# Simulation of Auger decay dynamics in the hard x-ray regime: HCl as showcase

Gildas Goldsztejn, Renaud Guillemin, Tatiana Marchenko, Oksana

Travnikova, Denis Céolin, Loïc Journel, Marc Simon, Maria Novella

Piancastelli, Ralph Püttner

► To cite this version:

Gildas Goldsztejn, Renaud Guillemin, Tatiana Marchenko, Oksana Travnikova, Denis Céolin, et al. Simulation of Auger decay dynamics in the hard x-ray regime: HCl as showcase. Physical Chemistry Chemical Physics, 2022, 24 (11), pp.6590-6604. 10.1039/D1CP05662J. hal-03856950

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Submitted on 17 Nov 2022

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# Simulation of Auger decay dynamics in the hard x-ray regime: HCl as showcase

G. Goldsztejn,¹ R. Guillemin,² T. Marchenko,² I. Ismail,² O. Travnikova,²
D. Céolin,³ L. Journel,² M. Simon,² M. N. Piancastelli,²,⁴ and R. Püttner⁵

¹Institut des Sciences Moléculaires d'Orsay (ISMO), CNRS,
Univ. Paris-Sud, Université Paris-Saclay, F-91405 Orsay (France)

²Sorbonne Université, CNRS, Laboratoire de Chimie Physique-Matière
et Rayonnement, LCPMR, F-75005 Paris Cedex 05, France

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

⁴Department of Physics and Astronomy, Uppsala University, SE-75120 Uppsala, Sweden

⁵Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany.

After photoexcitation or photoemission of an electron from a deep inner shell in the hard x-ray regime, an isolated system (atom or molecule) becomes unstable and rapidly relaxes by emitting photons (radiative decay) or electrons (nonradiative or Auger decay). The nonradiative path, i.e. the Auger decay, can be rather complex, implying a multitude of phenomena such as multiple-step cascades, post-collision interaction (PCI), and electronic state-lifetime interference. Furthermore, in a molecule nuclear motion can also be triggered. The high complexity of such decay patterns makes the interpretation of Auger decay spectra rather complicated. Here we discuss a comprehensive theoretical method which allows us to analyze in great detail Auger spectra measured around an inner-shell threshold. The investigated showcase molecule is HCl, photoexcited or photoionized around the deep Cl 1s threshold. Our method is able to calculated Auger cross sections considering the nature of the ground, intermediate and final states (bound or dissociative), and the evolution of the relaxation process, including both electron and nuclear dynamics. In particular, we show that with our method we can understand and measured a so-called experimental 2D map, consisting of a series of resonant Auger spectra obtained at different photon energies, therefore obtaining a detailed picture of all above-mentioned dynamical phenomena at once.

## I. INTRODUCTION

The investigation of photoexcitation and photoionization processes involving deep inner shells implies a high level of complexity, due to the multiplicity of dynamical phenomena which can play a role (see e.g. [1] for a recent review). Photoabsorption of an x-ray photon by an isolated atomic or molecular system can induce an electronic transition from a deep shell, either to an empty orbital or into the ionization continuum. The core-excited or core-ionized state thus created with a deep electron vacancy is highly unstable and relaxes on a very short time scale, of the order of one femtosecond ($10^{-15}$ s) or even less. The relaxation processes can proceed through multiple pathways, including a first branching between radiative and nonradiative decay. The nonradiative path, i.e. the Auger decay, can in turn be rather complex, implying multiple-step cascades (see e.g. [2–4], shake-up and shake-off transitions involving one or more electrons [5], post-collision interaction (PCI) phenomena [6, 7], etc. Furthermore, due to the short lifetime of states with a deep core hole, interference phenomena such as electronic state-lifetime interference can also play a role [11, 12]. Another effect to yet consider is that, if the intermediate state reached after photoexcitation is dissociative, in a molecule nuclear motion can be triggered in a time scale of few femtoseconds or even subfemtoseconds [2, 3, 8]. The high complexity of such decay patterns makes the interpretation of Auger decay spectra rather complicated.

In particular, the overall decay dynamics is governed by the interplay of the potential curves of the ground, intermediate and final state, which can be either bound or dissociative; the cross section of a specific final state is sensitive to the possibility of shake-up, shake-down and shake-off processes; the dispersion behavior of the Auger final states as a function of the impinging photons can be influenced by the possibility that part of the absorbed energy is dissipated by the nuclear motion rather than the kinetic energy of the outgoing Auger electrons; post-collision interaction causes shift in energy and distortion of the Auger peaks, and finally electronic and vibrational states’ interferences influence the population of the final states.

Here we discuss a comprehensive theoretical method which allows us to analyze in great detail Auger spectra measured around an inner-shell threshold. The investigated showcase molecule is HCl, chosen because it is a diatomic system containing a heavy atom. The obtained experimental data are in the form of 2D maps, meaning that we measure Auger spectra by changing the photon energy in small steps. This procedure allows not only a first general overview of the dynamics, but also to gather information on final states which are not well separated in kinetic energy.

Our method is able to calculate Auger cross sections taking into account the nature of the ground, intermediate and final states (bound or dissociative), and the evolution of the relaxation process, including both electron and nuclear dynamics. In particular, electronic state-lifetime interference phenomena and signature of nuclear dynamics even in a very short timescale are investigated.
We consider our methodology as a step forward in the direction of achieving a deep insight of a full 2D map by careful fitting procedures and advanced theoretical modeling.

## II. SUMMARY OF THEORETICAL ASPECTS

In this section we will summarize the important equations, and their physical meanings, which are necessary to simulate 2D maps consisting of resonant and normal Auger spectra. For a complete derivation of the formulas, please see Supplementary Material.

### A. From the double differential cross section to resonant and normal Auger

In this part we shall describe the expected lineshapes for resonant and normal Auger spectra. Schematic pictures of these two processes are shown in Fig. 1, which also show relevant states and energies. Here we distinguish between two cases, namely a final state with an infinite lifetime and a final state with a finite lifetime.

Let us start with the situation of an infinite experimental resolution and a stable final state, i.e. with infinite lifetime. In this case the double differential cross section is described in its general form by [8]:

$$ \sigma(\omega, \omega') \propto \sum_f |F_f|^2 \cdot \delta(\omega - \omega' - \omega_{fo}) \quad (1) $$

with

$$ F_f = \sum_c \frac{\langle \Psi_f | \mathbf{Q} | \Psi_c \rangle \langle \Psi_c | \mathbf{D} | \Psi_o \rangle}{\omega - \omega_{co} + i\Gamma_c} \quad (2) $$

Here, $\omega$ is the energy of the incoming photon, $\omega'$ the energy of the outgoing photon or electron(s), $\omega_{co}$ the energy difference between the ground state $|o\rangle$ and the core-excited state $|c\rangle$, $\omega_{fo}$ the energy difference between the ground state $|o\rangle$ and the core-excited state $|f\rangle$, $\mathbf{D}$ the dipole operator for the excitation, $\mathbf{Q}$ the decay operator (Coulomb operator for Auger decay or dipole operator for RIXS) and $\Gamma_c$ the lifetime broadening in the units of the half width at half maximum (HWHM) of the core-hole state.

Equation 1 can be applied to the resonant and normal Auger process. In the resonant Auger process in the first step an electron is excited from a core hole to an unoccupied orbital $n\ell$ below the ionization threshold. This neutral excited state can be considered as consisting of a cation $A^+$ with an electron in the orbital $n\ell$ bound to it. In the next step the cation $A^+$ undergoes an Auger decay and forms a dication $A^{2+}$ while the excited electron remains in a bound orbital. This process can be described as g.s. $\rightarrow A^+n\ell \rightarrow A^{2+}n\ell\epsilon'\ell'$ with g.s. being the ground state and $\epsilon'\ell'$ being the Auger electron. In contrast to this, to observe the normal Auger process, the core electron is promoted in the first step into the continuum $\ell$. After this, the cation decays to a dication and emits an Auger electron. Here the process can be described as g.s. $\rightarrow A^+\epsilon\ell \rightarrow A^{2+}\epsilon\ell\epsilon'\ell'$.

In the following we shall first focus on the resonant Auger process with only one intermediate state $|c\rangle$. In this case, in eqn. 2 the sum over $c$ drops and we obtain the partial differential cross section:

$$ \sigma(\omega, \omega') \propto \frac{\Gamma_c |\mathbf{Q}_{\text{res}}\mathbf{D}_{\text{res}}|^2}{\pi (\omega - \omega_{\text{co}})^2 + \Gamma_c^2} \cdot \delta(\omega - \omega' - \omega_{\text{fo}}) \quad (3) $$

In this equation the $\delta$ function represents the energy conservation and defines the kinetic energy $\omega'$ of the resonant Auger electron as a function of the incoming photon, see panel (a) of Fig. 1, and $\mathbf{Q}_{\text{res}} = \langle \Psi_f | \mathbf{Q} | \Psi_c \rangle$ as well as $\mathbf{D}_{\text{res}} = \langle \Psi_c | \mathbf{D} | \Psi_o \rangle$. By neglecting experimental broadenings, i.e. photon bandwidth and analyser resolution, the spectrum consists of a $\delta$-like peak at the energy $\omega' = \omega - \omega_{fo}$. Figure 2 displays in a comprehensive way the observable lineshapes for resonant and normal Auger spectra, both for final states with finite and infinite lifetime. The above-mentioned $\delta$-peak corresponds to the black vertical lines in panel (a) of Fig. 2. The matrix element $\mathbf{D}_{\text{res}}$ in eqn. 3 describes the resonant excitation and the matrix element $\mathbf{Q}_{\text{res}}$ the Auger decay. Due to the finite lifetime $\Gamma_c$ the process can resonate over a larger energy range. The intensity of the $\delta$-like peak varies with $\omega$ and the variation is described with a Lorentzian function, see Fig. 1(a) and as red curve in Fig. 2(a), which is derived from the denominator of eqn. 3. In summary, $|F_f|^2$ gives the probability of the process as a function of the photon energy $\omega$ and the $\delta$-function defines the energy $\omega'$ of the outgoing Auger electron as a function of the incoming photon.

In the next step we shall consider the normal Auger process subsequent to a photoionization process. In Fig. 1(b) it can be seen that the photon energy $\omega$ is not in the Lorentzian distribution of the core-hole state $|c\rangle$, i.e. the
<figure><img src="image_2.png" /><figcaption>FIG. 1: Schematics of a resonant Auger (a) and a normal Auger (b) process. $|o\rangle$, $|c\rangle$, and $|f\rangle$ indicate the ground state, the core-hole state, and the final state, respectively. $\omega_{co}$ ($\omega_{fo}$) describes the energy difference between the core-hole state (final state) and ground state. $\omega$ represents the photon energy and $\omega'$ the energy of the outgoing electrons. Note that for the resonant Auger process $\omega'$ is equal to the kinetic energy of the Auger electron, $\omega_A$, while in case of a normal Auger process $\omega'$ is shared between the photoelectron and the Auger electron, i.e. $\omega' = \omega_P + \omega_A$, with $\omega_P$ being the kinetic energy of the photoelectron. For both processes the position of the ionization potential IP is also indicated.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>FIG. 2: Schematic line profiles for Auger transitions. (a) Resonant Auger spectra without lifetime broadening of the final state. The spectral features are $\delta$-functions indicated in black. The red Lorentzian curves with a width of $2\Gamma_c$ indicate the intensity of the $\delta$-functions as a function of the detuning $\omega - \omega_{co}$. (b) Resonant Auger spectra with lifetime broadening of the final states. The black Lorentzian curves with a broadening of $2\Gamma_f$ represent the spectral features. For the red curve, see (a). (c) Normal Auger spectra without lifetime broadening of the final state. The black Lorentzian curves with a broadening of $2\Gamma_c$ represent the spectral features. (d) Normal Auger spectra with lifetime broadening of the final state. The black Lorentzian curves with a broadening of $2\Gamma_t = 2(\Gamma_c + \Gamma_f)$ represent the spectral features.</figcaption></figure>

process is non-resonant. Instead, the process leads to two outgoing electrons, namely the photoelectron with a kinetic energy $\omega_P$ and the Auger electron with a kinetic energy $\omega_A$. Note that $\omega_P$ and $\omega_A$ are the actual energies of the emitted electrons, but not the peak maxima in the photoelectron and the Auger spectrum. This leads to the relation $\omega' = \omega_P + \omega_A$, which can only be verified in high-resolution photoelectron-Auger electron coincidence spectra, see e.g. [10]. As shown in detail in the Supplementary Material we obtain

$$ \sigma(\omega, \omega_A) \propto \frac{\Gamma_c |\mathbf{Q}_{\text{nor}}\mathbf{D}_{\text{nor}}|^2}{\pi (\omega_A - \omega_{cf})^2 + \Gamma_c^2} \quad (4) $$

with $\omega_{cf} = \omega_{co} - \omega_{fo}$. Obviously, $\sigma(\omega, \omega_A)$ is described by a Lorentzian function around the energy difference $\omega_{fc}$
between the core hole and the final state, see Fig. 2(c). Contrary to the resonant Auger decay, here the Lorentzian function describes the spectral lineshape and not the intensity variation as a function of the photon energy. In contrast to eqn. 3, eqn. 4 does not depend on the photon energy $\omega$, i.e. normal Auger spectra are independent from the photon energy while resonant Auger spectra change with the photon energy.

Up to now we have assumed that the Auger final state has an infinite lifetime. This is a reasonable approximation for the Auger final states subsequent to the decay of shallow core-hole states, since in this case the final states undergo fluorescence decay which leads to much longer lifetimes. Contrary to this, the final states after the decay of deeper core levels (e.g. $Ar~1s^{-1} \rightarrow 2p^{-2}$) possess non-negligible lifetime broadenings which have to be taken into account. As a result, the accessible energy levels show a Lorentzian-like distribution with a width $\Gamma_f$ around the energy of the final state; here $\Gamma_f$ is the lifetime broadening of the final state (HWHM). Once again, we have to treat resonant and normal Auger separately.

For the resonant Auger case we obtain

$$ \sigma(\omega, \omega') \propto \frac{\Gamma_c |\mathbf{Q}_{\text{res}}\mathbf{D}_{\text{res}}|^2}{\pi (\omega - \omega_{co})^2 + \Gamma_c^2} \times \frac{\Gamma_f}{\pi ((\omega_{fo} - \omega - \omega')^2 + \Gamma_f^2)}. \quad (5) $$

Obviously, the obtained result is a product of the intensity factor already present in eqn. 3 and a Lorentzian function with a width $\Gamma_f$. From this follows that the kinetic energy of the Auger electron, $\omega'$, can be described by a Lorentzian function with a maximum at $\omega_{fo} - \omega$, i.e. the maximum depends on the photon energy, and a width of $\Gamma_f$, see black curve in Fig. 2(c). The first part on the right-hand side of the equation defines the red Lorentzian lineshape in Fig. 2(c) which indicates the intensity of the black Lorentzian curve.

In the next step we shall consider the normal Auger decay. Here we obtain

$$ \sigma(\omega, \omega_A) \propto \frac{\Gamma_t |\mathbf{Q}_{\text{nor}}\mathbf{D}_{\text{nor}}|^2}{\pi ((\omega_A - \omega_{cf})^2 + \Gamma_t^2)}. \quad (6) $$

As displayed in Fig. 2(d) the result of the convolution is also a Lorentzian function, however, with a larger width $\Gamma_t = \Gamma_f + \Gamma_c$, i.e. the sum of the lifetime broadening of the core-hole and the final state.

## B. The molecular case

In this part we shall discuss eqn. 1 for the case of molecules with electronic and nuclear degrees of freedom, with the focus of the discussion on the nuclear degree of freedom. In the Born-Oppenheimer's approximation, we can factorize the electronic and nuclear degrees of freedom such that $\Psi_i$ from the equation 1 can be rewritten as $|\Psi_i\rangle = |\Phi_i\rangle|\chi_i\rangle$. $|\Phi_i\rangle$ represents the electronic wavefunction as defined previously and $|\chi_i\rangle$ is the nuclear wavefunction. In this approximation eqn. 1 can be rewritten as

$$ \sigma(\omega, \omega') \propto \sum_f \left| \sum_c \frac{\langle \Phi_f | Q | \Phi_c \rangle \langle \chi_f | \chi_c \rangle \langle \Phi_c | D | \Phi_o \rangle \langle \chi_o | \chi_o \rangle}{\omega - \omega_{co} + i\Gamma_c} \right|^2 \Delta(\omega - \omega' - \omega_{fo}, \Gamma_f). \quad (7) $$

In this context $\Delta(\omega - \omega' - \omega_{fo}, \Gamma_f)$ describes the spectral features and has to be derived in line with the arguments above according to the situation, i.e. resonant or normal Auger as well as the lifetime of the final states.

Let us now focus on the overlap integrals and rewrite eqn. 7 with $\mathbf{D}_{\text{el}} = \langle \Phi_c | D | \Phi_o \rangle$ and $\mathbf{Q}_{\text{el}} = \langle \Phi_f | Q | \Phi_c \rangle$ as

$$ \sigma(\omega, \omega') \propto \sum_f |\mathbf{Q}_{\text{el}}|^2 |\mathbf{D}_{\text{el}}|^2 \left| \sum_c \frac{\langle \chi_f | \chi_c \rangle \langle \chi_c | \chi_o \rangle}{\omega - \omega_{co} + i\Gamma_c} \right|^2 \Delta(\omega - \omega' - \omega_{fo}, \Gamma_f). \quad (8) $$

Before we continue with the discussion we want to point out that the absolute value of the sum over $c$ can be written as

$$ \left| \sum_c \frac{\langle \chi_f | \chi_c \rangle \langle \chi_c | \chi_o \rangle}{\omega - \omega_{co} + i\Gamma_c} \right|^2 = \sum_c \frac{|\langle \chi_f | \chi_c \rangle|^2 |\langle \chi_c | \chi_o \rangle|^2}{(\omega - \omega_{co})^2 + \Gamma_c^2} + \sum_{c \neq c'} \frac{\langle \chi_f | \chi_c \rangle \langle \chi_c | \chi_o \rangle \langle \chi_o | \chi_{c'} \rangle \langle \chi_{c'} | \chi_f \rangle}{(\omega - \omega_{co} + i\Gamma_c)(\omega - \omega_{c'o} + i\Gamma_c')} \quad (9) $$

The first term on the right side of the equation are the so-called direct terms, which are sufficient when the excitation and the decay process are considered independent. For the case that excitation and decay are considered as one process, the second term, the so-called lifetime interference terms, have to be taken into account. Here we specify
<figure><img src="image_4.png" /></figure>

FIG. 3: Potential energy curves in black and wavefunctions in red for a bound-dissociative transition. The lower potential energy curve and wavefunction represent the ground state of HCl in the harmonic approximation. The hypothetical upper linear potential curve has at the ground state equilibrium distance $R_0$ an energy of $U_c(R_0) \approx 5.7$ eV and a slope of -15 eV/Å. The upper wavefunction (Airy function) is calculated for HCl based on the hypothetical potential energy curve. For more details, see text.

the direct terms and lifetime interference terms for the nuclear part of the wavefunctions, however, such specification are also possible for the electronic part of the wavefunctions. The most general case with vibrational and electronic lifetime interference contributions is discussed in Ref. [9].

To apply eqn. 8 for the vibrational states of electronic transitions, the nuclear wavefunctions $\chi_i$ have to be described according to the case in which they describe bound or dissociative molecular states. In principle, the transitions described by the matrix elements $\langle\chi_f|\chi_c\rangle$ and $\langle\chi_c|\chi_o\rangle$ can have different characters depending on the bond character of the potential energy curves.

In the following we shall first discuss the vibrational progressions of transitions between two states, where we have to distinguish between bound-bound transitions, bound-dissociative transitions and dissociative-dissociative transitions. After this we will discuss the entire excitation and decay process, which includes three different states. Since we only consider processes which involve the ground state, we here have to distinguish four cases, namely bound-bound-bound transitions, bound-bound-dissociative transitions, bound-dissociative-bound transitions and bound-dissociative-dissociative transitions.

### C. Transitions between two states

#### 1. Bound-bound transitions

We will start the discussion of the transitions between two states for the case of bound-bound transitions. Here the potential energy curves can be described with Morse potentials described with three parameters: equilibrium distance $R_0$, vibrational energy $\hbar\omega$, and anharmonicity $x\hbar\omega$. The simple case of an harmonic oscillator is obtained with $x\hbar\omega = 0$. In case of Morse potentials the vibrational overlap matrix elements $\langle\chi_f|\chi_c\rangle$ and $\langle\chi_c|\chi_o\rangle$ can be obtained by following the methods of Ory, Gittleman, and Maddox [13] as well as Halmann and Laulich [14]. Details of a corresponding fit approach are discussed in Ref. [15].

#### 2. Bound-dissociative transitions

In the following discussion of bound-dissociative transitions we have to calculate the Franck-Condon factor $\langle\chi_c|\chi_0\rangle$ where $|\chi_c\rangle$ is the continuum wavefunction for the energy $E_c$ and $|\chi_0\rangle$ the wavefunction of the vibrational ground state. The potential energy curves and the nuclear wavefunctions are presented in Fig. 3.

For the calculation we assume that the slope of the dissociative potential energy curve is constant. Moreover, we assume that the bound state is the initial state and that only the vibrational ground state of the bound state is
populated. By applying an harmonic oscillator potential for the bound state, the vibrational wavefunction is given by

$$ \chi_0(x) = \left(\frac{1}{\pi a_0^2}\right)^{1/4} \exp\left[-\frac{1}{2}\left(\frac{x}{a_0}\right)^2\right] \quad (10) $$

with $a_0 = (\frac{\hbar}{\mu\omega_0})^{1/2}$. Here, $\mu$ is the reduced mass, $\omega_0$ the vibrational frequency, and $a_0$ the average deviation of $x = R - R_0$ with $R_0$ being the equilibrium distance.

As stated above, we approximate the dissociative potential energy curve with a linear function, i.e. $V(x) = -Fx$; The potential and the corresponding solution for one energy is displayed in the upper part of Fig. 3. The corresponding Schrdinger equation to be solved is given by:

$$ \frac{d^2\Psi}{dx^2} + \frac{\mu}{\hbar^2}(E + Fx)\Psi = 0, \quad (11) $$

with $\mu$ being the reduced mass of the molecule. The eigenfunctions of this Schrdinger equation are given by [16]

$$ \Psi_E(x) = \frac{1}{\alpha F^{1/2}} A_i \left[ \frac{x+a}{\alpha} \right]. \quad (12) $$

with $a = \frac{E}{F}$, $\alpha = (-\frac{(2\mu F)^{1/3}}{\hbar^{2/3}})^{-1}$ and $A_i(x)$ being the Airy function, which is discussed in more detail in the Supplementary Material. For $\frac{x+a}{\alpha} < 0$ the Airy function decreases quickly and for $\frac{x+a}{\alpha} > 0$ it oscillates strongly. Based on this, the Franck-Condon factor can be approximated to

$$ \langle \chi_c | \chi_0 \rangle \approx \left( \frac{2\mu\alpha^3}{\sqrt{\pi}a_0} \right)^{1/2} \times \exp \left[ -\frac{1}{2} \left( \frac{\Delta E_c}{\gamma_c} \right)^2 \right] \quad (13) $$

with $\Delta E_c = E - F_c\alpha$ and $\gamma_c = F_c a_0$ [8]. Here $E$ is the the energy relative to $U_c(R_0)$ as defined in the Fig. 3.

Obviously, the dependence of the Franck-Condon factors with the energy $E$ can be described with a Gaussian distribution which is due to the Gaussian nuclear wavefunction in the electronic ground state. Because of this, the behavior of the Franck-Condon factors can be obtained by approximating the Airy functions by $\delta$ functions, which peak at the classical turning point. Note that in this way the energy shift $F\alpha$ can not be reproduced. One should also mention that the approximation of the Airy function by a $\delta$ function is not obvious since the width of the first oscillation of the Airy function is comparable to the width of the nuclear ground state, see Fig. 3. However, it has been shown that the Franck-Condon factors of the exact solution and the approximation deviate only very slightly, see Herzberg [17]. The approximation of the Airy functions with $\delta$ functions allows also a derivation of the Franck-Condon factors for higher vibrational states in the bound potential, see e.g. [18].

### 3. Dissociative-dissociative-transitions

In the following we want to consider dissociative-dissociative transitions and in particular the calculation of the vibrational matrix elements $\langle\chi_f|\chi_i\rangle$ where $i$ and $f$ refer to the initial and final state of a given transition. In this case both, $|\chi_i\rangle$ and $|\chi_f\rangle$ can be described with Airy functions, which allows to use [19]

$$ \frac{1}{|\alpha\beta|} \int_{-\infty}^{\infty} A_i \left[ \frac{x+a}{\alpha} \right] A_i \left[ \frac{x+b}{\beta} \right] dx = \begin{cases} \frac{\delta(b-a)}{|\beta^3 - \alpha^3|^{1/3}} & \text{if } \alpha = \beta \\ \frac{1}{|\beta^3 - \alpha^3|^{1/3}} A_i \left[ \frac{b-a}{(\beta^3 - \alpha^3)^{1/3}} \right] & \text{if } \beta > \alpha \end{cases} \quad (14) $$

to calculate the Franck-Condon-factors for the dissociative-dissociative transitions. Note that $\alpha = \beta$ if the slopes of the two potential energy curves involved are equal. This assumption is generally assumed to be valid for RIXS spectra. In this case a nuclear wavefunction in the electronic final state is populated via one nuclear wavefunction in the initial state. As a result of a excitation and decay process the intermediate nuclear state is always exactly known so that no vibrational lifetime interference occurs.
## D. The entire excitation and decay process

### 1. The bound-bound-bound case

In the case of bound-bound-bound transitions one can start with equation 8 by taking into account that the vibrational levels are all discrete. In this case the vibrational matrix elements $\langle\chi_f|\chi_c\rangle$ and $\langle\chi_c|\chi_o\rangle$ can be calculated by the approach given in section II C 1. Details on the simulation of bound-bound-bound transitions can be found in Refs. [9, 20].

### 2. The bound-dissociative-dissociative case

In the following we will discuss the transition from a bound ground state to a dissociative core-hole state and from there to a dissociative final state. Here we have to distinguish between two cases, namely that the slopes of the two dissociative potential curves can be identical or different. The first case is realized in Resonant Inelastic X-ray Scattering (RIXS) experiments. In this case vibrational lifetime interference can be strictly ruled out because of the first line in eqn. 14. Since this case is not important for the present study it is only discussed in the Supplementary Material. The second case with different slopes is realized in the present case of the resonant Auger decay and will be discussed in the following. Here, due to the second line in eqn. 14 vibrational lifetime interference cannot generally be ruled out.

To specify the partial cross section for the given case of resonant Auger decay one has to start with eqn. 8 and can use eqn. 13 to replace $\langle\chi_0|\chi_c\rangle$. As discussed above, it was shown that for a bound-dissociative transition the Airy function can be replaced by a $\delta$ function localized at the classical turning point. We now assume for the calculation of the overlap integral $\langle\chi_c|\chi_f\rangle$ that this also works well for a dissociative-dissociative transition with different slopes for the potential energy curves, i.e.

$$|\chi_c\rangle = \delta(R - (R_0 - \frac{\Delta E_c}{F_c})) \quad \text{and} \quad |\chi_f\rangle = \delta(R - (R_0 - \frac{\Delta E_f}{F_f})), \qquad (15)$$

with $E_f$ and $\Delta E_f$ being the same quantities for the final state as $E_c$ and $\Delta E_c$, which are defined below eqn. 13. By considering also the finite lifetime of the final state we obtain as cross section

$$\sigma(\omega, \omega') \propto \int d\Delta E_f \frac{\exp\left[-\left(\frac{\Delta E_f}{\gamma_f}\right)^2\right]}{(\omega - \omega_{co}(R_0) - \frac{F_c}{F_f}\Delta E_f)^2 + \Gamma_c^2} \times \frac{\Gamma_f}{\pi((\omega - \omega' - \omega_{fo}(R_0) - \Delta E_f)^2 + \Gamma_f^2)}. \qquad (16)$$

Here $\omega_{co}(R_0)$ and $\omega_{fo}(R_0)$ represent the energy differences between the states at the equilibrium distance $R_0$. The integration over the different final states is represented by the integration $\int d\Delta E_f$. Note that eqn. 16 was already given in [4], however with the $F_c = F_f$.

### 3. The bound-dissociative-bound case

In the following we discuss the exotic case of a bound ground state, a bound final state and a dissociative intermediate state. This case is only realized in the shake-up process during the Auger decay. To describe this process we start again with equation 8 and assume a finite lifetime of the final state. Moreover, we approximate the core-hole state with $|\chi_c\rangle = \delta(R - (R_0 - \frac{\Delta E_c}{F_c}))$. In this way we obtain

$$\sigma(\omega, \omega') \propto \sum_f \int d\Delta E_c \int d\Delta \tilde{E}_c \frac{\chi_f(R_0 - \frac{\Delta E_c}{F_c})\chi_0(R_0 - \frac{\Delta E_c}{F_c})}{[\omega - \omega_{co}(R_0) - \Delta E_c + i\Gamma_c]} \times \frac{\chi_f(R_0 - \frac{\Delta \tilde{E}_c}{F_c})\chi_0(R_0 - \frac{\Delta \tilde{E}_c}{F_c})}{[\omega - \omega_{co}(R_0) - \Delta \tilde{E}_c - i\Gamma_c]} \times \frac{1}{(\omega - \omega' - \omega_{fo})^2 + \Gamma_f^2}. \qquad (17)$$

Here $\chi_0(R)$ and $\chi_f(R)$ are the vibrational wavefunctions of the bound initial and final state. The integrals $\int d\Delta E_c$ and $\int d\Delta \tilde{E}_c$ ensure that all possible intermediate core-excited states are taken into account.
### 4. The bound-bound-dissociative case

The last and also exotic case we want to discuss is the bound-bound-dissociative case. By already assuming a finite lifetime for the final state and the arguments given above we obtain

$$ \sigma(\omega, \omega') \propto \int d\Delta E_f \left| \sum_c \frac{\langle \chi_f | \chi_c \rangle \langle \chi_c | \chi_o \rangle}{\omega - \omega_{co} + i\Gamma_c} \right|^2 \times \frac{1}{(\omega - \omega' - \omega_{fo})^2 + \Gamma_f^2} \quad (18) $$

Detailed information for calculating the overlap matrix elements $\langle \chi_c | \chi_o \rangle$ and $\langle \chi_f | 0_c \rangle$ can be found in sections II C 1 and section II C 2, respectively, as well as in the Supplementary Material.

## III. THE INFLUENCE OF THE EXPERIMENTAL RESOLUTION

<figure><img src="image_5.png" /><figcaption>FIG. 4: The simulated (left) and the experimental (right) 2D-map. Above each map, the corresponding absorption spectra are given, which also allows to identify the core-hole states. For the identification of the final states of the different spectral contributions, see Figs. 6 to 10 below. Modified figure still missing. In one of the partial electron yield spectra on top of each map assignments of the spectral features will be given.</figcaption></figure>

In Figure 4 2D maps of resonant and normal Auger spectra of HCl close to the Cl 1s ionization threshold are displayed. On the left side the simulations and on the right side the experimental results are shown. On the x-axis the photon energy $\omega$ and on the y-axis the kinetic energy of the emitted electron $\omega'$ are displayed. In addition, the intensities of the emitted electrons is displayed by the color code. The ionization threshold can be found at $\approx 2829$ eV, i.e. below this photon energy resonant Auger spectra and above this value normal Auger spectra as well as electron
recapture processes can be observed. An integration along the kinetic-energy axis leads to a partial-electron-yield spectrum, which represents an absorption spectrum in this energy region. The corresponding spectra are shown above each map. A direct comparison of the experimental and the theoretical absorption spectrum is presented in the Supplementary Material. The absorption spectra allow identifying the intermediate core-hole states. The final states of the displayed transitions can be identified by comparing with Figs. 6 to 10 below.

In the maps, the normal Auger transition $Cl~1s^{-1} \rightarrow 2p^{-2}(1D)$ is represented by the almost horizontal line at 2373 eV kinetic energy. The resonant Auger transitions below threshold mostly show inclined lines, which represent resonant Auger transitions to stable or metastable final states. The strong horizontal line around a photon energy of 2824 eV and a kinetic energy of 2383 eV is due to the $Cl~1s^{-1}6\sigma^* \rightarrow 2p^{-2}(1D)6\sigma^*$. Here both states are strongly dissociative and the horizontal orientation of the spectral feature is a signature of ultrafast nuclear dynamics [29].

The spectral features can be described by the formulas given above, which, however, do not take into account the experimental resolution consisting of the photon bandwidth and the detector resolution. To obtain simulated maps that can be compared with the experimentally observed 2D maps, the theoretical maps have to be convoluted with the photon bandwidth along the photon energy axis and with the detector resolution along the electron kinetic energy.

In the following we shall discuss how these convolutions along different directions contribute to the spectra, which are present or can be derived from the 2D map. As already stated above, an integration along the kinetic-energy axis results in a photoabsorption spectrum. In this case, and because the photon bandwidth and instrumental resolution can be defined by normalized Gaussian functions, the detector resolution cancels out so that only the convolution with the photon bandwidth contribute to the absorption spectrum. Normal Auger spectra far above threshold lead to horizontal lines, i.e. do not depend on the photon energy. Because of this only the detector resolution has to be taken into account.

As discussed in Guillemin et al. [7] the situation becomes more complicated close to the ionization threshold. In this region the energy position and the lineshape are influenced by post collision interaction (PCI) and depend on the photon energy. This can be seen by the almost vertical line visible above threshold at a kinetic energy of $\approx 2373$ eV, which is slightly bent towards higher kinetic energies while approaching the threshold. As a consequence, the Auger spectra become dependent from the photon bandwidth. Therefore, in a rigorous treatment, the simulated Auger lines have to be convoluted as discussed above. For normal Auger spectra close to threshold a simplified way is described in Guillemin et al. [7]. In this case the PCI lineshape can be convoluted with the detector resolution and a modified photon bandwidth $\gamma' = |m| \times \gamma$ with $m = \frac{d\Delta E_{PCI}}{dE_{ex}}$. Here $\Delta E_{PCI}$ is the PCI-shift and $E_{ex}$ the kinetic energy of the photoelectron. Note that $m$ is usually small and the resulting resolution can be neglected.

Although not visible in the present case, an emission of a photoelectron causes a diagonal line with a slope of $m = 1$ in a 2D map. Because of this, a photoelectron line has to be convoluted with both the photon bandwidth and the detector resolution. Finally, we shall discuss the lines caused by resonant Auger transitions to different final states, which also show a dispersion with photon energy. This dispersion is linear with a slope of $m = 1$, i.e. results in diagonal lines, with the exception of dissociative final states in the vicinity of the resonance position. Although in this case the slope is identical to that for a photoemission process, the resonant Auger lines have to be treated differently because of two reasons. First, the core-hole lifetime does not contribute to the lineshape, see above, and second, the cross section varies strongly with the photon energy, see Fig. 2. Therefore, the lineshape in a resonant Auger spectrum can be obtained by multiplying the Lorentzian curve in Fig. 2 with the Gaussian distribution for the photon bandwidth; note that this can lead in the case of broad photon bandwidths to asymmetric lineshapes [22, 23]. Finally, the result has to be convoluted with the detector resolution.

# IV. RESULTS AND DISCUSSION

## A. Computation of the potential energy curves

The potential energy curves shown in Fig. 5 were computed using the MOLPRO package [24] at the CASSCF level. We chose the Gaussian basis set aug-cc-pCVTZ [25, 26], in order to take into account core-core and core-valence correlation effects, and included relativistic corrections using the Douglas-Kroll Hamiltonian [27, 28].

In the present study, the 15 lowest singly excited $Cl~1s^{-1}$ states were calculated and only those states which were observed experimentally in Ref. [12] are represented in Fig. 5. In this figure, three distinct regions corresponding to the different types of electronic states populated in the entire excitation and resonant Auger process are visible: the ground state at low energies, the $K^{-1}V$ region around 2824 eV, which corresponds to the core-excited states and the $L^{-2}V$ region around 450 eV, which corresponds to the Auger final states. In addition, the potential curve of the ionized $K^{-1}$ state is represented by the black line and the Franck-Condon (FC) region is indicated by a red-shaded area.
<figure><img src="image_6.png" /><figcaption>FIG. 5: Calculated potential energy curves for the ground state, the intermediate Cl K⁻¹V core-hole states and Cl L₂,₃(¹D)V double core-hole final states. The potential energy curve of the K⁻¹ ionic state is represented in black. The Franck-Condon region is indicated by the shaded vertical region.</figcaption></figure>

The lowest potential energy curve in each of the regions K⁻¹V and L⁻²V are highly dissociative and correspond to the states K⁻¹6σ* and L⁻²6σ*, respectively. These states have attracted substantial interest in the literature [2, 3, 12, 29] since they lead to ultrafast nuclear motion and even show indication of ultrafast dissociation.

Also of interest is the bond character of the L⁻²Ryd states where Ryd indicates a Rydberg orbital. In first approximation the bond character of these states can be estimated on the exotic Z + 2 molecule HK²⁺, which suggests a dissociative character. Nevertheless, some of these final states are meta-stable with a shallow potential well like L⁻²4sσ or L⁻²3dσ and other (L⁻²4pσ, π and L⁻²4pσ, π) are dissociative. A similar behavior was found for the O K⁻²Ryd states in CO [30], which can be compared with the molecule CNe²⁺. From these results we conclude that double-core-hole Rydberg excitations tend to be close to the borderline between meta-stable and dissociative so that each state has to be individually considered.

To obtain simple analytical expressions, which allow analytical treatments of the vibrational part of the transitions, the potential energy curves were fitted using the model potential functions as described in Gortel et al. [31]. All potential energy curves with a minimum in the Franck-Condon region were fitted to a Morse potential. Note that in particular the L⁻²V states show at large distances strong deviations from the Morse potential. Because of this, the fits of the potential were only performed close to the Franck-Condon region, which is relevant for the transitions.

The dissociative states were fitted by the exponential decay form

$$V(R) = V e^{(V'/V)(R-R_0)} + V^{\infty} \qquad (19)$$

where R₀ is the equilibrium distance of the ground state, V' = -F the slope of the dissociative potential at R₀, V∞ the asymptotic value, i.e., the binding energy of the free ion, and V the difference of the potential energy curve at R₀ and R∞, i.e. V = V(R₀) - V∞.

## B. Parameters used in the simulations

The energy positions of the various excited states relative to the ground state and the Auger energies for the transitions to the studied final states are summarized in table I. As mentioned in Ref. [12], we observe resonant Auger transitions to two groups of final states, namely the 2p⁻²(¹D)V and 2p⁻²(¹S)V. The notation corresponds to the spectroscopic term for the equivalent atomic notation with two holes in the 2p shell and one electron in a high-energy molecular orbital, which is denoted by V and is unoccupied in the ground state. The transitions to the 2p⁻²(³P)V final states are too weak to be observed.

To compare the obtained simulations better with the experimental data of the Ref. [12], we convoluted our results with two normalized Gaussian functions with a full width at half maximum (FWHM) of 0.22 eV and 0.30 eV, in order to simulate the detector resolution and photon bandwidth, respectively. The lifetime broadening (FWHM) were chosen as Γc = 0.65 eV for the 1s⁻¹V excited states [34] and Γf = 0.24 eV for the 2p⁻²V final states [2].

Finally, for our simulations, we applied for the Auger transitions to the 2p⁻²(¹D)V and the 2p⁻²(¹S)V states the same parameters, with two exceptions. First, we used different kinetic energies for the Auger electrons as stated in table I. And second, we applied a 2p⁻²(¹S)V to 2p⁻²(¹D)V intensity ratio of 0.1; this ratio has been
observed for the diagram lines in argon [33] and it is expected to vary only slightly with the atomic number Z so that the value for argon can be used as good approximation for Cl.

<table><caption>TABLE I: Summarized values of the energies of the various intermediate $1s^{-1}V$ states relative to the ground state as well as the kinetic energies for the Auger electrons of spectator decays to the the $2p^{-2}V$ final states, i.e. for example $1s^{-1}6\sigma^* \rightarrow 2p^{-2}(^1D)6\sigma^*$.</caption><thead><tr><th>Excited electronic state</th><th>Energy (eV)</th></tr></thead><tbody><tr><td>1s<sup>-1</sup>6σ<sup>*</sup></td><td>2823.90</td></tr><tr><td>1s<sup>-1</sup>4sσ</td><td>2827.05</td></tr><tr><td>1s<sup>-1</sup>4pσ</td><td>2827.75</td></tr><tr><td>1s<sup>-1</sup>5pσ</td><td>2828.90</td></tr><tr><td>1s<sup>-1</sup></td><td>IP = 2829.00</td></tr></tbody><thead><tr><th>Final electronic state</th><th>Kinetic energy for spectator decays (eV)</th></tr></thead><tbody><tr><td>2p<sup>-2</sup>(<sup>1</sup>D)6σ<sup>*</sup></td><td>2382.9</td></tr><tr><td>2p<sup>-2</sup>(<sup>1</sup>D)4sσ</td><td>2379.5</td></tr><tr><td>2p<sup>-2</sup>(<sup>1</sup>D)4pσ, π</td><td>2377.3</td></tr><tr><td>2p<sup>-2</sup>(<sup>1</sup>D)5pσ, π</td><td>2374.5</td></tr><tr><td>2p<sup>-2</sup>(<sup>1</sup>D)</td><td>2372.3</td></tr></tbody><tbody><tr><td>2p<sup>-2</sup>(<sup>1</sup>S)6σ<sup>*</sup></td><td>2374.2</td></tr><tr><td>2p<sup>-2</sup>(<sup>1</sup>S)4sσ</td><td>2370.3</td></tr><tr><td>2p<sup>-2</sup>(<sup>1</sup>S)4pσ, π</td><td>2368.1</td></tr><tr><td>2p<sup>-2</sup>(<sup>1</sup>S)5pσ, π</td><td>2365.3</td></tr><tr><td>2p<sup>-2</sup>(<sup>1</sup>S)</td><td>2363.3</td></tr></tbody></table>

### C. Details of the simulations of the bound-bound-bound, the bound-bound-dissociative and bound-dissociative-bound transitions

In the simulations, some of the excitation and decay processes were treated as bound-bound-bound transitions, see section II D 1. The parameters $R_0$, $\hbar\omega$, and $x\hbar\omega$ for the ground state of the molecule can be obtained from literature. The parameters for the core-hole state and the final state can either be derived from a fit of the photoelectron and the Auger spectrum, respectively, or from a fit of the calculated potential energy curves to the Morse formula, see section IV A The final states are not really stable but meta-stable since for large distances the Coulomb repulsion becomes important. However, if the inner well of such a potential is deep enough, a Morse potential is a good approximation, see e.g. [35, 36].

In the following we shall discuss the calculations for HCl in more detail. Here we are interested in the Cl $1s^{-1}$ photoabsorption process and the subsequent resonant Auger decay to the $2p^{-2}nl$ final states. As discussed above, for this we have to calculate in principle the overlap matrix elements $\langle\chi_f|\chi_c\rangle$ and $\langle\chi_c|\chi_o\rangle$. As can be seen from Fig. 5, the Cl $1s^{-1}$Ryd states and the Cl $1s^{-1}$ ionic state are stable according to our calculations. This is not surprising since potential energy curves are expected to be very similar to the HCl $2p$-hole states. This is due to the fact that potential energy curves are determined by the valence electrons, not by the core holes. The $2p^{-1}$ states are known to be bound also from experiment, see e.g. for photoabsorption Ref. [37] and for photoemission Ref. [38]. The spectra in the latter two references show practically only transitions to the vibrational ground state of the core-hole state, while transitions to higher vibrational substates are almost absent. From this follows that matrix element $|\langle 0_c|0_o\rangle|^2 \cong 1$ while the matrix elements $|\langle n_c|0_o\rangle|^2 \cong 0$ for $n > 0$. Because of this, only the vibrational ground states in the core-hole states are taken in our simulations into account. This also leads to the fact that vibrational lifetime interference can be neglected.

To calculate the Franck-Condon factors $|\langle\chi_f|0_c\rangle|^2$ for the resonant Auger transitions, the potential energy curves of the Cl $1s^{-1}$ core-hole and the Cl $2p^{-2}$ final states were fitted to a Morse potential. Since the Cl $1s^{-1}$Ryd states are all stable the fit is performed over the entire region of internuclear distances. Contrary to this the $2p^{-2}$Ryd are only meta-stable. Because of this we use for the fit only the region of 1 to 2 Å for the internuclear distances in order to approximate the states with a Morse potential.
In case of large changes in the internuclear distances and/or very shallow wells for the Morse potentials transitions to the continuum wavefunctions for the nuclear motion are possible, i.e. in these cases both bound-bound and bound-dissociative transitions, occur. Since for Auger final states the dissociation energy of Morse potentials is normally larger than the potential energy barrier of the potential energy curve, see [35, 36] we utilized the above-described approach only for these transitions, for which we obtained that at least 70 % of the transitions end in a bound vibrational state. All other transitions are described as with bound-dissociative transitions.

In the next step we shall discuss the bound-bound-dissociative transitions. These transitions are relevant for some of the Cl $1s^{-1}$Ryd → $2p^{-2}$Ryd transitions. Due to the fact that $|\langle 0_c|0_o \rangle|^2 \cong 1$ we can simplify eqn. 18 with eqn. 10 to

$$ \sigma(\omega, \omega') \propto \int d\Delta E_f \frac{\exp\left[-\left(\frac{\Delta E_f}{\gamma_f}\right)^2\right]}{(\omega - \omega_{co})^2 + \Gamma_c^2} \times \frac{1}{(\omega - \omega' - \omega_{fo} - \Delta E_f)^2 + \Gamma_f^2} \quad (20) $$

Now let us turn to the bound-dissociative-bound transitions, which are realized for the Cl $1s^{-1}$6σ* → $2p^{-2}$4sσ and Cl $1s^{-1}$6σ* → $2p^{-2}$3dσ shake-up transitions. For these transitions, eqn. 17 has to be applied. Here the vibrational wavefunction for the electronic ground state $\chi_0(R)$ can be described in good approximation with that of the harmonic oscillator.

## D. Results of the simulations and comparison with experiment

In the following, we will describe the results on each final state. Actually, we performed two types of simulations, namely first the simulations of the entire map, see Fig. 4, and with this the contributions of each individual final state to the map, see below. Second, we simulated final-state cross sections, which give the probability for a population of a given final state of the type $2p^{-2}V$ as a function of the photon energy. Experimental final-state cross sections can be obtained by plotting the intensities along the diagonal lines in the measured 2D-map as a function of the photon energy. Details on the analysis and simulation of final-state cross sections including electronic state lifetime interference are presented in Ref. [11, 12]. The present experimental data are identical to those presented in Goldsztejn et al. [12], however, two of the final-state cross sections, namely those for the $2p^{-2}$5pσ, π and the $2p^{-2}$3dσ are presented for the first time. In the following we will first present the results for the dissociative final states and then for the bound final states.

### 1. Dissociative final states

For resonant Auger decay after Cl 1s excitations, we simulated the cross sections for the population of three dissociative final states, namely $2p^{-2}$6σ*, $4pσ$, π and $5pσ$, π. The first two cross sections are the two most intense features in the 2D-map. The third one to the final state $2p^{-2}$5pσ, π is also clearly visible in the 2D-map. However, the spectral features are close to those of other final electronic states, which made it difficult to extract the experimental cross section accurately from the map. Nevertheless, the experimental and the simulated cross sections are in fair agreement.

#### a. Final state $2p^{-2}$6σ*

To simulate the final-state cross section of $2p^{-2}$6σ*, we considered the relaxation to this state from the $1s^{-1}$6σ* intermediate state, which corresponds to the spectator decay, shake-downs from the orbitals $4sσ$, $4pσ$, π and $5pσ$, π as well as a recapture after 1s ionization. Following Ref. [32], one can define the modulus squared of the overlap integral $\langle\Phi_f|\epsilon\rangle \approx \sqrt{1 - \exp(-\frac{\Gamma}{E_{exc}^{3/2}}\beta)}$ as a probability of recapture, where β is a slowly varying function of energy and $E_{exc}$ is the excess energy.

The result of our simulation is shown on the left side in Fig. 6 as a red solid line, together with black dots corresponding to the experimental data points. The dashed lines below the graph indicate the individual contributions from the different $1s^{-1}$V excited states.

Both the $1s^{-1}$6σ* intermediate and the $2p^{-2}$6σ* final state are dissociative so that we simulated this transition with Eqn. 16. As clearly visible in Fig. 6, this transition is highly dominant in this final-state cross section. Its broad Gaussian-like shape originates from the highly dissociative character of the potential curve of the excited state with a slope of $F_c = 9.27$ eV/Å at the ground state equilibrium distance $R_0 = 1.27$Å. The only other visible contribution from a shake-down from the $4sσ$ orbital. The $1s^{-1}$4sσ excited state is bound and we therefore used for the simulation eqn. 20, which describes bound-bound-dissociative transition.
<figure><img src="image_7.png" /><figcaption>FIG. 6: Cross section of the $2p^{-2}6\sigma^*$ (left) as a function of the photon energy, the black dots are the experimental points and the solid line is the result of our simulation. 2D-contribution of this final state alone displayed as a function of the photon energy.</figcaption></figure>

The slope of the potential energy curve of the intermediate state at the equilibrium distance is relevant to fit the final-state cross section, see above. However, to reproduce well also the resonant Auger spectra, and with this the shape in the 2D map, we had to use a slope of the potential energy curve of $F_f = 11.77$ eV/Å. This slope has been calculated for an internuclear distance of $\approx R_0 + 0.1$Å and takes into account the propagation of the wavepacket in the intermediate state prior to its projection on the final-state nuclear wavefunctions.

We show the result of this simulation also on the right side of Fig. 6, where one can observe two spectral contributions. One shows a linear dispersion with the photon energy, which is due to the kinetic energy conservation part of the Lorentzian function of the equation 16 as discussed in detail above. The other contribution displays in the resonance region a non-linear dispersion, which corresponds to strongly dissociative intermediate and final states. The exact shape of this non-linear dispersion depends on the ratio of the two potential slopes, as can be understood from the equation 16. These two contributions are also present in the experimental 2D-map, which is visible on the right side of the Fig.4.

b. Final states $2p^{-2}4p\sigma, \pi$

The left side of figure 7 displays the experimental (black data points) and simulated (red line) cross sections of the final electronic states $2p^{-2}4p\sigma, \pi$. The dashed black lines below the cross sections represent the direct terms of the dominant $1s^{-1}4p\sigma, \pi \rightarrow 2p^{-2}4p\sigma, \pi$ spectator Auger decay as well as shake transitions from other intermediate states. The intermediate $1s^{-1}4p\sigma, \pi$ electronic state is bound, i.e. the entire excitation and decay process is of the type bound-bound-dissociative and can be described with eqn. 20. Since this main decay channel implies a bound intermediate state, no nuclear dynamics can be observed in the Auger spectra and only linearly dispersive contribution are visible in the respective 2D-map, see right hand side of Fig. 7. The second clearly visible contributions to the $2p^{-2}4p\sigma, \pi$ cross section originates from a transition $1s^{-1}6\sigma^*$ intermediate state and includes a shake-up process of the excited electron. The corresponding process is of the type bound-dissociative-dissociative. Because of this, nuclear dynamics can occur in the intermediate state. This is reflected in the 2D-map of Fig. 7 in the energy range around the $1s^{-1}6\sigma^*$ resonance energy by a slope slightly different from one.

The final-state cross section also shows a small contribution from a shake-down process involving the $1s^{-1}5p\sigma, \pi$
<figure><img src="image_8.png" /><figcaption>FIG. 7: Cross section of the $2p^{-2}4p\sigma, \pi$ (left) as a function of the photon energy, the black dots are the experimental points and the solid line is the result of our simulation. 2D-contribution of this final state alone displayed as a function of the photon energy.</figcaption></figure>

electronic state. This contribution is responsible for the weakly pronounced minimum of the cross section at a photon energy of $\approx 2828.5$ eV and can also be seen in the experimental results with an increase of the intensity with the last data point. This minimum is even more clearly visible in the 2D-map where one observes an intensity “hole” slightly below the photon energy of $h\nu=2829$ eV. Finally there are two more very weak contributions of approximately the same peak intensity, namely a shake-up after the passage through the $1s^{-1}4s\sigma$ intermediate state and from a recapture of the photoelectron emitted during $1s$ ionization.

### c. Final states $2p^{-2}5p\sigma, \pi$

The cross sections of the final states $2p^{-2}5p\sigma, \pi$ are shown on the left-hand side Fig. 8 and reveal contributions from three different intermediate states. The most intense one is due to a shake-up from the $1s^{-1}4p\sigma, \pi$ intermediate electronic state. This is due to the fact that the increasing positive charge seen by the excited state orbital, namely +1 in the core-hole state and +2 in the Auger final state, leads to a shrinking of the orbitals and, therefore, to strong shake-up transitions, see e.g. Ref. [39]. The second most intense transition comes from a recapture of a photoelectron emitted during the ionization process. This recapture depends on the overlap of the continuum wavefunction of the photoelectron and the excited final-state orbital (here $5p\sigma, \pi$). The third contribution to this final-state cross sections is the spectator decay corresponding to a $1s^{-1}5p\sigma, \pi \rightarrow 2p^{-2}5p\sigma, \pi$ transition. All of these transitions are bound-bound-dissociative.

## 2. Bound final states

In the present study, we simulated the transitions to the two bound final states $2p^{-2}4s\sigma$ and $2p^{-2}3d\sigma$. The resulting spectral features will be discussed in the following.

### a. Final state $2p^{-2}4s\sigma$
<figure><img src="image_9.png" /><figcaption>FIG. 8: Cross section of the $2p^{-2}5p\sigma, \pi$ (left) as a function of the photon energy, the black dots are the experimental points and the solid line is the result of our simulation. 2D-contribution of this final state alone displayed as a function of the photon energy.</figcaption></figure>

On the left side of Fig. 9, we represented in black the experimental data points and as red solid lines the simulated cross sections of the $2p^{-2}4s\sigma$ final state, along with the direct terms contributing to the cross section, which are indicated by dashed black lines. In the final-state cross section three main contributions are visible, namely the spectator decay of the $1s^{-1}4s\sigma$ state, a shake-down process from the $1s^{-1}4p\sigma$ state and a shake-up process after excitation to the $1s^{-1}6\sigma^*$ excited state. In the first two cases, the excited states are bound so that the entire process can be considered bound-bound-bound as described in section II D 1. The situation is more complex for the case of the shake-up from the LUMO orbital that involves a dissociative intermediate state.

As described in the section II D 3, to calculate the overlap matrix element for the nuclear wavefunctions of the dissociative intermediate and the bound final state, generalized Laguerre polynomials have to be applied. Therefore, the cross section has to be calculated in a time-consuming manner numerically by applying a double integration over the vibrational states populated. Because of this we used a simplified approach of eqn. 17 by making two approximations. First, we neglected vibrational lifetime interference by using only the direct terms and second, we assumed a harmonic oscillator potential for the final state and populated only the vibrational ground state. Actually, the final state has several vibrational levels and we shifted the energy position to the center of the positions of the first five vibrational levels. In this way we obtained the simplified cross section

$$d\sigma(\omega, \omega') \propto \int_{\Delta E_c} d\Delta E_c \frac{\exp\left[-\frac{1}{2a_0^2}\left(\frac{\Delta E_c}{F_c}\right)^2 - \frac{1}{2a_f^2}\left(\frac{\Delta E_c}{F_c} + \epsilon\right)^2\right]^2}{(\omega - \omega_{co} - \Delta E_c)^2 + \Gamma_c^2} \times \frac{1}{(\omega - \omega' - \omega_{fo})^2 + \Gamma_f^2/4} \quad (21)$$

where $\epsilon$ is the difference between the equilibrium distances $R_0$ and $R_f$. This simplified equation works rather well. However, the simulated cross section is narrower than the experimental one and does not reproduce the asymmetrical shape of the cross section, see final-state cross section below $h\nu = 2824$ eV on the left hand side of the Fig. 9. We assume that these differences between experiment and theory are due to this simplification, since a restriction to the lowest vibrational level of the final state is expected to lead to a peak narrowing. The other energy regions of the simulated final-state cross sections are in good agreement with the experiment. In the 2D-map visible on the right hand side of Fig. 9, we observe the expected linear dispersion with the photon energy of the $2p^{-2}4s\sigma$ final state. There is a very slight discrepancy to this dispersion across the $1s^{-1}6\sigma^*$ resonance, due to the dissociative character of its
<figure><img src="image_10.png" /><figcaption>FIG. 9: Cross section of the $2p^{-2}4s\sigma$ (left) as a function of the photon energy, the black dots are the experimental points and the solid line is the result of our simulation. 2D-contribution of this final state alone displayed as a function of the photon energy.</figcaption></figure>

potential energy curve. The most intense contribution in the final-state cross sections originates from the spectator decay, followed by the shake-down process from the $4p\sigma, \pi$ orbital and the shake-up one from the $6\sigma^*$ orbital, respectively.

b. Final state $2p^{-2}3d\sigma$

The experimental and simulated cross sections of the $2p^{-2}3d\sigma$ final state are shown on the left-hand side of Fig. 10 as black dots and red line, respectively. Interestingly, this final state can only be accessed through shake processes but not by a spectator decay via the $1s^{-1}3d\sigma$ state since such excitations are not present in the absorption spectrum; this excitation is expected to be located approximately 1.5 eV above the $1s^{-1}4s\sigma$ state [41], but no spectral features are visible in this region of the cross section. This observation may be explained as follows: in the core-excited state the $3d\sigma$ Rydberg orbital is still rather atomic-like so that the dipole selection rules for atoms explain the absence of this excitation in the absorption spectrum. In the final state the $3d\sigma$ orbital becomes smaller due to the higher charge caused by the two core holes. In this way it obtains a stronger valence character and mixes with other orbitals of the same symmetry. In this way the monopole selection rules for shake transitions can explain the population of this final state.

The main contributions to this final state are indicated by dashed vertical lines below the cross section, see Fig. 10). Two of the contributions are caused by shake-up processes, namely from the $1s^{-1}6\sigma^*$ and $1s^{-1}4s\sigma$ excited states, and a two more by a shake-down process from the $1s^{-1}5p\sigma$ excited state and a recapture from the singly ionized state. Due to its weak intensity and its proximity to other final states in the 2D-map, see Fig. 4, it was very challenging to extract the experimental cross section from the map. However, we believe that the overall agreement between the experimental and the simulated cross section is fairly good. The largest discrepancies occur at $\approx 2828$ eV, which is the region with the strongest overlap with the more intense transitions to the final state $2p^{-2}5p\sigma, \pi$, see Fig. 4, so that the extraction of the $2p^{-2}3d\sigma$ cross section was very difficult and not highly accurate.

The two most intense contributions to this final state cross section are due to the shake processes from the $1s^{-1}5p\sigma, \pi$ and $1s^{-1}4s\sigma$ intermediate states. Both these intermediate states are bound and therefore the entire process is described as a bound-bound-bound one. For the contributions via the $1s^{-1}6\sigma^*$ intermediate state we applied again equation 21 since the entire process represents a bound-dissociative-bound case. Finally, although the
<figure><img src="image_11.png" /><figcaption>FIG. 10: Cross section of the $2p^{-2}3d\sigma$ (left) as a function of the photon energy, the black dots are the experimental points and the solid line is the result of our simulation. 2D-contribution of this final state alone are displayed as a function of the photon energy.</figcaption></figure>

intensity of the direct term involving a recapture of an emitted photoelectron onto the $3d\sigma$ orbital is not strong, its interference terms, in particular with the terms involving the $4s\sigma$ and $5p\sigma, \pi$ orbitals are very important to lower the intensity of the simulated curve in the energy region between the $1s^{-1}4s\sigma$ and the $1s^{-1}5p\sigma, \pi$.

### 3. Resonant and normal Auger spectra

As already mentioned, the strength of this 2-dimensional simulation allows not only to have access to the cross sections of resonant and normal Auger features, but to their spectra as well. This has already proven important to assess in case of bound-dissociative-dissociative processes the slope of the final state, as already mentioned above. In turn, it will be essential to derive fine effects on the spectra and extract the utterly important parameters such as lifetime broadening, slopes of potential curves or to characterize the resolutions for instance. We show two examples of these simulations in the Fig. 11, namely on the left-hand side the resonant Auger spectrum recorded at the resonance energy of the $1s^{-1}6\sigma^*$ transition and on the right hand side a normal Auger spectrum using a photon energy of three eV higher than the ionization potential. In both spectra, the black dots represent the experimental and the red solid lines the simulated spectra; both spectra are normalized to 1.

## V. SUMMARY AND CONCLUSIONS

In previous studies, experimental 2D map consisting of resonant Auger spectra obtained at different photon energies were mostly investigated by considering only partial aspects. In this context, often the core hole to LUMO excitations are in the focus of publications since they allow to investigate ultrafast nuclear motion along dissociative potential energy curves. In contrast to this, in the present work we analyzed the 2D-map as a whole and paved the road towards a comprehensive understanding of such 2D maps. For this purpose, we compiled the present understanding of all theoretical aspects that are necessary to compendium and simulate an experimental 2D map consisting of resonant
<figure><img src="image_12.png" /><figcaption>FIG. 11: Left: Resonant Auger spectrum recorded at the resonance energy of the $1s^{-1}6\sigma^*$ transition. The black dots represent the experimental data and the red solid line the simulation. Right: Experimental normal Auger spectrum measured at $\hbar\omega = IP + 3$ eV (black dots) along with the corresponding simulated spectrum (red line). In both cases, the spectra were normalized to one.</figcaption></figure>

Auger spectra obtained at different photon energies.

We applied the presented theory to simulate the 2D map of HCl close to the Cl $1s^{-1}$ ionization threshold and found good agreement with experimental data. Such an approach allows to separate overlapping spectral contributions and gives, therefore, deeper understanding of the electronic and nuclear dynamics reflected in the maps. In the present case we found generally good agreement between simulations and experiment. However, we also found disagreement in some minor points, which indicates that the spectrum is not yet understood in full detail. These differences might be due to simplifications in the applied theoretical framework. So for example, in the Auger case, i.e. in bound-dissociative-dissociative processes with different slopes for the intermediate and the final state, the contributions of vibrational lifetime interference was neglected. Since such contributions cannot strictly be ruled out it might be necessary to investigate possible influences of vibrational lifetime interference in more detail. However, the differences might also be due to effects, which are not yet known to be present in such 2D maps. But these topics are beyond the scope of the present work.

The next natural step would be an implementation of the theoretical framework in a 2D-fit approach. This would allow to investigate regions with strongly overlapping contributions, like in the present case close to threshold, or for other molecules with more complex electronic structures. In the near future we also expect 2D-maps with significantly higher experimental resolution since new high-resolution monochromators in the HAXPES regime are now available, providing a photon energy resolutions of $E/\Delta E$ of almost 50.000 at 7 keV [40]. 2D-Maps measured with such high experimental resolution will reveal in particular in the resonant Auger region much finer details and will provide a benchmark for theoretical descriptions.

In summary, we believe that the present approach is an important step towards a more effective data analysis of 2D maps of Auger spectra close to deep core holes, both for atoms and molecules.

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