# Anisotropic molecular photoemission dynamics: Interpreting and accounting for the nuclear motion

Antoine Desrier, Morgan Berkane, Camille Lévêque, Richard Taïeb, Jérémie Caillat

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Antoine Desrier, Morgan Berkane, Camille Lévêque, Richard Taïeb, Jérémie Caillat. Anisotropic molecular photoemission dynamics: Interpreting and accounting for the nuclear motion. Physical Review A, 2024, 109 (5), pp.053106. 10.1103/PhysRevA.109.053106. hal-04607319

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# Anisotropic molecular photoemission dynamics: Interpreting and accounting for the nuclear motion

Antoine Desrier, Morgan Berkane, Camille Lévêque, Richard Taïeb, Jérémie Caillat

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

To cite this version:

Antoine Desrier, Morgan Berkane, Camille Lévêque, Richard Taïeb, Jérémie Caillat. Anisotropic molecular photoemission dynamics: Interpreting and accounting for the nuclear motion. Physical Review A, 2024, 109 (5), pp.053106. 10.1103/PhysRevA.109.053106. hal-04607319

HAL Id: hal-04607319

https://hal.science/hal-04607319v1

Submitted on 10 Jun 2024

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_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Anisotropic molecular photoemission dynamics: Interpreting and accounting for the nuclear motion

2

3

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

5

(Dated: April 21, 2024)

We investigate how vibration affects molecular photoemission dynamics, through simulations on two-dimension asymmetric model molecules including the electronic and nuclear motions in a fully correlated way. We show that a slight anisotropy in the electron-ion momentum sharing is sufficient to prevent one from unambiguously characterizing the vibrationally averaged photoemission dynamics in terms of stereo Wigner delays. We further show that vibrational resolution can be retrieved in fixed-nuclei simulations, using effective molecular conformations that are specific to each vibrational channel. The optimal internuclear distances found empirically in 1-photon processes can be identified a priori using simple physical arguments. They also turn out to be efficient to simulate vibrationally-resolved RABBIT measurements and to account for interchannel coherences in broadband 1-photon ionization.

6

## I. INTRODUCTION

From its early emergence as an applied branch of at- 7 tosecond sciences to its most recent and ongoing de- 8 velopments, attochemistry offers and envisions unprece- 9 dented ways of exploring fundamental dynamics occur- 10 ring in molecules on their natural time scale [1–7]. In the 11 broader perspective of attosecond time-resolved spectro- 12 scopies, photoemission takes a singular role, either as a 13 characterization tool [8, 9] or as the probed process it- 14 self [10–12].

Revisiting a process as essential as molecular photoem- 16 mission in the time domain is made possible only through 17 intertwined, long term interactions between theory and 18 experiments, that tackle and exploit the complexities and 19 richness specific to molecular systems both in the con- 20 ception and in the interpretation of simulations or mea- 21 surements. It was pioneered by investigating experimen- 22 tally the intricate dynamics of resonant ionization in N₂ 23 with vibrational resolution [13], using the ‘reconstruction 24 of attosecond beatings by interferences of two-photon 25 transitions’ (RABBIT) scheme [8, 14]. The interpreta- 26 tion of the measured channel-selective RABBIT phases 27 in terms of transition delays was subsequently clari- 28 fied by means of simulations on low-dimensional model 29 molecules [15, 16]. The experimental results were fur- 30 thermore reproduced and expanded recently in elaborate 31 simulations aiming at a quantitative agreement [17]. The 32 anisotropy of ionization dynamics, which is another es- 33 sential feature of molecular photoemission, was consid- 34 ered first in simulations of attosecond ‘streaking’ mea- 35 surements [9, 18] on a model CO molecule [19] and then 36 explored experimentally, also on CO, using RABBIT [20]. 37 38 Further experimental and theoretical investigations ad- 39 dressing anisotropy have since then been reported on sys- 40 tems of increasing complexities [21–26], see also [27] and 41 references therein.

The imprints of nuclear motion on photemission dy- 42 namics in polyatomic molecules is the subject of an in- 43 creasing number of theoretical and experimental inves- 44 tigations. Among the most recent studies, the authors 45 of Ref. [28] investigate theoretically how nuclear motion 46 affects photoelectron spectra in the context of RABBIT 47 measurements. In this context, they establish an expres- 48 sion of molecular RABBIT spectra in terms of a con- 49 volution between a nuclear autocorrelation function and 50 two-photon electronic transition matrix elements, follow- 51 ing an approach initially derived to simulate molecular 52 streaking [29]. In [30], the imprints of nuclear motion 53 on attosecond photoemission delays is investigated ex- 54 perimentally on the methane molecule and its deutered 55 counterpart, using RABBIT and an advanced theoreti- 56 cal support. The authors find no significant isotopic ef- 57 fect, but a ~ 20 as difference between dissociative and 58 non-dissociative channels. A closely related study on the 59 same polyatomic molecules further highlighted how nu- 60 clear motion impacts the coherence of the photoemission 61 process [31], which is an essential issue for most of the in- 62 terferometric attosecond resolved pump-probe schemes. 63

Nonetheless, investigations on smaller – diatomic – 64 benchmark molecules [32–44] remain of great importance 65 to gradually and comprehensively explore the interplay of 66 the molecular degrees of freedom from a time-dependent 67 perspective. On the theory side, simulations on simple 68 models with limited degrees of freedom [45] are essential 69 to guide and interpret experiments and elaborate simula- 70 tions in terms of intuitive pictures and practical notions. 71

In the present article, we investigate numerically 72 how vibration imprints the attosecond dynamics of 73 anisotropic molecular photoemission even when the vi- 74 bronic couplings are minimal – notably far from any vi- 75 bronic resonance<fnref n="1" />. Our study is based on low dimen- 76 sional model molecules allowing for extensive numerically 77

<footnote>* jeremie.caillat@sorbonne-universite.fr</footnote>

<footnote n="1">¹ It comes as a companion paper to Ref. [27], which is focused on the definition and RABBIT measurement of anisotropic molecular photoemission delays, regardless of the nuclear motion.</footnote>
# Anisotropic molecular photoemission dynamics: Interpreting and accounting for the nuclear motion

2

3

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

5

(Dated: April 21, 2024)

We investigate how vibration affects molecular photoemission dynamics, through simulations on two-dimension asymmetric model molecules including the electronic and nuclear motions in a fully correlated way. We show that a slight anisotropy in the electron-ion momentum sharing is sufficient to prevent one from unambiguously characterizing the vibrationally averaged photoemission dynamics in terms of stereo Wigner delays. We further show that vibrational resolution can be retrieved in fixed-nuclei simulations, using effective molecular conformations that are specific to each vibrational channel. The optimal internuclear distances found empirically in 1-photon processes can be identified a priori using simple physical arguments. They also turn out to be efficient to simulate vibrationally-resolved RABBIT measurements and to account for interchannel coherences in broadband 1-photon ionization.

6

## I. INTRODUCTION

From its early emergence as an applied branch of at- tosecond sciences to its most recent and ongoing de- velopments, attochemistry offers and envisions unprece- dented ways of exploring fundamental dynamics occur- ring in molecules on their natural time scale [1–7]. In the broader perspective of attosecond time-resolved spectro- scopies, photoemission takes a singular role, either as a characterization tool [8, 9] or as the probed process it- self [10–12].

Revisiting a process as essential as molecular photoe- mission in the time domain is made possible only through intertwined, long term interactions between theory and experiments, that tackle and exploit the complexities and richness specific to molecular systems both in the con- ception and in the interpretation of simulations or mea- surements. It was pioneered by investigating experimen- tally the intricate dynamics of resonant ionization in N₂ with vibrational resolution [13], using the ‘reconstruction of attosecond beatings by interferences of two-photon transitions’ (RABBIT) scheme [8, 14]. The interpreta- tion of the measured channel-selective RABBIT phases in terms of transition delays was subsequently clari- fied by means of simulations on low-dimensional model molecules [15, 16]. The experimental results were fur- thermore reproduced and expanded recently in elaborate simulations aiming at a quantitative agreement [17]. The anisotropy of ionization dynamics, which is another es- sential feature of molecular photoemission, was consid- ered first in simulations of attosecond ‘streaking’ mea- surements [9, 18] on a model CO molecule [19] and then explored experimentally, also on CO, using RABBIT [20]. Further experimental and theoretical investigations ad- dressing anisotropy have since then been reported on sys- tems of increasing complexities [21–26], see also [27] and references therein.

The imprints of nuclear motion on photemission dy- namics in polyatomic molecules is the subject of an in- creasing number of theoretical and experimental inves- tigations. Among the most recent studies, the authors of Ref. [28] investigate theoretically how nuclear motion affects photoelectron spectra in the context of RABBIT measurements. In this context, they establish an expres- sion of molecular RABBIT spectra in terms of a con- volution between a nuclear autocorrelation function and two-photon electronic transition matrix elements, follow- ing an approach initially derived to simulate molecular streaking [29]. In [30], the imprints of nuclear motion on attosecond photoemission delays is investigated ex- perimentally on the methane molecule and its deutered counterpart, using RABBIT and an advanced theoreti- cal support. The authors find no significant isotopic ef- fect, but a ~20 as difference between dissociative and non-dissociative channels. A closely related study on the same polyatomic molecules further highlighted how nu- clear motion impacts the coherence of the photoemission process [31], which is an essential issue for most of the in- terferometric attosecond resolved pump-probe schemes.

Nonetheless, investigations on smaller – diatomic – benchmark molecules [32–44] remain of great importance to gradually and comprehensively explore the interplay of the molecular degrees of freedom from a time-dependent perspective. On the theory side, simulations on simple models with limited degrees of freedom [45] are essential to guide and interpret experiments and elaborate simula- tions in terms of intuitive pictures and practical notions.

In the present article, we investigate numerically how vibration imprints the attosecond dynamics of anisotropic molecular photoemission even when the vi- bronic couplings are minimal – notably far from any vi- bronic resonance<fnref n="1" />. Our study is based on low dimen- sional model molecules allowing for extensive numerically

<footnote>* jeremie.caillat@sorbonne-universite.fr</footnote>

<footnote n="1">¹ It comes as a companion paper to Ref. [27], which is focused on the definition and RABBIT measurement of anisotropic molecular photoemission delays, regardless of the nuclear motion.</footnote>
<figure><img src="image_2.png" /><figcaption>FIG. 1. (Color online) Born-Oppenheimer energies of the model molecules A [frame (a)], B [frame (b)] and C [frame (c)] designed for the simulations, as a function of the internuclear distance R. The energy scale refers here to the overall ground states of the neutral molecules. For each molecule, the full grey and black curves correspond to the neutral and ionic electronic ground states, respectively. In frames (b) and (c), the dashed grey curve is an image of the neutral ground curve vertically shifted towards the ionization threshold, displayed for comparison purpose. The vertical dotted lines indicate the equilibrium distance of the neutral R<sub>eq</sub>. The vibrational levels v presenting a significant overlap with the ground state [|F(v)|<sup>2</sup> > 10<sup>-2</sup>, see Eq. (21) and Fig. 7] are indicated in the ionic curve of each molecule.</figcaption></figure>

78 exact simulations. The objectives are two-fold: i) to 79 study the interplay of molecular asymmetry and nuclear 80 motion on attosecond time-resolved photoemission, and 81 ii) to identify a way to account for the vibrational resolu- 82 tion in standard theoretical approaches with fixed nuclei 83 – keeping in mind that only the latter can be routinely 84 applied to the simulations of more realistic models.

85 The paper is organized as follows. Section II presents 86 the model molecules and the overall methodology fol- 87 lowed in our study. In section III, we benchmark the 88 signatures of molecular asymmetry on 1-photon ioniza- 89 tion dynamics with a model displaying no effective nu- 90 clear motion during and upon photoemission. It partly 91 reproduces and expands the results presented and com- 92 mented in [27]. In section IV, we consider photoemis- 93 sion occurring with sensible nuclear motion. We ques- 94 tion the relevance of defining an orientation-resolved pho- 95 toemission delay for the photoelectron wave packet av- 96 eraged over the open vibrational channels. Then we 97 consider vibrationally-resolved photoemission dynamics, 98 and investigate the possibility to reproduce it in a fixed- 99 nuclei approach. In Section IV C we address the capac- 100 ities of the fixed nuclei approach to reproduce coherent, 101 vibrationally-resolved, photoemission beyond the context 102 in which it was designed. The conclusions are presented 103 in section V.

104 Unless stated otherwise, the equations are displayed in 105 atomic units (a.u.) all through the paper.

## II. THEORETICAL TOOLBOX

### A. Model molecules

108 Our simulations were performed on single-active elec- 109 tron diatomic model molecules including correlated elec-

110 tronic and internuclear motions each in 1D. They are 111 similar to the ones used eg in [15, 46, 47].

#### 1. Generic hamiltonian

113 The generic field-free hamiltonian of the model 114 molecules reads

$$H_0 = -\frac{1}{2\mu} \underbrace{\frac{\partial^2}{\partial R^2}}_{H_N} + V_{\text{N-N}}(R) - \frac{1}{2} \underbrace{\frac{\partial^2}{\partial x^2}}_{H_e} + V_{\text{N-e}}(x, R) \quad (1)$$

115 where the two coordinates x and R are the electron posi- 116 tion and the internuclear distance, respectively, µ is the 117 nuclei reduced mass, V<sub>N-N</sub>(R) is the interatomic potential 118 in absence of the active electron, and V<sub>N-e</sub>(x, R) is the 119 interaction potential between the active electron and the 120 molecular ion.

121 The interatomic potential V<sub>N-N</sub> for each molecule is 122 defined numerically on a discretized R-grid, while the 123 electron-nuclei potential V<sub>N-e</sub> is defined as an asymmetric 124 two-center soft-Coulomb potential,

$$V_{\text{N-e}}(x, R) = \sum_{j=1,2} -\frac{q_j}{\sqrt{(x - X_j)^2 + a(R)}} \quad (2)$$

125 where

$$X_j = (-1)^j \frac{\mu}{M_j} R \quad (3)$$

126 are the positions of each nucleus (j = 1, 2), of mass M<sub>j</sub>, 127 with respect to their center of mass. The V<sub>N-e</sub> potential 128 is adjustable through the effective charges q<sub>j</sub> > 0 (with 129 the constraint q<sub>1</sub> + q<sub>2</sub> = 1) and the screening parameter 130 a(R) > 0 defined numerically over the same R-grid as 131 V<sub>N-N</sub>. One should note that V<sub>N-e</sub> accounts not only for
<figure><img src="image_3.png" /><figcaption>FIG. 1. (Color online) Born-Oppenheimer energies of the model molecules A [frame (a)], B [frame (b)] and C [frame (c)] designed for the simulations, as a function of the internuclear distance R. The energy scale refers here to the overall ground states of the neutral molecules. For each molecule, the full grey and black curves correspond to the neutral and ionic electronic ground states, respectively. In frames (b) and (c), the dashed grey curve is an image of the neutral ground curve vertically shifted towards the ionization threshold, displayed for comparison purpose. The vertical dotted lines indicate the equilibrium distance of the neutral R<sub>eq</sub>. The vibrational levels v presenting a significant overlap with the ground state [|F(v)|<sup>2</sup> > 10<sup>-2</sup>, see Eq. (21) and Fig. 7] are indicated in the ionic curve of each molecule.</figcaption></figure>

78 exact simulations. The objectives are two-fold: i) to 79 study the interplay of molecular asymmetry and nuclear 80 motion on attosecond time-resolved photoemission, and 81 ii) to identify a way to account for the vibrational resolu- 82 tion in standard theoretical approaches with fixed nuclei 83 – keeping in mind that only the latter can be routinely 84 applied to the simulations of more realistic models.

85 The paper is organized as follows. Section II presents 86 the model molecules and the overall methodology fol- 87 lowed in our study. In section III, we benchmark the 88 signatures of molecular asymmetry on 1-photon ioniza- 89 tion dynamics with a model displaying no effective nu- 90 clear motion during and upon photoemission. It partly 91 reproduces and expands the results presented and com- 92 mented in [27]. In section IV, we consider photoemis- 93 sion occurring with sensible nuclear motion. We ques- 94 tion the relevance of defining an orientation-resolved pho- 95 toemission delay for the photoelectron wave packet av- 96 eraged over the open vibrational channels. Then we 97 consider vibrationally-resolved photoemission dynamics, 98 and investigate the possibility to reproduce it in a fixed- 99 nuclei approach. In Section IV C we address the capac- 100 ities of the fixed nuclei approach to reproduce coherent, 101 vibrationally-resolved, photoemission beyond the context 102 in which it was designed. The conclusions are presented 103 in section V.

104 Unless stated otherwise, the equations are displayed in 105 atomic units (a.u.) all through the paper.

## II. THEORETICAL TOOLBOX

### A. Model molecules

108 Our simulations were performed on single-active elec- 109 tron diatomic model molecules including correlated elec-

110 tronic and internuclear motions each in 1D. They are 111 similar to the ones used eg in [15, 46, 47].

### 1. Generic hamiltonian

113 The generic field-free hamiltonian of the model 114 molecules reads

$$H_0 = -\frac{1}{2\mu} \underbrace{\frac{\partial^2}{\partial R^2}}_{H_N} + V_{N-N}(R) - \frac{1}{2} \underbrace{\frac{\partial^2}{\partial x^2}}_{H_e} + V_{N-e}(x, R) \quad (1)$$

115 where the two coordinates x and R are the electron posi- 116 tion and the internuclear distance, respectively, µ is the 117 nuclei reduced mass, V<sub>N-N</sub>(R) is the interatomic potential 118 in absence of the active electron, and V<sub>N-e</sub>(x, R) is the 119 interaction potential between the active electron and the 120 molecular ion.

121 The interatomic potential V<sub>N-N</sub> for each molecule is 122 defined numerically on a discretized R-grid, while the 123 electron-nuclei potential V<sub>N-e</sub> is defined as an asymmetric 124 two-center soft-Coulomb potential,

$$V_{N-e}(x, R) = \sum_{j=1,2} -\frac{q_j}{\sqrt{(x - X_j)^2 + a(R)}} \quad (2)$$

125 where

$$X_j = (-1)^j \frac{\mu}{M_j} R \quad (3)$$

126 are the positions of each nucleus (j = 1, 2), of mass M<sub>j</sub>, 127 with respect to their center of mass. The V<sub>N-e</sub> potential 128 is adjustable through the effective charges q<sub>j</sub> > 0 (with 129 the constraint q<sub>1</sub> + q<sub>2</sub> = 1) and the screening parameter 130 a(R) > 0 defined numerically over the same R-grid as 131 V<sub>N-N</sub>. One should note that V<sub>N-e</sub> accounts not only for
the asymmetry related to the electronegativity difference (through $q_j$) but also for the mass asymmetry (through $X_j$).

In practice, we adjusted $V_{N-N}$ and $V_{N-e}$ empirically [45] for the energies of the model molecule to match some desired energy curves within the Born-Oppenheimer (BO) approximation: $V_{N-N}$ is directly the molecular ion energy curve, while the negative eigen-energies of $H_e + V_{N-N}$ computed at each $R$ [see partition of $H_0$ in Eq. (1)] provide the electronic ground and excited curves of the neutral. Within the BO framework, the ground state energy $\varepsilon_0(R)$ of the electronic hamiltonian $H_e$ is the opposite of the vertical ionization potential for each $R$,

$$ \tilde{E}_I(R) = -\varepsilon_0(R), \quad (4) $$

i.e., the energy gap between the ionic and the neutral ground state curves.

## 2. Model parameters

We considered three model molecules, hereafter referred to as A, B and C for simplicity. For each of them, the effective charges were set to $q_1 = 0.33$ a.u. and $q_2 = 0.67$ a.u, and the nuclei masses to $M_1 = 20.53$ u and $M_2 = 7.47$ u. The chosen masses are such that $M_1 + M_2 = 28$ u, i.e., the mass of CO – a prototypical hereronuclear diatomics on which orientation- and time-resolved photoemission simulations [19] and experiments [20] have already been performed. However, our models are only loosely based on CO, as we, among other liberties, increased the mass ratio $M_2/M_1$ in order to emphasize the consequences of mass asymmetry on photoemission dynamics. The potentials were adjusted to obtain the BO curves displayed in Fig. 1. The three molecular systems share the same ground state energy curve, which mimics the one of CO [48], with an equilibrium distance $R_{eq} = 1.115$ Å.

In molecule A, the ionic curve is a vertically shifted image of the ground state. In molecule B (resp. C), it is also a replica of the ground state curve, translated towards smaller (resp. larger) values of $R$ with its minimum at 1.029 Å (resp. 1.201 Å). The energy shifts of the ionic curves were adjusted to achieve $\tilde{E}_I(R_{eq}) \approx 30$ eV for all molecules (29.81 eV, 29.18 eV and 29.30 eV for molecules A, B and C, respectively). As an illustration, the x-dependency of the electron-core potential for molecule A at the equilibrium distance $R = R_{eq}$ can be found in Fig. 4(a).

Due to the relative positions of these energy curves, photoionization takes place with no effective nuclear motion in molecule A, while it initiates a bond contraction in molecule B and an elongation in molecule C.

## B. Methodology overview

We considered two different approaches for the simulations. The first one consists in simulating the ‘complete’ vibronic dynamics of the model molecules. To this end, we performed numerically converged, vibrationally correlated, simulations based on solving the time-dependent Schrödinger equation (TDSE). The second approach consists in adopting a standard simplified treatment with fixed nuclei, based on the Born-Oppenheimer (BO) approximation, where the photoemission dynamics are encoded in the continuum solutions of the electronic time-independent Schrödinger equation (TISE).

Our motivation is two-fold. On the one hand, the ‘complete’ vibronic simulations highlight the impact of vibronic couplings on the dynamics of molecular photoemission investigated with attosecond resolution. On the other hand, by confronting these ‘exact’ dynamics to the ones obtained in the BO framework, we assess the relevance of fixed-nuclei approaches to investigate molecular photoemission process (see eg [19, 23, 49–52]), when the latter are performed with well-chosen molecular conformations. In all the simulations, the $x$ origin was set a priori to coincide with the center of mass of the molecule, consistently with the definition of the $V_{N-e}$ potential [see Eqs. (2) and (3)].

The central quantities used in this work to characterize anisotropic photoionization dynamics are orientation-resolved yields and stereo ionization delays, which can be computed indifferently in both approaches.

### 1. ‘Complete’ simulations

The complete dynamics of the molecules were simulated by solving numerically the vibronic TDSE

$$ i \frac{\partial \Phi(x, R, t)}{\partial t} = [H_0 + W(t)] \Phi(x, R, t). \quad (5) $$

Here $\Phi(x, R, t)$ is the propagated vibronic wave function and $W(t)$ represents the dipole interaction of the molecule with the ionizing pulse, implemented in the velocity gauge. The initial state is the ground vibronic state $\Phi_0(x, R)$, obtained by imaginary time propagation, in all the simulations. Solving numerically the TDSE [Eq. (5)] was performed using a split operator propagation based on a grid representation of the $x$ coordinate, combined with an expansion over a set of eigenvectors of $H_N$ for the $R$ coordinate, as detailed in [15, 45]. The vector potentials of the ionizing pulses were assigned $\sin^2$ temporal envelopes with central photon energies $\omega_{xuv}$ corresponding to harmonics of a Ti:Sapphire laser ($\lambda_0 = 800$ nm, $\hbar\omega_0 = 1.55$ eV) in the extreme ultraviolet (XUV). Case specific pulse parameters are indicated further in the text. Simulations on molecule A were performed by expanding the wave function on a single vibrational state ($v = 0$ is the only open channel). For molecules B and C,
the asymmetry related to the electronegativity difference (through $q_j$) but also for the mass asymmetry (through $X_j$).

In practice, we adjusted $V_{N-N}$ and $V_{N-e}$ empirically [45] for the energies of the model molecule to match some desired energy curves within the Born-Oppenheimer (BO) approximation: $V_{N-N}$ is directly the molecular ion energy curve, while the negative eigen-energies of $H_e + V_{N-N}$ computed at each $R$ [see partition of $H_0$ in Eq. (1)] provide the electronic ground and excited curves of the neutral. Within the BO framework, the ground state energy $\varepsilon_0(R)$ of the electronic hamiltonian $H_e$ is the opposite of the vertical ionization potential for each $R$,

$$ \tilde{E}_I(R) = -\varepsilon_0(R), \quad (4) $$

i.e., the energy gap between the ionic and the neutral ground state curves.

## 2. Model parameters

We considered three model molecules, hereafter referred to as A, B and C for simplicity. For each of them, the effective charges were set to $q_1 = 0.33$ a.u. and $q_2 = 0.67$ a.u, and the nuclei masses to $M_1 = 20.53$ u and $M_2 = 7.47$ u. The chosen masses are such that $M_1 + M_2 = 28$ u, i.e., the mass of CO – a prototypical hereronuclear diatomics on which orientation- and time-resolved photoemission simulations [19] and experiments [20] have already been performed. However, our models are only loosely based on CO, as we, among other liberties, increased the mass ratio $M_2/M_1$ in order to emphasize the consequences of mass asymmetry on photoemission dynamics. The potentials were adjusted to obtain the BO curves displayed in Fig. 1. The three molecular systems share the same ground state energy curve, which mimics the one of CO [48], with an equilibrium distance $R_{eq} = 1.115$ Å.

In molecule A, the ionic curve is a vertically shifted image of the ground state. In molecule B (resp. C), it is also a replica of the ground state curve, translated towards smaller (resp. larger) values of $R$ with its minimum at 1.029 Å (resp. 1.201 Å). The energy shifts of the ionic curves were adjusted to achieve $\tilde{E}_I(R_{eq}) \approx 30$ eV for all molecules (29.81 eV, 29.18 eV and 29.30 eV for molecules A, B and C, respectively). As an illustration, the x-dependency of the electron-core potential for molecule A at the equilibrium distance $R = R_{eq}$ can be found in Fig. 4(a).

Due to the relative positions of these energy curves, photoionization takes place with no effective nuclear motion in molecule A, while it initiates a bond contraction in molecule B and an elongation in molecule C.

## B. Methodology overview

We considered two different approaches for the simulations. The first one consists in simulating the ‘complete’ vibronic dynamics of the model molecules. To this end, we performed numerically converged, vibrationally correlated, simulations based on solving the time-dependent Schrödinger equation (TDSE). The second approach consists in adopting a standard simplified treatment with fixed nuclei, based on the Born-Oppenheimer (BO) approximation, where the photoemission dynamics are encoded in the continuum solutions of the electronic time-independent Schrödinger equation (TISE).

Our motivation is two-fold. On the one hand, the ‘complete’ vibronic simulations highlight the impact of vibronic couplings on the dynamics of molecular photoemission investigated with attosecond resolution. On the other hand, by confronting these ‘exact’ dynamics to the ones obtained in the BO framework, we assess the relevance of fixed-nuclei approaches to investigate molecular photoemission process (see eg [19, 23, 49–52]), when the latter are performed with well-chosen molecular conformations. In all the simulations, the $x$ origin was set a priori to coincide with the center of mass of the molecule, consistently with the definition of the $V_{N-e}$ potential [see Eqs. (2) and (3)].

The central quantities used in this work to characterize anisotropic photoionization dynamics are orientation-resolved yields and stereo ionization delays, which can be computed indifferently in both approaches.

### 1. ‘Complete’ simulations

The complete dynamics of the molecules were simulated by solving numerically the vibronic TDSE

$$ i \frac{\partial \Phi(x, R, t)}{\partial t} = [H_0 + W(t)] \Phi(x, R, t). \quad (5) $$

Here $\Phi(x, R, t)$ is the propagated vibronic wave function and $W(t)$ represents the dipole interaction of the molecule with the ionizing pulse, implemented in the velocity gauge. The initial state is the ground vibronic state $\Phi_0(x, R)$, obtained by imaginary time propagation, in all the simulations. Solving numerically the TDSE [Eq. (5)] was performed using a split operator propagation based on a grid representation of the $x$ coordinate, combined with an expansion over a set of eigenvectors of $H_N$ for the $R$ coordinate, as detailed in [15, 45]. The vector potentials of the ionizing pulses were assigned $\sin^2$ temporal envelopes with central photon energies $\omega_{xuv}$ corresponding to harmonics of a Ti:Sapphire laser ($\lambda_0 = 800$ nm, $\hbar\omega_0 = 1.55$ eV) in the extreme ultraviolet (XUV). Case specific pulse parameters are indicated further in the text. Simulations on molecule A were performed by expanding the wave function on a single vibrational state ($v = 0$ is the only open channel). For molecules B and C,
the results shown in this paper were obtained by including all vibrational channels up to $v = 8$ and 17, respectively, which safely ensures convergence.

The analysis of these complete time-dependent simulations are mostly based on the outgoing electron flux computed in the asymptotic $x$ region, where the short-range components of $V_{N-e}$ (including the vibronic couplings) vanish.

## 2. Simulations at fixed internuclear distances

In the BO framework, we treated 1-photon ionization by analyzing electronic continuum wave functions computed at fixed internuclear distances. Among the solutions of the TISE

$$H_e \psi_{\varepsilon}(x) = \varepsilon \psi_{\varepsilon}(x) \quad (6)$$

in the degenerate continuum (at energies $\varepsilon > 0$), we worked with the wave functions specifically selected by 1-photon transitions starting from the ground state, hereafter referred to as SCWF for 'selected continuum wave functions'. These real-valued continuum wave functions are defined unambiguously, and their analysis and interpretation are independent from their definition and computation. The SCWF formalism was introduced in [53] and later used in [27, 37]. Note that the electronic hamiltonian $H_e$ [as defined in Eq. (1)], its eigenfunctions $\psi_{\varepsilon}(x)$ and eigenvalues $\varepsilon$ depend parametrically on the internuclear distance $R$ – although it does not appear explicitly in Eq. (6) for the sake of readability.

Within this framework, the dynamics of photoemission are encoded in the asymptotic features of the SCWF.

More computational details are provided along with the presentation of the results.

## III. PHOTOEMISSION FROM MOLECULE A: SIGNATURES OF THE POTENTIAL ASYMMETRY

We present here the results obtained for molecule A, which was designed to benchmark our methodological approach. We first detail the analysis of the time-dependent vibronic simulations, and then verify that the latter are consistent with time-independent BO simulations at equilibrium distance.

### A. 'Complete' simulations

In these simulations, the XUV pulse intensities were set to $10^{12}$ W/cm$^2$, i.e., low enough to avoid any significant multi-photon process in 1-color photoionization. The total durations of the $\sin^2$ pulse envelopes were set to 16 fs, which corresponds to 6 fundamental periods ($T_0 = 2\pi/\omega_0 = 2.67$ fs).

<figure><img src="image_3.png" /><figcaption>FIG. 2. Outgoing electron flux $j(\theta_k, r_d, t)$ computed at $r_d = 800$ a.u. on the left ($\theta_k = 180^\circ$, red full curve) and right ($\theta_k = 0^\circ$, green dashed curve) sides of molecule A ionized by a $\omega_{xuv} = 35.67$ eV pulse, as a function of propagation time $t$. The flux is normalized to 1 at the overall maximum reached on the left side. The temporal envelope of the ionizing pulse is represented by the blue filled curve.</figcaption></figure>

### 1. Photoelectron flux

In the complete time-dependent simulations, we characterized the ionization dynamics through the outgoing photoelectron flux computed at a given detection distance $r_d$ from the $x$-origin, on either side of the molecule and averaged over $R$:

$$j(\theta_k, r_d, t) = \cos \theta_k \times \quad (7)$$

$$\mathfrak{Im} \int_0^\infty \Phi^*(r_d \cos \theta_k, R, t) \Phi'(r_d \cos \theta_k, R, t) dR$$

where $\Phi^*(x, R, t)$ is the complex conjugate of the vibronic wave function propagated according to Eq. (5), $\Phi'(x, R, t)$ its derivative with respect to $x$. Here and all through the paper, $\theta_k$ represents the direction of photoemission, restricted to two discrete values: $\theta_k = 0^\circ$ (emission towards the right, $x > 0$) and $180^\circ$ (towards the left, $x < 0$).

As an illustration, Figure 2 shows the flux computed at $r_d = 800$ a.u. (423 Å) on the left and right sides of molecule A ionized by a pulse of central frequency $\omega_{xuv} = 23 \times \omega_0 = 35.67$ eV. The flux profiles follow the ionizing pulse envelope (also shown), shifted by $\approx 30$ fs, which is consistent with the time needed for a (nearly) free electron with the energy $\omega_{xuv} - \tilde{E}_I(R_{eq}) = 5.86$ eV to cover the distance $r_d$. The flux maximum $\sim 1.5$ times larger on the left side than on the right side is a clear signature of the photoemission anisotropy.

### 2. Orientation-resolved yields and delays

We then computed the orientation-resolved yields,

$$\mathcal{Y}(\theta_k) = \int j(\theta_k, r_d, t) dt, \quad (8)$$
230 the results shown in this paper were obtained by including all vibrational channels up to $v = 8$ and 17, respectively, which safely ensures convergence.

233 The analysis of these complete time-dependent simulations are mostly based on the outgoing electron flux computed in the asymptotic $x$ region, where the short-range components of $V_{N-e}$ (including the vibronic couplings) vanish.

## 2. Simulations at fixed internuclear distances

239 In the BO framework, we treated 1-photon ionization by analyzing electronic continuum wave functions computed at fixed internuclear distances. Among the solutions of the TISE

$$H_e \psi_e(x) = \varepsilon \psi_e(x) \quad (6)$$

243 in the degenerate continuum (at energies $\varepsilon > 0$), we worked with the wave functions specifically selected by 245 1-photon transitions starting from the ground state, here- 246 after referred to as SCWF for ‘selected continuum wave 247 functions’. These real-valued continuum wave functions 248 are defined unambiguously, and their analysis and interpretation are independent from their definition and computation. The SCWF formalism was introduced in [53] 250 and later used in [27, 37]. Note that the electronic hamiltonian $H_e$ [as defined in Eq. (1)], its eigenfunctions $\psi_e(x)$ 252 and eigenvalues $\varepsilon$ depend parametrically on the internuclear distance $R$ – although it does not appear explicitly 254 in Eq. (6) for the sake of readability.

256 Within this framework, the dynamics of photoemission are encoded in the asymptotic features of the SCWF.

258 More computational details are provided along with the 260 presentation of the results.

## III. PHOTOEMISSION FROM MOLECULE A: SIGNATURES OF THE POTENTIAL ASYMMETRY

264 We present here the results obtained for molecule A, 265 which was designed to benchmark our methodological approach. We first detail the analysis of the time-dependent 266 vibronic simulations, and then verify that the latter 268 are consistent with time-independent BO simulations at 269 equilibrium distance.

### A. ‘Complete’ simulations

271 In these simulations, the XUV pulse intensities were 270 set to $10^{12}$ W/cm$^2$, i.e., low enough to avoid any significant multi-photon process in 1-color photoionization. 274 The total durations of the $\sin^2$ pulse envelopes were set 275 to 16 fs, which corresponds to 6 fundamental periods 276 ($T_0 = 2\pi/\omega_0 = 2.67$ fs).

<figure><img src="image_4.png" /><figcaption>FIG. 2. Outgoing electron flux $j(\theta_k, r_d, t)$ computed at $r_d = 800$ a.u. on the left ($\theta_k = 180^\circ$, red full curve) and right ($\theta_k = 0^\circ$, green dashed curve) sides of molecule A ionized by a $\omega_{xuv} = 35.67$ eV pulse, as a function of propagation time $t$. The flux is normalized to 1 at the overall maximum reached on the left side. The temporal envelope of the ionizing pulse is represented by the blue filled curve.</figcaption></figure>

### 1. Photoelectron flux

278 In the complete time-dependent simulations, we characterized the ionization dynamics through the outgoing 279 photoelectron flux computed at a given detection distance $r_d$ from the $x$-origin, on either side of the molecule 281 and averaged over $R$:

$$j(\theta_k, r_d, t) = \cos \theta_k \times \qquad (7) \\ \mathfrak{Im} \int_0^\infty \Phi^*(r_d \cos \theta_k, R, t) \Phi'(r_d \cos \theta_k, R, t) dR$$

283 where $\Phi^*(x, R, t)$ is the complex conjugate of the vibronic wave function propagated according to Eq. (5), 284 $\Phi'(x, R, t)$ its derivative with respect to $x$. Here and all 286 through the paper, $\theta_k$ represents the direction of photoemission, restricted to two discrete values: $\theta_k = 0^\circ$ 287 (emission towards the right, $x > 0$) and $180^\circ$ (towards 288 the left, $x < 0$).

290 As an illustration, Figure 2 shows the flux computed 291 at $r_d = 800$ a.u. (423 Å) on the left and right sides 292 of molecule A ionized by a pulse of central frequency 293 $\omega_{xuv} = 23 \times \omega_0 = 35.67$ eV. The flux profiles follow the 294 ionizing pulse envelope (also shown), shifted by $\approx 30$ fs, 295 which is consistent with the time needed for a (nearly) 296 free electron with the energy $\omega_{xuv} - \bar{E}_I(R_{eq}) = 5.86$ eV 297 to cover the distance $r_d$. The flux maximum $\sim 1.5$ times 298 larger on the left side than on the right side is a clear 299 signature of the photoemission anisotropy.

### 2. Orientation-resolved yields and delays

We then computed the orientation-resolved yields,

$$\mathcal{Y}(\theta_k) = \int j(\theta_k, r_d, t) dt, \quad (8)$$
<figure><img src="image_4.png" /><figcaption>FIG. 3. Anisotropic photoemission from molecule A. (a) left/right ionization yield ratio computed in the fully correlated simulations and in the BO framework at $R_{eq}$, resp. $\mathcal{R}$ [Eq (9)] (line) and $\mathcal{R}$ [Eq (14)] (circles); (b) stereo ionization delays computed in the fully correlated simulations and in the BO framework at $R_{eq}$, resp. $\Delta\tau$ [Eq (11)] (line) and $\Delta\tau_w$ [Eq (16)] (circles). All data plotted as functions of the photoelectron energy $\epsilon$.</figcaption></figure>

302 in a series of simulations with field frequencies corresponding to harmonics 21 to 28, i.e., $\omega_{xuv}$ ranging from 303 32.57 eV to 43.43 eV by 1.55 eV steps. The corresponding yield ratio 304

$$ \mathcal{R} = \frac{\mathcal{Y}(180^{\circ})}{\mathcal{Y}(0^{\circ})} \qquad (9) $$

306 is displayed in Fig. 3(a) (coloured circles) against the photoelectron energy $\epsilon$. It evolves smoothly slightly above 307 1.4, indicating that photoemission is sensitive to the 308 asymmetry of the ionic potential, with little variations 309 over the energy range covered in the simulations.

310 To characterize the asymmetry of photoemission in 311 the time domain, we used the average “time of flight” 312 (TOF) [15, 16, 53] towards the detector in each direction 313 as 314

$$ \tau(\theta_k) = \frac{\int t \times j(\theta_k, r_d, t) dt}{\int j(\theta_k, r_d, t) dt} \qquad (10) $$

315 To further investigate the angular variations of the TOF 316 computed as such, one must take into consideration the 317 arbitrary origin $x = 0$ on either sides of which the virtual 318 detectors are located, and which does not coincide with 319 the average initial position of the electron. In the fol-

320 lowing, we will therefore characterize these angular variations with the origin-corrected TOF difference, 321

$$ \Delta\tau = \tau(180^{\circ}) - \tau(0^{\circ}) - 2\frac{\langle x \rangle_0}{\sqrt{2\epsilon}}, \qquad (11) $$

322 where $\langle x \rangle_0$ is the average initial electron position in the 323 ground state of the molecule $\Phi_0(x, R)$,

$$ \langle x \rangle_0 = \langle \Phi_0 | x | \Phi_0 \rangle. \qquad (12) $$

324 The role of the last term on the r.h.s of Eq. (11) is discussed in details in the companion paper [27] in the context of time-independent approaches. Its justifications 325 identically holds for time-dependent simulations. It compensates a spurious shift appearing when computing ion- 326 ization delays with an arbitrary origin – while the photo- 327 electron in average originates from $\langle x \rangle_0$. We verified numerically that including this term (after computing the 328 TOFs) is equivalent to shifting the potential such that 329 $\langle x \rangle_0 = 0$ (prior to solving the TDSE). In the present simulations, as well as in molecules B and C, $\langle x \rangle_0 = -0.160$ 330 331

332 The stereo ionization delays $\Delta\tau$ obtained for the considered set of XUV frequencies are plotted in Fig. 3(b). 333 Starting from ~ -40 as just below $\epsilon = 2$ eV, its magnitude decays smoothly while the photon energy increases, 334 as could be expected in absence of significantly structured continuum. Note that such attosecond delays cannot be resolved visually in Fig. 2, since they are extremely 335 small compared to the temporal spread of the photoelectron wave-packet at detection (~ 16 fs, approximately 336 the XUV pulse duration). They are nonetheless significant and we have ensured their numerical convergence. 337

338 Among a series of standard numerical checks, we verified that the measured stereo delays $\Delta\tau$ do not depend 339 on the detection distance $r_d$ (as long as it lies far enough 340 from the ionic core), which is a fundamental property of 341 the short-range scattering delays. 342

## B. Simulations at fixed internuclear distances

351 Here, we consider the molecule A at its equilibrium 352 internuclear distance, see Fig. 4(a). It corresponds to the 353 1D model molecule used for the numerical experiments 354 presented and analysed in [27]. 355

### 1. Selected continuum wave function

356 The BO approach relies on the analysis of the electronic continuum wave functions $\psi_{\epsilon}(x)$ associated with 357 the photoemission processes. We thus computed the 358 SCWF of molecule A by solving Eq. (6) over the same 359 energy range as in the full time-dependent simulations, 360 with a fixed internuclear distance set to $R_{eq}$. 361

362 The SCWF computed at the illustrative energy $\epsilon = 363 5.86$ eV (corresponding to $\omega_{xuv} = 23 \times \omega_0 = 35.67$ eV, as
<figure><img src="image_5.png" /><figcaption>FIG. 3. Anisotropic photoemission from molecule A. (a) left/right ionization yield ratio computed in the fully correlated simulations and in the BO framework at $R_{eq}$, resp. $\mathcal{R}$ [Eq (9)] (line) and $\mathcal{R}$ [Eq (14)] (circles); (b) stereo ionization delays computed in the fully correlated simulations and in the BO framework at $R_{eq}$, resp. $\Delta\tau$ [Eq (11)] (line) and $\Delta\tau_w$ [Eq (16)] (circles). All data plotted as functions of the photoelectron energy $\epsilon$.</figcaption></figure>

302 in a series of simulations with field frequencies corresponding to harmonics 21 to 28, i.e., $\omega_{xuv}$ ranging from 303 32.57 eV to 43.43 eV by 1.55 eV steps. The corresponding yield ratio 304 305

$$ \mathcal{R} = \frac{\mathcal{Y}(180^{\circ})}{\mathcal{Y}(0^{\circ})} \qquad (9) $$

306 is displayed in Fig. 3(a) (coloured circles) against the photoelectron energy $\epsilon$. It evolves smoothly slightly above 307 1.4, indicating that photoemission is sensitive to the 308 asymmetry of the ionic potential, with little variations 309 over the energy range covered in the simulations.

310 To characterize the asymmetry of photoemission in 311 the time domain, we used the average “time of flight” 312 (TOF) [15, 16, 53] towards the detector in each direction 313 314

$$ \tau(\theta_k) = \frac{\int t \times j(\theta_k, r_d, t) dt}{\int j(\theta_k, r_d, t) dt}. \qquad (10) $$

315 To further investigate the angular variations of the TOF 316 computed as such, one must take into consideration the 317 arbitrary origin $x=0$ on either sides of which the virtual 318 detectors are located, and which does not coincide with 319 the average initial position of the electron. In the fol-

320 lowing, we will therefore characterize these angular variations with the origin-corrected TOF difference, 321

$$ \Delta\tau = \tau(180^{\circ}) - \tau(0^{\circ}) - 2\frac{\langle x \rangle_0}{\sqrt{2\epsilon}}, \qquad (11) $$

322 where $\langle x \rangle_0$ is the average initial electron position in the 323 ground state of the molecule $\Phi_0(x, R)$,

$$ \langle x \rangle_0 = \langle \Phi_0 | x | \Phi_0 \rangle. \qquad (12) $$

324 The role of the last term on the r.h.s of Eq. (11) is discussed in details in the companion paper [27] in the context of time-independent approaches. Its justifications 325 326 identically holds for time-dependent simulations. It compensates a spurious shift appearing when computing ion- 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351

The stereo ionization delays $\Delta\tau$ obtained for the considered set of XUV frequencies are plotted in Fig. 3(b). 352 Starting from $\sim -40$ as just below $\epsilon = 2$ eV, its magnitude decays smoothly while the photon energy increases, 353 354 355 356 as could be expected in absence of significantly structured continuum. Note that such attosecond delays cannot be resolved visually in Fig. 2, since they are extremely 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000

## B. Simulations at fixed internuclear distances

Here, we consider the molecule A at its equilibrium 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000

### 1. Selected continuum wave function

The BO approach relies on the analysis of the electronic continuum wave functions $\psi_{\epsilon}(x)$ associated with the photoemission processes. We thus computed the SCWF of molecule A by solving Eq. (6) over the same energy range as in the full time-dependent simulations, with a fixed internuclear distance set to $R_{eq}$.

The SCWF computed at the illustrative energy $\epsilon = 5.86$ eV (corresponding to $\omega_{xuv} = 23 \times \omega_0 = 35.67$ eV, as
<figure><img src="image_5.png" /><figcaption>FIG. 4. Molecule A in the BO framework (equilibrium inter-nuclear distance). (a) Electron-nuclei potential $V_{N-e}(x, R_{eq})$ as a function of the electron position $x$ (dark yellow full curve). The ground state electronic wave function is also shown (light blue filled curve). (b) Electronic continuum wave function (full curve) selected at the energy $\epsilon = 5.86$ eV by a 1-photon transition from the electronic ground state. Odd-parity reference wave function (dashed curve) used to define and compute the orientation-dependent phase-shifts. The displayed continuum wave functions are normalized such that their amplitudes asymptotically converge to 1 on the left-hand side of the molecule. In this Figure, the left/right discrimination and the parity refer to the arbitrary $x = 0$ origin (indicated by a vertical dotted line).</figcaption></figure>

366 in Fig. 2) is shown in Fig. 4(b) (full curve). The pseudo-367 period of the oscillations on both sides is consistent with 368 a kinetic energy asymptotically converging to 5.86 eV. 369 Here, the anisotropy of photoemission is clearly visible in 370 the asymmetry of the amplitudes of the SCWF on either 371 sides of the molecule.

## 2. Orientation-resolved yields and delays

373 For each pulse frequency considered in the full-fledged 374 approach, an alternative evaluation of the orientation- 375 dependent ionization yields is provided, up to a global 376 factor, as

$$ \tilde{\mathcal{Y}}(\theta_k) = |A(R_{eq}; \theta_k)|^2, \quad (13) $$

377 where $A(R_{eq}; \theta_k)$ is the asymptotic amplitude of the 378 SCWF computed with $R = R_{eq}$, at the average photo-379 electron energy, on each side of the molecule. We eval- 380 uated these amplitudes using Strömgren's normalization 381 procedure (see [54] and references therein).

382 The corresponding yield ratio

$$ \tilde{\mathcal{R}} = \frac{\tilde{\mathcal{Y}}(180^{\circ})}{\tilde{\mathcal{Y}}(0^{\circ})} \quad (14) $$

383 computed over the considered energy range is displayed 384 as a full line in Fig. 3(a). The perfect agreement with the 385 ratio $\mathcal{R}$ computed in the vibronic time-dependent simu-386 lations [Eq. (9)] constitutes a first numerical validation 387 of our comparative approach.

388 From the time domain perspective, we analyzed the 389 SCWF in terms of orientation resolved Wigner delays [55], as detailed in [27]. In few words, the Wigner 390 delay is defined as the spectral derivative

$$ \tau_w(\theta_k) = \frac{\partial \eta(\theta_k)}{\partial \epsilon} \quad (15) $$

391 of the asymptotic phase shifts of the SCWF computed 392 on each side of the model molecule, with respect to an 393 arbitrary intermediate reference (here, radial Coulomb 394 s-waves centered at $x = 0$). As explained in [27], and 395 consistently with the 'complete' vibronic simulations, we 396 characterized the angular variations of the photoemis- 397 sion dynamics in the BO framework through the origin- 398 corrected stereo Wigner delay

$$ \Delta\tau_w = \tau_w(180^{\circ}) - \tau_w(0^{\circ}) - 2\frac{\langle x \rangle_0}{\sqrt{2\epsilon}}. \quad (16) $$

400 Since the exact ground state of the molecule is accurately 401 modelled by its BO counterpart, the value of the initial 402 average position $\langle x \rangle_0$ is the same here as in the 'complete' 403 simulations. The stereo Wigner delays computed in the 404 considered energy range, already shown in Fig. 3 of [27], 405 are reported in Fig. 3(b) (full line). They perfectly agree 406 with the stereo delays computed in the 'complete' time- 407 dependent simulations.

408 Apart from benchmarking our comparative approach, 409 the results obtained with molecule A put forwards the 410 signatures of the left-right electronegativity asymmetry 411 of the molecule on the photoelectron dynamics. The per-412 fect agreement observed in Fig. 3 between the 'complete' 413 and the fixed nuclei approach comes as no surprise since 414 (i) the equivalence between the time-dependent and time-415 independent approaches has already been put forward by 416 Wigner when interpreting in the time-domain the group 417 delay associated with a scattering phase shift [55], and 418 (ii) photoemission of molecule A occurs with no effective 419 nuclear motion, i.e., the electron and nuclear degrees of 420 freedom, $x$ and $R$, are de facto uncoupled due to the rel-421 ative positions of the ground state and ionic curves [see 422 Fig. 1(a)].

423 We now move on to the results obtained with the B 424 and C molecules, for which photoemission occurs along 425 with sensible nuclear dynamics.

## IV. PHOTOEMISSION FROM MOLECULES B AND C: SIGNATURES OF THE VIBRONIC COUPLINGS

429 The necessity to take the nuclear motion into consid-430 eration in near-threshold photoemission from molecules 431 B and C is a consequence of the relative position shifts of
<figure><img src="image_6.png" /><figcaption>FIG. 4. Molecule A in the BO framework (equilibrium inter-nuclear distance). (a) Electron-nuclei potential $V_{N-e}(x, R_{eq})$ as a function of the electron position $x$ (dark yellow full curve). The ground state electronic wave function is also shown (light blue filled curve). (b) Electronic continuum wave function (full curve) selected at the energy $\epsilon = 5.86$ eV by a 1-photon transition from the electronic ground state. Odd-parity reference wave function (dashed curve) used to define and compute the orientation-dependent phase-shifts. The displayed continuum wave functions are normalized such that their amplitudes asymptotically converge to 1 on the left-hand side of the molecule. In this Figure, the left/right discrimination and the parity refer to the arbitrary $x = 0$ origin (indicated by a vertical dotted line).</figcaption></figure>

366 in Fig. 2) is shown in Fig. 4(b) (full curve). The pseudo-367 period of the oscillations on both sides is consistent with 368 a kinetic energy asymptotically converging to 5.86 eV. 369 Here, the anisotropy of photoemission is clearly visible in 370 the asymmetry of the amplitudes of the SCWF on either 371 sides of the molecule.

## 2. Orientation-resolved yields and delays

373 For each pulse frequency considered in the full-fledged 374 approach, an alternative evaluation of the orientation- 375 dependent ionization yields is provided, up to a global 376 factor, as

$$ \tilde{\mathcal{Y}}(\theta_k) = |A(R_{eq}; \theta_k)|^2, \quad (13) $$

377 where $A(R_{eq}; \theta_k)$ is the asymptotic amplitude of the 378 SCWF computed with $R = R_{eq}$, at the average photo-379 electron energy, on each side of the molecule. We eval- 380 uated these amplitudes using Strömgren's normalization 381 procedure (see [54] and references therein).

382 The corresponding yield ratio

$$ \tilde{\mathcal{R}} = \frac{\tilde{\mathcal{Y}}(180^{\circ})}{\tilde{\mathcal{Y}}(0^{\circ})} \quad (14) $$

383 computed over the considered energy range is displayed 384 as a full line in Fig. 3(a). The perfect agreement with the 385 ratio $\mathcal{R}$ computed in the vibronic time-dependent simu-386 lations [Eq. (9)] constitutes a first numerical validation 387 of our comparative approach.

388 From the time domain perspective, we analyzed the 389 SCWF in terms of orientation resolved Wigner delays [55], as detailed in [27]. In few words, the Wigner 390 delay is defined as the spectral derivative

$$ \tau_w(\theta_k) = \frac{\partial \eta(\theta_k)}{\partial \epsilon} \quad (15) $$

391 of the asymptotic phase shifts of the SCWF computed 392 on each side of the model molecule, with respect to an 393 arbitrary intermediate reference (here, radial Coulomb 394 s-waves centered at $x = 0$). As explained in [27], and 395 consistently with the 'complete' vibronic simulations, we 396 characterized the angular variations of the photoemis- 397 sion dynamics in the BO framework through the origin- 398 corrected stereo Wigner delay

$$ \Delta\tau_w = \tau_w(180^{\circ}) - \tau_w(0^{\circ}) - 2\frac{\langle x \rangle_0}{\sqrt{2\epsilon}}. \quad (16) $$

400 Since the exact ground state of the molecule is accurately 401 modelled by its BO counterpart, the value of the initial 402 average position $\langle x \rangle_0$ is the same here as in the 'complete' 403 simulations. The stereo Wigner delays computed in the 404 considered energy range, already shown in Fig. 3 of [27], 405 are reported in Fig. 3(b) (full line). They perfectly agree 406 with the stereo delays computed in the 'complete' time- 407 dependent simulations.

408 Apart from benchmarking our comparative approach, 409 the results obtained with molecule A put forwards the 410 signatures of the left-right electronegativity asymmetry 411 of the molecule on the photoelectron dynamics. The per-412 fect agreement observed in Fig. 3 between the 'complete' 413 and the fixed nuclei approach comes as no surprise since 414 (i) the equivalence between the time-dependent and time-415 independent approaches has already been put forward by 416 Wigner when interpreting in the time-domain the group 417 delay associated with a scattering phase shift [55], and 418 (ii) photoemission of molecule A occurs with no effective 419 nuclear motion, i.e., the electron and nuclear degrees of 420 freedom, $x$ and $R$, are de facto uncoupled due to the rel-421 ative positions of the ground state and ionic curves [see 422 Fig. 1(a)].

423 We now move on to the results obtained with the B 424 and C molecules, for which photoemission occurs along 425 with sensible nuclear dynamics.

## IV. PHOTOEMISSION FROM MOLECULES B AND C: SIGNATURES OF THE VIBRONIC COUPLINGS

429 The necessity to take the nuclear motion into consid-430 eration in near-threshold photoemission from molecules 431 B and C is a consequence of the relative position shifts of
<table><caption>TABLE I. Stereo delay $\Delta\tau$ computed according to Eq. (11) in molecule B for an illustrative set of detection distances $r_d$ and photon frequencies $\omega_{xuv}$.</caption><thead><tr><th>ω<sub>xuv</sub></th><th>r<sub>d</sub></th><th>Δτ</th></tr></thead><tbody><tr><td rowspan="2">32.6 eV</td><td>600 a.u.</td><td>119 as</td></tr><tr><td>800 a.u.</td><td>153 as</td></tr><tr><td rowspan="2">34.1 eV</td><td>800 a.u.</td><td>74 as</td></tr><tr><td>1000 a.u.</td><td>89 as</td></tr></tbody></table>

432 their ground states and ionic BO curves [see Fig. 1(b,c)].
433 We investigate how it affects the photoemission dynamics in the present section by presenting and analyzing the
434 orientation-resolved photoemission yields and stereo delays in these two model molecules, and addressing the
435 relevance of fixed-nuclei simulations in this context.

A. ‘Complete’ simulations

439 We used here the same pulse parameters as in the simulations with molecule A (Sec. III A).

1. *Signatures of electron-nuclei couplings*

442 A striking signature of the electron-nuclei couplings in
443 the photoemission dynamics of B shows up in the de-
444 lays inferred from the outgoing flux. In contrast with the
445 previous case, the stereo delays measured according to
446 Eqs. (10)–(11) strongly depend on the detection distance
447 $r_d$. This is illustrated in Table I with few representative
448 cases. Not only does it evidence a rather dramatic depen-
449 dency of $\Delta\tau$ with respect to $r_d$, well above the numerical
450 accuracy of the simulations, but it turns out that $\Delta\tau$
451 monotonically diverges with increasing $r_d$ (not shown).
452 This tells us that, the delays computed as such are irrel-
453 evant for an objective characterization of the ionization
454 dynamics.

455 Nevertheless their $r_d$-dependency bear a crucial infor-
456 mation: it suggests that the electron wave-packets travel
457 on each side of the molecule with slightly different aver-
458 age velocities. This is the signature of an anisotropic
459 momentum sharing between the nuclei and the active
460 electron during the *concerted* ionization/contraction of
461 the molecular ion. It involves the nuclear mass asymme-
462 try since the lighter the nucleus, the more the photoelec-
463 tron is prone to share kinetic energy with it. The prin-
464 ciple, which applies to the average velocity, is sketched
465 in Fig. 5. Assuming that only the lighter nucleus (on
466 the left) significantly moves, the photoelectron ends up
467 with a smaller velocity if it exits in the direction of mo-
468 tion of that nucleus (towards the right) than in the other
469 direction.

473 Properly accounting for this momentum sharing hence
474 requires giving up global characterization and rather
475 considering vibrationally-resolved observables. In the

<figure><img src="image_6.png" /><figcaption>FIG. 5. Sketch of the asymmetric average momentum sharing between the photoelectron (empty circle) and the nuclei (full circles) during the photoionization/early contraction of molecule B. For the sake of clarity, the velocity asymmetries have been exaggerated compared to the ones obtained in the actual simulations.</figcaption></figure>

476 present context, they are derived from the channel-
477 resolved wave-packets

$$
\varphi_v(x, t) = \int_0^\infty \chi_v(R) \Phi(x, R, t) \, dR \quad (17)
$$

478 in the vibronic TDSE simulations. In Eq. (17), $\chi_v(R)$
479 are the vibrational eigenstates of the molecular ionic core
480 hamiltonian $H_N$ [see Eq. (1) and Appendix A]. The in-
481 dividual channel functions are relevant as soon as they
482 are uncoupled from each other, i.e., in the asymptotic
483 $x$ region where $V_{N-e}$ no longer depends significantly on $R$
484 (typically beyond few 100 a.u.). Each channel is assigned
485 a specific ionization potential

$$
E_I(v) = \mathcal{E}_v^{(+)} - \mathcal{E}_0^{(0)}, \quad (18)
$$

486 where $\mathcal{E}_v^{(+)}$ is the energy of the corresponding ionization
487 threshold (in the BO framework, that is the eigenvalue
488 of $H_N$ associated with $\chi_v$) and $\mathcal{E}_0^{(0)}$ the vibronic ground
489 energy of the neutral molecule.

490 The key time-dependent observable becomes the
491 vibrationally-resolved electron flux

$$
j_v(\theta_k, r_d, t) = \cos \theta_k \times \tag{19} \\
\mathfrak{J}\{\varphi_v^*(r_d \cos \theta_k, t) \varphi_v'(r_d \cos \theta_k, t)\}
$$

492 computed at a distance $r_d$ on each side of the molecule.
493 Vibrationally-resolved anisotropic yields and delays can
494 be computed from the flux $j_v(\theta_k, r_d, t)$ in a similar fashion
495 to the integrated case, see Section III.

496 The flux computed at the distance $r_d = 800$ a.u. on the
497 left (a) and right (b) sides of the B molecule submitted
498 to a $\omega_{xuv} = 35.67$ eV pulse is shown in Fig. 6, for the
499 first few populated channels. The flux maxima of the
500 different channels are time-shifted from each other, which
501 is a manifestation of the energy conservation law

$$
\epsilon = \omega_{xuv} - E_I(v). \quad (20)
$$

502 The larger $E_I(v)$, the lower the photoelectron energy $\epsilon$
503 and the larger the time needed to reach the detector. As
<table><caption>TABLE I. Stereo delay $\Delta\tau$ computed according to Eq. (11) in molecule B for an illustrative set of detection distances $r_d$ and photon frequencies $\omega_{xuv}$.</caption><thead><tr><th>ω<sub>xuv</sub></th><th>r<sub>d</sub></th><th>Δτ</th></tr></thead><tbody><tr><td rowspan="2">32.6 eV</td><td>600 a.u.</td><td>119 as</td></tr><tr><td>800 a.u.</td><td>153 as</td></tr><tr><td rowspan="2">34.1 eV</td><td>800 a.u.</td><td>74 as</td></tr><tr><td>1000 a.u.</td><td>89 as</td></tr></tbody></table>

432 their ground states and ionic BO curves [see Fig. 1(b,c)].
433 We investigate how it affects the photoemission dynamics in the present section by presenting and analyzing the
434 orientation-resolved photoemission yields and stereo delays in these two model molecules, and addressing the
435 relevance of fixed-nuclei simulations in this context.
436
437

A. ‘Complete’ simulations

439 We used here the same pulse parameters as in the simulations with molecule A (Sec. III A).
440

1. *Signatures of electron-nuclei couplings*

442 A striking signature of the electron-nuclei couplings in
443 the photoemission dynamics of B shows up in the de-
444 lays inferred from the outgoing flux. In contrast with the
445 previous case, the stereo delays measured according to
446 Eqs. (10)–(11) strongly depend on the detection distance
447 $r_d$. This is illustrated in Table I with few representative
448 cases. Not only does it evidence a rather dramatic depen-
449 dency of $\Delta\tau$ with respect to $r_d$, well above the numerical
450 accuracy of the simulations, but it turns out that $\Delta\tau$
451 monotonically diverges with increasing $r_d$ (not shown).
452 This tells us that, the delays computed as such are irrel-
453 evant for an objective characterization of the ionization
454 dynamics.
455

455 Nevertheless their $r_d$-dependency bear a crucial infor-
456 mation: it suggests that the electron wave-packets travel
457 on each side of the molecule with slightly different aver-
458 age velocities. This is the signature of an anisotropic
459 momentum sharing between the nuclei and the active
460 electron during the *concerted* ionization/contraction of
461 the molecular ion. It involves the nuclear mass asymme-
462 try since the lighter the nucleus, the more the photoelec-
463 tron is prone to share kinetic energy with it. The prin-
464 ciple, which applies to the average velocity, is sketched
465 in Fig. 5. Assuming that only the lighter nucleus (on
466 the left) significantly moves, the photoelectron ends up
467 with a smaller velocity if it exits in the direction of mo-
468 tion of that nucleus (towards the right) than in the other
469 direction.
470

473 Properly accounting for this momentum sharing hence
474 requires giving up global characterization and rather
475 considering vibrationally-resolved observables. In the

<caption>FIG. 5. Sketch of the asymmetric average momentum sharing between the photoelectron (empty circle) and the nuclei (full circles) during the photoionization/early contraction of molecule B. For the sake of clarity, the velocity asymmetries have been exagerated compared to the ones obtained in the actual simulations.</caption>

476 present context, they are derived from the channel-
477 resolved wave-packets

$$
\varphi_v(x, t) = \int_0^\infty \chi_v(R) \Phi(x, R, t) \, dR \quad (17)
$$

478 in the vibronic TDSE simulations. In Eq. (17), $\chi_v(R)$
479 are the vibrational eigenstates of the molecular ionic core
480 hamiltonian $H_N$ [see Eq. (1) and Appendix A]. The in-
481 dividual channel functions are relevant as soon as they
482 are uncoupled from each other, i.e., in the asymptotic
483 $x$ region where $V_{N-e}$ no longer depends significantly on $R$
484 (typically beyond few 100 a.u.). Each channel is assigned
485 a specific ionization potential

$$
E_I(v) = \mathcal{E}_v^{(+)} - \mathcal{E}_0^{(0)}, \quad (18)
$$

486 where $\mathcal{E}_v^{(+)}$ is the energy of the corresponding ionization
487 threshold (in the BO framework, that is the eigenvalue
488 of $H_N$ associated with $\chi_v$) and $\mathcal{E}_0^{(0)}$ the vibronic ground
489 energy of the neutral molecule.
490

490 The key time-dependent observable becomes the
491 vibrationally-resolved electron flux

$$
j_v(\theta_k, r_d, t) = \cos \theta_k \times \tag{19} \\
\mathfrak{J}\{\varphi_v^*(r_d \cos \theta_k, t) \varphi_v'(r_d \cos \theta_k, t)\}
$$

492 computed at a distance $r_d$ on each side of the molecule.
493 Vibrationally-resolved anisotropic yields and delays can
494 be computed from the flux $j_v(\theta_k, r_d, t)$ in a similar fashion
495 to the integrated case, see Section III.
496

496 The flux computed at the distance $r_d = 800$ a.u. on the
497 left (a) and right (b) sides of the B molecule submitted
498 to a $\omega_{xuv} = 35.67$ eV pulse is shown in Fig. 6, for the
499 first few populated channels. The flux maxima of the
500 different channels are time-shifted from each other, which
501 is a manifestation of the energy conservation law

$$
\epsilon = \omega_{xuv} - E_I(v). \quad (20)
$$

502 The larger $E_I(v)$, the lower the photoelectron energy $\epsilon$
503 and the larger the time needed to reach the detector. As

<figure><img src="image_7.png" /></figure>
<figure><img src="image_7.png" /><figcaption>FIG. 6. Vibrationally resolved outgoing electron flux $j_v(\theta_k, r_d, t)$ computed, in the fully correlated simulations, at $r_d = 800$ a.u. on the left [$\theta_k = 180^\circ$, frame (a)] and right [$\theta_k = 0^\circ$, frame (b)] sides of molecule B ionized by $\omega_{XUV} = 35.67$ eV pulse, as a function of propagation time $t$, for the 5 first dominant channels. Each curve is labelled according to its corresponding channel ($v = 0-4$). The flux is normalized to 1 at the overall maximum reached in the $v = 2$ channel, on the left side. The temporal envelope of the ionizing pulse is represented by the blue filled curve.</figcaption></figure>

for molecule A, the asymmetry of molecule B results in anisotropic ionization yields, visible when comparing the flux magnitudes on the left and on the right. Similar results were obtained with molecule C (not shown).

To get a better insight on the vibrational distributions upon ionization, we display in Fig. 7, column i, the Franck-Condon (FC) factors for molecules B (frame a) and C (frame b), defined as

$$F(v) = \left| \int_{0}^{\infty} \chi_{v}(R) \xi_{0}(R) dR \right|^{2}, \quad (21)$$

where $\xi_0(R)$ is the ground vibrational wave function of the neutral molecule treated in the BO framework. The FC factors peak at $v = 1$ for both model molecules, and extend significantly up to $v = 4$ and 8 for molecules B and C, respectively. They qualitatively reproduce the actual $v$-resolved yields

$$\mathcal{Y}(v; \theta_k) = \int j_v(\theta_k, r_d, t) dt, \quad (22)$$

computed on the left (columns ii) and right (columns iii) sides of the molecules ionized by the illustrative $\omega_{XUV} = 35.67$ eV pulse. For the sake of comparison, the displayed data are normalized such that each set sums up to 1.

A feature that cannot be included in the FC factors alone is obviously the orientation dependency. In particular for molecule B, the vibrational distributions of the yields in Fig. 7(a) display slight yet significant differences on the left and right emission sides. In spite of being small, these discrepancies are sufficient to result in different average photoelectron velocities, and to the above-mentioned $r_d$ dependence of the integrated left-right delay $\Delta\tau$. We emphasize that the data in Fig. 7 are normalized to highlight the orientation-resolved vibrational distributions, and therefore discards the vibrationally-resolved magnitude of the ionization probability on each side of the molecules.

<figure><img src="image_8.png" /><figcaption>FIG. 7. Vibrational distribution of photoemission from molecules B (a) and C (b) with a $\omega_{XUV} = 35.67$ eV pulse. (i): Franck-Condon factors $F(v)$ [Eq. (21)]; (ii) and (iii): normalized yields computed in the fully correlated simulations, resp. $\mathcal{Y}(v; 180^\circ)$ and $\mathcal{Y}(v; 0^\circ)$ [Eq. (22)]; (iv) and (v): normalized yields computed in the BO framework at the optimal internuclear distances $R_{opt}(v)$, resp. $\tilde{\mathcal{Y}}(v; 180^\circ)$ and $\tilde{\mathcal{Y}}(v; 0^\circ)$ [Eq. (26)]. The displayed data are normalized such that each set sums up to 1, for comparison purpose.</figcaption></figure>

## 2. Photoemission yields and delays

A finer insight on the $v$-resolved photoemission anisotropy is provided in Fig. 8 which shows, for
<figure><img src="image_8.png" /><figcaption>FIG. 6. Vibrationally resolved outgoing electron flux $j_v(\theta_k, r_d, t)$ computed, in the fully correlated simulations, at $r_d = 800$ a.u. on the left [$\theta_k = 180^\circ$, frame (a)] and right [$\theta_k = 0^\circ$, frame (b)] sides of molecule B ionized by $\omega_{XUV} = 35.67$ eV pulse, as a function of propagation time $t$, for the 5 first dominant channels. Each curve is labelled according to its corresponding channel ($v = 0-4$). The flux is normalized to 1 at the overall maximum reached in the $v = 2$ channel, on the left side. The temporal envelope of the ionizing pulse is represented by the blue filled curve.</figcaption></figure>

for molecule A, the asymmetry of molecule B results in anisotropic ionization yields, visible when comparing the flux magnitudes on the left and on the right. Similar results were obtained with molecule C (not shown).

To get a better insight on the vibrational distributions upon ionization, we display in Fig. 7, column i, the Franck-Condon (FC) factors for molecules B (frame a) and C (frame b), defined as

$$F(v) = \left| \int_{0}^{\infty} \chi_{v}(R) \xi_{0}(R) dR \right|^{2}, \qquad (21)$$

where $\xi_0(R)$ is the ground vibrational wave function of the neutral molecule treated in the BO framework. The FC factors peak at $v = 1$ for both model molecules, and extend significantly up to $v = 4$ and 8 for molecules B and C, respectively. They qualitatively reproduce the actual $v$-resolved yields

$$\mathcal{Y}(v; \theta_k) = \int j_v(\theta_k, r_d, t) dt, \qquad (22)$$

computed on the left (columns ii) and right (columns iii) sides of the molecules ionized by the illustrative $\omega_{XUV} = 35.67$ eV pulse. For the sake of comparison, the displayed data are normalized such that each set sums up to 1.

A feature that cannot be included in the FC factors alone is obviously the orientation dependency. In particular for molecule B, the vibrational distributions of the yields in Fig. 7(a) display slight yet significant differences on the left and right emission sides. In spite of being small, these discrepancies are sufficient to result in different average photoelectron velocities, and to the above-mentioned $r_d$ dependence of the integrated left-right delay $\Delta\tau$. We emphasize that the data in Fig. 7 are normalized to highlight the orientation-resolved vibrational distributions, and therefore discards the vibrationally-resolved magnitude of the ionization probability on each side of the molecules.

<figure><img src="image_9.png" /><figcaption>FIG. 7. Vibrational distribution of photoemission from molecules B (a) and C (b) with a $\omega_{XUV} = 35.67$ eV pulse. (i): Franck-Condon factors $F(v)$ [Eq. (21)]; (ii) and (iii): normalized yields computed in the fully correlated simulations, resp. $\mathcal{Y}(v; 180^\circ)$ and $\mathcal{Y}(v; 0^\circ)$ [Eq. (22)]; (iv) and (v): normalized yields computed in the BO framework at the optimal internuclear distances $R_{opt}(v)$, resp. $\tilde{\mathcal{Y}}(v; 180^\circ)$ and $\tilde{\mathcal{Y}}(v; 0^\circ)$ [Eq. (26)]. The displayed data are normalized such that each set sums up to 1, for comparison purpose.</figcaption></figure>

## 2. Photoemission yields and delays

A finer insight on the $v$-resolved photoemission anisotropy is provided in Fig. 8 which shows, for
<figure><img src="image_9.png" /><figcaption>FIG. 8. Vibrationally-resolved anisotropic photoemission from molecule B (left) and C (right). All data are plotted as functions of the photoelectron energy $\epsilon$. Left/right yield ratio (top) computed in the fully correlated simulations and in the BO framework, resp. $R(v)$ (symbols) and $\tilde{R}(v)$ (lines); stereo delays (bottom) computed in the fully correlated simulations and in the BO framework, resp. $\Delta\tau(v)$ (symbols) and $\Delta\tau_w(v)$ (lines). In all frames, the vibronic data are displayed for the first few significant vibrational channels, see inset keys for details. The displayed BO data were computed at a set of v-dependent optimal distances [$R_{opt}(v)$] for molecule B, and of 'vertical' distances [$R_{vrt}(v)$] for molecule C (see text and Tab. II). The BO data computed at the equilibrium distance ($R_{eq}$) are also shown for both molecules.</figcaption></figure>

538 molecules B [frame (a)] and C [frame (b)], the yield ratio

$$ \mathcal{R}(v) = \frac{\mathcal{Y}(v; 180^{\circ})}{\mathcal{Y}(v; 0^{\circ})} \qquad (23) $$

539 obtained with several values of $\omega_{xuv}$ in the 'complete' 540 simulations (symbols), as a function of the photoelectron 541 energy $\epsilon$. Their values, comprised between ~ 1.2 and 542 ~ 1.7, follow for each v the global trend observed with 543 molecule A, see Fig. 3(a). However, they display a clear 544 additional v-dependency for both molecules B and C.

545 We characterized the dynamics revealed in these 546 simulations consistently with the unresolved case, see 547 Eq. (11). Here, the channel-resolved stereo delays are 548 defined as

$$ \Delta\tau(v) = \tau(v; 180^{\circ}) - \tau(v; 0^{\circ}) - 2\frac{\langle x \rangle_0}{\sqrt{2\epsilon}} \qquad (24) $$

549 where the numerical TOF towards the virtual detector

$$ \tau(v; \theta_k) = \frac{\int t \times j_v(\theta_k, r_d, t) dt}{\int j_v(\theta_k, r_d, t) dt}, \qquad (25) $$

550 is now computed in each v-channel. In Eq. (24), one 551 should keep in mind that $\epsilon$ implicitly depends on v 552 through Eq. (20), for a given $\omega_{xuv}$. We have no- 553 tably checked that the vibrationally resolved stereo de- 554 lays $\Delta\tau(v)$ do not depend on $r_d$, in contrast with the

555 v-integrated data commented before for molecule B (Ta- 556 ble I). Indeed, for each vibrational channel, energy con- 557 servation ensures a symmetric asymptotic electron veloc- 558 ity.

561 The obtained v-dependent stereo delays are plotted as 562 symbols in Fig. 8 for molecules B [frame (c)] and C [frame 563 (d)]. We first note that they globally behave like the 564 stereo delays reported for molecule A in Fig. 3(b), with 565 a comparable order of magnitude which decays smoothly 566 towards 0 when the photoelectron energy $\epsilon$ increases. Be- 567 sides, a marked v-dependency is observed in molecule B, 568 reminiscent of the behavior of the corresponding yield 569 ratios [frame (a)]. In molecule C however, the values of 570 $\Delta\tau(v)$ appear to depend mostly on $\epsilon$ regardless of the 571 vibrational channel, since all of them follow a common 572 spectral evolution. This is consistent with the normalized 573 vibrational distributions displayed in Fig. 7(b), which are 574 almost identical in the two directions, as pointed out be- 575 fore.

## B. Simulations at fixed internuclear distances

577 In this section, we investigate the possibilities to re- 578 trieve the orientation-dependent data, including their v-
<figure><img src="image_10.png" /><figcaption>FIG. 8. Vibrationally-resolved anisotropic photoemission from molecule B (left) and C (right). All data are plotted as functions of the photoelectron energy $\epsilon$. Left/right yield ratio (top) computed in the fully correlated simulations and in the BO framework, resp. $R(v)$ (symbols) and $\tilde{R}(v)$ (lines); stereo delays (bottom) computed in the fully correlated simulations and in the BO framework, resp. $\Delta\tau(v)$ (symbols) and $\Delta\tau_w(v)$ (lines). In all frames, the vibronic data are displayed for the first few significant vibrational channels, see inset keys for details. The displayed BO data were computed at a set of v-dependent optimal distances [$R_{opt}(v)$] for molecule B, and of 'vertical' distances [$R_{vrt}(v)$] for molecule C (see text and Tab. II). The BO data computed at the equilibrium distance ($R_{eq}$) are also shown for both molecules.</figcaption></figure>

538 molecules B [frame (a)] and C [frame (b)], the yield ratio

$$R(v) = \frac{\mathcal{Y}(v; 180^{\circ})}{\mathcal{Y}(v; 0^{\circ})} \qquad (23)$$

539 obtained with several values of $\omega_{xuv}$ in the 'complete' 540 simulations (symbols), as a function of the photoelectron 541 energy $\epsilon$. Their values, comprised between ~ 1.2 and 542 ~ 1.7, follow for each v the global trend observed with 543 molecule A, see Fig. 3(a). However, they display a clear 544 additional v-dependency for both molecules B and C.

545 We characterized the dynamics revealed in these 546 simulations consistently with the unresolved case, see 547 Eq. (11). Here, the channel-resolved stereo delays are 548 defined as

$$\Delta\tau(v) = \tau(v; 180^{\circ}) - \tau(v; 0^{\circ}) - 2\frac{\langle x \rangle_0}{\sqrt{2\epsilon}} \qquad (24)$$

549 where the numerical TOF towards the virtual detector

$$\tau(v; \theta_k) = \frac{\int t \times j_v(\theta_k, r_d, t) dt}{\int j_v(\theta_k, r_d, t) dt}, \qquad (25)$$

550 is now computed in each v-channel. In Eq. (24), one 551 should keep in mind that $\epsilon$ implicitly depends on v 552 through Eq. (20), for a given $\omega_{xuv}$. We have no- 553 tably checked that the vibrationally resolved stereo de- 554 lays $\Delta\tau(v)$ do not depend on $r_d$, in contrast with the

555 v-integrated data commented before for molecule B (Ta- 556 ble I). Indeed, for each vibrational channel, energy con- 557 servation ensures a symmetric asymptotic electron veloc- 558 ity.

561 The obtained v-dependent stereo delays are plotted as 562 symbols in Fig. 8 for molecules B [frame (c)] and C [frame 563 (d)]. We first note that they globally behave like the 564 stereo delays reported for molecule A in Fig. 3(b), with 565 a comparable order of magnitude which decays smoothly 566 towards 0 when the photoelectron energy $\epsilon$ increases. Be- 567 sides, a marked v-dependency is observed in molecule B, 568 reminiscent of the behavior of the corresponding yield 569 ratios [frame (a)]. In molecule C however, the values of 570 $\Delta\tau(v)$ appear to depend mostly on $\epsilon$ regardless of the 571 vibrational channel, since all of them follow a common 572 spectral evolution. This is consistent with the normalized 573 vibrational distributions displayed in Fig. 7(b), which are 574 almost identical in the two directions, as pointed out be- 575 fore.

## B. Simulations at fixed internuclear distances

576 In this section, we investigate the possibilities to re- 577 trieve the orientation-dependent data, including their v-
<table><caption>TABLE II. Channel-dependent ionization energy $E_I$ [Eq. (18)] and representative internuclear distances $R_{vrt}$, and $R_{opt}$ [see text and Eq. (27)] for molecules *B* and *C* in the main vibrational ionization channels *v*.</caption><thead><tr><th colspan="5">(a) molecule <i>B</i></th><th colspan="8">(b) molecule <i>C</i></th></tr><tr><th><i>v</i></th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th></tr></thead><tbody><tr><td><i>E</i><sub>I</sub> (eV)</td><td>29.17</td><td>29.48</td><td>29.78</td><td>30.10</td><td>30.41</td><td>29.30</td><td>29.61</td><td>29.92</td><td>30.23</td><td>30.53</td><td>30.83</td><td>31.13</td><td>31.42</td><td>31.71</td></tr><tr><td><i>R</i><sub>vrt</sub> (Å)</td><td>1.076</td><td>1.106</td><td>1.145</td><td>1.201</td><td>1.319</td><td>1.160</td><td>1.135</td><td>1.115</td><td>1.098</td><td>1.082</td><td>1.066</td><td>1.055</td><td>1.041</td><td>1.031</td></tr><tr><td><i>R</i><sub>opt</sub> (Å)</td><td>1.084</td><td>1.113</td><td>1.146</td><td>1.203</td><td>1.248</td><td>1.174</td><td>1.148</td><td>1.124</td><td>1.105</td><td>1.091</td><td>1.079</td><td>1.069</td><td>1.055</td><td>1.038</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>FIG. 9. Vertical ionization potential $\tilde{E}_I(R)$ of molecules *B* (a) and *C* (b) as a function of *R* (blue full curve), used to identify the internuclear distances $R_{vrt}(v)$ fulfilling Eq. (27) out of the *v*-dependent ionization potentials $E_I(v)$.</figcaption></figure>

579 dependencies, by analyzing electronic continuum wave functions obtained in simulations with fixed nuclei.

581 The yield ratio and delays computed for both molecules in the BO framework at $R_{eq}$ according to Eq. (14) are displayed as a full black lines in Fig. 8. They lie near the data obtained for the dominant channels in the full TDSE simulations (between $v = 1$ and 2), and very close to the equivalent results obtained with *A* [see Fig. 3(a)]. Nevertheless, and quite obviously, simulations performed at a single fixed internuclear distance cannot reproduce the non-trivial *v*-dependency observed in the full simulations for the yield ratio and stereo delays of molecule *B*, and for the yield ratio of molecule *C*.

592 Below, we use the simulations performed on molecule *B* to identify and interpret *v*-dependent geometries that allow reproducing the results of the vibronic simulations with the fixed nuclei approach. We then assess our interpretation by applying it to fixed nuclei simulations on molecule *C*.

## 1. Optimal conformations for molecule *B*

599 By scanning through the support of the initial vibrational state $\xi_0(R)$ of molecule *B* ($R \sim 1.0 - 1.3$ Å), we identified empirically a set of channel-specific optimal distances $R_{opt}(v)$ for which both the yield ratio and the stereo delays in the BO approach reproduce the ones of the complete vibronic simulations. The values of $R_{opt}(v)$ obtained are reported in Tab. II(a). The orientation- and channel-resolved yields, now computed in the BO frame-

607 work at a given optimal conformation for each channel,

$$ \tilde{\mathcal{Y}}(v; \theta_k) = |A(R_{\text{opt}}(v); \theta_k)|^2 \times F(v), \quad (26) $$

608 are displayed for the left and right sides in columns iv and *v* of Fig. 7(a), respectively. They reproduce very well the asymmetric *v*-dependency of the actual yields $\mathcal{Y}(v; \theta_k)$ (columns ii and iii). The corresponding yield ratio $\tilde{R}(v)$ and stereo delays $\Delta\tau_w(v)$, extracted from the SCWF computed at the optimal conformation for each *v* are displayed as full lines in frames (a) and (c) of Fig. 8. They are in excellent agreement with the *v*-dependent data obtained in the fully correlated simulations for both molecules, apart from the $v = 4$, as will be discussed below.

## 2. Physical interpretation

620 The optimal molecular conformations can be interpreted by looking at the *R*-dependent ionization potential $\tilde{E}_I(R)$ [Eq. (4)], which for molecule *B* increases monotonically with *R* within the support of $\xi_0(R)$, see Fig. 9(a). On the same figure, we indicate the internuclear distances $R_{vrt}(v)$ at which $\tilde{E}_I$ matches the *v*-dependent ionization potential [Eq. (18)],

$$ \tilde{E}_I[R_{vrt}(v)] = E_I(v). \quad (27) $$

627 These “vertical” internuclear distances are reported in Table II(a), for comparison with the optimal distances found empirically. It turns out that $R_{opt}$ matches $R_{vrt}$
<table><caption>TABLE II. Channel-dependent ionization energy $E_I$ [Eq. (18)] and representative internuclear distances $R_{vrt}$, and $R_{opt}$ [see text and Eq. (27)] for molecules *B* and *C* in the main vibrational ionization channels *v*.</caption><thead><tr><th colspan="5">(a) molecule <i>B</i></th><th colspan="8">(b) molecule <i>C</i></th></tr><tr><th><i>v</i></th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th></tr></thead><tbody><tr><td><i>E</i><sub>I</sub> (eV)</td><td>29.17</td><td>29.48</td><td>29.78</td><td>30.10</td><td>30.41</td><td>29.30</td><td>29.61</td><td>29.92</td><td>30.23</td><td>30.53</td><td>30.83</td><td>31.13</td><td>31.42</td><td>31.71</td></tr><tr><td><i>R</i><sub>vrt</sub> (Å)</td><td>1.076</td><td>1.106</td><td>1.145</td><td>1.201</td><td>1.319</td><td>1.160</td><td>1.135</td><td>1.115</td><td>1.098</td><td>1.082</td><td>1.066</td><td>1.055</td><td>1.041</td><td>1.031</td></tr><tr><td><i>R</i><sub>opt</sub> (Å)</td><td>1.084</td><td>1.113</td><td>1.146</td><td>1.203</td><td>1.248</td><td>1.174</td><td>1.148</td><td>1.124</td><td>1.105</td><td>1.091</td><td>1.079</td><td>1.069</td><td>1.055</td><td>1.038</td></tr></tbody></table>

<figure><img src="image_11.png" /><figcaption>FIG. 9. Vertical ionization potential $\tilde{E}_I(R)$ of molecules *B* (a) and *C* (b) as a function of *R* (blue full curve), used to identify the internuclear distances $R_{vrt}(v)$ fulfilling Eq. (27) out of the *v*-dependent ionization potentials $E_I(v)$.</figcaption></figure>

579 dependencies, by analyzing electronic continuum wave functions obtained in simulations with fixed nuclei.

581 The yield ratio and delays computed for both molecules in the BO framework at $R_{eq}$ according to Eq. (14) are displayed as a full black lines in Fig. 8. They lie near the data obtained for the dominant channels in the full TDSE simulations (between $v = 1$ and 2), and very close to the equivalent results obtained with *A* [see Fig. 3(a)]. Nevertheless, and quite obviously, simulations performed at a single fixed internuclear distance cannot reproduce the non-trivial *v*-dependency observed in the full simulations for the yield ratio and stereo delays of molecule *B*, and for the yield ratio of molecule *C*.

592 Below, we use the simulations performed on molecule *B* to identify and interpret *v*-dependent geometries that allow reproducing the results of the vibronic simulations with the fixed nuclei approach. We then assess our interpretation by applying it to fixed nuclei simulations on molecule *C*.

## 1. Optimal conformations for molecule *B*

599 By scanning through the support of the initial vibrational state $\xi_0(R)$ of molecule *B* ($R \sim 1.0 - 1.3$ Å), we identified empirically a set of channel-specific optimal distances $R_{opt}(v)$ for which both the yield ratio and the stereo delays in the BO approach reproduce the ones of the complete vibronic simulations. The values of $R_{opt}(v)$ obtained are reported in Tab. II(a). The orientation- and channel-resolved yields, now computed in the BO frame-

607 work at a given optimal conformation for each channel,

$$ \tilde{\mathcal{Y}}(v; \theta_k) = |A(R_{\text{opt}}(v); \theta_k)|^2 \times F(v), \quad (26) $$

608 are displayed for the left and right sides in columns iv and *v* of Fig. 7(a), respectively. They reproduce very well the asymmetric *v*-dependency of the actual yields $\mathcal{Y}(v; \theta_k)$ (columns ii and iii). The corresponding yield ratio $\tilde{R}(v)$ and stereo delays $\Delta\tau_w(v)$, extracted from the SCWF computed at the optimal conformation for each *v* are displayed as full lines in frames (a) and (c) of Fig. 8. They are in excellent agreement with the *v*-dependent data obtained in the fully correlated simulations for both molecules, apart from the $v = 4$, as will be discussed below.

## 2. Physical interpretation

620 The optimal molecular conformations can be interpreted by looking at the *R*-dependent ionization potential $\tilde{E}_I(R)$ [Eq. (4)], which for molecule *B* increases monotonically with *R* within the support of $\xi_0(R)$, see Fig. 9(a). On the same figure, we indicate the internuclear distances $R_{vrt}(v)$ at which $\tilde{E}_I$ matches the *v*-dependent ionization potential [Eq. (18)],

$$ \tilde{E}_I[R_{vrt}(v)] = E_I(v). \quad (27) $$

627 These “vertical” internuclear distances are reported in Table II(a), for comparison with the optimal distances 628 found empirically. It turns out that $R_{opt}$ matches $R_{vrt}$
within ~1% in all the considered channels, except $v = 4$ (~5%).

Alternative insight from the wave function perspective can be gained by expressing the channel-resolved photoemission probability as

$$d_v(\theta_k) = \int \tilde{d}(R; \theta_k) O_v(R) dR \quad (28)$$

where $\tilde{d}(R; \theta_k)$ is the orientation-resolved electronic dipole computed at each $R$, and $O_v(R) = \chi_v(R)\xi_0(R)$ is the nuclear overlap function introduced in Eq. (21). Our approach consists in approximating the $R$-integration as $d_v(\theta_k) \approx \tilde{d}(R_{opt}(v); \theta_k)O_v(R_{opt}(v))$. We found that in each channel but $v = 4$, both $R_{opt}$ and $R_{vrt}$ lie in the vicinity of $R_{ctp}$, the ionic classical turning point nearest to $R_{eq}$ (see Appendix A). By essence, this turning point approximately fullfills Eq. (27) within or near the FC region, i.e., in the main open channels. Here, $R_{opt}(v) \approx R_{ctp}(v)$ therefore appears as a characteristic distance for the vibrational wave function $\chi_v(R)$ within the support of the initial state $\xi_0(R)$ while the $v$-independent dipole smoothly evolves with $R$. See Ref. [56] for a full treatment, articulated around Eq. (28), of anisotropic photoemission with vibrational resolution.

We repeated the same procedure for molecule C. The optimal internuclear distances, $R_{opt}(v)$, that we found empirically are reported in Table II(b), together with the ‘vertical’ ones, $R_{vrt}(v)$, extracted from the ionization potential displayed in Fig. 9(b). It is here also a monotonous function of $R$, with opposite variations than for molecule B (as mentioned earlier, molecule B contracts upon ionization, while molecule C expands upon ionization). For this molecule, a very good agreement is found between $R_{opt}(v)$, $R_{vrt}(v)$ and $R_{ctp}(v)$ in all open channels, see Appendix A. The BO data displayed as lines in Fig. 8(b) and (d) for molecule C were directly obtained with $R_{vrt}(v)$. The differences with the data obtained using $R_{opt}(v)$, not shown, would be hardly discernible on that figure. The agreement between the ‘complete’ simulations and the fixed-nuclei approach is here excellent in all considered channels – including the ionization ratios [frame (b)] which follow spectral trends that significantly depend on $v$.

These results provide a simple physical criterion for selecting a priori a set of optimal conformations which allow reproducing, in a fixed-nuclei framework, the details of $v$-dependent molecular photoionization dynamics. It is nevertheless likely that these specific distances can be representative of the $v$-dependent photoemission dynamics only when $\tilde{E}_I(R)$ varies monotonically with $R$. In particular for molecule B the $\tilde{E}_I(R)$ variations are less pronounced as $R$ grows towards the $R_{vrt}$ of the highest channels, within the studied range. Consistently, the data computed in the BO framework for molecule B turned out to be less sensitive to small $R$ variations beyond $R_{opt}(v = 3)$ – to the extent that we could not find an optimal distance for the $v = 4$ channel with the same accuracy as for other channels. This is particu-

<figure><img src="image_11.png" /><figcaption>FIG. 10. Photoelectron spectra recorded in RABBIT simulations, on the left (a) and right (b) sides of molecule B. The spectral range is focused on sideband 24, and each peak is associated with a specific ionization channel $v = 0-4$ (see labels). The different spectra were obtained with different values of the pump-probe delay $\tau_{xuv-ir}$ (see inset keys, where $T_0 = 2 \times \pi/\omega_0$ is the laser period).</figcaption></figure>

larly visible on the yield ratio [Fig. 8(a)], where the BO data never reach the ‘exact’ ones obtained for the highest ($v = 4$) channel, the displayed BO data corresponding to $R_{opt}(v = 4)$ being the closest one could get. This however concerns a minor channel [see Fig. 7(a)]. For molecule C, the narrower distributions of the data can be directly related to the sharper monotonic decay of $\tilde{E}_I$ when $R$ increases, which implies a narrower dispersion of $R_{vrt}(v)$ than for molecule B.

### C. Illustrative applications of the optimal conformations

In this last section, we address the relevance of the optimal conformations beyond the context in which they were identified. We investigate their relevance first in simulations of interferometric RABBIT measurements, and then to include electron-ion coherences during broadband 1-photon ionization.

The RABBIT scheme, together with the attosecond streaking approach, provide indirect ways to access photoemission dynamics<fnref n="2" />, the direct time domain approaches used in our simulations having no experimental equivalent with attosecond resolution (eg magnetic bottles have typical deadtimes of few 10 ns [57] and time-to-digit convertor resolutions in the 100 ps range [58]).

These are interferometric techniques, where the time domain information is extracted from the spectral variations of measured phases. Coherence is thus an essential issue in the design and exploitation of these approaches, the purpose of which is to reveal the fundamental dynamics of essentially quantum processes.

<footnote n="2">² See [27] and references therein for discussions dedicated to the links between RABBIT measurements and fundamental photoemission delays.</footnote>
within ~1% in all the considered channels, except $v = 4$ (~5%).

Alternative insight from the wave function perspective can be gained by expressing the channel-resolved photoemission probability as

$$d_v(\theta_k) = \int \tilde{d}(R; \theta_k) O_v(R) dR \quad (28)$$

where $\tilde{d}(R; \theta_k)$ is the orientation-resolved electronic dipole computed at each $R$, and $O_v(R) = \chi_v(R)\xi_0(R)$ is the nuclear overlap function introduced in Eq. (21). Our approach consists in approximating the $R$-integration as $d_v(\theta_k) \approx \tilde{d}(R_{opt}(v); \theta_k)O_v(R_{opt}(v))$. We found that in each channel but $v = 4$, both $R_{opt}$ and $R_{vrt}$ lie in the vicinity of $R_{ctp}$, the ionic classical turning point nearest to $R_{eq}$ (see Appendix A). By essence, this turning point approximately fullfills Eq. (27) within or near the FC region, i.e., in the main open channels. Here, $R_{opt}(v) \approx R_{ctp}(v)$ therefore appears as a characteristic distance for the vibrational wave function $\chi_v(R)$ within the support of the initial state $\xi_0(R)$ while the $v$-independent dipole smoothly evolves with $R$. See Ref. [56] for a full treatment, articulated around Eq. (28), of anisotropic photoemission with vibrational resolution.

We repeated the same procedure for molecule C. The optimal internuclear distances, $R_{opt}(v)$, that we found empirically are reported in Table II(b), together with the ‘vertical’ ones, $R_{vrt}(v)$, extracted from the ionization potential displayed in Fig. 9(b). It is here also a monotonous function of $R$, with opposite variations than for molecule B (as mentioned earlier, molecule B contracts upon ionization, while molecule C expands upon ionization). For this molecule, a very good agreement is found between $R_{opt}(v)$, $R_{vrt}(v)$ and $R_{ctp}(v)$ in all open channels, see Appendix A. The BO data displayed as lines in Fig. 8(b) and (d) for molecule C were directly obtained with $R_{vrt}(v)$. The differences with the data obtained using $R_{opt}(v)$, not shown, would be hardly discernible on that figure. The agreement between the ‘complete’ simulations and the fixed-nuclei approach is here excellent in all considered channels – including the ionization ratios [frame (b)] which follow spectral trends that significantly depend on $v$.

These results provide a simple physical criterion for selecting a priori a set of optimal conformations which allow reproducing, in a fixed-nuclei framework, the details of $v$-dependent molecular photoionization dynamics. It is nevertheless likely that these specific distances can be representative of the $v$-dependent photoemission dynamics only when $\tilde{E}_I(R)$ varies monotonically with $R$. In particular for molecule B the $\tilde{E}_I(R)$ variations are less pronounced as $R$ grows towards the $R_{vrt}$ of the highest channels, within the studied range. Consistently, the data computed in the BO framework for molecule B turned out to be less sensitive to small $R$ variations beyond $R_{opt}(v = 3)$ – to the extent that we could not find an optimal distance for the $v = 4$ channel with the same accuracy as for other channels. This is particu-

<figure><img src="image_12.png" /><figcaption>FIG. 10. Photoelectron spectra recorded in RABBIT simulations, on the left (a) and right (b) sides of molecule B. The spectral range is focused on sideband 24, and each peak is associated with a specific ionization channel $v = 0-4$ (see labels). The different spectra were obtained with different values of the pump-probe delay $\tau_{xuv-ir}$ (see inset keys, where $T_0 = 2 \times \pi/\omega_0$ is the laser period).</figcaption></figure>

larly visible on the yield ratio [Fig. 8(a)], where the BO data never reach the ‘exact’ ones obtained for the highest ($v = 4$) channel, the displayed BO data corresponding to $R_{opt}(v = 4)$ being the closest one could get. This however concerns a minor channel [see Fig. 7(a)]. For molecule C, the narrower distributions of the data can be directly related to the sharper monotonic decay of $\tilde{E}_I$ when $R$ increases, which implies a narrower dispersion of $R_{vrt}(v)$ than for molecule B.

### C. Illustrative applications of the optimal conformations

In this last section, we address the relevance of the optimal conformations beyond the context in which they were identified. We investigate their relevance first in simulations of interferometric RABBIT measurements, and then to include electron-ion coherences during broadband 1-photon ionization.

The RABBIT scheme, together with the attosecond streaking approach, provide indirect ways to access photoemission dynamics<fnref n="2" />, the direct time domain approaches used in our simulations having no experimental equivalent with attosecond resolution (eg magnetic bottles have typical deadtimes of few 10 ns [57] and time-to-digit convertor resolutions in the 100 ps range [58]).

These are interferometric techniques, where the time domain information is extracted from the spectral variations of measured phases. Coherence is thus an essential issue in the design and exploitation of these approaches, the purpose of which is to reveal the fundamental dynamics of essentially quantum processes.

<footnote n="2">² See [27] and references therein for discussions dedicated to the links between RABBIT measurements and fundamental photoemission delays.</footnote>
<figure><img src="image_12.png" /><figcaption>FIG. 11. Vibrationally-resolved anisotropic RABBIT simulations in molecule A (left), B (center) and C (right). Data derived from the analysis of sidebands 22 to 28 (left to right) plotted as functions of the photoelectron energy $\epsilon$. Left-right yield ratio (top) computed in the fully correlated simulations and in the BO framework, resp. $R_{\text{rab}}(v)$ (symbols) and $\tilde{R}_{\text{rab}}(v)$ (crosses+guidelines); stereo molecular delays (bottom) computed in the fully correlated simulations and in the BO framework, resp. $\Delta\tau_{\text{mol}}(v)$ (symbols) and $\Delta\tilde{\tau}_{\text{mol}}(v)$ (crosses+guideline). In each frame, the data are displayed for the first few significant vibrational channels, see inset keys for details. Consistently with the results shown in Fig. 8, the displayed BO data were obtained with the set of $v$-dependent optimal distances [$R_{\text{opt}}(v)$] identified in 1-photon simulations for molecule B, and of ‘vertical’ distances [$R_{\text{vrt}}(v)$] for molecule C (see text and Tab. II). The BO data computed at the equilibrium distance ($R_{\text{eq}}$) are also shown for both molecules. The data displayed in frame (f) are all multiplied by 4 for a better readability.</figcaption></figure>

## 1. Intra-channel coherences: RABBIT interferometry

Following a standard 800-nm RABBIT scheme [8, 14], we simulated photoemission from the three model molecules in presence of the fundamental IR field and a comb of its odd harmonics $HA_q$ in the XUV domain (orders $q$ from 21 to 29), with an adjustable pump-probe delay $\tau_{\text{xuv-ir}}$. We computed vibrationally- and orientation-resolved photoelectron spectra

$$ \sigma(v; \epsilon, \theta_k) = |a(v; \epsilon, \theta_k)|^2 \quad (29) $$

out of the final amplitudes corresponding to the ionized molecule, $a(v; \epsilon, \theta_k)$. The latter were obtained in the time-dependent simulations through a position-to-momentum Fourier transform of the channel wave packets $\varphi_v(x, t)$ accumulated in the asymptotic region at each time $t$ of the propagation (see Appendix C of [59]).

As expected, we obtained sideband (SB) peaks (orders 22 to 28) resulting from 2-photon XUV ± IR transitions, the intensities of which oscillate when $\tau_{\text{xuv-ir}}$ is tuned [60]. We used long enough pulses (80 fs for the XUV and for the IR) to resolve the vibrational channels in each sideband. This is illustrated in Fig. 10, which shows the vibrationally-structured, orientation

and delay-dependent SB$_{24}$ obtained with molecule B. Similarly to the 1-photon case, the relative peak intensities evidence a significantly larger photoemission probability towards the left (a) than towards the right (b). Note however that the orientation-dependency of the SB oscillations are too subtle to be resolved visually in this figure. This is consistent with the ultrashort timescale of the simulated non-resonant photoemission processes [27]. We thus performed a RABBIT analysis consisting in fitting the generic function [60]

$$ f(\tau_{\text{xuv-ir}}) = \alpha + \beta \cos(2\omega_0 \tau_{\text{xuv-ir}} - \phi) \quad (30) $$

to the $\tau_{\text{xuv-ir}}$ evolution of each sideband peak associated with a given vibrational channel $v$, and a given direction $\theta_k$. The fitting procedure gives access to the orientation and channel-resolved phases of the peak oscillations $\vartheta(v; \theta_k) = \phi$, as well as to the $\tau_{\text{xuv-ir}}$-averaged photoelectron yields $P(v; \theta_k) = \alpha$. Since the XUV components in our simulations were all synchronized (i.e., they carry no attochirp), $\vartheta(v; \theta_k)$ directly corresponds to the so-called “molecular phase”, see [27] and references therein.

We performed simulations both in the fully correlated approach, and in the BO framework. The main results are shown in Fig. 11, for SB$_{22}$ to SB$_{28}$ in molecules A
<figure><img src="image_13.png" /><figcaption>FIG. 11. Vibrationally-resolved anisotropic RABBIT simulations in molecule A (left), B (center) and C (right). Data derived from the analysis of sidebands 22 to 28 (left to right) plotted as functions of the photoelectron energy $\epsilon$. Left-right yield ratio (top) computed in the fully correlated simulations and in the BO framework, resp. $R_{\text{rab}}(v)$ (symbols) and $\tilde{R}_{\text{rab}}(v)$ (crosses+guidelines); stereo molecular delays (bottom) computed in the fully correlated simulations and in the BO framework, resp. $\Delta\tau_{\text{mol}}(v)$ (symbols) and $\Delta\tilde{\tau}_{\text{mol}}(v)$ (crosses+guideline). In each frame, the data are displayed for the first few significant vibrational channels, see inset keys for details. Consistently with the results shown in Fig. 8, the displayed BO data were obtained with the set of v-dependent optimal distances [$R_{\text{opt}}(v)$] identified in 1-photon simulations for molecule B, and of 'vertical' distances [$R_{\text{vrt}}(v)$] for molecule C (see text and Tab. II). The BO data computed at the equilibrium distance ($R_{\text{eq}}$) are also shown for both molecules. The data displayed in frame (f) are all multiplied by 4 for a better readability.</figcaption></figure>

## 1. Intra-channel coherences: RABBIT interferometry

Following a standard 800-nm RABBIT scheme [8, 14], we simulated photoemission from the three model molecules in presence of the fundamental IR field and a comb of its odd harmonics $HA_q$ in the XUV domain (orders $q$ from 21 to 29), with an adjustable pump-probe delay $\tau_{\text{xuv-ir}}$. We computed vibrationally- and orientation-resolved photoelectron spectra

$$ \sigma(v; \epsilon, \theta_k) = |a(v; \epsilon, \theta_k)|^2 \quad (29) $$

out of the final amplitudes corresponding to the ionized molecule, $a(v; \epsilon, \theta_k)$. The latter were obtained in the time-dependent simulations through a position-to-momentum Fourier transform of the channel wave packets $\varphi_v(x, t)$ accumulated in the asymptotic region at each time $t$ of the propagation (see Appendix C of [59]).

As expected, we obtained sideband (SB) peaks (orders 22 to 28) resulting from 2-photon XUV ± IR transitions, the intensities of which oscillate when $\tau_{\text{xuv-ir}}$ is tuned [60]. We used long enough pulses (80 fs for the XUV and for the IR) to resolve the vibrational channels in each sideband. This is illustrated in Fig. 10, which shows the vibrationally-structured, orientation

and delay-dependent SB$_{24}$ obtained with molecule B. Similarly to the 1-photon case, the relative peak intensities evidence a significantly larger photoemission probability towards the left (a) than towards the right (b). Note however that the orientation-dependency of the SB oscillations are too subtle to be resolved visually in this figure. This is consistent with the ultrashort timescale of the simulated non-resonant photoemission processes [27]. We thus performed a RABBIT analysis consisting in fitting the generic function [60]

$$ f(\tau_{\text{xuv-ir}}) = \alpha + \beta \cos(2\omega_0 \tau_{\text{xuv-ir}} - \phi) \quad (30) $$

to the $\tau_{\text{xuv-ir}}$ evolution of each sideband peak associated with a given vibrational channel $v$, and a given direction $\theta_k$. The fitting procedure gives access to the orientation and channel-resolved phases of the peak oscillations $\vartheta(v; \theta_k) = \phi$, as well as to the $\tau_{\text{xuv-ir}}$-averaged photoelectron yields $P(v; \theta_k) = \alpha$. Since the XUV components in our simulations were all synchronized (i.e., they carry no attochirp), $\vartheta(v; \theta_k)$ directly corresponds to the so-called "molecular phase", see [27] and references therein.

We performed simulations both in the fully correlated approach, and in the BO framework. The main results are shown in Fig. 11, for SB$_{22}$ to SB$_{28}$ in molecules A
758 (left), B (center) and C (right). The disconnected symbols in the upper and lower frames respectively correspond to the ratio of the $\tau_{xuv-ir}$-averaged yields,

$$ \mathcal{R}_{\text{rab}}(v) = \frac{P(v; 180^{\circ})}{P(v; 0^{\circ})} \qquad (31) $$

761 and the “stereo molecular delay” [12, 19]

$$ \Delta\tau_{\text{mol}}(v) = \frac{\vartheta(v; 180^{\circ})}{2 \times \omega_0} - \frac{\vartheta(v; 0^{\circ})}{2 \times \omega_0} \qquad (32) $$

762 obtained in the fully correlated approach for the main v channels. As in the 1-photon case, these quantities 763 follow standard general trends which can be summarized 764 by looking at the single-channel results of molecule A. Its 765 yield ratio $\mathcal{R}_{\text{rab}}(0)$ decays slowly, from ~ 1.4 down to ~ 766 1.3 over the covered energy range, which is reminiscent of 767 the 1-photon counterpart shown in Fig. 3(a). The stereo 768 molecular delay $\Delta\tau_{\text{mol}}(0)$ decays in magnitude, from -10 769 as at $\text{SB}_{22}$ down to near 0 as at $\text{SB}_{28}$. The relationship 770 between the latter and the 1-photon ionization delay is 771 the subject of the companion paper [27] and will not be 772 further discussed here. Beyond the trends commented 773 above, the results for molecule B and C displays clear v- 774 dependencies, both in the yield ratios and in the stereo 775 molecular delays, that are slightly more pronounced than 776 in the 1-photon case. 777

778 The crosses with linear guidelines correspond to the 779 equivalent quantities, resp. $\tilde{\mathcal{R}}_{\text{rab}}(v)$ and $\Delta\tilde{\tau}_{\text{mol}}(v)$, ob- 780 tained in time-dependent simulations at fixed internu- 781 clear distances. For molecule A, we used again the equi- 782 librium distance $R_{\text{eq}}$, which unsurprislingly perfectly re- 783 produces the results of the ‘complete’ simulations. Note 784 that these fixed nuclei data appear in Fig. 6 of [27]. 785 For molecule B, we additionally show the BO results ob- 786 tained with the v-dependent optimal distances $R_{\text{opt}}$ iden- 787 tified in the 1-photon simulations (see Table II), while for 788 molecule C we used the vertical internuclear distances 789 $R_{\text{vrt}}(v)$. The data obtained at $R_{\text{eq}}$ both with molecules B 790 and C are similar to the ones obtained with molecule A. 791 Regarding the results provided at optimal conformations, 792 one clearly sees that the BO framework fails to reproduce 793 the yield ratio for molecule B in the minority $v = 4$ chan- 794 nel [frame (b)], already discussed in the 1-photon case. 795 Apart from this, the data obtained in the BO frame- 796 work, including the stereo molecular delays in all chan- 797 nels for both molecules are in excellent agreement with 798 the ones obtained in the ‘complete’ simulations. This 799 illustrates the capacities of BO approaches with appro- 800 priate internuclear distances to simulate vibrationally- 801 resolved RABBIT measurements with high fidelity.

## 2. Inter-channel coherences: broadband photoemission

802 As a last case, we now highlight the capacities of fixed- 803 nuclei approaches to keep track of inter-channel coher- 804 ences, when reconstructing the complete vibronic wave- 805 packet out of a set of simulations performed at the v-

<figure><img src="image_13.png" /><figcaption>FIG. 12. Photoelectron spectrum recorded on the left (a) and right (b) sides of molecule B, upon single-photon ionization with a 8 fs pulse of 37.23 eV central frequency ($\text{HA}_{24}$ of a 800 nm field). Thick coloured lines: v-resolved spectra ($v = 0-4$, see labels); Thin grey lines: v-integrated spectra.</figcaption></figure>

807 dependent optimal conformations. To this end, we sim- 808 ulated the photoionization of molecules B and C with a 809 broad XUV pulse of central frequency $\omega_{\text{XUV}} = 37.23$ eV 810 ($\text{HA}_{24}$ of a 800 nm laser) and a duration of 8 fs (3 funda- 811 mental IR cycles), in the 1-photon perturbative regime. 812 The bandwidth of this pulse overlaps few vibrational 813 channels in the photoelectron spectra, as shown in Fig. 12 814 for molecule B. One therefore partially loses the vibra- 815 tional resolution when looking at the v-integrated spec- 816 tra, in contrast to the results obtained with narrower 817 pulses, see for instance Fig. 10.

818 From the final channel-selective amplitudes in the con- 819 tinuum $a(v; \varepsilon, \theta_k)$, we computed the reduced density ma- 820 trix (RDM) of the ion in the final state as

$$ \rho_{\text{ion}}(v, v') = \sum_{\theta_k=0^{\circ},180^{\circ}} \int_{\Delta} d\varepsilon [a(v; \varepsilon, \theta_k)]^* a(v'; \varepsilon, \theta_k) \qquad (33) $$

821 where $[ ]^*$ denotes the complex conjugate and the spec- 822 tral integration range $\Delta$ is restricted to the overall sup- 823 port of the photoelectron spectrum. The modulus of 824 the RDM computed in the complete simulations for 825 molecules B and C in the main vibrational channels are 826 respectively shown in frames (a) and (c) of Fig. 13. For 827 both molecules, we observe non vanishing off-diagonal el- 828 ements which are signatures of coherences between over- 829 lapping channels ($|v' - v| \lesssim 3$).

830 In the fixed nuclei simulations, we reconstructed the fi- 831 nal RDM of the ion out of the amplitudes $\tilde{a}(R_{\text{opt}}(v); \varepsilon, \theta_k)$ 832 obtained in time-dependent simulations using the v- 833 dependent vertical internuclear distances $R_{\text{vrt}}(v)$ for both 834 molecules. The results are shown in frames (b) and (d) 835 of Fig. 13 for molecules B and C respectively. The over- 836 all agreement with the complete simulations is excellent 837 both for the populations (diagonal elements) and the co- 838 herences. To further evaluate the quality of these recon- 839 structed vibrational RDM, we computed in each simula-
758 (left), B (center) and C (right). The disconnected symbols in the upper and lower frames respectively correspond to the ratio of the $\tau_{xuv-ir}$-averaged yields,

$$ \mathcal{R}_{\text{rab}}(v) = \frac{P(v; 180^{\circ})}{P(v; 0^{\circ})} \qquad (31) $$

761 and the “stereo molecular delay” [12, 19]

$$ \Delta\tau_{\text{mol}}(v) = \frac{\vartheta(v; 180^{\circ})}{2 \times \omega_0} - \frac{\vartheta(v; 0^{\circ})}{2 \times \omega_0} \qquad (32) $$

762 obtained in the fully correlated approach for the main v channels. As in the 1-photon case, these quantities 763 follow standard general trends which can be summarized 764 by looking at the single-channel results of molecule A. Its 765 yield ratio $\mathcal{R}_{\text{rab}}(0)$ decays slowly, from ~ 1.4 down to ~ 766 1.3 over the covered energy range, which is reminiscent of 767 the 1-photon counterpart shown in Fig. 3(a). The stereo 768 molecular delay $\Delta\tau_{\text{mol}}(0)$ decays in magnitude, from ~10 769 as at $\text{SB}_{22}$ down to near 0 as at $\text{SB}_{28}$. The relationship 770 between the latter and the 1-photon ionization delay is 771 the subject of the companion paper [27] and will not be 772 further discussed here. Beyond the trends commented 773 above, the results for molecule B and C displays clear v- 774 dependencies, both in the yield ratios and in the stereo 775 molecular delays, that are slightly more pronounced than 776 in the 1-photon case. 777

778 The crosses with linear guidelines correspond to the 779 equivalent quantities, resp. $\tilde{\mathcal{R}}_{\text{rab}}(v)$ and $\Delta\tilde{\tau}_{\text{mol}}(v)$, ob- 780 tained in time-dependent simulations at fixed internu- 781 clear distances. For molecule A, we used again the equi- 782 librium distance $R_{\text{eq}}$, which unsurprislingly perfectly re- 783 produces the results of the ‘complete’ simulations. Note 784 that these fixed nuclei data appear in Fig. 6 of [27]. 785 For molecule B, we additionally show the BO results ob- 786 tained with the v-dependent optimal distances $R_{\text{opt}}$ iden- 787 tified in the 1-photon simulations (see Table II), while for 788 molecule C we used the vertical internuclear distances 789 $R_{\text{vrt}}(v)$. The data obtained at $R_{\text{eq}}$ both with molecules B 790 and C are similar to the ones obtained with molecule A. 791 Regarding the results provided at optimal conformations, 792 one clearly sees that the BO framework fails to reproduce 793 the yield ratio for molecule B in the minority $v = 4$ chan- 794 nel [frame (b)], already discussed in the 1-photon case. 795 Apart from this, the data obtained in the BO frame- 796 work, including the stereo molecular delays in all chan- 797 nels for both molecules are in excellent agreement with 798 the ones obtained in the ‘complete’ simulations. This 799 illustrates the capacities of BO approaches with appro- 800 priate internuclear distances to simulate vibrationally- 801 resolved RABBIT measurements with high fidelity.

## 2. Inter-channel coherences: broadband photoemission

802 As a last case, we now highlight the capacities of fixed- 803 nuclei approaches to keep track of inter-channel coher- 804 ences, when reconstructing the complete vibronic wave- 805 packet out of a set of simulations performed at the v-

<figure><img src="image_14.png" /><figcaption>FIG. 12. Photoelectron spectrum recorded on the left (a) and right (b) sides of molecule B, upon single-photon ionization with a 8 fs pulse of 37.23 eV central frequency ($\text{HA}_{24}$ of a 800 nm field). Thick coloured lines: v-resolved spectra ($v = 0-4$, see labels); Thin grey lines: v-integrated spectra.</figcaption></figure>

807 dependent optimal conformations. To this end, we sim- 808 ulated the photoionization of molecules B and C with a 809 broad XUV pulse of central frequency $\omega_{\text{XUV}} = 37.23$ eV 810 ($\text{HA}_{24}$ of a 800 nm laser) and a duration of 8 fs (3 funda- 811 mental IR cycles), in the 1-photon perturbative regime. 812 The bandwidth of this pulse overlaps few vibrational 813 channels in the photoelectron spectra, as shown in Fig. 12 814 for molecule B. One therefore partially loses the vibra- 815 tional resolution when looking at the v-integrated spec- 816 tra, in contrast to the results obtained with narrower 817 pulses, see for instance Fig. 10.

818 From the final channel-selective amplitudes in the con- 819 tinuum $a(v; \varepsilon, \theta_k)$, we computed the reduced density ma- 820 trix (RDM) of the ion in the final state as

$$ \rho_{\text{ion}}(v, v') = \sum_{\theta_k=0^{\circ},180^{\circ}} \int_{\Delta} d\varepsilon [a(v; \varepsilon, \theta_k)]^* a(v'; \varepsilon, \theta_k) \qquad (33) $$

821 where $[ ]^*$ denotes the complex conjugate and the spec- 822 tral integration range $\Delta$ is restricted to the overall sup- 823 port of the photoelectron spectrum. The modulus of 824 the RDM computed in the complete simulations for 825 molecules B and C in the main vibrational channels are 826 respectively shown in frames (a) and (c) of Fig. 13. For 827 both molecules, we observe non vanishing off-diagonal el- 828 ements which are signatures of coherences between over- 829 lapping channels ($|v' - v| \lesssim 3$).

830 In the fixed nuclei simulations, we reconstructed the fi- 831 nal RDM of the ion out of the amplitudes $\tilde{a}(R_{\text{opt}}(v); \varepsilon, \theta_k)$ 832 obtained in time-dependent simulations using the v- 833 dependent vertical internuclear distances $R_{\text{vrt}}(v)$ for both 834 molecules. The results are shown in frames (b) and (d) 835 of Fig. 13 for molecules B and C respectively. The over- 836 all agreement with the complete simulations is excellent 837 both for the populations (diagonal elements) and the co- 838 herences. To further evaluate the quality of these recon- 839 structed vibrational RDM, we computed in each simula-
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>FIG. 13. Photoionization of molecule B (top) and C (bottom) with a broadband XUV pulse (same simulations as Fig. 12). Modulus of the final ion's reduced density matrix $ρ_{ion}(v, v')$ [Eq. (33)] in the complete simulations (left) and in simulations at fixed conformations $R_{vrt}(v)$ (right). The values of the matrix elements' modulii are provided in the lower triangles (where the compact notation $m^y$ stands for $m \times 10^y$). The matrices are displayed for the main open channels. Each one is normalized to have a trace equal to 1, consistently with the data displayed in Fig. 7. The value of the purity $p$ [Eq. (34)] is indicated at the top of each matrix.</figcaption></figure>

tion the purity of the final state

$$p = \frac{\text{tr}(\rho_{\text{ion}}^2)}{\text{tr}(\rho_{\text{ion}})^2}, \qquad (34)$$

840 where tr() denotes the trace operation. The purity values

841 are indicated in Fig. 13 at the top of each matrix<fnref n="3" />. The

842 purity in fixed-nuclei results accurately match the ‘ex-

843 act’ ones, within less than 1%, for both molecules. It is

844 worth noting that, when using the $R_{opt}(v)$ conformations

845 rather than the $R_{vrt}(v)$ ones, the fixed-nuclei simulations

847 agree with the ‘exact’ ones with a ~ 10% accuracy only

848 (results not shown). On the one hand, this underlines

849 the sensitivity of the computed purity with respect to

850 the conformations used in the fixed nuclei approach. On

851 the other hand, it emphasizes the remarkable efficiency

852 of the combined vertical conformations to reproduce the

853 exact interchannel coherent dynamics.

## V. SUMMARY AND CONCLUSION

855 We studied numerically how nuclear motion affects the

856 dynamics of orientation-resolved photoemission in asym-

857 metric diatomic model molecules presenting minimal vi-

858 bronic couplings. We considered near-threshold photoe-

859 mission in absence of any resonance, where the so-called

860 stereo Wigner delays amount to few tens of attoseconds

861 or less.

<footnote n="3">³ We verified that the normalized matrices and the purities remain practically unchanged when looking at the $\theta$-dependent RDM obtained as in Eq. (33) but without angular integration (results not shown). This is consistent with the weak $\theta_k$-dependency of the normalized probabilities displayed in Fig. 7.</footnote>
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>FIG. 13. Photoionization of molecule B (top) and C (bottom) with a broadband XUV pulse (same simulations as Fig. 12). Modulus of the final ion's reduced density matrix $ρ_{ion}(v, v')$ [Eq. (33)] in the complete simulations (left) and in simulations at fixed conformations $R_{vrt}(v)$ (right). The values of the matrix elements' modulii are provided in the lower triangles (where the compact notation $m^y$ stands for $m \times 10^y$). The matrices are displayed for the main open channels. Each one is normalized to have a trace equal to 1, consistently with the data displayed in Fig. 7. The value of the purity $p$ [Eq. (34)] is indicated at the top of each matrix.</figcaption></figure>

tion the purity of the final state

$$p = \frac{\text{tr}(\rho_{\text{ion}}^2)}{\text{tr}(\rho_{\text{ion}})^2}, \qquad (34)$$

840 where tr() denotes the trace operation. The purity values

841 are indicated in Fig. 13 at the top of each matrix<fnref n="3" />. The

842 purity in fixed-nuclei results accurately match the ‘ex-

843 act’ ones, within less than 1%, for both molecules. It is

844 worth noting that, when using the $R_{opt}(v)$ conformations

845 rather than the $R_{vrt}(v)$ ones, the fixed-nuclei simulations

847 agree with the ‘exact’ ones with a ~ 10% accuracy only

848 (results not shown). On the one hand, this underlines

849 the sensitivity of the computed purity with respect to

850 the conformations used in the fixed nuclei approach. On

851 the other hand, it emphasizes the remarkable efficiency

852 of the combined vertical conformations to reproduce the

853 exact interchannel coherent dynamics.

## V. SUMMARY AND CONCLUSION

855 We studied numerically how nuclear motion affects the

856 dynamics of orientation-resolved photoemission in asym-

857 metric diatomic model molecules presenting minimal vi-

858 bronic couplings. We considered near-threshold photoe-

859 mission in absence of any resonance, where the so-called

860 stereo Wigner delays amount to few tens of attoseconds

861 or less.

<footnote n="3">³ We verified that the normalized matrices and the purities remain practically unchanged when looking at the $θ$-dependent RDM obtained as in Eq. (33) but without angular integration (results not shown). This is consistent with the weak $θ_k$-dependency of the normalized probabilities displayed in Fig. 7.</footnote>
We have shown that the intrinsic molecular asymmetry results in an anisotropic electron-ion momentum sharing which, as slight as it may be, may prevent us from assigning unambiguously a stereo Wigner delay to the channel-averaged photoemission process. Indeed, a small asymmetry in the average photoelectron energy leads to stereo delay values that diverge when the (virtual) detection distance increases. This is circumvented by considering vibrationally-resolved photoemission. However, on the theory side, comprehensive time-dependent vibronic simulations of molecules interacting with external fields are restricted to smaller molecules such as H₂ with a limited range of physical and numerical parameters, or to simplified low dimensional model molecules such as the ones used in the present work.

Therefore, we investigated ways of retrieving the vibrationally-resolved photoemission dynamics revealed by the 'complete' vibronic simulations, out of more standard and broadly applicable time-independent approaches with fixed nuclei. We found empirically that each vibrational channel could be assigned an effective internuclear distance that reproduces the channel-resolved orientation-dependent photoemission yields and delays with a good accuracy. Furthermore, we identified a physical criterion, relying on the molecule's ionization energies, that allows selecting a priori the v-dependent effective internuclear distances: It corresponds to the distance for which the vertical ionization potential matches the exact channel-dependent ionization potential. Identifying such unique effective molecular conformations is expected to work efficiently as long as the vertical ionization potential varies significantly when the molecular conformation is changed. Retrieving photoemission dynamics with vibrational resolution out of fixed nuclei simulation is also expected to work as long as photoemission takes place with little vibronic correlation, typically in smooth continua with sufficiently separated ionic vibrational levels.

Finally, we assessed the relevance of the fixed-nuclei approach beyond the context in which we identified the effective internuclear distances. We first showed that it could be used to accurately simulate anisotropic vibrationally-resolved RABBIT interferometry, both in terms of phase and amplitudes. Then, we investigated the fixed nuclei approach capacities to account for interchannel coherences in broadband photoionization of our model molecules, where the different channels significantly overlap. We showed that the fixed-nuclei approach could satisfactorily reproduce both the ion's reduced density matrix and purity in the final state. This approach is thus of particular interest to model attosecond resolved photoemission dynamics in benchmark molecules without the need to resort to full-fledged vibronic approaches [56]. It comes as a complement to existing approaches tackling the issues of anisotropy in molecular photoemission, but where the de-

tails of the photoelectron-ion interactions are neglected, see eg [61, 62]. It could be applied to simulate attosecond time-resolved interferometry, highly non-linear processes such as strong field ionization [63] or molecular high-order harmonic generation [46, 64–67], and to investigate ultrafast decoherence processes in molecules [35, 38, 44] of crucial importance in attochemistry.

## ACKNOWLEDGMENTS

This research received the financial support of the French National Research Agency through Grants No. ANR-15-CE30-0001-CIMBAAD and ANR-20-CE30-0007-DECAP.

## Appendix A: Vibrational states

### 1. Vibrational basis

Figure 14 shows the ionic vibrational wave functions $\chi_v(R)$ of the main open channels in single-photon ionization of model molecules $B$ [frame (a)] and $C$ [frame (b)]. These states (with additional minor channels) serve as basis functions for the numerical implementation of the TDSE solver in the complete vibronic simulations, see Section IIIB1. They are also physically relevant states for the study of vibrationally resolved photoemission. The ground vibrational wave functions of the neutral molecules, $\xi_0(R)$, are also shown.

### 2. Characteristic internuclear distances

In each channel, the optimal distance $R_{opt}$ (see Section IVB) is indicated with a solid vertical line. The 'vertical' distance $R_{vrt}$ fullfilling Eq. (27) is indicated with a dotted line. Moreover, the distance $R_{ctp}(v)$ lying in the FC region and fullfilling

$$V_{N-N}(R_{ctp}(v)) = \mathcal{E}^{+}(v), \quad (A1)$$

i.e., the classical turning point closest to $R_{eq}$, is indicated with a vertical dashed line. The agreement between the three sets of characteristic distances is very good (except for molecule $B$, $v = 4$, as discussed in the main text).

Given the definitions of $E_I(v)$ [Eq. (18)] and $\tilde{E}_I(R)$ [Eq. (4)] on the one hand, and the equations fullfilled by $R_{vrt}$ and $R_{ctp}$ respectively on the other hand, we can expect these two characteristic distances to be nearly equal when the potential energy in the neutral is close to the vibrational ground state, $V_{N-N}(R) + \epsilon_0(R) \approx \mathcal{E}_0^{(0)}$, corresponding to the FC region. Therefore $R_{vrt} \approx R_{ctp}$ should apply to the main open ionization channels, as observed empirically in our simulations.
We have shown that the intrinsic molecular asymmetry results in an anisotropic electron-ion momentum sharing which, as slight as it may be, may prevent us from assigning unambiguously a stereo Wigner delay to the channel-averaged photoemission process. Indeed, a small asymmetry in the average photoelectron energy leads to stereo delay values that diverge when the (virtual) detection distance increases. This is circumvented by considering vibrationally-resolved photoemission. However, on the theory side, comprehensive time-dependent vibronic simulations of molecules interacting with external fields are restricted to smaller molecules such as H₂ with a limited range of physical and numerical parameters, or to simplified low dimensional model molecules such as the ones used in the present work.

Therefore, we investigated ways of retrieving the vibrationally-resolved photoemission dynamics revealed by the 'complete' vibronic simulations, out of more standard and broadly applicable time-independent approaches with fixed nuclei. We found empirically that each vibrational channel could be assigned an effective internuclear distance that reproduces the channel-resolved orientation-dependent photoemission yields and delays with a good accuracy. Furthermore, we identified a physical criterion, relying on the molecule's ionization energies, that allows selecting a priori the v-dependent effective internuclear distances: It corresponds to the distance for which the vertical ionization potential matches the exact channel-dependent ionization potential. Identifying such unique effective molecular conformations is expected to work efficiently as long as the vertical ionization potential varies significantly when the molecular conformation is changed. Retrieving photoemission dynamics with vibrational resolution out of fixed nuclei simulation is also expected to work as long as photoemission takes place with little vibronic correlation, typically in smooth continua with sufficiently separated ionic vibrational levels.

Finally, we assessed the relevance of the fixed-nuclei approach beyond the context in which we identified the effective internuclear distances. We first showed that it could be used to accurately simulate anisotropic vibrationally-resolved RABBIT interferometry, both in terms of phase and amplitudes. Then, we investigated the fixed nuclei approach capacities to account for interchannel coherences in broadband photoionization of our model molecules, where the different channels significantly overlap. We showed that the fixed-nuclei approach could satisfactorily reproduce both the ion's reduced density matrix and purity in the final state. This approach is thus of particular interest to model attosecond resolved photoemission dynamics in benchmark molecules without the need to resort to full-fledged vibronic approaches [56]. It comes as a complement to existing approaches tackling the issues of anisotropy in molecular photoemission, but where the de-

tails of the photoelectron-ion interactions are neglected, see eg [61, 62]. It could be applied to simulate attosecond time-resolved interferometry, highly non-linear processes such as strong field ionization [63] or molecular high-order harmonic generation [46, 64–67], and to investigate ultrafast decoherence processes in molecules [35, 38, 44] of crucial importance in attochemistry.

## ACKNOWLEDGMENTS

This research received the financial support of the French National Research Agency through Grants No. ANR-15-CE30-0001-CIMBAAD and ANR-20-CE30-0007-DECAP.

## Appendix A: Vibrational states

### 1. Vibrational basis

Figure 14 shows the ionic vibrational wave functions $\chi_v(R)$ of the main open channels in single-photon ionization of model molecules $B$ [frame (a)] and $C$ [frame (b)]. These states (with additional minor channels) serve as basis functions for the numerical implementation of the TDSE solver in the complete vibronic simulations, see Section IIIB1. They are also physically relevant states for the study of vibrationally resolved photoemission. The ground vibrational wave functions of the neutral molecules, $\xi_0(R)$, are also shown.

### 2. Characteristic internuclear distances

In each channel, the optimal distance $R_{\text{opt}}$ (see Section IVB) is indicated with a solid vertical line. The 'vertical' distance $R_{\text{vrt}}$ fullfilling Eq. (27) is indicated with a dotted line. Moreover, the distance $R_{\text{ctp}}(v)$ lying in the FC region and fullfilling

$$V_{\text{N-N}}(R_{\text{ctp}}(v)) = \mathcal{E}^{+}(v), \quad (\text{A1})$$

i.e., the classical turning point closest to $R_{\text{eq}}$, is indicated with a vertical dashed line. The agreement between the three sets of characteristic distances is very good (except for molecule $B$, $v = 4$, as discussed in the main text).

Given the definitions of $E_I(v)$ [Eq. (18)] and $\tilde{E}_I(R)$ [Eq. (4)] on the one hand, and the equations fullfilled by $R_{\text{vrt}}$ and $R_{\text{ctp}}$ respectively on the other hand, we can expect these two characteristic distances to be nearly equal when the potential energy in the neutral is close to the vibrational ground state, $V_{\text{N-N}}(R) + \varepsilon_0(R) \approx \mathcal{E}_0^{(0)}$, corresponding to the FC region. Therefore $R_{\text{vrt}} \approx R_{\text{ctp}}$ should apply to the main open ionization channels, as observed empirically in our simulations.
<figure><img src="image_16.png" /><figcaption>FIG. 14. Final vibrational wave functions $\chi_v(R)$ in the main ionization channel, as a function of the internuclear distance $R$, for model molecules $\mathcal{B}$ (a) and $\mathcal{C}$ (b). The vibrational wave function $\xi_0(R)$ of the neutral ground state is also shown, see labels. In each channel, the distances $R_{\text{opt}}(v)$, $R_{\text{vrt}}(v)$, and $R_{\text{ctp}}(v)$ are indicated by a solid line, a dotted line and a dashed line, respectively.</figcaption></figure>

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