Rio de Janeiro, 21941-972 Rio de Janeiro, RJ, Brazil

⁷Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany

<figure><img src="image_2.png" /></figure>

(Received 3 October 2024; accepted 3 December 2024; published 17 December 2024)

In this study, we integrate experimental observations and theoretical models to elucidate the complex phenom-
ena observed in the resonant S K-edge KLL Auger scattering spectra of the SF₆ molecule. A two-dimensional
spectral map, constructed of incident photon energy and kinetic energy of the emitted Auger electron, is
shown to be a versatile tool for understanding a character of the core-excited potential energy surface and
change of the molecular geometry. Our findings reveal how the distinct dispersion behavior of multiple spectral
lines enables mapping of ultrafast dynamics within the short-lived core-excited states. Our results confirm the
presence of nuclear dynamics in the S1s⁻¹6a¹ᵢg and S1s⁻¹6t¹ᵢu core-excited states, while dynamics is absent in
the S1s⁻¹t¹ᵢu state. Using a combination of ab initio analysis, simulations with Coulomb model potentials,
and a simple analytical approximation, we qualitatively demonstrate how the varying characteristics of spectral
dispersion—classified as Raman, non-Raman, and anti-Raman—mirror the relative gradients of the intermediate
and final states in the resonant x-ray scattering process. This insight allows for the effective mapping of molecular
potential energy curves, offering a prospective tool on the underlying mechanisms of resonant Auger scattering
and its potential for probing molecular dynamics.

DOI: 10.1103/PhysRevA.110.062814

# I. INTRODUCTION

An accurate description of intramolecular interaction is
crucial in chemistry, physics, and materials science. The
potential energy curve (PEC) is a fundamental concept for
understanding the kinetics and mechanisms of chemical reac-
tions in both ground and excited electronic states. However,
the accuracy of *ab initio* PECs for highly excited states in
polyatomic systems is often limited, posing challenges for
modern spectroscopy in directly determining PECs from ex-
perimental data. In this context, x-ray spectroscopy techniques
are invaluable due to their inherent high element and site
selectivity, as well as their ability to probe highly excited core-
hole states [1,2]. Techniques such as resonant inelastic x-ray

scattering (RIXS) and resonant Auger spectroscopy (RAS)
have recently been successfully applied to map PECs and
the corresponding vibrational wave functions [2]. The short
lifetime of the core-excited states serves as a natural clock
to measure the duration of ultrafast processes [3–7]. More-
over, using spectral detuning of the incoming photon energy
from the resonance with a core-excited state, an advanced
control over the duration of the resonant x-ray Raman scat-
tering process is achieved, making RIXS and RAS techniques
powerful tools for temporal resolution of ultrafast nuclear dy-
namics [1,2]. Different dispersion relations of different bands
in resonant x-ray scattering spectra allow one to distinguish
molecular (Raman dispersion) and fragment (non-Raman dis-
persion) lines of the ultrafast dissociation in the core-excited
states [2,8–10]. Due to the ultrashort lifetime of deep core
holes, such as S 1s⁻¹ or Cl 1s⁻¹, which is of the order of 1
femtosecond, there is typically insufficient time for significant
bond elongation that would result in bond breaking within
the core-hole lifetime; however, triggered bond elongation
significantly affects the RIXS and RAS profiles even in this
case [10,11]. Moreover, the system can acquire substantial
initial kinetic energy due to the steep potential gradients of
the core-excited states, leading to ultrafast dissociation during

Published by the American Physical Society under the terms of the
Creative Commons Attribution 4.0 International license. Further
distribution of this work must maintain attribution to the author(s)
and the published article's title, journal citation, and DOI. Funded
by Bibsam.

<footnote>*Contact author: kimberg@kth.se</footnote>

<footnote>†Contact author: oksana.travnikova@sorbonne-universite.fr</footnote>
subsequent stages of the Auger cascade decay [11,12]. A detailed understanding of this ultrafast electronic and nuclear dynamics may disclose information about the interatomic interactions and molecular PECs at different stages of the Auger cascade process. It is also worth noting that the dispersions of RIXS and RAS resonances also play an important role in charge transfer studies [13,14].

There are several important studies related to resonant and nonresonant Auger spectroscopy at S K-edge excitation of molecules. Nonresonant KLL Auger spectra of the SF₆, H₂S, and SO₂ molecules were studied by Asplund *et al.* [15]. The resonances S K-edge excitation and their decay properties was addressed in Ref. [16]. Photoabsorption measurements of KL-edge double-core-hole states [17] and K⁻²V double-core-hole valence-excited states of SF₆ [18] were reported. Theoretically, the calculations of the SF₆ molecular orbitals was performed using the discrete variational X_α method [19]. The coincidence studies of SF₆ using excitation around the S K edge [20] show strong fragmentation selectivity in symmetry-forbidden vibrational transitions. In our previous study [21], the manifestation of core-hole-clock spectroscopy was given in the framework of RAS from a SF₆ molecule near the S K edge (2484-2488 eV), in particular focusing on the S1s⁻¹⁶a₁g core-excited state.

In this paper, we present a combined theoretical-experimental study of RAS via three core-excited states at the S K edge of SF₆, S1s⁻¹⁶a₁g, S1s⁻¹⁶t₁u, and S1s⁻¹⁷t₁u, providing a method for PEC gradient reconstruction based on the analysis of the dispersion laws for different spectral bands. For a scattering process in a general three-level system (e.g., in an atom), the Raman-Stokes dispersion law is fulfilled in RAS,

$$E = \omega - \omega_{f0} = \omega_{cf} + \Omega, \quad (1)$$

where ω and E are the energies of an incoming photon and outgoing Auger electron, respectively, ωf₀ and ωcf are the transition energies between the ground 0, intermediate core-excited c, and final f states, and Ω = ω - ωc₀ is the spectral detuning from the x-ray absorption resonance. However, it was shown that the nonlinear dispersion law can be observed even in an atomic three-level system under broadband excitation, when the spectral function of the incoming photons is large or comparable to the lifetime broadening of the core-excited state [22-24]. In molecular systems, where the vibrational broadening of transitions plays an important role, a strong correlation of the dispersion law on the shape of the bound PECs of the ground, core-excited, and final states was established [1,24]. Variation of the dispersion dependence in resonant x-ray scattering on dissociative PECs was previously addressed at the Cl K edge of HCl [10] and at the I L edge of CH₃I [25]. However, a simple relationship between the gradients of dissociative potential and the dispersion law was not yet established. In the present paper, we address these questions using core-excited states at the S K edge of SF₆, where the dynamics of the nuclear wave packet is limited to the region near the ground-state equilibrium due to a very short lifetime. Comparing accurate *ab initio* simulations with the experiment, we explain an intriguing change of the dispersion law in RAS due to a strong shift of the S1s⁻¹⁶t₁u core-excited PEC. Further theoretical simulations employing the

Coulomb potentials model and simple analytical expression show that the character of the dispersion law (Raman, non-Raman, or anti-Raman) allows one to qualitatively predict the ratio of the intermediate and the final states' gradients near the equilibrium.

The paper is organized as follows. In Sec. II, we present experimental results and discuss the basics of the studied process; the details of the theoretical methodology of quantum chemical simulations of the electronic structure and quantum nuclear dynamics calculations are given in Secs. III A and III B, respectively. In Sec. IV, we discuss the main features of the S K-edge x-ray absorption spectra (Sec. IV A) and formation of RAS spectra (Sec. IV B), as well as variation of the dispersion observed in the experiment and theory (Sec. IV C). The results of the model simulations for the dispersion behavior in RAS are discussed in Sec. V. Our findings are summarized in Sec. VI.

## II. EXPERIMENTAL RESULTS

The experimental results (see Ref. [21] for details) were obtained at the French synchrotron facility SOLEIL, on the GALAXIES beam line, with the gas-phase end station dedicated to hard-x-ray photoelectron spectroscopy (HAXPES) [26]. Linearly polarized photon beams from 2480 to 2490 eV were produced by a U20 undulator. The beam energy was selected and monochromatized by using a Si(111) double-crystal monochromator. The Auger-electron spectra were acquired by using a specifically designed SCIENTA EW4000 analyzer, with collection along the axis parallel to the beam polarization vector, but not at the magic angle. However, as discussed in Ref. [21], for SF₆, no strong influence of the detection direction on the Auger spectrum is expected. It is worth noting that polarization sensitivity of the RAS and RIXS spectra is different. Polarization dependence was commonly observed in RIXS and RAS of various molecular systems [1,2,9,27]. However, it does not mask the nonlinear dispersion phenomena discussed in the present paper. For example, a RIXS study of many chlorine-containing molecules at the Cl 1s threshold show significant anisotropy effects, together with a well-pronounced non-Raman dispersion curve [28]. Strong anisotropy was also observed in the x-ray emission spectra of SF₆ at the F K edge [29].

Experimental results are summarized in Fig. 1 as a two-dimensional (2D) map showing the dependence of the Auger electron kinetic energy versus the excitation x-ray photon energy. The excitation energy is scanned across S 1s pre-edge resonances 1a₁g → 6a₁g, 1a₁g → 6t₁u, and 1a₁g → 7t₁u [the 1a₁g molecular orbital (MO) corresponds to the S 1s core orbital of SF₆]. The lowest transition 1a₁g → 6a₁g is dipole forbidden in the Oₕ symmetry, which sufficiently decreases the intensity of the transition (the x-ray absorption spectrum is shown on a logarithmic scale in the panel on the left-hand side), compared to the dipole-allowed higher resonances 6t₁u and 7t₁u. The 1a₁g → 6a₁g take place through the commonly named “intensity-borrowing mechanism” [21]. The same holds for the weak S1s⁻¹⁴s¹ (7a₁g) excitation, which is not discussed in this work; its overlap in its excitation energy with the strongest S1s⁻¹⁶t₁u resonance shows, however, a different Auger energy (see the feature at kinetic energy 2106 eV
<figure><img src="image_3.png" /><figcaption>FIG. 1. Experimental 2D map (excitation photon energy vs electron kinetic energy) of KLL RAS in SF$_{6}$. Resonant excitations $S1s \rightarrow 6a_{1g}$, $6t_{1u}$, and $7t_{1u}$ are marked by horizontal dashed lines.</figcaption></figure>

in Fig. 1). This overlap can also be clearly seen in the $2p^{-1}$ photoabsorption spectrum of Hudson et al. [30], where, due to different selection rules, $a_{1g}$ final-state orbitals are allowed and $t_{1u}$ orbitals are forbidden.

The RAS process studied here consists of the core exci-
tation of a $1a_{1g}$ electron to unoccupied levels $6a_{1g}$, $6t_{1u}$, and $7t_{1u}$, followed by the transition $2p \rightarrow 1s$ with ejection of the Auger electron from the same shell, $2p \rightarrow \psi_E$, as illustrated in Fig. 2. The dipole-forbidden KLL RAS channel via the dissociative core-excited state $S1s \rightarrow 6a_{1g}$ was recently stud-
ied in Ref. [21]. In the present paper, we additionally focus on KLL RAS under core excitation to the higher unoccupied MO's triple degenerate $6t_{1u}$ and $7t_{1u}$ states. As we show below, RAS via these resonances provides important information about the molecular dynamics in the core-excited states and sheds light on the PEC gradients of the intermediate and final

<figure><img src="image_4.png" /><figcaption>FIG. 2. Scheme of S K-edge x-ray absorption spectra (XAS) (blue arrows) and KLL RAS (red arrows) electronic transitions in SF$_{6}$.</figcaption></figure>

states. Indeed, the 2D map (Fig. 1) shows linear Raman and non-Raman dispersion for different spectral lines resulting from distinct shapes of the PECs.

### III. THEORETICAL DESCRIPTION

#### A. Quantum-chemical calculations of the potential energy curves

All quantum-chemical calculations were performed us-
ing the density functional theory (DFT) method [31] with
the generalized gradient approximation GGA B3LYP hy-
brid functional [32–34]. Geometry optimization of the SF₆
molecule was carried out with the use of the def2-TZVP basis
set [35] in the GAMESS software [36] under Oₕ symmetry
constraint. The use of the octahedral symmetry is a natu-
ral choice for SF₆ geometry optimization; it also allows for
an accurate assignment of molecular orbital symmetries. For
potential energy curve (PEC) simulations, Oₕ symmetry ap-
parently breaks down and thus this constraint was waived.
Calculations of PECs of the intermediate and final states along
the elongation of the SF₅-F chemical bond were performed in
the framework of approximation, realized in the XES module
of the ORCA software [37,38] when transition energies are
estimated as differences of the orbital energies received from
ground-state DFT simulations. For this purpose, we used the
ZORA-def2-TZVP basis set (relativistically recontracted version
of the all-electron def2-TZVP basis set), the zero-order regular
approximation (ZORA) [39], the resolution-of-identity (RI)
approximation [40–45], and the corresponding Segmented
All-electron Relativistically Contracted auxiliary (SARC/J)
basis set [46]. At this theory level, PECs of the ground and
intermediate states were calculated for a neutral SF₆ molecule,
whereas for the calculation of the final states, we considered
a SF₆⁺ cation. The results of the PEC simulations are sum-
marized in Fig. 3. To take into account the energy difference
between the ground state of the neutral molecule and the
cation, the PECs of the final states were shifted by 158 eV.

The calculations and experiment [47,48] show that PECs
with the core hole in *K* and *L* shells are almost insensitive
to the electronic structure of deep core-hole states. In fact,
this constitutes the background of the widely used Z+1 ap-
proximation in calculations of the core-hole states' potentials
[49]. Due to this, we use a reasonable approximation assuming
that the shape of PECs for the two-hole final states ¹S₀, ¹D₂,
and ³P₂.₀ (see Fig. 1) are the same, neglecting the spin-orbit
(SO) interaction and using the |2pₓ⁻¹2pₓ⁻¹⟩ configuration for
the PEC's simulation. This assumption is also verified by the
same dispersion behavior of the transitions to the 2p⁻² (¹S₀,
¹D₂, ³P₀.₂) states displayed in Fig. 1. The formation of the
final states is discussed in more detail in Sec. IV B. In our
representation, we fixed the orientation of the 6t₁u and 7t₁u
MOs along the *z* axis and performed a scan with elongation of
the SF₅-F bond along the *x*, *y*, and *z* coordinates (Fig. 4). Due
to this, the PECs for the S-F bond along *z* differ from the PECs
for the *x* and *y* directions. In other words, Fig. 3 shows the cuts
through the multidimensional potential energy surface along
the *x*, *y*, and *z* stretching modes for a selected state. We also
perform geometry optimization for the S1s⁻¹6t₁u core-excited
state varying all molecular coordinates, which clearly showed
<figure><img src="image_5.png" /><figcaption>FIG. 3. Potential energy curves of the ground, intermediate core-excited S1s⁻¹, and final two-hole S2p⁻² cationic states of SF₆ along a single S-F bond elongation. The orientation of 6t₁u and 7t₁u MOs is fixed along the z axis. Therefore, the PECs for intermediate (S1s⁻¹6t₁u, S1s⁻¹7t₁u) and final (S2p⁻²6t₁u, S2p⁻²7t₁u) states that are the same for the x and y S-F bonds qualitatively differ from the PECs along the z axis (see Fig. 4). The notation 6t₁u(x, y) and 6t₁u(z) means the PEC along the x, y, and z axes, respectively. The PECs for the ground, S1s⁻¹6a₁g intermediate, and S2p⁻²6a₁g final states do not depend on the orientation of the S-F bond. The energies of the core-excited PECs are given according to the quantum-chemical calculations; the final-state PECs are shifted by 158 eV to fit the experimental 6t₁u(¹D₂) peak near 2102 eV.</figcaption></figure>

SF₅-F bond elongation along the chosen MO, while all F-S-F angles remained equal to 90° in the optimized core-excited state, which is the same as in the ground-state geometry. As we will see below, this bond elongation plays a crucial role in the formation of the RAS spectra.

## B. Wave-packet calculations of RAS spectra

The RAS cross section $\sigma(E, \omega)$ is simulated using the time-dependent representation of the Kramers-Heisenberg equation [1,2,50,51], where $\sigma(E, \omega)$ is written in terms of

<figure><img src="image_6.png" /><figcaption>FIG. 4. For the selected final state (S2p⁻²6t₁u or S2p⁻²7t₁u) with the t²₁u MO oriented along the z axis, there are three vibrational modes, corresponding to the elongation along x, y, and z axes, as shown by the dashed lines.</figcaption></figure>

nuclear wave packets $|\psi_c(t)\rangle, |\Psi(0)\rangle$, and $|\Psi(t)\rangle$,

$$ \begin{aligned} \sigma(E, \omega) &= \frac{1}{\pi} \mathrm{Re} \int_0^\infty dt \, \sigma(t) e^{i(\omega-E-\omega_{c0}+i\Gamma_f)t}, \\ \sigma(t) &= \langle\Psi(0)|\Psi(t)\rangle, \quad |\Psi(t)\rangle = e^{-i\hbar f_t} |\Psi(0)\rangle, \\ |\Psi(0)\rangle &= Q_{fc} \int_0^\infty dt \, e^{-t/\tau} |\psi_c(t)\rangle, \\ |\psi_c(t)\rangle &= e^{-i\hbar c t} d_{c0}|0\rangle, \quad \tau = \frac{1}{\Gamma - i\Omega}, \end{aligned} \quad (2) $$

where $Q_{fc}$ is the Coulomb amplitude of the Auger decay and $d_{c0}$ is the transition dipole moment on the core-excitation transition, which were considered geometry independent in our simulations; the intensity of the decay channels was fitted from comparison with the experiment. Here, $|\tau| = 1/\sqrt{\Omega^2 + \Gamma^2}$ is the scattering duration time [52] that depends on the detuning $\Omega = \omega - \omega_{c0}$, $\omega$ is the frequency of the incoming photon, and $\omega_{c0}$ and $\omega_{f0}$ are the vertical electronic transition energies from the ground to core-excited and final state, respectively. $\Gamma$ and $\Gamma_f$ are the lifetime broadening (HWHM) of the intermediate c and final f (double-2p-hole) states, respectively. In the simulations, if not stated otherwise, we use $2\Gamma = 0.55$ and $2\Gamma_f = 0.14$ eV [21]. The dynamics of the wave packets is defined by the nuclear Hamiltonians $h_c$ and $h_f$ of the core-excited and final states, respectively. In what follows, we consider one-dimensional simulations of the time-dependent wave-packet dynamics along the z molecular coordinate. The PECs of the ground, core-excited, and final states along the x and y coordinates are parallel for the 6t₁u and 7t₁u states, which makes those vibrational modes inactive in the Franck-Condon approximation.

The time-dependent representation of the RAS spectra has two advantages. First, it allows us to get insight in the dynamical aspects of the resonant Raman scattering process. The second advantage of this approach is the numerical benefit for
the case when the dissociative or quasidissociative states are involved in the process [53]. Indeed, the stationary Kramers-Heisenberg equation faces serious computational costs in this case due to the high density of the nuclear continuum states.

IV. RESULTS AND DISCUSSIONS

A. Main spectral features of S K-edge XAS

In the process studied here, an S 1s electron (1a1g MO) is first promoted to an unoccupied virtual MO, as schematically shown in Fig. 2. The high-symmetry O_h group of the SF₆ prevents molecular field splitting of the core levels, while the SO interaction splits the S 2p levels (2p → 2p₃/₂, 2p₁/₂) with the spacing ΔSO ~ 1.1 eV. Core excitations to the unoccupied MOs 6a1g, 6t1u, 7t1u, 2t2g, 4e_g form the S K- and L₂,₃-edge x-ray absorption spectra. Due to the electronic selection rules, transitions S 1s → 6t1u, 7t1u are allowed, while S1s → 6a1g, 2t2g, 4e_g core excitations are dipole forbidden and thus have rather small intensity in S K-edge XAS (Fig. 1). However, KLL RAS via the dipole-forbidden excitation channel S1s → 6a1g has appreciable intensity (Fig. 1) due to a vibronic coupling [54,55] according to our recent study [21]. The picture is opposite in the S L₂,₃-edge XAS, where S 2p(t₁u) → 6a1g, 2t2g, 4e_g are allowed, while 2p(t₁u) → 6t1u, 7t1u excitations are forbidden [30]. These selection rules are confirmed by experimental S K [16,56] and S L₂,₃ [30,57] x-ray absorption spectra, as well as theoretical calculations [58], where excited states at the K and L edges were compared against each other.

The electronic transitions S1s → 6a1g, 6t1u, 7t1u are lying below the S K-edge ionization threshold (2490 eV), while the S1s → 2t2g, 4e_g transitions are lying above the core-ionization threshold and not considered in our study (Fig. 2). In the case of S L₂,₃ XAS, those transitions result in strong and narrow spectral features [57]. The physical reason for those so-called shape resonances [59] lying in the electronic continuum is explained in detail in [58] using the multiple scattering theory, which was previously used for a theoretical description of shape resonances in smaller molecules [60,61].

The S K-edge XAS of SF₆ discussed here (Fig. 1, left panel) is dominated by the strong S1s → 6t1u peak near 2486 eV, having two side structures: a weak symmetry forbidden S1s → 6a1g dissociative resonance below the main peak (2482 eV) and a feature at the high-energy shoulder of the main peak near 2488 eV, which we assigned to the S 1s → 7t1u excitation. Our DFT simulations show (Fig. 3) about 4 eV energy spacing at the ground-state equilibrium geometry between the S 1s → 6t1u and S 1s → 7t1u excitations, which is twice as much as observed in the experiment (≈2 eV) (Fig. 1). Our analysis shows sufficient sensitivity of the theoretical energy separation between 6t1u and 7t1u core-excited states on the presence of the diffuse functions in the bases set (e.g., aug-cc-pVTZ versus cc-pVTZ), which decreases the spacing by up to 1 eV. However, the use of the augmented basis set increases the relative intensity of the 7t1u state, in disagreement with the experimental observations. Nevertheless, the energy position of the unoccupied MOs in our simulations are in qualitative agreement with the earlier calculations of Gianturco et al. [62], which also show that the 7t1u peak is lying between 6t1u and 2t2g resonances. Moreover,

Hudson et al. [30] identifies the 7t1u state in the L₂,₃ spectrum as an excitation to a 4p Rydberg orbital, which shows a relatively short vibrational progression, meaning that the excited state is stable with a bond distance close to the one of the ground state, in full agreement with our ab initio calculations (Fig. 3).

B. Formation of KLL RAS from SF₆

The final states in the studied RAS process have two core holes in S 2p shell. The parent 2p double-core-hole state results in the two main bands ¹S₀ and ¹D₂ in the LS-coupling scheme (see Fig. 1), while the forbidden ³P₂,₀ band borrows intensity from the ¹D₂ and ¹S₀ bands, respectively [21]. The energy position of the S - P - D triplet is sensitive to the 6a1g, 6t1u, 7t1u scattering channels due to large shifts of the double-core-hole 2p⁻² final-state orbitals. Moreover, due to the change of the molecular charge in the Auger process, these orbital energies are very sensitive to the charge distribution. Indeed, Fig. 1 shows a large shift of about 7.5 eV for the 7t1u channel to lower kinetic energies and about 4 eV blueshift for the 6a1g relative to the main 6t1u resonance. It is worthwhile to note that the situation is qualitatively different in the case of Kα x-ray emission spectra, where the decay transitions between the states of the same charge (single-hole 1s⁻¹ and 2p⁻¹ states) occur. In this case, the energy shift may only result from the change in the charge distribution of the holes, which is negligibly small for the core-hole orbitals.

Apparently, the electrostatic repulsion of the electrons in the open 2p shell of the configurations,

$$|2p_x^{-1}2p_x^{-1}\rangle, \quad |2p_x^{-1}2p_y^{-1}\rangle, \qquad (3)$$

is not the same, resulting from the difference between the Coulomb Δ = J(2pₓ2pₓ) - J(2pₓ2pᵧ) and exchange integrals. This explains the large energy separation between the ¹S₀, ¹D₂, and ³P₂,₀ bands, observed in the experimental RAS spectrum (Fig. 1). Since Δ ≫ ΔSO, we can use the LS-coupling scheme as a good approximation. In this case, the energy of the term ²S⁺¹Lⱼ is characterized by the total spin S, total angular momentum L, and the total momentum J = S + L. This allows us to get the following expression for the ratio of the intervals between the position of the ¹S₀, ¹D₂ and ¹D₂, ³P₂,₀ bands in the RAS spectrum for the atomic case [63,64]:

$$k = \frac{E(^1D) - E(^1S)}{E(^3P) - E(^1D)} = \frac{3}{2}, \qquad (4)$$

where E is the energy of the Auger electron. The above expression overestimates the experimental ratio kₐₗₚ ≈ 1.1 (Fig. 1), due to deviation from the LS-coupling scheme, which shifts the Auger transitions to the ¹S₀ and ¹D₀ states to lower kinetic energies and those to the ³P₀,₂ states to higher kinetic energies. A similar effect was observed for the nonresonant KLL spectra of H₂S and SO₂, where the k ≈ 1.2 and 1.1, respectively [21].

C. Dispersion laws in RAS

The experimental RAS map (Fig. 1) clearly shows the different dispersion relations in the vicinity of the resonance
<figure><img src="image_7.png" /><figcaption>(b) Experiment</figcaption></figure>

<figure><img src="image_8.png" /></figure>

FIG. 5. (a) Theoretical simulation compared against (b) experimental results for the $6t_{1u}(^1S_0)$ and $7t_{1u}(^1D_2)$ RAS bands. The simulations are performed using the wave-packet approach (Sec. III B). The peak position and relative intensities are fitted using experimental data, while no adjustments were made for the description of the detuning dependence (dispersion law).

for the scattering via $S1s^{-1}6t_{1u}^1$ and $S1s^{-1}7t_{1u}^1$ core-excited states. It is especially visible for the partially overlapped $6t_{1u}(^1S_0)$ and $7t_{1u}(^1D_2)$ bands. Our numerical simulations of these bands, where the nuclear dynamics was fully included, given in Fig. 5(a), reproduces the experimental observation [see Fig. 5(b)] with remarkable agreement.

Let us now analyze in detail the formation of these features by taking a separate look at RAS for each core-excited state and discuss the observed effect. A dependence of the theoretical RAS spectra on the variation of the excitation energy in the vicinity of the three core-excited resonances $S1s^{-1}6a_{1g}$, $S1s^{-1}6t_{1u}$, $S1s^{-1}7t_{1u}$ is illustrated in Figs. 6(a)–6(c), respectively. When the excitation energy is tuned far from the resonance, the dispersion follows the Raman law $E = \omega + \text{const}$ in all three considered cases. However, in the vicinity

of the resonance, one can clearly see a qualitative difference in the dispersion behavior: (a) minor deviation from the Raman dispersion $E = 0.60 \times \omega + \text{const}$; (b) nearly non-Raman dispersion $E = 0.19 \times \omega + \text{const}$; and (c) strictly Raman dispersion law $E = \omega + \text{const}$. The nonlinear dispersion laws obtained from the *ab initio* simulations are in good agreement with those extracted from the experiment (Fig. 1): $E \approx 0.42 \times \omega + \text{const}$ ($S1s^{-1}6a_{1g}$), $E = 0.19 \times \omega + \text{const}$ ($S1s^{-1}6t_{1u}$), and $E = \omega + \text{const}$ ($S1s^{-1}7t_{1u}$).

Let us first consider the RAS channel via the $S1s^{-1}7t_{1u}$ state [Fig. 6(c)]. For this scattering channel, the PEC’s minima of the intermediate (core-excited) and final states along the *x*, *y*, and *z* molecular coordinates are parallel to the ground-state PEC (Fig. 3), which makes this RAS channel vibrationally inactive. As noted above, this is fully in line with the $1s \rightarrow 4p$

<figure><img src="image_9.png" /></figure>

FIG. 6. Excitation photon energy $\omega$ dependence of the theoretical RAS spectra for the (a) $2p^{-2}6a_{1g}$, (b) $2p^{-2}6t_{1u}$, and (c) $2p^{-2}7t_{1u}$ final states. All RAS spectra are normalized on the peak maximum. Photon energies and corresponding normalization coefficients are given at the right- and left-hand sides of the plots, respectively. Simulations are done using the PECs of Fig. 3 with resonant excitation energy (a) 2428.7, (b) 2433.5, and (c) 2437.5 eV.
excitation of the L$_{2,3}$ spectrum of Hudson et al. [30], where the $v = 0$ vibrational transition is dominant, clearly indicating a very small change in the equilibrium geometry. Indeed, according to the Frank-Condon principle, only the lowest vibrational states $v = 0$ are excited in the intermediate and final states, making the scattering process equivalent to the simple three-level system, where the Raman dispersion law is preserved. Since the core-hole lifetime broadening $\Gamma \gg \Gamma_f$, with $\Gamma_f = 0.07$ eV being the lifetime broadening of the $2p^{-2}$ final state, and the photon bandwidth 0.3 eV is smaller than the lifetime broadening of the core-excited state $2\Gamma = 0.55$ eV, one can expect strictly linear Raman dispersion here for the molecular system as well [24], in full agreement with the experiment (Fig. 1) and wave-packet simulations [Fig. 6(c)]. The same explanation is applied for the vibrational modes related to the $S1s^{-1}6t_{1u}(x, y)$ core-excited PECs (Fig. 3), which are parallel to the ground and final electronic states' PECs, resulting in linear Raman dispersion behavior (not shown here). As discussed earlier, these modes are vibrationally inactive and thus give no contribution to the RAS spectrum.

The situation is qualitatively changing for the case of the $S1s^{-1}6t_{1u}(z)$ PEC [Fig. 6(b)], which is sufficiently shifted compared to the ground electronic state, providing nuclear dynamics and resulting in almost non-Raman dispersion for this scattering channel. One also can see sufficient broadening of the RAS profile due to vibrational excitation [compare to the narrow emission line for the $S1s^{-1}7t_{1u}$ channel in Fig. 6(c)]. In the case of the $S1s^{-1}6a_{1g}(z)$ RAS channel [Fig. 6(a)], one can see an intermediate situation when a minor deviation from the linear Raman dispersion law is observed. This is explained by a smaller gradient of the $S1s^{-1}6a_{1g}(z)$ core-excited PEC as compared to the $S1s^{-1}6t_{1u}(z)$ PEC (see Fig. 3, upper panel), thus providing weaker vibrational dynamics in the $S1s^{-1}6a_{1g}$ core-excited state.

This qualitative dynamical discussion is confirmed with simulation of the time-integrated core-excited nuclear wave packet $|\Psi(0)\rangle$ [Eq. (2)] for the $S1s^{-1}7t_{1u}$, $S1s^{-1}6t_{1u}$, and $S1s^{-1}6a_{1g}$ states along the $z$ direction (see Fig. 7). In spite of a rather short lifetime $1/2\Gamma \approx 1$ fs of the intermediate states with the core hole in the $S1s$ shell, the nuclear dynamics is clearly traceable. The core-excited nuclear dynamics is strongest for the $S1s^{-1}6t_{1u}$ state, is moderate for the $S1s^{-1}6a_{1g}$ state, and is negligibly small for the $S1s^{-1}7t_{1u}$ state. Quantitative evaluation of the nuclear dynamics can be obtained from evaluation of the overlap of the integral core-excited wave packet $\Gamma|\Psi(0)\rangle$ with the ground-state vibrational wave function $|0\rangle$ (shaded area in Fig. 7): $\Gamma|\langle0|\Psi(0)\rangle| \approx 0.97, 0.71$, and 0.86 for the $S1s^{-1}7t_{1u}$, $S1s^{-1}6t_{1u}$, and $S1s^{-1}6a_{1g}$ core-excited states, respectively. The lowest value of the overlap integral of 0.71 corresponds to the most extensive nuclear dynamics in the $S1s^{-1}6t_{1u}$ excited state.

It is also useful to provide an estimation of the “characteristic” time needed for the core-excited wave packet to experience a sufficient displacement of the size of the ground-state wave function $a$ (see Fig. 7), given by the equation $\tau_{wp} \approx 0.7\sqrt{\mu a/|F_c|}$, where $a \approx 0.13$ a.u., $\mu = 30346.8$ a.u. is the reduced mass of the SF$_5$-F system, and $F_c$ is the slope of the potential energy curve at the equilibrium distance of the ground state along the $z$ direction. For the case of the $S1s^{-1}6t_{1u}$ channel in SF$_6$ with $|F_c| \approx 0.83$ a.u., the

<figure><img src="image_10.png" /><figcaption>FIG. 7. Illustration for the dynamics in the $S1s$ core-excited states, showing the time-integrated wave packet $|\Gamma\Psi(0)|$ in the $S1s^{-1}6a_{1g}$ state (dashed red), $S1s^{-1}6t_{1u}$ state (solid blue), and $S1s^{-1}7t_{1u}$ state (dash-dotted green), computed for $\Omega = 0$ (excitation to the top of each XAS resonance), where the scattering duration takes its maximal value $\tau = 1/\Gamma$. The initial wave packet of the ground state $|0\rangle$ (shaded area) is shown for comparison.</figcaption></figure>

characteristic propagation time $\tau_{wp} \approx 1.2$ fs is comparable to the core-hole lifetime $1/2\Gamma \approx 1.2$ fs, which ensures considerable nuclear dynamics in this case (Fig. 7). In the $S1s^{-1}6a_{1g}$ core-excited state, the gradient $|F_c| \approx 0.42$ a.u. and $\tau_{wp} \approx 1.6$ fs is significantly larger compared to one in the $S1s^{-1}6t_{1u}$ state, resulting in less pronounced nuclear dynamics in this case (see dashed red line in Fig. 7). Nevertheless, the core-excited wave packet cannot propagate far from the vertical transition point in all scattering channels due to the extremely short lifetime of the $S1s^{-1}$ core hole.

V. MODEL DISPERSION LAWS DEPENDENCE ON THE POTENTIALS' GRADIENT

Up to now, the presented dispersion relations (see Fig. 5 and 6) have been based on the calculated potential energy curves given in Fig. 3. There is a notable relation between the dispersion relations and the PEC' gradients for different scattering channels. Indeed, the *ab initio* gradients for the core-excited $F_c$ and final $F_f$ PEC are zero for the $S1s^{-1}7t_{1u}$ channel, resulting in linear dispersion. The gradients for the $S1s^{-1}6a_{1g}$ channel ($F_c \approx -6.0$ eV/Å, $F_f \approx -5.7$ eV/Å) are relatively smaller compared to the $S1s^{-1}6t_{1u}$ channel ($F_c \approx -12.0$ eV/Å, $F_f \approx -14.0$ eV/Å), providing a smaller deviation from the Raman law in the former case.

In the present section, we want to use simplified potentials, which allow one to derive general statements about the involved potential energy curves. The short lifetime of the core-excited state limits the wave-packet dynamics in a rather small coordinate region near the ground-state equilibrium position, as illustrated in Fig. 7. This allows us to suggest that
the nonlinear dispersion relation of the RAS spectra can be addressed by the value of the gradients of the core-excited and final states near the vertical transition point from the ground-state equilibrium, which we discuss in Sec. VB using analytic derivation. However, let us first discuss a more general approach based on a Coulomb repulsion potential in Sec. VA. This approach is more realistic for longer core-hole lifetimes and steep gradients since it describes significant changes in the internuclear distance much better. Moreover, the dispersion relation does not only depend on the potential energy curves' slopes, but also on the lifetime, as we discuss below.

A. Coulomb potential model

In order to explore the gradient dependence of the RAS dispersion laws, we performed a number of model simulations using Coulomb potentials for the core-excited and final states (in a.u.):

$$U_i(R) = \frac{Z_i}{R} + U_i(\infty) \quad (i = c, f). \qquad (5)$$

Note that $Z_i$ does not represent a real charge, but it is a parameter that allows one to vary the slope of the PEC at the internuclear distance of interest. In fact, the change of the slope with $R$ becomes relevant only if the core-hole lifetime becomes larger than $\tau_{wp}$. In the present case of the SF$_6$, the wave-packet propagation is rather limited (see Fig. 7) and a linear potential of the form $U(R) = -\text{const} \times R$ would produce similar results. The use of the Coulomb potentials (5), however, provides better convergence for the Schrödinger equation solution (2) and thus were applied here. Figure 8 presents theoretical RAS spectra obtained within the quantum wave-packet simulations [Eqs. (2)] for the Coulomb PECs (5) of the core-excited and final states with $Z_c = 2$ and $Z_f = 3$. The corresponding PEC's gradients $F_i(R_0) = -Z_i/R_0^2$ at the equilibrium distance are $F_c = -11.4$ and $F_f = -17.1$ eV/Å; these values are close to those for the $S1s^{-1}6t_{1u}$ channel in the SF$_6$. The transition energies and lifetime broadening constants used in this model simulation are adapted from the KLL RAS process in the SF$_6$ molecule, studied in the previous section. Simulations of Fig. 8 clearly show non-Raman dispersion behavior in the vicinity of the resonant excitation, which is rather similar to the situation observed for the $S1s^{-1}6t_{1u}$ RAS channel with ab initio PECs [Fig. 6(b)].

Using the Coulomb PEC model, we performed a set of simulations varying the ratio of the gradients of the final and core-excited states $F_f/F_c = Z_f/Z_c$, as well as the core-hole lifetime broadening $\Gamma$. The results of simulations using the lifetime of the $S1s^{-1}$ core hole are summarized in Fig. 9, where the dispersion laws following the RAS peak maximum [Fig. 9(a)] and the spectral center of gravity [Figs. 9(b) and 9(c)] for various values of $F_f/F_c$ [Fig. 9(b)] as well as for $F_f/F_c = 1$, however, with different values of slopes [Fig. 9(c)], are shown; for details, see the figure legends. These curves clearly identify a dramatic dependence of the dispersion law on the relative gradient's values of the core-excited and final states. One can see a transition from nearly Raman law for $F_f/F_c = 2/3$ through non-Raman ($F_f/F_c = 3/2$) towards anti-Raman dispersion with further increase of the $F_f/F_c$ value. The dependencies are sharper for the

<figure><img src="image_11.png" /><figcaption>FIG. 8. RAS dispersion simulations using the model Coulomb potentials (5) with $Z_c = 2$ and $Z_f = 3$ a.u. The RAS spectrum on top of the resonance excitation is marked with the thick black line. All RAS spectra are normalized to the peak maximum; the excitation energies and normalization coefficients are given at the right- and left-hand side of the plot, respectively.</figcaption></figure>

dispersion of the RAS peak maximum [Fig. 9(a)] compared to the dispersion of the spectral center of gravity [Fig. 9(b)], as an effect of the presence of a minor vibrational structure (see Fig. 8).

The present Coulomb PEC model simulations are in good qualitative agreement with the results obtained using the ab initio PECs for the SF$_6$. Indeed, in the case of the $S1s^{-1}6a_{1g}$ channel, the gradients' ratio is $F_f/F_c \approx 0.9 < 1$, and this channel shows nearly Raman dispersion behavior [Fig. 6(a)], similar to the case $F_f/F_c = 2/3 < 1$ of the Coulomb PEC model [Figs. 9(a) and 9(b)]. In contrast, the $S1s^{-1}6t_{1u}$ channel shows clearly non-Raman dispersion [Fig. 6(b)] with the gradients' ratio $F_f/F_c \approx 1.2 > 1$, similar to the case of $F_f/F_c = 3/2$ of the Coulomb PEC model.

A special comment is in order for the case where the gradients of the intermediate and final states are the same. Previously, this situation was addressed in relation to ultrafast dissociation in RIXS of water [65] and methanol [66], where the dissociative PECs of the lowest $O1s^{-1}$ core-excited and
<figure><img src="image_12.png" /><figcaption>FIG. 9. Dispersion laws for (a) the maximum of the RAS peak and (b),(c) the center of gravity of the RAS spectra computed using the model Coulomb potentials with gradients of the final $F_f$ and core-excited $F_c$ states shown in the legends. The lifetime broadening and transition energies are adapted from KLL RAS in the SF$_6$.</figcaption></figure>

of the first valence-excited states are nearly parallel. In this case, the emission energy does not depend on the internuclear distance forming a narrow nondispersive spectral feature, denoted as the “pseudo-atomic” peak [65,66]. Similar observations have also been discussed in tender RIXS from HCl [10]. Figure 9(c) presents numerical simulations for the parallel Coulomb PECs of the intermediate and final states, while the absolute values of the gradients near the classical turning point increase from $Z_c = Z_f = 1$ up to $Z_c = Z_f = 5$. One can see that non-Raman dispersion behavior appears when the absolute values of the gradients become large enough ($Z_f = Z_c = 5$) to trigger sufficient nuclear dynamics within the short $S1s^{-1}$ core-hole lifetime $\Gamma$. For this purpose, we define the core-excited state $\Delta\epsilon$, which is the product of the width of the Franck-Condon region a (see Fig. 7), and the absolute value of the slope of the potential energy curve, $|Z_c/R_0^2|$. Note that $\Delta\epsilon$ represents the width of the spectral resonance caused by the electronic excitation to the dissociative state. If $\Gamma \gg \Delta\epsilon$, the nuclear dynamics in the core-excited state is almost absent and the amplitude of RAS can be described as a direct jump from the ground state $|0\rangle$ to the vertical transition point of the final-state potential,

$$ \frac{\langle 0|\epsilon_c \rangle \langle \epsilon_c|\epsilon_f \rangle}{\omega - \omega_{c0} - \epsilon_c + i\Gamma} \approx \frac{\langle 0|\epsilon_f \rangle}{i\Gamma}, \quad (6) $$

where $|\epsilon_c\rangle$ and $|\epsilon_f\rangle$ are the vibrational wave functions in the core-excited and final states, respectively. Hence, in this case, the Raman dispersion $E = \omega - \omega_{f0}$ is preserved, with $\omega_{f0}$ corresponding to the vertical transition energy from the ground-state equilibrium. This situation is illustrated in the case of the small gradients, $Z_f = Z_c = 1$ [Fig. 9(c)]. However, when the PEC gradients increase, so that $\Delta\epsilon \gtrsim \Gamma$, the core-excited dynamics is sufficient even for the large-$\Gamma$ values, which results in a nondispersive spectral feature [see $Z_f = Z_c = 5$ in Fig. 9(c)], similar to the pseudo-atomic peak discussed previously [65,66].

As discussed above, the relation between the gradients of the dissociative PECs and the core-hole lifetime plays a significant role in the spectral dispersion. Due to this, it is instructive to also analyze the change of the RAS dispersion when the core-excited lifetime broadening $\Gamma$ alters. The

results of the simulations for the RAS spectrum center-of-gravity dispersion curves with variation of $\Gamma$ are shown in Fig. 10. One can see that an increase of the $\Gamma$ [Fig. 10(a)] decreases the slopes of the dispersion curves, leading to a smaller deviation from the Raman dispersion law, as a result of the suppressed core-excited dynamics [see Eq. (6)]. In contrast, the decrease of the $\Gamma$ [Figs. 10(b) and 10(c)] makes the gradients' influence on the dispersion curves much more pronounced. Within a longer lifetime, the core-excited wave packet spreads along larger internuclear distances $R$ with changing gradient value, which results in a more complex dynamic effects. It is worthwhile to note that a decrease of the final-state lifetime broadening $\Gamma_f$ does not play any significant role for the dispersion dependencies (not shown here).

## B. Simple analytic estimation of the gradient ratio from the RAS dispersion law

In the present example of SF$_6$, we study the case of a short lifetime of the intermediate core-excited state where the dynamics of the nuclear wave packet is limited by the region near the ground-state equilibrium $R_0$. In this case, the dissociative or quasidissociative PECs of the core-excited and final states can be approximately described by their gradients $F_i = -dE_i(R)/dR$ ($i = c, f$) at the equilibrium $R = R_0$ (Fig. 11),

$$ E_i(R) \approx E_i(R_0) - F_i x, $$

where $x = R - R_0$. The nuclear state with the kinetic energy $\epsilon_i$ at the equilibrium is characterized by the wave function $\psi_{\epsilon_i}(R)$ which absolute value has its maximum in the vicinity of the classical turning point $|x_i| = \epsilon_i/F_i$ (Fig. 11). Apparently, the RAS cross section reaches its maximum when the Frank-Condon factor $|\langle\psi_{\epsilon_f}|\psi_{\epsilon_c}\rangle|^2$ of the decay transition is maximal, which happens when the overlap of the core-excited and final-state nuclear wave functions is maximal, i.e., when the classical turning points of the core-excited and final states coincide $|x_f| = |x_c|$. This requirement, together with the energy conservation laws for XAS and RAS,

$$ \frac{\epsilon_c}{F_c} = \frac{\epsilon_f}{F_f}, \quad \omega = \omega_{c0} + \epsilon_c, \quad \omega = E + \omega_{f0} + \epsilon_f, $$
<figure><img src="image_13.png" /><figcaption>FIG. 10. Dispersion laws for the center of gravity of the RAS spectra for the same $F_f/F_c$ as in Figs. 9(a) and 9(b), but varying the lifetime broadening of the core-excited state $\Gamma$ as (a) 0.55, (b) 0.15, and (c) 0.05 eV.</figcaption></figure>

allow us to obtain a remarkably simple dependence of the energy of the Auger electron $E$ on the photon energy in the vicinity of a XAS resonance, where the detuning $\Omega = \omega - \omega_{c0}$ is small,

$$E \approx \omega \left( 1 - \frac{F_f}{F_c} \right) + \text{const.} \quad (7)$$

and where $\text{const} = -\omega_{f0} + \omega_{c0}F_f/F_c$ and the transition energies are defined at the equilibrium $R_0$ as $\omega_{i0} = E_i(R_0) - E_0(R_0)$, $i = f, c$. The dispersion law of Eq. (7) is derived in an assumption of a sufficient nuclear dynamics near the XAS resonance where the scattering duration $\tau = 1/\sqrt{\Omega^2 + \Gamma^2} \approx 1/\Gamma$ reaches its maximum. Due to this, the non-Raman dispersion predicted by (7) for $F_c = F_f$ could be observed in the case of significant gradient values or small $\Gamma$, as discussed above in connection to Fig. 9(c). The dispersion law qualitatively changes far from the resonance where the scattering duration tends to zero, $\tau = 1/\sqrt{\Omega^2 + \Gamma^2} \approx 1/|\Omega| \to 0$. In that limit

<figure><img src="image_14.png" /><figcaption>FIG. 11. Schematic picture of the nuclear wave functions of the core-excited $\psi_c$ and final $\psi_f$ electronic states, indicated in blue. The PECs are approximated by their gradients at the ground-state equilibrium distance $R_0$.</figcaption></figure>

of the fast scattering, the energy of the Auger electron follows the Raman dispersion law, $E = \omega - \omega_{f0}$; see Eq. (6).

This very simple estimation (7) for the gradient dependence of the RAS dispersion law is nevertheless in qualitative agreement with quantum wave-packet simulations in the Coulomb PEC model. The slopes of the RAS dispersion curves [Fig. 9(a)] in the vicinity of resonant excitation from Coulomb model simulation are compared against values estimated using Eq. (7) in Table I. In both cases, the tendency towards anti-Raman dispersion is clearly seen when the factor $F_f/F_c$ becomes larger than 1. This shows the applicability of Eq. (7) for a rough estimation of the relative gradients of the core-excited and final states in RAS and RIXS directly from the experimental 2D map (Fig. 1) assuming dissociative or quasidissociative PECs on the considered transition and a relatively short core-hole lifetime, limiting the nuclear dynamics in the region near the equilibrium.

It is worthwhile to note that for diatomic or quasidiatomic (e.g., CH$_3$-F) molecules, one can directly extract the core-excited gradient $F_c$ from the width of the XAS resonance $\Delta\epsilon$ and the Franck-Condon width $a$, which can be estimated from the vibrational energies of the corresponding vibrational mode [67]. The 2D map, such as the one shown in Fig. 1, also contains a photoabsorption profile, and thus information on both $F_c$ and $F_f$ gradients can be extracted. For polyatomic molecules, such as SF$_6$ considered here, the situation is more complex as several modes may give contribution to the vibrational broadening of the XAS resonances. Due to this, one has to perform calculations of PECs, such as those given in Fig. 3,

<table><caption>TABLE I. Spectral dispersion slope values fitted from the RAS dispersion within the Coulomb PEC model [Fig. 9(a)] (Numeric) and those computed using analytic estimation (7) (Analytic).</caption><thead><tr><th>F<sub>f</sub>/F<sub>c</sub></th><th>Numeric</th><th>Analytic</th></tr></thead><tbody><tr><td>2/3</td><td>0.47</td><td>0.5</td></tr><tr><td>3/2</td><td>-0.06</td><td>-0.5</td></tr><tr><td>5/2</td><td>-0.65</td><td>-1.5</td></tr><tr><td>7/2</td><td>-1.2</td><td>-2.5</td></tr></tbody></table>
to estimate the excitation of different vibrational modes and their contribution to the XAS broadening.

The analytic model (7) assumes that a sufficient dynamic in the core-excited state has happened, so that the maximum of the wave function is located at the classical turning point (Fig. 11). On the other hand, extended dynamics in the core-excited state allows the wave packet to reach the region with large values of *R*, where the linear PEC approximation used in the derivation of the analytic equation sufficiently deviates from the Coulomb PEC shape. This explains the quantitative disagreement of Eq. (7) with the Coulomb model simulations (Table I). It is worthwhile to note that for simulations with smaller values of the core-hole lifetime broadening $\Gamma = 0.15$ and $\Gamma = 0.05$ eV [Figs. 10(b) and 10(c)], the slope of the $F_f/F_c = 7/2$ dispersion curve becomes about -1.5 and -2.8, respectively, which is closer to the value predicted from the analytic estimation (-2.5). This is explained by dynamic effects. Indeed, as discussed earlier, the longer core-hole lifetime allows the nuclear wave packet to reach the classical turning point of the PEC, making the analytical estimation (7) more robust. On the other hand, when the propagation time is long (e.g., $\Gamma = 0.05$ eV), the wave packet may reach the region where the Coulomb PEC strongly deviates from the linear gradient approximation, so the validity of the analytical expression is reduced. Nevertheless, estimation (7) is rather useful for the qualitative analysis of the PEC's gradients directly from the experimental dispersion measurements.

## VI. CONCLUSIONS

In summary, we experimentally and theoretically study nonlinear dispersion laws in S KLL RAS spectra from SF₆. Using advanced *ab initio* simulations of the potential energy curves of the core-excited and final states, and quantum modeling of the nuclear wave-packet dynamics, we explain intriguing features in RAS dispersion behavior observed in the experiment. The nondispersing character of the RAS channel

via the $S1s^{-1}6r_{1u}$ core-excited state was attributed to sufficient elongation of the SF₅-F distance in both intermediate core-excited $S1s^{-1}$ and final $S2p^{-2}$ states.

Spectral detuning from the resonance in RAS is a powerful tool to trace nuclear dynamics using the scattering duration control concept. The nonlinear dispersion in this case is a clear indication of a nontrivial nuclear dynamics. A set of model simulations with Coulomb potentials varying gradients of the intermediate and final states was performed. We show that the dispersion character of RAS bands—Raman, non-Raman, or anti-Raman—can be applied for an advanced study of the potential’s gradient near the equilibrium. A simple analytical equation, describing the dispersion slope as a simple function of the intermediate and final states’ gradients, that provides qualitative agreement with the simulations is given. Established in RAS in the present study, our findings may be fully applied for the case of RIXS, when dissociative or quasidisociative states are involved in the x-ray scattering process. Analysis of intramolecular potentials using the spectral dispersion behavior suggested here is a general technique and may be applied to the study of other polyatomic molecules.

## ACKNOWLEDGMENTS

Experiments were performed on the GALAXIES beam line at SOLEIL Synchrotron, France (Proposal No. 20170368). The research leading to the obtained results has received funding from the Swedish Research Council (Project No. 2019-03470), the Ministry of Science and Higher Education of Russian Federation (Project No. FSRZ 2023-0006), and STINT Mobility Grants for Internationalisation (Project No. MG2021-9085). The computations were enabled by resources provided by the National Academic Infrastructure for Supercomputing in Sweden (NAISS Projects No. 2024/22-543 and No. 2024/23-254) (NAISS) at PDC and NSC partially funded by the Swedish Research Council through Grants No. 2022-06725 and No. 2018-05973.

[1] F. Gel'mukhanov and H. Ågren, Resonant x-ray Raman scattering, Phys. Rep. **312**, 87 (1999).

[2] F. Gel'mukhanov, M. Odelius, S. P. Polyutov, A. Föhlisch, and V. Kimberg, Dynamics of resonant x-ray and Auger scattering, Rev. Mod. Phys. **93**, 035001 (2021).

[3] W. Schuelke, Electron Dynamics by Inelastic X-Ray Scattering (Oxford University Press, Oxford, 2007).

[4] F. D. Groot and A. Kotani, Core Level Spectroscopy of Solids (CRC Press, Boca Raton, FL, 2008).

[5] L. J. P. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill, and J. van den Brink, Resonant inelastic x-ray scattering studies of elementary excitations, Rev. Mod. Phys. **83**, 705 (2011).

[6] M. N. Piancastelli, T. Marchenko, R. Guillemin, L. Journel, O. Travnikova, I. Ismail, and M. Simon, Hard x-ray spectroscopy and dynamics of isolated atoms and molecules: A review, Rep. Prog. Phys. **83**, 016401 (2020).

[7] M. Kavčič, M. Žitnik, K. Bučar, A. Mihelič, S. Carniato, L. Journel, R. Guillemin, and M. Simon, Electronic state

interferences in resonant x-ray emission after K-shell excitation in HCl, Phys. Rev. Lett. **105**, 113004 (2010).

[8] O. Travnikova, V. Kimberg, R. Flammini, X.-J. Liu, M. Patanen, C. Nicolas, S. Svensson, and C. Miron, On routes to ultrafast dissociation of polyatomic molecules, J. Phys. Chem. Lett. **4**, 2361 (2013).

[9] O. Travnikova, E. Kukk, F. Hosseini, S. Granroth, E. Itälä, T. Marchenko, R. Guillemin, I. Ismail, R. Moussaoui, L. Journel et al., Ultrafast dissociation of ammonia: Auger Doppler effect and redistribution of the internal energy, Phys. Chem. Chem. Phys. **24**, 5842 (2022).

[10] M. Simon, L. Journel, R. Guillemin, W. Stolte, I. Minkov, F. Gel'mukhanov, P. Sałek, H. Ågren, S. Carniato, R. Taieb et al., Femtosecond nuclear motion of HCl probed by resonant x-ray Raman scattering in the Cl 1s region, Phys. Rev. A **73**, 020706 (2006).

[11] O. Travnikova, T. Marchenko, G. Goldsztejn, K. Jänkälä, N. Sisourat, S. Carniato, R. Guillemin, L. Journel, D. Céolin, R. Püttner, H. Iwayama, E. Shigemasa, M. N.
Piancastelli, and M. Simon, Hard-x-ray-induced multistep ultrafast dissociation, Phys. Rev. Lett. **116**, 213001 (2016).

[12] O. Travnikova, N. Sisourat, T. Marchenko, G. Goldsztejn, R. Guillemin, L. Journel, D. Céolin, I. Ismail, A. F. Lago, R. Püttner *et al.*, Subfemtosecond control of molecular fragmentation by hard x-ray photons, Phys. Rev. Lett. **118**, 213001 (2017).

[13] F. O. L. Johansson, E. Berggren, L. M. Cornetta, D. Céolin, M. Fondell, H. Ågren, and A. Lindblad, Resonant Auger spectroscopy on solid xenon on gold, silver, and copper substrates, Phys. Rev. A **107**, 032802 (2023).

[14] N. Velasquez, F. B. Nunes, O. Travnikova, I. Ismail, R. Guillemin, J. B. Martins, D. Céolin, L. Journel, L. Fillaud, D. Koulentianos *et al.*, X-ray induced ultrafast charge transfer in thiophene-based conjugated polymers controlled by core-hole clock spectroscopy, Phys. Chem. Chem. Phys. **26**, 1234 (2024).

[15] L. Asplund, P. Kelfve, B. Blomster, H. Siegbahn, K. Siegbahn, R. L. Lozes, and U. I. Wahlgren, Molecular Auger electron spectra of second row elements. Sulfur compounds, Phys. Scr. **16**, 273 (2015).

[16] T. A. Ferrett, D. W. Lindle, P. A. Heimann, H. G. Kerkhoff, U. E. Becker, and D. A. Shirley, Sulfur 1s core-level photoionization of SF₆, Phys. Rev. A **34**, 1916 (1986).

[17] C. Reynaud, M.-A. Gaveau, K. Bisson, I. Millié, P. Nenner, S. Bodeur, P. Archirel, and B. Lévy, Double-core ionization and excitation above the sulphur *K*-edge in H₂S, S₂O and SF₆, J. Phys. B: At. Mol. Opt. Phys. **29**, 5403 (1996).

[18] R. Feifel, J. H. D. Eland, S. Carniato, P. Selles, R. Püttner, D. Koulentianos, T. Marchenko, L. Journel, R. Guillemin, G. Goldsztejn, O. Travnikova, I. Ismail, B. C. de Miranda, A. F. Lago, D. Céolin, P. Lablanquie, F. Penent, M. N. Piancastelli, and M. Simon, Cationic double *K*-hole pre-edge states of CS₂ and SF₆, Sci. Rep. **7**, 13317 (2017).

[19] H. Nakamatsu and T. Mukoyama, Theoretical x-ray absorption spectra of SF₆ and H₂S, J. Chem. Phys. **95**, 3167 (1991).

[20] A. Gomes, R. Oliveira, A. Rocha, W. Wolff, K. Alcantara, G. Sigaud, and A. Santos, Strong selectivity in symmetry forbidden vibronic transitions in deep core ionic photofragmentation of the SF₆ molecule, Intl. J. Mass Spectrom. **388**, 9 (2015).

[21] A. C. F. Santos, O. Travnikova, N. Boudjemia, T. Marchenko, R. Guillemin, I. Ismail, D. Koulentianos, D. Céolin, F. Gel'mukhanov, M. Simon, M. N. Piancastelli, and R. Püttner, Resonant Auger decay induced by the symmetry-forbidden $1a_{1g} \rightarrow 6a_{1g}$ transition of the SF₆ molecule, J. Vac. Sci. Technol. A **40**, 042801 (2022).

[22] G. B. Armen and H. Wang, Incident-photon energy-distribution effects on radiationless resonant Raman scattering, Phys. Rev. A **51**, 1241 (1995).

[23] E. Kukk, S. Aksela, and H. Aksela, Features of the Auger resonant Raman effect in experimental spectra, Phys. Rev. A **53**, 3271 (1996).

[24] F. Gel'mukhanov and H. Ågren, Raman, non-Raman, and anti-Raman dispersion in resonant x-ray scattering spectra of molecules, Phys. Rev. A **54**, 3960 (1996).

[25] T. Marchenko, L. Journel, T. Marin, R. Guillemin, S. Carniato, M. Žitnik, M. Kavčič, K. Bučar, A. Mihelič, J. Hoszowska *et al.*, Resonant inelastic x-ray scattering at the limit of subfemtosecond natural lifetime, J. Chem. Phys. **134**, 144308 (2011).

[26] J.-P. Rueff, J. M. Ablett, D. Céolin, D. Prieur, T. Moreno, V. Balédent, B. Lassalle-Kaiser, J. E. Rault, M. Simon, and A.

Shukla, The GALAXIES beam line at the SOLEIL synchrotron: Inelastic x-ray scattering and photoelectron spectroscopy in the hard x-ray range, J. Synchrotron Rad. **22**, 175 (2015).

[27] C. Miron, V. Kimberg, P. Morin, C. Nicolas, N. Kosugi, S. Gavrilyuk, and F. Gel'mukhanov, Vibrational scattering anisotropy generated by multichannel quantum interference, Phys. Rev. Lett. **105**, 093002 (2010).

[28] R. Guillemin, W. C. Stolte, L. Journel, S. Carniato, M. N. Piancastelli, D. W. Lindle, and M. Simon, Angular and dynamical properties in resonant inelastic x-ray scattering: Case study of chlorine-containing molecules, Phys. Rev. A **86**, 013407 (2012).

[29] V. Ekholm, G. Chiuzbčžian, C. Săthe, A. Nicolaou, M. Guarise, M. Simon, N. Jaouen, J. Lüning, C. Hague, F. Gel'mukhanov *et al.*, Core-hole localization and ultra-fast dissociation in SF₆, J. Phys. B: At. Mol. Opt. Phys. **53**, 185101 (2020).

[30] E. Hudson, D. A. Shirley, M. Domke, G. Remmers, A. Puschmann, T. Mandel, C. Xue, and G. Kaindl, High-resolution measurements of near-edge resonances in the core-level photoionization spectra of SF₆, Phys. Rev. A **47**, 361 (1993).

[31] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. **136**, B864 (1964).

[32] A. D. Becke, Density-functional thermochemistry. III. The role of exact exchange, J. Chem. Phys. **98**, 5648 (1993).

[33] C. Lee, W. Yang, and R. G. Parr, Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density, Phys. Rev. B **37**, 785 (1988).

[34] S. H. Vosko, L. Wilk, and M. Nusair, Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis, Can. J. Phys. **58**, 1200 (1980).

[35] F. Weigend and R. Ahlrichs, Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy, Phys. Chem. Chem. Phys. **7**, 3297 (2005).

[36] M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. Su, T. L. Windus, M. Dupuis, and J. A. Montgomery, General atomic and molecular electronic structure system, J. Comput. Chem. **14**, 1347 (1993).

[37] F. Neese, The ORCA program system, WIREs Comput. Mol. Sci. **2**, 73 (2012).

[38] F. Neese, Software update: The ORCA program system, version 4.0, WIREs Comput. Mol. Sci. **8**, e1327 (2018).

[39] E. van Lenthe, J. G. Snijders, and E. J. Baerends, The zero-order regular approximation for relativistic effects: The effect of spin-orbit coupling in closed shell molecules, J. Chem. Phys. **105**, 6505 (1996).

[40] E. J. Baerends, D. E. Ellis, and P. Ros, Self-consistent molecular Hartree-Fock-Slater calculations I. The computational procedure, Chem. Phys. **2**, 41 (1973).

[41] B. I. Dunlap, J. W. D. Connolly, and J. R. Sabin, On some approximations in applications of Xα theory, J. Chem. Phys. **71**, 3396 (1979).

[42] C. V. Alsenoy, Ab initio calculations on large molecules: The multiplicative integral approximation, J. Comp. Chem. **9**, 620 (1988).

[43] R. A. Kendall and H. A. Früchtl, The impact of the resolution of the identity approximate integral method on modern ab initio algorithm development, Theor. Chem. Acc. **97**, 158 (1997).
[44] K. Eichkorn, O. Treutler, and H. Öhm, Auxiliary basis sets to approximate Coulomb potentials, *Chem. Phys. Lett.* **240**, 283 (1995).

[45] K. Eichkorn, F. Weigend, and O. Treutler, Auxiliary basis sets for main row atoms and transition metals and their use to approximate Coulomb potentials, *Theor. Chem. Acc.* **97**, 119 (1997).

[46] F. Weigend, Accurate Coulomb-fitting basis sets for H to Rn, *Phys. Chem. Chem. Phys.* **8**, 1057 (2006).

[47] T. Marchenko, G. Goldsztejn, K. Jänkälä, O. Travnikova, L. Journel, R. Guillemin, N. Sisourat, D. Céolin, M. Žitnik, M. Kavčič, K. Bučar, A. Mihelič, B. C. de Miranda, I. Ismail, A. F. Lago, F. Gel'mukhanov, R. Püttner, M. N. Piancastelli, and M. Simon, Potential energy surface reconstruction and lifetime determination of molecular double-core-hole states in the hard x-ray regime, *Phys. Rev. Lett.* **119**, 133001 (2017).

[48] L. Barreau, A. D. Ross, V. Kimberg, P. Krasnov, S. Blinov, D. M. Neumark, and S. R. Leone, Core-excited states of SF₆ probed with soft-x-ray femtosecond transient absorption of vibrational wave packets, *Phys. Rev. A* **108**, 012805 (2023).

[49] O. Plashkevych, T. Privalov, H. Ågren, V. Carravetta, and K. Ruud, On the validity of the equivalent cores approximation for computing x-ray photoemission and photoabsorption spectral bands, *Chem. Phys.* **260**, 11 (2000).

[50] F. K. Gel'mukhanov, L. N. Mazalov, and A. V. Kondratenko, A theory of vibrational structure in the x-ray spectra of molecules, *Chem. Phys. Lett.* **46**, 133 (1977).

[51] P. Safek, F. Gel'mekhanov, and H. Ågren, Wave-packet dynamics of resonant x-ray Raman scattering: Excitation near the Cl L<sub>II,III</sub> edge of HCl, *Phys. Rev. A* **59**, 1147 (1999).

[52] F. Gel'mukhanov, P. Safek, T. Privalov, and H. Ågren, Duration of x-ray Raman scattering, *Phys. Rev. A* **59**, 380 (1999).

[53] F. Gel'mukhanov and H. Ågren, X-ray resonant scattering involving dissociative states, *Phys. Rev. A* **54**, 379 (1996).

[54] H. Köuppel, W. Domcke, and L. S. Cederbaum, Multimode molecular dynamics beyond the Born-Oppenheimer approximation, *Adv. Chem. Phys.* **57**, 59 (1984).

[55] P. Skytt, P. Glans, J.-H. Guo, K. Gunnelin, C. Såthe, J. Nordgren, F. K. Gel'mukhanov, A. Cesar, and H. Ågren, Quenching of symmetry breaking in resonant inelastic x-ray scattering by detuned excitation, *Phys. Rev. Lett.* **77**, 5035 (1996).

[56] R. E. LaVilla and R. D. Deslattes, K-absorption fine structures of sulfur in gaseous SF₆, *J. Chem. Phys.* **44**, 4399 (1966).

[57] T. M. Zimkina and V. A. Fomichev, Absorption spectrum of sulphur hexafluoride in the region of ultra-soft x-ray emission, *Dokl. Akad. Nauk* **169**, 1304 (1966).

[58] R. V. Vedrinskii and V. L. Kraizman, Theory of x-ray absorption spectra of central atoms in high-symmetry molecules and complexes, *Soviet Phys. JETP* **47**, 637 (1978).

[59] M. N. Piancastelli, The neverending story of shape resonances, *J. Electron Spectrosc. Relat. Phenom.* **100**, 167 (1999).

[60] L. N. Mazalov, F. K. Gel'mukhanov, and V. M. Chermoshentsev, Effects of multiple scattering in the x-ray absorption spectra of molecules and solids, *Zh. Strukt. Khim.* **15**, 975 (1974).

[61] J. L. Dehmer and D. Dill, Shape resonances in K-shell photoionization of diatomic molecules, *Phys. Rev. Lett.* **35**, 213 (1975).

[62] F. A. Gianturco, C. Guidotti, and U. Lamanna, Electronic properties of sulphur hexafluoride. II. Molecular orbital interpretation of its x-ray absorption spectra, *J. Chem. Phys.* **57**, 840 (1972).

[63] E. U. Condon and G. H. Shortley, *The Theory of Atomic Spectra* (Cambridge University Press, Cambridge, 1959).

[64] I. I. Sobel'man, *Introduction to the Theory of Atomic Spectra* (Pergamon Press, Oxford, 1972).

[65] E. Ertan, V. Savchenko, N. Ignatova, V. V. da Cruz, R. C. Couto, S. Eckert, M. Fondell, M. Dantz, B. Kennedy, T. Schmitt et al., Ultrafast dissociation features in RIXS spectra of the water molecule, *Phys. Chem. Chem. Phys.* **20**, 14384 (2018).

[66] V. Vaz da Cruz, N. Ignatova, R. C. Couto, D. A. Fedotov, D. R. Rehn, V. Savchenko, P. Norman, H. Ågren, S. Polyutov, J. Niskanen et al., Nuclear dynamics in resonant inelastic x-ray scattering and x-ray absorption of methanol, *J. Chem. Phys.* **150**, 234301 (2019).

[67] D. Koulentianos, R. Püttner, G. Goldsztejn, T. Marchenko, O. Travnikova, L. Journel, R. Guillemin, D. Céolin, M. N. Piancastelli, M. Simon et al., KL double core hole pre-edge states of HCl, *Phys. Chem. Chem. Phys.* **20**, 2724 (2018).