Advances in modeling attosecond electron dynamics in molecular photoionization

The dramatic progress of experimental attosecond science has called for the development of new theoretical and computational tools capable of accurately model the correlated electron dynamics triggered by attosecond molecular photoionization. We describe the nature and the main outcome of this development, with particular focus on the B‐spline ADC and RCS‐ADC ab initio methods.


| INTRODUCTION
Molecular photoionization is the process during which, upon interaction with radiation, the many-electron quantum system breaks up and a photoelectron drifts away from the parent ion.Theoretical modeling of molecular photoionization is fundamental to our understanding of several physical and chemical phenomena ranging from radiation damage 1,2 and high-order harmonic generation (HHG) 3 to atmospheric and interstellar plasma formation. 4Moreover, in recent years the detailed understanding of molecular photoionization dynamics has proven key to attosecond science, an emergent and rapidly growing field of study dealing with electron dynamics in matter on the attosecond (10 À18 s) timescale. 5In fact, the availability of ionizing attosecond laser pulses, based on the impressive development of novel coherent light sources, such as HHG 6 and x-ray free electron lasers (FELs), 7 over the last two decades, now allows one to photo-ionize a molecular system over a broad range of energies and produce, in principle arbitrary, coherent linear superpositions of electronic eigenstates in the molecular cation.The creation of such nonstationary electronic states upon photoionization results in novel ultrafast coherent many-electron dynamics: the kind that was not previously accessible with more traditional photoionization experiments with synchrotron light, that mostly addressed the properties of individual, incoherently populated eigenstates of the ionized system.Study of these coherent superpositions by means of attosecond pump-probe spectroscopy techniques [8][9][10] is at the heart of modern attosecond science.One particularly important example of such ultrafast dynamics initiated by molecular photoionization is the so-called molecular hole migration, 11 a process due to electron correlation where the positive charge created in the cation migrates across the molecular backbone, on a sub-to few-femtosecond (fs) timescale. 12On longer timescales, this purely electronic dynamics is potentially subject to decoherence 13,14 due to the couplings of the electrons to the slower motion of the nuclei, which act as a bath and eventually leads to the final localization of the positive hole charge.Since molecular bonds are more likely to break where the electronic hole is localized, ultrafast charge redistribution underpinned by electronic coherence in the cation opens the way to photoionization-induced, charge-directed reactivity on attosecond timescales, and ultimately to the possibility of steering the molecular reaction dynamics at a very early stage of its quantum evolution by using the electronic degrees of freedom before the dephasing occurs.
The rapid progress of attosecond science has been driven, on the one hand, by formidable development of cuttingedge attosecond pump-probe spectroscopy experimental techniques, and, on the other hand, it has been supported by a matching increase in our theoretical capabilities.A close integration of theoretical and experimental efforts is indeed essential to realize the full potential impact of attosecond science.Because of the complexity of the photoionization dynamics triggered in molecular systems, the capability to model laser-induced many-electron dynamics from first principles is absolutely key to our ability to predict new physical phenomena.Theoretical guidance is also crucial to design attosecond experiments, whose interpretation also relies heavily on theoretical support.This has fuelled the development, in the last decade, of a variety of new theoretical and computational methods for modeling ultrafast photoionization electron dynamics in molecular systems.The ultimate goal of this class of methods is to predict, as accurately as possible, the state of the molecular cation produced by attosecond photoionization, as well as the ensuing ultrafast dynamics characterized by the onset and decay of the quantum electronic coherences forming the superposition states.In order to fully achieve this goal, the new advanced computational tools need to combine modern many-electron theory with the description of the electronic continuum as well as, in principle, nuclear dynamics theory.In this paper we review the recent theoretical advances in describing the attosecond many-electron dynamics in photoionized molecular systems, with focus on the modeling of purely electronic dynamics (thus with no inclusion of the nuclear dynamics) and on the B-spline ADC 15,16 and B-spline RCS-ADC 17 methods, and we also outline possible future directions of methodological development.

| Key goal quantities for theoretical molecular attosecond science
Prediction of the quantum state of the parent ion resulting from attosecond molecular photoionization requires one to notice that, differently from the total N-electron molecular system, the cationic sub-system, as well as the photoelectron sub-system, can not necessarily be described by a wave function, but it is in general represented by a reduced density matrix obtained by tracing the degrees of freedom pertaining to the photoelectron sub-system out from the total density matrix of the N-electron system.
The trace operation of Equation ( 1) yields a reduced ionic density matrix (R-IDM) ranging in principle from a completely mixed state, in the fully entangled case, to a pure quantum state in the case of maximum coherence.In any given basis set of states describing the (N-1)-electron cationic system, diagonal matrix elements describe the populations of each state , while the degrees of quantum electronic coherence, 0 ≤ G m,n ≤ 1, between any two of such states is related to the corresponding off-diagonal matrix element: Theoretical prediction of the quantum coherences of Equation ( 2) 18 is essential for our fundamental understanding of the physics underlying photochemical transformations at ultrashort timescales, as it allows us to reconstruct the ultrafast coherent electron dynamics that can be triggered by the photoionization process in the parent molecular ion.Key theoretical targets also include the unveiling of the specific mechanisms that govern the onset of ionic coherence in ultrafast outer-and inner-shell ionization of molecular systems, as well as a deeper understanding of the level of entanglement produced in the photoionization process and the consequent role of measurement on the observation of the ensuing dynamics.In order to accurately calculate the R-IDM of Equation ( 1), the novel theoretical methods need to explicitly model the photoionization process and provide a representation of the full N-electron system, starting from the initial state of the neutral molecule and including a description of the photoelectron degrees of freedom.This can be done, for example, by tracking the time evolution of the N-electron wavefunction, whose ionized part can be written in general as where j Φ NÀ1 j e a i is a product state of stationary eigenstates of the ionic j Φ NÀ1 j i and photoelectron j e a i sub-systems, with indices j,k,l,… and a,b,c,… for their quantum numbers, respectively.

| Theoretical challenges to overcome
Until approximately 10 years ago, general ab initio frameworks capable of answering fundamental questions on coherence and entanglement in ultrafast photoionization dynamics were lacking.In fact, despite the importance of ionphotoelectron entanglement had already been highlighted by a few pioneering experimental works such as, 19 which had shown that the observation of hole localization or delocalization upon inner-shell ionization of molecular nitrogen depends on how the entangled Bell state created by Auger decay is detected by the measurement, the photoelectron degree of freedom had been neglected in all the early studies of molecular hole migration.This is because of the major challenge given by the multi-center (with respect to the single-center atomic case) and multi-channel nature of the many-electron problem, which also has to be solved in the presence of an external time-dependent laser field.The effects of the external field and of electron correlation (here we refer to the electron correlation, as standard in quantum chemistry, as to the effect of electron-electron repulsion that is not accounted for within the Hartree-Fock (selfconsistent field) approximation 20 ), both contributing to the formation of coherence in the molecular cation, need to be taken into account across a wide energy window, which is defined by the interaction with the ionizing pulse, leading one to face several computational difficulties.
The first one consists of providing an accurate prediction of the R-IDM matrix elements corresponding to each of the (most physically-relevant) many-electron states of the parent ion (ionization channels), as a function of the parameters (e.g., frequency ω E and intensity I E ) of the ionizing laser pulse.
Photoionization with high-energy photons in the XUV and soft x-ray energy range requires an accurate treatment of electron correlation effects, such as shakeup states, which dominate in the inner-valence energy region of the ionic spectrum, and of the electronic relaxation that happens due to the change in the mean-field potential that each electron experiences upon creation of a hole in a deeply-bound orbital.Moreover, things are made complicated also by the high density of states in the region close to the double-ionization threshold and by the fact that each of these states of the parent ion has to be coupled to a continuum of states for the photoelectron sub-system.The second is given by the necessity of having a time-dependent description of the photoionization process in order to capture the ultrafast formation and loss of quantum coherence.Accurate solution of the many-electron time-dependent Schrödinger equation (TDSE) also requires one to account for the effect of the inter-channel couplings between the emitted electron and the parent ion, that is, for correlation effects in the continuum, which cause the loss of coherence in the parent-ion during the departure of the emitted electron.This type of residual interaction between the parent-ion and the photoelectron is different with respect to the so-called intra-channel coupling, where the photoelectron experiences the potential created by the charge distribution of the specific fixed cationic state, insofar as the state of the parent-ion is also allowed to change as a result of it.
In addition, things are made more complicated by the fact that standard quantum chemistry methods of electronic structure calculation make use of Gaussian-type orbitals (GTOs) single-electron basis functions, which are poorly adapted to the description of continuum electrons.Moreover, attosecond (HHG and pump-probe) experiments typically involve either tunnel/multi-photon ionization or (sequential) double ionization, which are even more difficult to take into account.Finally, molecular systems are characterized by a larger size, a higher number of degrees of freedom and, in general, a lower symmetry with respect to atoms, which poses additional demands on the computation.As a result, until recently, the density matrix characterizing the ionic state emerging from attosecond ionization could not be predicted theoretically or characterized experimentally in the general molecular case-but only in some atomic cases, [21][22][23] for which a very recent experimental work has also reported the experimental reconstruction of the density matrix of the photoelectron sub-system. 24

| Overview of the different families of theoretical approaches to attosecond electron dynamics in molecules
These challenges have been tackled by the recent methodological development of a variety of new methods to simulate ultrafast correlated many-electron processes in molecular attosecond physics. 25Most of the approaches that have been developed in the field of theoretical attosecond physics to tame the complexity of simulating the TDSE with the manyelectron Hamiltonian of molecular systems interacting with ionizing laser fields, differentiate from each other both via the level of approximation they employ and by the ranges of applicability where they are accurate.Every approach must face a challenging trade-off between accuracy in capturing the relevant many-body effects, on the one hand, and the computational cost that it requires, on the other.The key difficulty is the handling of electron-correlation effects, which are difficult to manage at full rigor.Because of this, many methods adopt an "intermediate" approach, which allows for lower computational expense, while at the same time limiting the accuracy of the physical description.
Numerical methods thus form a hierarchy, with rising accuracy as more electron correlation effects are included: single-active-electron (SAE) approaches are the simplest numerical approaches to the TDSE 26 and require model potentials to mimic larger systems.8][29][30] Density functional theory (DFT) allows an effective single-particle description 31 which still can include some electron correlation effects through the use of a suitable exchange-correlation functional. 32Within attosecond science, examples of approaches which specifically target attosecond molecular ionization dynamics include real-time time-dependent (TD)-DFT [33][34][35][36] and TD first-order perturbation theory static-exchange DFT. 37In addition, nonequilibrium Green's function theory also allows one to describe the many-body problem from first principles by using effectively-single-particle approaches. 38,39inally, quantum-chemistry approaches go beyond the SAE approximation and TD-DFT to include, more directly, the effects of electron correlation.The starting point for this is generally the Hartree-Fock (HF) mean-field approach, though this is rarely sufficient on its own.Because of this, quantum-chemistry methods climb the ladder all the way to the full configuration interaction (CI) limit, a complete description of electron correlation (which is generally so computationally intensive that it is out of reach in practice).Most of the standard approaches of quantum chemistry were developed to describe bound states of molecular systems, 20,40 and they have also proven to be successful for modeling band structures in solid-state systems. 41Nevertheless, they often require significant extensions to work well in attosecond science, particularly regarding how the ionization continuum is handled.In this respect the choice of the basis set used to formulate and solve the TDSE is a crucial part of the methodological development as it determines the subspace of the solutions that we can reasonably explore, in turn influencing the physics that can be investigated with the method, the level of accuracy of the calculations and the computational cost required to reach convergence to a stable solution that captures the full physics of your problem.The basis sets most commonly used in traditional quantum chemistry, particularly GTOs, have indeed been highly optimized by tailored fitting procedures over many years and, as a result, they have enabled the flourishing of ab initio quantum chemical methods. 42There, the driving goal is to have accurate and fast numerical convergence for the physical quantities that interest quantum chemists, such as groundstate energies and electric polarizabilities.
However, these basis sets are generally poorly suited to describe free electrons in the continuum, and struggle when describing molecular ionization over a wide range of photoelectron kinetic energies. 43,44By extension, this limits our ability to describe general attosecond and strong-field physics.For attosecond science the key requirement is an accurate description of wavefunctions with highly oscillatory behavior far away from the molecular region, and this drives the development when existing basis sets are insufficient.

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The ab initio B-spline ADC method for molecular attosecond science and its major achievements One such sophisticated ab initio method is the time-dependent (TD) B-spline algebraic diagrammatic construction (ADC) 15,16 for many-electron wave packet dynamics in atoms and molecules.This was the first successful attempt to extend a traditional ab initio quantum chemical method to the realm of ultrafast ionization dynamics in molecules.It integrates a full description of the photoelectron continuum states with the accurate description of electron correlation given by the many-electron ADC(n) approach for electronic excitations. 40,45The ADC(n) approaches have been vastly successful as ab initio quantum chemical methods for calculating, using standard GTOs, the excitation energies of neutral closed shell 46 and open shell species, 47 as well as their single, 48,49 double 50 and triple 51 ionization energies.It has been applied to the calculation of properties of and transition moments between excited 52 singly ionized, 53 and doubly ionized 54 bound states, as well as for the calculations of the decay widths of resonance states 55 and of the interatomic coulombic decay rates. 56Similar to CI, a general ADC scheme involves an expansion in excitation classes, but unlike the CI expansion, an nth order scheme in the ADC(n) hierarchy of approximation 40 uses a perturbation-theoretically corrected (rather than Hartree-Fock) reference state, leading to a more compact, size-consistent representation.The first order TD B-spline ADC(1) method includes only single (one hole-one particle, 1h1p) excitations, while the second order ADC(2) and ADC (2)x extend the configuration space to double (two hole-two particle, 2h2p) electronic excitations as well. 40Within TD B-spline ADC the TDSE for an N-electron polyatomic molecule interacting with an ultrashort laser pulse is solved by expanding the time-dependent N-electron ADC state, describing both molecular excitation and ionization, in the basis of the so-called ADC "intermediate excited states," 1h1p and 2h2p configurations derived from the correlated ground state of the neutral molecule where the indexes ij and αβ refer to occupied and virtual HF molecular orbitals, respectively.The total time-dependent Hamiltonian of Equation ( 5) where a complex absorbing potential (CAP) 57 of the form b H À E 0 and the dipole operator b D, respectively.To accurately describe the behavior of photoelectron continuum wavefunctions over an arbitrarily large spatial range, and overcome the limitations inherent to the use of GTOs, the TD B-spline ADC technique relies on the use of a single-center expansion for the single-particle orbitals ψ r, θ, ϕ ð Þ in radial coordinates and of B-splines B i r ð Þ piecewise polynomial functions 15,16 to describe their radial dependence where the angular dependence is described by spherical harmonics Y l m θ, ϕ ð Þ of angular momentum quantum numbers l and m.This technique allows one to accurately describe molecular bound-continuum dynamics in a series of diverse scenarios and regimes of the light-matter interaction, [58][59][60][61] from single-and multi-photon perturbative to strong-field non-perturbative, and to model from first principles, with controlled systematic inclusion of electron correlation, laserinduced and laser-driven many-electron processes in molecules across multiple ionization stages.Major achievements of the TD B-spline ADC implementation include the first, fully ab initio, multi-channel simulation of molecular HHG 59 and the first complete, fully ab initio simulation a prototypical attosecond molecular pump-probe experiment. 60Inclusion of multichannel dynamics in the modeling of HHG in the CO 2 molecule 59 allowed us to shed light on and quantify the importance of the electron-electron Coulomb interactions in the ultrafast electron dynamics during molecular HHG and uncover the subtle effects of dipole couplings between the molecular ionic states.Moreover, we were able to show the limitations of the HHG spectroscopy as an imaging/attosecond probe tool.
In Reference [60] we described from first principles, with a full account of the continuum dynamics of the photoelectrons, sequential molecular double ionization by the pump and probe laser pulses with controlled time-delay.In particular, we used the TD B-spline ADC(1) method to simulate molecular strong-field ionization of the CO 2 molecule by intense infrared (IR) laser pulses, and to predict, as a function of the ionizing field parameters, the degrees of quantum electronic coherences between the populated internal states of the resulting (CO 2 ) + molecular cation (See Figure 1).In addition, we were able to simulate the interaction of the formed (CO 2 ) + cation with a second VUV probe pulse, predict the oscillatory dependence of the resulting double ionization signal ((CO 2 ) 2+ ) on the pumpprobe time-delay as well as the dependence of the (CO 2 ) 2+ -yield oscillation amplitude on the parameters of the pump field used to produce the cation (See Figure 1).The numerical results of Figure 1 show how the degree of coherence achieved at the pump stage can be inferred from the time-dependent oscillations amplitude of the final CO 2 2+ yield.

| The new ab initio B-spline RCS-ADC method for molecular attosecond science and its major achievements
In order to go beyond the limitations still present within the standard B-spline ADC implementation, such as the increasingly-high computational demand given by the size of the 2h2p configurations space in B-spline basis set, Ruberti M. has recently developed a new approach to the ionization dynamics of many-electron molecular systems, namely the time-dependent (TD) B-spline restricted correlation space (RCS)-algebraic diagrammatic construction (ADC) method. 17 oscillations.Reproduced from Reference [60] with permission from the American Chemical Society.
In Equation ( 9), j Ψ RCS 0 i n ½ is the n-th order RCS correlated ground state, while j ΨN I RCS ;ADC n ð Þ i and j Φ NÀ1,RCS m;ADC n ð Þ i indicate the excited intermediate states and the ionic eigenstates of the nth-order ADC(n) scheme, respectively, both built using the single-particle RCS basis.The total single-electron orbital space has been separated into correlation (RCS) and ionization (IS) spaces, b a † μ is a creation operator for the photoelectron represented in the IS orbital space, the index I RCS and the index m represent different bound excitations classes and different ADC ionic eigenstates, respectively.The ansatz of Equation ( 9) includes a full description of electron correlation effects, such as shakeup processes, breakdown of the molecular orbital (MO) picture and inter-channel couplings in the continuum, which can play an essential role both during the ionization event and the post-ionization charge dynamics.
The j Φ NÀ1,RCS m i are obtained by diagonalizing the ionic Hamiltonian calculated at the ADC (2)x level of theory and using the single-particle RCS basis set where the ionization energy is given by is the energy of the RCS ground state.Within the RCS-ADC(2)x scheme, the ionic states are expanded into 1h and 2h1p configurations derived from the correlated RCS ground state of the neutral molecule RCS-ADC extends the ab initio ADC approach by using a multi-center B-spline basis set 17 and goes beyond previously available theoretical studies where the ionization step was either completely neglected (using the so-called "sudden approximation picture" of photoionization) 62 or modeled by disregarding important electron correlation effects. 37he strength of this method is that it naturally bridges the gap between multi-configurational ab initio techniques and closed-coupling schemes based on a limited number of essential, physically relevant ionic states, combining the key advantages of both.As a result, it allows one to model the ionization dynamics of much larger polyatomic molecules than previously possible, at the same or higher level of accuracy.
In Figure 2 we show a comparison between the total photoionization cross-section of the CO 2 molecule calculated using different theoretical methods ranging from configuration interaction singles (CIS) to RCS-ADC (2)x, and the experimental result of. 63The RCS-ADC(2) and RCS-ADC (2)x cross-sections are obtained using a RCS consisting of all the virtual orbitals from a GTO cc-pVTZ HF calculation.From the results of Figure 2 it is possible to notice how the most refined RCS-ADC (2)x method yields results which are very close to the experimental ones.In general, within the B-spline RCS-ADC, the combination of the advanced treatment of electron correlation and of the representation of the continuum states of the photoelectron yields accurate photoionization cross-sections for molecular systems, with relative deviations from experimental values of the order to few % (See Figure 2).This type of new computational capability makes it possible not only to precisely calculate photoionization crosssections, but also to accurately predict the mixed quantum states of the ionized molecular system prepared by attosecond ionization, 64 crucially turning first-principles modeling of coherence and entanglement in photoionization of polyatomic molecular systems into a tractable problem.Complete theoretical characterization of the cationic state and of the photo-induced attosecond charge dynamics is achieved, going beyond previous works limited to atomic systems and to a less accurate description of electron correlation, 22,23 by calculating the R-IDM for the bipartite ion-photoelectron system, with inclusion of the description of the correlated ionic shakeup states, and fully capturing the onset and loss of the quantum electronic coherences generated in the ionic system during the laser-induced dynamics.This is a major milestone for the field, leading to a new level of theoretical description of attosecond phenomena such as molecular hole migration.As an illustrative example, in Figure 3 we show the degrees of quantum electronic coherence G m,n , as calculated using the TD B-spline RCS-ADC (2)x method, between the (N-1)-electron states of the (C 2 H 4 ) + cation populated by attosecond XUV photoionization of the ethylene neutral molecule. 64he power of this approach lies in the possibility of calculating the von Neumann entropy of the multi-dimensional entangled state, fully capture the ion-photoelectron entanglement, and the Schmidt decomposition of the R-IDM. 65In fact, the latter unveils the many-electron dynamics triggered by the pump and allows for the identification of the key pure-state channels involved in the quantum coherent many-electron dynamics. 656][67] This is because the hierarchies of both ADC and RCS-ADC methods extend across different ionization stages, thus allowing the simulation of the interaction of the cation with yet another ionizing laser pulse probing its dynamics.The ab initio simulations of the interaction between the molecular systems and the pump and probe laser pulses are performed by solving the TD Schrödinger for the neutral system and the von Neumann equation Upper panel: (a) spectral intensity of the ionic states (See Equation ( 11)) of the carbon dioxide cation included in the expansion of the wavefunction of Equation ( 9) and calculated using the ADC(2)x method and the cc-pVTZ Gaussian-type-orbitals basis set (red).The ionic states at which the convergence of the RCS-ADC(2)x total cross-section is achieved are in blue; (b) experimental and theoretical B-spline RCS-ADC total photoionization cross-section of the carbon dioxide molecule.The theoretical cross-section is calculated both using the configuration interaction singles (CIS) method and at different levels of accuracy within the ADC hierarchy of methods, from ADC(1) to RCS-ADC (2)x; Lower panel (c): deviations of the RCS-ADC theoretical cross-sections, calculated at different levels of accuracy within the ADC hierarchy of methods, from the experimentally-measured cross-sections averaged over a series of different molecular systems.Reproduced from Reference [17] with permission from the American Chemical Society.
for the parent ion, respectively.Simulation of attosecond measurements for the fully characterized attosecond ionized states is a key numerical experiment which allows one to unravel the mapping of the results of the measurements onto the nonzero elements of the R-IDM, thus shedding light on how the coherent physical phenomena triggered by attosecond x-ray ionization can be measured in the laboratory.
In this respect, we would like to highlight two examples of such numerical simulations of experimental observables in attosecond pump-probe experiments.In the first one, 65 an XUV pump pulse is used to photo-ionize the pyrazine molecule and a second, time-delayed x-ray pulse is used to probe the quantum-coherent electron dynamics, triggered in the pyrazine cation by the pump, by means of its (attosecond) transient absorption spectrum (ATAS).Such a complete XUV-pump/x-ray-probe attosecond experiment on the pyrazine molecule was simulated in a fully ab initio fashion using the TD B-spline RCS-ADC (2)x method in, 65 and it was demonstrated in detail how quantum electronic coherences prepared in the cation by attosecond photoionization of a polyatomic molecule, and predetermining its subsequent charge-directed reactivity, can be efficiently retrieved by using transient absorption spectroscopy with attosecond soft x-ray probe pulses.
In Figure 4 we show the main results of the aforementioned simulation.Attosecond XUV photoionization dynamics of the pyrazine molecule was calculated using the TD B-spline RCS-ADC (2)x method.The main output of this simulation is the R-IDM of the formed pyrazine cation and the degrees of quantum electronic coherence G m,n between the many-electron ionic states populated by the pump.The calculated G m,n of Equation ( 2) are shown in the left panel of Figure 4, between the ionic states of pyrazine calculated using the RCS-ADC (2)x method and a cc-pVTZ basis set.The presence of quantum coherence drives a many-electron dynamics in the pyrazine cation, which manifests itself in the time-dependence of the probe absorption spectrum around the Carbon K-edge absorption energy.The time-dependent cross-section observable shown in the right panel of Figure 4 is calculated by solving Equation ( 12) using the TD RCS-ADC (2)x method.The Fourier-transform of the ATAS along the pump-probe time-delay axis yields isolated peaks F I G U R E 3 Quantum electronic coherence G m,n between the many-electron ionic states populated by attosecond XUV photoionization of the C 2 H 4 molecule.The ionic states were calculated using the RCS-ADC (2)x method and a cc-pVTZ basis set.The photoionization dynamics of ethylene, interacting with an attosecond XUV pulse linearly polarized along the C-C axis on the molecular plane, was simulated using the time-dependent (TD) version of the B-spline RCS-ADC(2)x method.The populations upon photoionization of the various ionic states are also shown in the vertical and horizontal side panels; different stick colors correspond to ionic states of different molecular point group symmetry.Adapted from Reference [64] with permission from the Royal Society of Chemistry.
which correspond to the quantum beatings due to the pump-prepared quantum electronic coherences between ionic states of the pyrazine cation.In particular, in Figure 4 the ones corresponding to the coherences between the 1A g and 2A g states, as well as between the 3B 2u and 4B 2u states, respectively, are highlighted in the results of the RCS-ADC simulation.
In the second example that we would like to highlight, 66 the B-spline RCS-ADC method was used for the interpretation and theoretical modeling of a femtosecond x-ray pump-x-ray probe experiment performed at the FLASH X-FEL (Hamburg, Germany) on the photoionization dynamics of molecular glycine.In, Reference [66] an x-ray pump pulse is used to photo-ionize the molecular Glycine I conformer (Gly I) and a second time-delayed, identical x-ray pulse is used to probe the quantum-coherent electron dynamics triggered in the Gly I cation by the pump by looking at the timedelay dependent photoelectron yield.The photon energy of the probe (and pump) pulse was tuned to 274 eV, 66,67 that is, just below the C K-edge.
TD B spline RCS-ADC was used to simulate the whole pump-probe experiment.In Figure 5 we show the main results of the aforementioned simulation: the pump ionization process triggers a periodic hole-charge dynamics in the (Gly I) + cation, characterized by a $ 20 fs period (See panel a in Figure 5); the interaction of (Gly I) + with a timedelayed probe pulse produces a photoelectron yield that oscillates in time with different phases depending on the photoelectron's kinetic energy range (See panel b in Figure 5).The results of the calculations were in close agreement with the main result of the measurements, that is, the direct observation (i.e., through electronic observables) of femtosecond, coherent electron dynamics of molecular inner-valence ionized states.Indeed, in both the experimental and the theoretical results it was possible to observe at early times (for time-delays less than 50 fs) an oscillatory period of around 20 fs in the time-delay dependent photoelectron signal.The results of the simulation enabled us to connect the oscillations in the data to the coherent electron dynamics in the energy region of the inner-valence 10A' molecular orbital.
F I G U R E 4 Left panel (a): quantum electronic coherence G m,n between the many-electron ionic states populated by attosecond XUV photoionization of the pyrazine molecule.The ionic states were calculated using the RCS-ADC(2)x method and a cc-pVTZ basis set.The photoionization dynamics of pyrazine upon interaction with the pump pulse was simulated using the TD B-spline RCS-ADC(2)x method.The populations upon photoionization of the various ionic states are also shown in the vertical and horizontal side panels; different stick colors correspond to ionic states of different molecular point group symmetry.Right panel: (b) Carbon K-edge attosecond transient absorption spectrum of the 272 eV centered attosecond x-ray probe pulse by the pyrazine cation prepared by the XUV pump ionization; (c) Fourier-transform along the pump-probe time-delay axis of (b); the peaks correspond to the beatings due to the pump-prepared quantum electronic coherences between ionic states of the pyrazine cation.In particular, the ones corresponding to the coherences between the 1A g and 2A g states, as well as between the 3B 2u and 4B 2u states, respectively, are highlighted.Adapted from Reference [65] with permission from the Royal Society of Chemistry.

| More on new available methods for attosecond photoionization dynamics in molecules
The state-of-the-art TD B-spline ADC and RCS-ADC ab initio methods are not the only approaches that have been developed to tackle attosecond electron photoionization dynamics in molecules.Other examples of such approaches include methods based on multi-reference (MR)-CI, 68,69 multi-configuration TD Hartree, 70 TD Hartree-Fock 71 and restricted-active-space self-consistent-field (RAS-SCF), [72][73][74] as well as the molecular R-matrix with-time approach by Benda et al. 75,76 (In this review we give a non-exhaustive list of methods for the description of ultrafast electron dynamics happening on the attosecond timescale, for more details see 77 ).These methods also differentiate by their description of the electronic continuum.The latter ranges from B-spline functions, both on their own [15][16][17][64][65][66]78 and combined with Gaussian-type orbitals (GTOs), [72][73][74] finite-difference approaches, 75,76 finite-element discretevariable-representation functions, 79,80 grid-based methods, 81 and simple plane-waves. 82 In adtion to the B-spline ADC family of approaches, we would like to highlight three other promising new methods: the first one is the multi-reference extension (developed by Ponzi et al. 83 and applied to attosecond molecular ionization dynamics by Martin and collaborators 84 ), of the time-dependent first-order perturbation theory, staticexchange DFT method developed by P. Decleva and collaborators.8 The static-exchange DFT method provides an effective single-particle description of the photoionization dynamics, as well as a more basic description of electron correlation effects through the use of exchange-correlation functionals in the calculation of the ground-state Kohn-Sham (KS) orbitals.The multi-reference static-exchange scattering method 83 improves the description of electron correlation with respect to 8 by considering the multi-configurational character of the parent-ion wavefunction. However, both methods nlect inter-channel couplings in the electronic continuum and their first-order perturbation theory implementation is limited to description of weak-field, single-photon ionization processes.The second one is the XCHEM method by Martin and collaborators, 72 a wavefunction method that uses a Close-Coupling approach based on a RAS-SCF description of electron correlation and on the use of a mixed basis set comprising both Gaussian type orbitals and B-splines.This method, which, as well as the ADC-based methods, includes a description of electron correlation effects in the continuum, has so far been applied to the time-independent description of photoionization in small molecules.74 Finally, the third one is the real-time nonequilibrium Green's function methods (NEGF) as implemented Relative change of the probe-produced photoelectron yield as a function of x-ray pump-probe delay, in both the kinetic energy ranges above and below the phase jump point (kinetic energy value at which the phase of oscillations changes sign).Adapted from Reference [66] with permission from AAAS.
in the CHEERS code by Perfetto et al. 38 allows one to describe the many-body problem from first principles by using effectively-single-particle approaches.Dynamical correlation effects are added to the Hartree-Fock dynamics through a many-body self-energy.The real-time implementation of the method is potentially suitable for applications to describe non-perturbative ionization dynamics.The description of the photoelectron system is limited by the use of KS continuum states a in planewave or grid basis set.

| CONCLUSION
The methodological development reviewed in this article opens the way to the study and fundamental understanding of the physics underlying photo-chemical/physical transformations of matter at ultrashort timescales, fully exploring the role of quantum entanglement and coherence in these processes.
However, despite our new theoretical and computational capabilities provide us with extremely powerful tools for the analysis and understanding of complex photoemission processes, representing a major advance over previously existing approaches, the methods that are most sophisticated in describing the many-electron dynamics, such as B-spline RCS-ADC, are still implemented within the fixed-nuclei approximation, thus not including the possible decoherence effect of nuclear motion and the molecular nonadiabatic dynamics playing a critical role on the tens-offs timescale.To date, examples of coupling photoionization dynamics with a treatment of nuclear motion for attosecond dynamics include static-exchange DFT combined with trajectory surface hopping, 84 the SAE approximation combined with a partitioning technique 85 and the treatment of the photoelectron as a classical point charge in a mixed quantum-classical simulation of ultrafast molecular dynamics. 86In general, available studies either disregard electron correlation and/or nuclear quantum effects.Thus, future directions for further methodological development consist of coupling the state-of-the-art methods for the description of many-electron dynamics with a description of nuclear dynamics, thus allowing us to better quantitatively compare the theoretical predictions with the experimental attosecond measurements.
Þis used to eliminate wave packet reflections from the boundaries of the B-spline radial grid.The laser-molecule interaction driven by the pump electric field E Pump t ð Þ À Á is described within the dipole approximation in the length form, and b H N ADC and b D N ADC are the shifted field-free Hamiltonian b H ¼ b B-spline RCS-ADC is based on the following ansatz for the time-dependent many-electron wavefunction, expressed as a sum of bound/localized and ionization-channel components, F I G U R E 1 B-spline ADC simulation of IR-pump VUV-probe experiment in the carbon dioxide molecule.The strong-field ionization of neutral CO 2 molecules by an IR pump pulse produces a (CO 2 ) + cation characterized by some degree of quantum electronic coherence between its internal states of Σ symmetry G Σg,Σu À Á , as well as between its internal states of Π symmetry G Πg,Πu À Á : The dependence of G Σg,Σu (dotted yellow line) and G Πg,Πu (dotted green line) on the IR pump pulse parameters, namely on the IR pump peak intensity and on the IR carrier frequency is shown in the left and right panel, respectively.Moreover, the interaction of a time-delayed VUV probe pulse with the (CO 2 ) + cation gives rise to a (CO 2 ) 2+ double ionization yield which oscillates as a function of the time-delay between the probe and the pump pulse.The dependence of the amplitude of the time-delay dependent oscillation of the (CO 2 ) 2+ double-ionization yield resulting from the Σ symmetry (full yellow line) and the Π symmetry (full green line) parts of the (CO 2 ) + density matrix on the IR pump pulse parameters, namely on the IR pump peak intensity and on the IR carrier frequency, is also shown in the left and right panel, respectively.The numerical results show how the degree of coherence achieved at the pump stage (dotted lines) can be inferred from the amplitude of the CO 2 2+ yield

F
I G U R E 5 Upper panel (a): Hole-charge densities in the (Gly I) + cation from t = 1 fs to t = 21.4 fs after the pump ionization event.The change of this density reflects coherent electron dynamics in the energy region of the inner-valence 10A' molecular orbital.The density isosurfaces displayed are the ones with value 0.015, blue and red colors indicate positive and negative values of the hole density, respectively.Lower panel (b): TD B-spline RCS-ADC simulation of the interaction between the probe pulse and the pump-prepared (Gly I) + cation.