Attosecond Photoionization Dynamics in Neon

Attosecond photoionization dynamics in neon

Juan J. Omiste Chemical Physics Theory Group, Department of Chemistry, and Center for Quantum Information and Quantum Control, University of Toronto, Toronto, ON M5S 3H6, Canada

Lars Bojer Madsen Department of Physics and Astronomy, Aarhus University, 8000 Aarhus C, Denmark (Dated: September 11, 2018) We study the role of electron-electron correlation in the ground-state of Ne, as well as in photoionization dynamics induced by an attosecond XUV pulse. For a selection of central photon energies around 100 eV, we ﬁnd that while the mean-ﬁeld time-dependent Hartree-Fock method provides qualitatively correct results for the total ionization yield, the photoionization cross section, the photoelectron momentum distribution as well as for the time-delay in photoionization, electron-electron correlation is important for a quantitative description of these quantities.

I. INTRODUCTION mally updated in each time-step. The latter approach benefits not only from a reduction in the number of orbitals, but also from an implicit description of multiple Ever since the early days of atomic physics, the role of ionization events, and from its flexibility to identify the electron-electron correlation has been a key topic. Even most important configurations for an accurate descrip- though ground and singly-excited states can be described tion of the system [8, 10, 11, 16, 17]. We will study by mean-field theory, it was soon recognized that a cor- the ground-state energy and the photoionization dynam- related basis is beneficial for convergence to an accu- ics and their sensitivity to the active orbital space, i.e., rate ground-state energy [1, 2]. For multiply excited to the RAS scheme. Finally, we will study the impor- states, correlation is crucial and theory based on single- tance of electron correlation on the time-delay in pho- configurations breaks down [3]. In strong-field and at- toionization. Time-delay studies have attracted much tosecond physics, the theoretical description is typically attention because of available experimental data [18, 19] explicitly time-dependent and processes often involve and the development of powerful theoretical and com- continua wavepackets carrying the temporal information putational methods [20]. Time-delays have been evalu- encoded by ultrafast pump and probe pulses [4]. The ated in different scenarios, including in photoionization presence of multiple continua challenges theory tremen- from first principles [21], as a function of angle of the dously. Needless to say carrying out a mean-field time- ejected electron [22–24] or in strong-field ionization [25]. dependent Hartree-Fock calculation is possible for phys- In particular, the experimental measurement of the time- ical systems so large that fully correlated configuration- delay in Ne between the photoemission of electrons from interaction calculations are not. One of the tasks for the 2s and 2p shells [18] has provoked a lot of interest theory is therefore to establish which level of approxi- due to a disagreement in the magnitude of the mea- mation is sufficient for a qualitative correct description sured and calculated time-delays, see, e.g., Refs. [26–35]. of a given observable. This is one of the questions that Note that very recently, analysis of interferometric mea- we address in this work. To this end we need a the- surements [36] suggests that a shake-up process, not re- ory where we can control the level of approximation, solved and accounted for in the streaking experiment of and we need to investigate a system which is possi- Ref. [18], could affect the experimental result and possible to describe at such different levels. Thus, we de- bly bring the experimental time-delay in agreement with scribe the photoionization of the Ne ground-state [5– the many-body calculations [34]. In this work, we prop- 7] mediated by an ultrashort XUV pulse using the agate the photoelectron wavepacket to directly measure time-dependent restricted-active-space self-consistent- the time of emission from each electronic shell for a set field (TD-RASSCF) approach [8–11]. This method per- of active spaces to probe the role of the electronic corre- arXiv:1712.00625v1 [physics.atom-ph] 2 Dec 2017 mits the introduction of restrictions on the number of lation [16]. excitations in the active orbital space and is a gen- The paper is organized as follows. In Sec. II, we eralization of the multiconfigurational time-dependent summarize the computational approach. In Sec. III, we Hartree-Fock method [12]. In contrast to many-body present our results. First, we study the ground-state for methods based on time-independent single-particle or- different RAS schemes. Next, we analyze the ionization bitals such as time-dependent configuration-interaction dynamics induced by the laser. This analysis consists with singles (TD-CIS) [13], time-dependent restricted- of the calculation of the ionization cross section and the active-space configuration-interaction (TD-RASCI) [14] description of the main features of the single photoion- or time-dependent general-active-space configuration- ization, in particular, the contribution of each ioniza- interaction (TD-GASCI) [15], the TD-RASSCF is based tion channel obtained by considering the photoelectron on time-dependent single-particle orbitals that are opti- spectrum. Finally, we present computed time-delays be- 2 tween the electrons ejected from the 2s and 2p subshells the matrix elements are given by and compare with available experimental and theoretical p ∗ values. We conclude in Sec. IV. Atomic units are used hq (t) = d~rφp(~r, t)h(~r, t)φq(~r, t), (3) throughout unless indicated otherwise. Z φ∗(~r, t)φ∗(r~0, t)φ (~r, t)φ (r~0, t) pr ~0 p r q s vqs (t) = d~rdr .(4) ~r r~0 ZZ | − | The TD-RASSCF methodology is a generalization of II. SUMMARY OF THE TD-RASSCF METHOD MCTDHF [38, 39] in the sense that it includes the pos- sibility to impose restrictions on the excitations in the active space [8, 10], i. e., the many-body wavefunction In this section, we summarize the TD-RASSCF reads method used to propagate the many-electron wavefunction. We refer to previous works for details [8–10, 16]. Ψ(t) = CI (t) ΦI (t) , (5) | i | i I∈V We propagate the dynamics of an Ne-electron atom in X the laser field in the length gauge within the dipole ap- where the sum runs over the set of configurations , and proximation. The dynamics of this system is described V by the Hamiltonian not necessarily the full configuration space, and CI (t) and ΦI (t) are the amplitudes and Slater determinants of the| configurationi specified by the index I, which contains direct products of spin-up and spin-down strings, Ne p2 Z Ne Ne 1 j ~ i.e., I = I↑ I↓, each of them including the indices of the H = + E(t) ~rj + = ⊗ 2 − rj · ~rj ~rk spatial orbitals [40, 41]. Each Slater determinant is built j=1 ! j=1 k>j | − | X X X from M time-dependent spatial orbitals φ (~r, t) M , in N N N j j=1 e e e 1 the active orbital space . In the case{ of MCTDHF,} = h(~rj, t) + , (1) P ~rj ~rk FCI, that is, the full configuration space [39]. On j=1 j=1 k>j X X X | − | theV ≡ other V hand, in the TD-RASSCF, the configurations are taken from the restricted active space, RAS, which is defined as a subset of by imposingV ≡ V re- where the first sum is over one-body operators and the FCI strictions on the excitations in theV active space. In this second over two-body operators. The nuclear charge is method, the active orbital space is divided into 3 sub- denoted by Z and the external electric field of the laser P ~ spaces: 0, 1 and 2. 0 constitutes the core, and its pulse is E(t). To formulate and apply the TD-RASSCF orbitalsP areP fully occupied.P P All the different ways to theory, it is convenient to work in second quantization. form configurations by combination of orbitals in 1 are We work in the spin-restricted framework, which implies P allowed. The orbitals in 2 are filled with restrictions that a given Slater determinant, ΦI (t) , is formed by P | i by excitations from 1. The number of orbitals in 0, Ne/2 spatial orbitals for each spin specie. In second P P 1 and 2 are denoted by M0, M1 and M2, and the quantization, the Hamiltonian reads P P total number of orbitals equals M = M0 + M1 + M2. The single-particle Hilbert space is completed by the - Q 1 space such that the unit operator can be resolved as H(t) = hp(t)Eq + vpr(t)Eqs, (2) q p qs pr 1 = P (t) + Q(t), with P (t) = j φj(t) φj(t) and pq 2 pqrs | ih | Q(t) = φa(t) φa(t) , with φj(t) belonging to - X X a | ih | | P i P space and φa(t) to -space. In thisP work,| i we doQ not consider a core, i. e., we do where we use the spin-free excitation operators [37] q qs q † qs not have a 0 subspace. We apply the TD-RASSCF- Ep and Epr , defined as Ep = bpσbqσ, and Epr = P σ=↑,↓ D method, i. e., include double (D) excitations from the † † † P active space partition 1 to 2. The TD-RASSCF-D bpσbrγ bsσbqγ with bpσ and bpσ the creation and P P σ=↑,↓ γ=↑,↓ method was shown to be numerically efficient and stable annihilationP P operators of a single spin-orbital φp(t) σ in the case of photoionization of Be [16]. The equations and σ denoting the spin degree of freedom.| In Eq.i⊗| (2),i of motion (EOM) read [8] 3

i i j 1 ik jl iC˙ I (t) = h (t) iη (t) ΦI (t) E Ψ(t) + v (t) ΦI (t) E Ψ(t) , (6) j − j h | i | i 2 jl h | ik| i ij ijkl X X

j j k jl i Q(t) φ˙j(t) ρ (t) = Q(t) h(t) φj(t) ρ (t) + W (t) φj(t) ρ (t) , (7) | i i  | i i l | i ik  j j jkl X X X k00 k00 l0j00  j00m kl kl j00m  h 0 (t) iη 0 (t) A 00 0 (t) + v (t)ρ 0 (t) v 0 (t)ρ (t) = 0, (8) l − l k i kl i m − i m kl k00l0 klm X h i X h i with M i η (t) = φi(t) φ˙j(t) ,Q(t) = 1 P (t) = 1 φj(t) φj(t) (9) j h | i − − | i h | j=1 X 1 k ∗ ~0 ~0 ~0 j j Wl (~r, t) = φk(r , t) φl(r , t)dr , ρi (t) = Ψ(t) Ei Ψ(t) , (10) ~r r~0 h | | i Z | − | ρjl (t) = Ψ(t) Ejl Ψ(t) ,Alj (t) = Ψ(t) EjEl El Ej Ψ(t) , (11) ik h | ik| i ki h | i k − k i | i where the orbitals denoted by single and double prime tion dynamics including time-delay studies, induced by indexes belong to different partitions. The strategy to the interaction with an XUV linearly polarized attosec- propagate the EOM is as follows. To propagate Ψ(t) ond pulse. In our simulations, we set the maximum angu- | i we need expressions for the time-derivative for the or- lar momentum to `max = 3 and the maximum magnetic bitals φ˙j(t) and the time-derivative of the amplitudes quantum number of each orbital to mmax = 2. For the | i C˙ I (t). The time derivative of the orbital is split in ground-state studies, the localization of the wavefunction and space contributions φ˙j(t) = P (t) φ˙j(t) + near the nucleus allows us to confine the extend of the P− ˙ Q− ˙| i |˙ i radial box to the interval r [0, 31), where we use 12 Q(t) φj(t) = φi(t) φi(t) φj(t) + Q(t) φj(t) = ∈ | i i=1 | i h | i | i equidistant elements for 0 r 6 and complete the box i ˙ ≤ ≤ ηj φi(t) + QP(t) φj(t) . Equation (8) is used to de- with 10 equidistant elements up to r = 31. The descrip- i=1 | i | i tion of the photoionization process requires a larger box, i terminedP the η ’s. With these at hand the P (t) φ˙j(t) - which we build by adding 68 elements of 2.5 atomic units j | i part of the derivative of the orbital is determined. Equa- of length up to rmax = 201. Each element contains 8 tion (7) is then used to find Q(t) φ˙j(t) and finally nodes. The results has been checked against convergence | i C˙ I (t) is determined from Eq. (6). The bottleneck of with respect to the parameters of the primitive basis. the propagation lies in the update at every time step of the two-body operator. In order to speedup this update, we recently derived and described the coupled ba- A. Ground-State sis method [16]. This method consists of coupling the angular part of the single-electron orbitals. It is easy The TD-RASSCF method is developed to solve the to see that the angular momentum and magnetic quan- TDSE for problems that are so large that a diagonaliza- tum numbers of the coupled angular momenta are pre- tion of the Hamiltonian is impossible. As a consequence, served by the two-body operator, 1/ ~r r~0 , (~` + `~0)2 = the eigenstates of the system cannot in general be ob- | − | 0 0 0 0 tained straightforwardly. However, propagation in imag- 1/ ~r r~ , `z + ` = 0, where h~` (`~ ) and `z (` )i are | − | z z inary time of an initial guess function makes it possible hthe one-electron angulari momentum operators. This to obtain the ground-state, since contributions of excited conservation property, together with the description of states are removed after long enough propagation time. the radial part by a finite-element discrete-variable- The nonlinearity of the EOM, Eqs.(6)-(8), together with representation (FE-DVR), significantly reduces the num- the 3D nature of the system demands an appropriate se- ber of operations in the evaluation [16]. lection of the guess function to facilitate the convergence to the ground-state [16]. Specifically, we choose the partitions (M1,M2) = (5, 0), (5, 1), (6, 0), (5, 4), (1, 8), (9, 0) III. RESULTS and (5, 9), where the orbitals in the initial guess function are chosen as the hydrogenic orbitals 1s, 2s, 2p, 3s, 3p In this section, we describe the many-electron wave- and 3d with nuclear charge Z = 10. In addition, we im- function of the ground-state of Ne and the photoioniza- pose that only the set of configurations ΦI (t) with {| i} 4

Table I. Ground-state energies of Ne for several RAS schemes. M1 and M2 denote the numbers of orbitals in the active orbital subspaces P1 and P2, respectively. When all the orbitals are in P1 (M1 = M and M2 = 0), the TD-RASSCF-D method corresponds to MCTDHF with M1 orbitals. The M1 = M = Ne/2 case corresponds to TDHF.

Number of orbitals M1 M2 Number of ground-state M = M1 + M2 conﬁgurations energy (a.u.) 5 5 0 1 -128.548 6 5 1 26 -128.561 6 0 36 -128.561 9 5 4 521 -128.679 1 8 8036 -128.682 9 0 15876 -128.683 14 5 9 2746 -128.765

FIG. 1. Ne ground-state and predominant Ne+ and Ne++ total magnetic quantum number M = 0 contribute to channels for photons with 75 eV ≤ ω ≤ 115 eV. Note that L we only show the lowest energy state in each fine-structure Ψ(t) of Eq. (5). This choice leads to stable numeri- multiplet. The data are taken from Refs. [46–48]. cal| performance,i because the initial set of orbitals and the restriction in ML ensures the correct symmetry of the ground-state. In addition, these two constrains im- ply that the magnetic quantum number of each orbital laser pulse that is linearly polarized along the Z axis of is conserved by the EOM, as shown in Appendix A (see the laboratory frame and given by the vector potential 2 also Refs. [42–44]). A~(t) = A0zˆcos [ωt/(2np)] sin ωt, where ω and np are We show the ground-state energies obtained by imagi- the angular frequency and the number of cycles, and the nary time propagation in Table I for the considered RAS duration is given by T = 2πnp/ω. The pulse begins at partitions. The ground-state energy for the Hartree-Fock t = T/2 and ends at T/2. We set the intensity of the − 14 2 method (5, 0) coincides with previous calculations [45]. pulse to 10 W/cm and choose np = 10 for ω in the Note that for M = 6, the TD-RASSCF-D with (5, 1) range 75 to 115 eV. For this photon energy range, the e and (6, 0) are theoretically equivalent, because M = predominant ionization channels are Ne+(1s22s2p6)2S + 2 2 5 2 o Ne/2 + 1 = 6. Therefore, we obtain the same ground- and Ne (1s 2s 2p ) P [46–48], see Fig. 1. The double state energy and, as we will see in the next section, the ionization threshold is at 62.53 eV corresponding to the e photoionization dynamics induced by linearly polarized channel Ne++(1s22s22p4)3P . light is the same [8]. The strength of the TD-RASSCF We show the ionization yield as a function of method clearly manifests itself in the case of 9 orbitals. time, P1(t) for ω = 95, 105 and 115 eV, in The energy difference between (5, 4) and (9, 0) is approx- Fig. 2. The ionization yield is determined from imately 0.004 a.u., whereas the number of configurations the electron density in the outer region, P1(t) = j ∞ ∗ considered in the MCTDHF is 30 times larger than for ρ dΩ φi(~r, t) φj(~r, t)dr, with rout = 20 a.u. ij i Ω rout the TD-RASSCF-D. The ground-state energy for (5, 9) First, let us remark that the ionization yield calculated is better than in the case of (9, 0) although the number usingP TDHF,R R MCTDHF with 6 orbitals or the TD- of configurations is 6 times smaller. Let us remark that RASSCF-D method for the RAS (M1,M2) = (5, 1) give a smaller number of configurations does not necessarily the same result, showing that for linearly polarized lasers mean a smaller numerical effort for a given number of these approaches are equivalent as was the case in the orbitals, M, since the number of operations required to previous section for the ground-state studies. Therefore, calculate the two-body operator scales as O(M 4) [16]. ∼ in the rest of this paper we only present the TDHF results However, for a given number of orbitals, the MCT- to illustrate these three cases. DHF calculation requires much more memory for stor- In Fig. 2(a) we show the ionization yield for ω = 95 eV. ing the amplitudes than the TD-RASSCF method. For At t = 0, i. e., at the maximum of the laser field, the de- instance, MCTDHF with 14 orbitals consists of 4008004 tected ionization is close to zero, and increases monoton- configurations, approximately 1459 times more than the ically with time as the photoelectron wavepacket escapes (M ,M ) = (5, 9) case. 1 2 from the inner region. For all the RAS partitions used, at approximately t = 15 a.u., P1(t) reaches a plateau, which means that the electron wavefunction is beyond B. Ionization and photoelectron spectrum rout. We see that the ionization yield obtained by the TDHF method is higher than the yields obtained with In this section, we investigate the ionization dynamics the other methods, which include correlation by populat- of Ne induced by an XUV laser pulse. We consider a ing 2. In numbers, for TDHF, the plateau-value for the P 5

4 10− 6.5 50 × (5,0) (a) 95 eV 6 (6,0) (5,1) 40 5.5 (5,4) (5,9) 5 Marr et al. 30 Samson et al. (t)

(Mb) 4.5 1 1 P

σ 20 (M1,M2) 4 (5,0) (6,0) 3.5 10 (5,1) (5,4) 3 (5,9) 0 2.5 35 75 85 95 105 115 ω (eV) (b) 105 eV 30 FIG. 3. Total photoionization cross section calculated for a 25 10 cycles linearly pulse with peak intensity 1014 W/cm2 as a function of the central frequency ω for several RAS partitions 20 compared to the experimental data by Marr et al. [49] and (t)

1 Samson et al. [50]. The outcome for TDHF and RAS (5, 1) P 15 and (6, 0) are indistinguishable (see text). 10

5 Accordingly, the effect of electron correlation cannot be clearly identified from the inspection of total ionization 0 yields. A deeper understanding necessitates the study 25 of a more differential quantity such as the photoelectron (c) 115 eV spectrum (PES), as we will come back to below. First, 20 however, we compare the TD-RASSCF calculation of the ionization with the experimental cross section [49, 50], σ, for 75 eV ω 115 eV in Fig. 3. We remark that the 15 experimental≤ data≤ are very similar, although the data (t)

1 from Ref. [49] are systematically larger than those from P 10 Ref. [50]. To extract the single ionization cross section, we use the following expression [51] 5 14 2 σ1(Mb) = 1.032 10 ω P1/(npI0), (12) × 2 0 where I0 is the peak intensity of the laser pulse in W/cm 0 5 10 15 20 25 30 and P1 corresponds to P1(t) at a time when a constant t (a. u.) value is reached. Equation (12) is valid for one-photon processes, and estimates the effective interaction time to FIG. 2. Total ionization yield as a function of time, P (t), 1 be Teff = 3πnp/4ω [51, 52]. In Fig. 3, we see that the for a 10 cycle linearly polarized pulse with peak intensity TDHF method overestimates the cross section, although 1014 W/cm2 for (a) ω = 95, (b) 105 and (c) 115 eV as a the difference with the experiment decreases with increas- function of time for several RAS partitions. Note that the maximum of the pulse is at t = 0. ing photon energy. On the other hand, for ω = 75 eV, the results of the (M1,M2) = (5, 4) and (5, 9) calculations coincide with the experimental results of Ref. [50]. As we increase ω, TD-RASSCF with M2 = 4 underesti- −4 ionization probability is P1 47.18 10 , and it reduces mates the cross section up to 115 eV, where the difference −4 ≈ −4 × to 44.13 10 and 45.12 10 for (M1,M2) = (5, 4) with the experimental value decreases. The numerical × × and (5, 9), respectively. For ω = 105 eV [Fig. 2(b)], the calculation using M2 = 9 is in better agreement with pattern is the same, but P1(t) in the plateau region is the experimental result, which manifests the key role of smaller, 32.86 10−4 for TDHF, since the photon energy the electron correlation in the photoionization process of is further from× the ionization energy than ω = 95 eV. Ne [6]. The agreement is better than that obtained with We see in Fig. 2(c) that P1(t) for ω = 115 eV is much other methods which cannot fully account for correlation smaller for TD-RASSCF with (M1,M2) = (5, 4) than for effects [5, 53]. TDHF, whereas the partition (M1,M2) = (5, 9) results in Finally, we compare the PES for ω = 95, 105 and an ionization yield similar to that obtained with TDHF. 115 eV as a function of the energy of the ejected elec- 6

10 5 35 × − and (5, 9), and, moreover, the position of the peak is (a) 95 eV (M1,M2) located at lower energies for the TDHF method. For 30 (5,0) (5,4) ω = 95 eV, the dominant peak is quite similar for both 25 (5,9) TD-RASSCF schemes, but, as we increase the photon

energy the maximum height of the peak corresponding 1

− 20 to the (5, 9) partition becomes slightly larger than in eV 15 the case of (5, 4) and the position of the peak shifts to P E d d higher energies. Speciﬁcally, the peaks are located at 10 42.47 and 71.53 eV for TDHF, at 44.087 and 72.8 eV for (M1,M2) = (5, 4) and at 44.87 and 72.40 eV for 5 (M1,M2) = (5, 9), compared to the experimental val- 0 ues of 46.525 and 73.435 eV [46, 48]. Let us remark 20 that as we increase the photon energy, since we set the (b) 105 eV 17.5 number of cycles to np = 10, the distributions are wider due to the broadening of the spectral components of 15 the laser pulse (∆ω ω/np with ω the central angu-

∝

1 12.5 lar frequency). The tails of the distributions induced − by this broadening does, however, not aﬀect the posi-

eV 10 tion of the neighboring peaks for the frequencies used in P E d d 7.5 this work. For instance, for ω = 115 eV and using the 5 RAS partition (M1,M2) = (5, 9), the ionization threshold (extracted from the position of the peaks of the PES) 2.5 are 22.67 and 50.16 eV for Ne+ [(1s22s22p5)2Po] and 0 Ne+ [(1s22s2p6)2Se], respectively, which agree with the 12.5 (c) 115 eV thresholds extracted using the same RAS but ω = 95 eV.

10 C. Time-delay 1

− 7.5 eV

In this section we calculate the time-delay between P E d d 5 the ejection of electrons from the 2s and 2p subshells of Ne after the interaction with the pulse and compare 2.5 with the experimental value obtained using the streaking technique [18] and values from theory [26, 29, 31–35]. 0 Our strategy consists of extrapolating the streaking time- 20 40 60 80 100 120 delay, τ, from the eﬀective ionization time of the photo- E (eV) electron ejected from a given subshell, tCoul(t), that we can extract from the dynamics of the electrons in the FIG. 4. Photoelectron spectra as a function of the emitted electron energy for a 10 cycle linearly polarized pulse with outer region. It reads as peak intensity 1014 W/cm2 for (a) ω = 95, (b) 105 and (c) 115 eV as a function of time for several RAS partitions. r(t) tCoul(t) = t h i, (13) − k

with r(t) the expectation value of the position in the tron in Fig. 4. We obtain the PES by considering the outerh regioni at a given time t. Let us note that the projection of the wavefunction for rout 20 on Coulomb apparent ionization time-delay t (t) depends on time ≥ Coul waves [54] using a window function to bypass boundary t because in the presence of the Coulomb tail of the eﬀects associated with the inner region and the end of ion, the photoelectron cannot be described by a ﬁeld- the box. The procedure is explained in Ref. [16]. At a free wavepacket in the outer region. We can separate the given photon energy, the main peak corresponds to an dependence on t using the relation [55] electron ejected from the p shell of the ground-state of 2 2 6 1 e Ne [(1s 2s 2p ) S ], where the ion is left in the state tCoul(t) = τEWS + ∆tCoul(t) (14) Ne+ [(1s22s22p5)2Po] whose energy is 21.565 eV [Fig. 1]. The peak at lower energy corresponds to ionization into where τEWS is the Eisenbud-Wigner-Smith (EWS) time- + 2 6 2 e the channel Ne [(1s 2s2p ) S ], at 48.475 eV [Fig. 1]. delay, which corresponds to the time required to escape For all the photon energies considered in Fig. 4, the the potential without the interaction with the Coulomb TDHF method overestimates the height of the main peak Z 2 tail and ∆tCoul(t) = 1 ln(2k t) is the distortion with respect to the TD-RASSCF using (M1,M2) = (5, 4) k3 − 7

10 5 4 × − caused by the long range nature of the Coulomb poten- (a) t = 32.57 a.u.(θ,φ = 0) (b) t = 40.71 a.u. tial. In Eq. (14), k is the linear momentum of the pho- 3.5 θ = 0 3 θ = π/2 + 1 toelectron and Z = 1 is the charge of the remaining Ne − 2.5 sr ion. Due to the short duration of the ionizing pulse, 2 Ω P d 3 1.5 r d the photoelectron is described by a wavepacket in k, and d 1 hence k attains several values. It is possible to describe 0.5 2 1/2 0 the distribution over k, e. g., by k or k , where 20 denotes expectation value. In thish work,i we calculate k byhi (c) t = 32.57 a.u.ℓ (d) t = 40.71 a.u. 16 0 solving Eq. (14) for two diﬀerent times. In the streaking 1 2 2

| 12 )

experiments it is not τEWS that is measured directly, but r ( ℓ

ρ rather the streaking time-delay, τ, which may be written | 8 as 4

0 τ = τEWS + τCLC, (15) 50 70 90 110 130 50 70 90 110 130 r (a.u.) r (a.u.) where τCLC is the contribution due to the Coulomb-laser coupling [20], and corresponds to the interaction with the FIG. 5. Triple differential density at different times after the IR field used in the streaking scheme. The quantity τCLC peak of the pulse with ω = 105 eV (a) t = 32.57 a.u. and (b) can be extrapolated accurately by [20, 34] t = 40.71 a.u. and the radial density of the components with a given ` at (c) t = 32.57 a.u. and (d) t = 40.71 a.u. for the Z πk2 photoelectron wavepacket for the TD-RASSCF method with τCLC = 2 ln , (16) (M ,M ) = (5, 4). k3 − ω 1 2 IR where ωIR is the frequency of the IR pulse [20]. In our study, we do not include any streaking field, but to com- in the outgoing wavepacket in Figs. 5(a)-(b), where we pare with experiments τCLC has to be accounted for. In show the triple differential density (TDD) along the par- this work, typical values for τCLC range from 9 as for allel and perpendicular directions for two different times a photon energy of ω = 85 eV to 2.3 as for∼ a photon after the peak of the XUV pulse with a central frequency energy of ω = 125 eV. ∼ of ω = 105 eV. The distribution along the parallel di- Next, we discuss the numerical method used to obtain rection at t = 32.57 a.u. is dominated by the emission the relative time-delay between the photoionization from of s and d photoelectrons, and it is peaked at approx- 2s and 2p subshells, i. e., τ2p−2s = τ2p τ2s. The pho- imately r = 80.5 a.u. and presents a small shoulder toionization channels involved are, in terms− of dominant around 60 a.u., which corresponds to the p continuum configurations, electron contribution. At a later time, t = 40.71 a.u., this shoulder becomes more pronounced because the p Ne [(1s22s22p6)1Se] Ne+ [(1s22s22p5)2Po] + e−(s, d) → photoelectron is slower than the s and d electron provok- (17) ing the separation of the contributions, simply because Ne [(1s22s22p6)1Se] Ne+ [(1s22s2p6)2Se] + e−(p), of the differences in the ionization thresholds in Eqs. (17) → and (18). Consequently, to distinguish both channels in (18) this direction is not numerically efficient or even feasible where the angular momentum ` of the emitted electron, in the present case. In particular, because it would be e−, is restricted to s and d when the electron is removed necessary to i) propagate for longer times and ii) employ from the 2p subshell and to p in the case of ionization of a larger radial box to avoid boundary effects. Further- the 2s shell. We note that both channels in Eqs. (17)- more, even if we could meet these two demands, it may (18) only involve the change of a single orbital, and they be not possible to determine the expected position of the can hence both occur within the TDHF description. photoelectron, due to the spreading of the density as a We can benefit from the difference in ` in the final function of time. On the other hand, the TDD along the continuum states to distinguish between the ionization perpendicular direction corresponds to removing the 2p channels by calculating r(t) along the parallel (all the electron, therefore we could calculate directly r(t) for h i three channels contribute)h andi perpendicular direction this channel. (only the s and d contribute) to the polarization of the To overcome these difficulties imposed by determin- laser. This technique was successfully applied in the case ing time-delays by using angular distributions, we dis- of Be, and relied on the fact that the contribution of the tinguish between the different angular contributions to s and d photoelectrons is negligible with respect to that the photoelectron wavepacket by selecting the single or- of the p electron [16]. This is, however, not the case bital angular momentum ` of the many-body wavefunc- for Ne, since the cross section for photoionization from tion in the outer region. Thus, the different channels are the 2p shell (i. e., s and d continuum electrons) is much labeled by the angular momentum of the photoelectron larger than that from the 2s shell (i. e., p continuum elec- wavepacket ` uniquely, as we can see in Fig. 5(c)-(d), and trons [5]). We illustrate the implications of this difference we can isolate them to obtain r(t) corresponding to a h i 8 given `. Taking all this into account, to obtain the time- 26 (5,0) delay between the photoelectrons ejected from 2s and 2p 24 (5,4) we follow the steps: i) calculate r(t) for two different Feist et al. h i 22 times t and ` = 1 and 2; ii) obtain k for each channel Moore et al. 20 Dahlström et al. as described just above Eq. (15); iii) evaluate tCoul(t) 18 Schultze et al. using the expression (13); iv) calculate τ2s and τ2p us- (as) s ing the expressions (14) and (15); and finally, v) obtain 2 16 − p

τ2p−2s = τ2p τ2s. Let us note that a good description 2 14 − τ not only of the potential induced by the electrons but also 12 an accurate description of the photoelectron spectrum is mandatory for the accurate application of this approach. 10 For instance, using a two-electron model in a mean-field 8 potential [31] gives a value of τ2p−2s = 4.3 as for a cen- 6 tral frequency of 107 eV, which is below the expected 85 95 105 115 125 theoretical value [see Fig. 6] although the result qualita- ω (eV) tively captures that the electron in the 2p shell is emitted after the 2s. In Fig. 6 we show the streaking time- FIG. 6. Relative time-delay of ionization in Ne, τ2p−2s, as a function of the central frequency of the XUV pulse for a delay τ2p−2s assuming a 780 nm IR pulse, to account for 780 nm IR pulse for (M1,M2) = (5, 0) and (5, 4) together τCLC for the RAS scheme (M1,M2) = (5, 0) and (5, 4), with the calculations by Feist et al. [34], Moore et al. [29] together with calculations which use different methods and Dahlströmet al. [56] and the measurement by Schultze to account for the electronic correlation [29, 34, 56]. In et al. [18]. Ref. [29] the R-matrix method was employed to describe the streaking process, where the inner region is described using configuration interaction and only one electron is schemes, decreases as we increase ω and almost vanishes allowed in the outer region. The results of this method is at ω = 125 eV, revealing that the correlation becomes less in agreement with the results of the TDHF method, i. e., important with increasing ω for the considered case in Ne. (M1,M2) = (5, 0), from 90 eV ω 100 eV, except that Let us note that the EWS time-delay between the 2p and the R-matrix prediction is larger≤ for≤ω < 90 eV and lower 2s photoelectrons for 105 eV, τEWS,2p τEWS,2s = 7.1 for ω > 105 eV. As stressed in Ref. [29], for ω > 105 eV − and 5.8 as for the RAS (M1 = 5,M2 = 0) and (M1 = there are contributions from pseudoresonances∼ induced 5,M2 = 4), are in good agreement with the EWS delay by the expansion in the inner region which may alter 6.4 as reported in Ref. [18] and obtained using the state- the result of the calculation. Furthermore, the sensitiv- specific approach for ω = 106 eV, but are significantly ity of the time-delay to the ionization energy of each higher than the 4.0 as calculated using multiconfigura- channel may also interfere for low ω’s. We can also tional Hartree-Fock method, also in Ref. [18]. conclude that part of the correlation can be described using a single configuration, as in the TDHF method, due to the flexibility in the propagation provided by the IV. CONCLUSION time-dependent orbitals. This situation is markedly different from the case we considered in Be where ionization of the Be[(1s22s2)1Se] ground-state into the chan- We have applied the TD-RASSCF-D method to inves- nel Be+[(1s22p)2Po] + e−(s or d) can not be described tigate the role of electron correlation in the ground-state by TDHF, since, contrary to the case in Eqs. (17)-(18), of Ne, as well as in photoionization processes induced more than a single orbital in the dominant configuration by attosecond XUV linearly polarized laser pulses. We is changing [16]. have shown that the TD-RASSCF-D method provides accurate results for the ground-state energy, converging The TD-RASSCF method with (M1,M2) = (5, 4) pro- to the MCTDHF method as we increase the electronic vides a smaller τ2p−2s, which is in agreement with the re- correlation, i. e., the number of accessible configurations. sults of Ref. [34]. In that work, the ground-state and scat- Most importantly, we also found that it is possible to tering states of Ne are built using up to 38 states to de- obtain a better ground-state by increasing the number scribe the atom and the Ne+ ion to extract the phase shift of orbitals while keeping the number of configurations corresponding to the long- and the short-range interac- manageable by design of the RAS scheme. The decisive ∂ tion, σ`(E)+δ`(E), and hence τEWS = ∂E [σ`(E)+δ`(E)]. role of the number of orbitals compared with the number Then, as in our case, the total time-delay is obtained by of configurations, i.e., the importance of having access adding τCLC [34]. In contrast, in Ref. [56], the time- to a few rather highly excited configurations in the SCF delay τ is fully extracted by studying the streaking pro- expansion of the total wave function, was also found in cess in a time-independent diagrammatic approach. The cold atom physics [17]. This finding shows the potential resulting time-delay is τ = 12 as for ω 105 eV, a bit of the TD-RASSCF approach for application to systems higher than the TDHF result. Finally, we∼ note that the where the MCTDHF can not be practically applied due difference between the τ2p-2s for the two different RAS to the dimension of the problem. For the photoionization 9 dynamics, we describe the ionization threshold energy of termined by the - and -space equations, Eqs. (7) + 2 6 2 e 2 2 5 2 o Q P i the channels Ne [(1s 2s2p ) S ] and [(1s 2s 2p ) P ]. and (8), respectively. By setting ηj(t) either to be 0 or We also obtained results for the angular distribution of hi (t), the only contribution in -space which may mix j Q the electron ejected by the XUV pulse. Moreover, we ob- the magnetic quantum number m in the orbital φi(t) tained numerical cross sections which are in agreement −1 n jl k | i is n(ρ (t))i jkl Q(t)ρnk(t)Wl (t) φj(t) . We first with the available experimental data. Finally, we cal- n | i prove that ρ−1(t) is non-zero only for m = m . From culated the time-delay between the propagated electrons P P i n i the definition of the one-body density operator ejected from the 2p and 2s shells by taking advantage of the angular momentum decomposition of the photoelec- n ∗ n ρ (t) = CI (t) CJ (t) ΦI (t) E ΦJ (t) , trons to measure independently these two channels. For i h | i | i I,J ω = 105 eV, we obtain τ2p−2s = 9.9 as in agreement with X n other theory works [26, 29–34, 56] and with the very re- we see that the contribution ΦI (t) E ΦJ (t) = 0 h | i | i 6 cent interferometric experimental measurements [36], but only if ΦI (t) and ΦJ (t) only differ in the orbitals | i | i in disagreement with the experimental value of 21 as φn(t) and φi(t) , respectively. Since ML is the same ∼ | i | i n for a photon energy of 106 eV [18]. For the present study for all the configurations, ρi (t) = 0 ML mi = n 6 ⇒ − in Ne, the channels considered are both accessible by the ML mn mi = mn. Thus, ρi (t) is block diagonal in change of a single orbital from the dominant ground- the single-electron− ⇒ magnetic quantum number. This im- state configuration following single-photon absorption. plies that ρ−1(t) is also block diagonal in m, and, there- We found that in this case, the TDHF method gives a −1 n fore, ρ (t) i = 0 only if mn = mi. A similar argument qualitatively correct estimate of the time-delay in pho- 6 jl can be applied for ρ (t) which reads as toionization. In systems that are too complicated to be nk investigated by any theory beyond mean-field, a compar- jl jl ρ (t) = C (t)∗C (t) Φ (t) E Φ (t) ison between, e.g., experimental time-delay data and the nk I J I nk J I,J D E results from TDHF could then serve to isolate presence X or absence of correlations effects. and it is non-zero only if mj + ml = m n + mk. Finally, k since Wl (t) φj(t) is a function with m = ml + mj mk and the projector| iQ(t) preserves the magnetic quantum− ACKNOWLEDGMENTS jl k number, Q(t)ρ W (t) φj(t) = 0 only if mn = ml + nk l | i 6 mj mk = mi, which ensures the conservation of m by This work was supported by the Villum Kann Ras- the − -space equation. mussen (VKR) Center of Excellence QUSCOPE. J.J.O. Now,Q we prove that the -space equation (8) satisfies was supported by NSERC Canada (via a grant to Prof. P. k00 P ηl0 = 0 if ml0 = mk00 . First, we evaluate the last sum- Brumer). The numerical results presented in this work mation which involves6 the two-body elements. It is easy were obtained at the Centre for Scientific Computing, j00m to prove that vkl (t) = 0 mk + ml = mm + mj00 , and Aarhus. 6 ⇒ kl using similar arguments of the previous proof, ρ 0 (t) = i m 6 0 ml + mk = mi0 + mm. Equating these two expres- ⇒ sions, we obtain that m 00 = m 0 , which also holds for Appendix A: Conservation of orbital m j i the second term of the summation. On the other hand, 0 00 00 00 00 l j j l0 l0 j j l0 Ak00i0 (t) = Ψ(t) Ei0 Ek00 Ek00 Ei0 Ψ(t) = δk00 ρi0 00 In this appendix, we prove that the magnetic quantum l0 j h | − 0 0| i − number m of each orbital is conserved if each orbital is δi0 ρk00 = 0 mj00 = mk00 with i = l and/or mi0 = ml0 with j006 = k⇒00. Taking all this into account when solving labelled with a well-defined m and the wavefunction is a k00 k00 this equation, we get that h 0 (t) iη 0 (t) = 0 only if linear combination of configurations ΦI (t) with the l l {| i} k00− 6 same total magnetic quantum number ML. ml0 = mk00 . Finally, using that hl0 (t) fulfils ml0 = mk00 k00 The propagation of the single-electron orbitals is de- it follows that η 0 = 0 ml0 = mk00 . l 6 ⇒

[1] Egil A Hylleraas, “Uber¨ den Grundzustand des Heliu- [5] D. Hochstuhl, C. Hinz, and M. Bonitz, “Time-dependent matoms,” Zeitschrift fürPhys. 48, 469–494 (1928). multiconfiguration methods for the numerical simulation [2] Egil A. Hylleraas, “Neue Berechnung der Energie des He- of photoionization processes of many-electron atoms,” liums im Grundzustande, sowie des tiefsten Terms von Eur. Phys. J. Spec. Top. 223, 177–336 (2014). Ortho-Helium,” Zeitschrift fürPhys. 54, 347–366 (1929). [6] Vinay Pramod Majety and Armin Scrinzi, “Photo- [3] U. Fano, “Effects of Configuration Interaction on In- Ionization of Noble Gases: A Demonstration of Hybrid tensities and Phase Shifts,” Phys. Rev. 124, 1866–1878 Coupled Channels Approach,” Photonics 2, 93 (2015). (1961). [7] Carlos Marante, Markus Klinker, Tor Kjellsson, Eva Lin- [4] F. Krausz and M. Ivanov, “Attosecond physics,” Rev. droth, JesúsGonzález-Vázquez,Luca Argenti, and Fer- Mod. Phys. 81, 163–234 (2009). nando Mart´ın, “Photoionization using the xchem ap- 10

proach: Total and partial cross sections of Ne and res- M. Sabbar, R. Boge, M. Lucchini, L. Gallmann, I. A. onance parameters above the 2s22p5 threshold,” Phys. Ivanov, A. S. Kheifets, J. M. Dahlström,E. Lindroth, Rev. A 96, 022507 (2017). L. Argenti, F. Mart´ın, and U. Keller, “Angular depen- [8] H. Miyagi and L. B. Madsen, “Time-dependent dence of photoemission time delay in helium,” Phys. Rev. restricted-active-space self-consistent-field theory for A 94, 63409 (2016). laser-driven many-electron dynamics,” Phys. Rev. A 87, [23] I. A. Ivanov and A. S. Kheifets, “Angle-dependent time 062511 (2013). delay in two-color XUV+IR photoemission of He and [9] H. Miyagi and L. B. Madsen, “Time-dependent Ne,” Phys. Rev. A 96, 13408 (2017). restricted-active-space self-consistent-field singles [24] S. Banerjee, P. C. Deshmukh, A. Kheifets, V. K. Dolma- method for many-electron dynamics.” J. Chem. Phys. tov, and S. T. Manson, “Study of angular dependence of 140, 164309 (2014). photoionization time delay in (n − 1)d → εf channels for [10] H. Miyagi and L. B. Madsen, “Time-dependent Zn, Cd and Hg,” J. Phys. Conf. Ser. 875, 022025 (2017). restricted-active-space self-consistent-field theory for [25] Nicolas Camus, Enderalp Yakaboylu, Lutz Fechner, laser-driven many-electron dynamics. II. Extended for- Michael Klaiber, Martin Laux, Yonghao Mi, Karen Z. mulation and numerical analysis,” Phys. Rev. A 89, Hatsagortsyan, Thomas Pfeifer, Christoph H. Keitel, 063416 (2014). and Robert Moshammer, “Experimental evidence for [11] H. Miyagi and L. B. Madsen, “Time-dependent Wigner’s tunneling time,” Phys. Rev. Lett. 119, 023201 restricted-active-space self-consistent-field theory with (2017). space partition,” Phys. Rev. A 95, 023415 (2017). [26] A. S. Kheifets and I. A. Ivanov, “Delay in atomic pho- [12] J. Caillat, J. Zanghellini, M. Kitzler, O. Koch, toionization,” Phys. Rev. Lett. 105, 233002 (2010). W. Kreuzer, and A. Scrinzi, “Correlated multielectron [27] J. C. Baggesen and L. B. Madsen, “Polarization effects systems in strong laser fields: A multiconfiguration time- in attosecond photoelectron spectroscopy,” Phys. Rev. dependent Hartree-Fock approach,” Phys. Rev. A 71, Lett. 104, 043602 (2010). 012712 (2005). [28] Yannis Komninos, Theodoros Mercouris, and Clean- [13] Stefan Pabst, L. Greenman, D. A. Mazziotti, and thes A. Nicolaides, “Regular series of doubly excited R. Santra, “Impact of multichannel and multipole ef- states inside two-electron continua: Application to 2s2- fects on the Cooper minimum in the high-order-harmonic hole states in neon above the Ne2+ 1s22s22p4 and spectrum of argon,” Phys. Rev. A 85, 023411 (2012). 1s22s22p5 thresholds,” Phys. Rev. A 83, 022501 (2011). [14] D. Hochstuhl and M. Bonitz, “Time-dependent [29] L. R. Moore, M. A. Lysaght, J. S. Parker, H. W. van der restricted-active-space configuration-interaction method Hart, and K. T. Taylor, “Time delay between photoe- for the photoionization of many-electron atoms,” Phys. mission from the 2p and 2s subshells of neon,” Phys. Rev. Rev. A 86, 053424 (2012). A 84, 061404 (2011). [15] S. Bauch, L. K. Sørensen, and L. B. Madsen, [30] M. Ivanov and O. Smirnova, “How accurate is the at- “Time-dependent generalized-active-space configuration- tosecond streak camera?” Phys. Rev. Lett. 107, 213605 interaction approach to photoionization dynamics of (2011). atoms and molecules,” Phys. Rev. A 90, 062508 (2014). [31] S. Nagele, R. Pazourek, J. Feist, and Joachim [16] Juan J. Omiste, Wenliang Li, and Lars Bojer Madsen, Burgdörfer,“Time shifts in photoemission from a fully “Electron correlation in beryllium: Effects in the ground correlated two-electron model system,” Phys. Rev. A 85, state, short-pulse photoionization, and time-delay stud- 033401 (2012). ies,” Phys. Rev. A 95, 053422 (2017). [32] J. Su, H. Ni, A. Jarón-Becker, and A. Becker, “Time [17] Camille Lévêqueand L. B. Madsen, “Time-dependent delays in two-photon ionization,” Phys. Rev. Lett. 113, restricted-active-space self-consistent-field theory for 263002 (2014). bosonic many-body systems,” New J. Phys. 19, 043007 [33] H. Wei, T. Morishita, and C. D. Lin, “Critical evalua- (2017). tion of attosecond time delays retrieved from photoelec- [18] M. Schultze, M. Fieß, N. Karpowicz, J. Gagnon, tron streaking measurements,” Phys. Rev. A 93, 053412 M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, (2016). Y. Komninos, Th. Mercouris, C. A. Nicolaides, R. Pa- [34] J. Feist, O. Zatsarinny, S. Nagele, R. Pazourek, Joachim zourek, S. Nagele, J. Feist, Joachim Burgdörfer,A. M. Burgdörfer,X. Guan, K. Bartschat, and B. I. Schneider, Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, “Time delays for attosecond streaking in photoionization E. Goulielmakis, F. Krausz, and V. S. Yakovlev, “Delay of neon,” Phys. Rev. A 89, 033417 (2014). in Photoemission,” Science 328, 1658–1662 (2010). [35] R. Pazourek, M. Reduzzi, P. A. Carpeggiani, G. San- [19] K. Klünder, J. M. Dahlström, M. Gisselbrecht, sone, M. Gaarde, and K. Schafer, “Ionization delays in T. Fordell, M. Swoboda, D. Guénot, P. Johnsson, few-cycle-pulse multiphoton quantum-beat spectroscopy J. Caillat, J. Mauritsson, A. Maquet, R. Ta¨ıeb, and in helium,” Phys. Rev. A 93, 023420 (2016). A. L’Huillier, “Probing single-photon ionization on the [36] M. Isinger, R. J. Squibb, D. Busto, S. Zhong, A. Harth, attosecond time scale,” Phys. Rev. Lett. 106, 143002 D. Kroon, S. Nandi, C. L. Arnold, M. Miranda, J. M. (2011). Dahlström,E. Lindroth, R. Feifel, M. Gisselbrecht, and [20] Renate Pazourek, S. Nagele, and Joachim Burgdörfer, A. L’Huillier, “Photoionization in the time and frequency “Attosecond chronoscopy of photoemission,” Rev. Mod. domain,” Science 358, 893–896 (2017). Phys. 87, 765–802 (2015). [37] T. Helgaker, P. Jørgensen, and J. Olsen, Mol. Electron. [21] Jing Su, Hongcheng Ni, Andreas Becker, and Agnieszka Theory, john wiley ed. (2000). Jaroń-Becker, “Numerical simulation of time delays in [38] M. Beck, A. Jäckle, G. A. Worth, and H.-D. Meyer, “The light-induced ionization,” Phys. Rev. A 87, 33420 (2013). multiconfiguration time-dependent Hartree (MCTDH) [22] S. Heuser, A. JiménezGalán,C. Cirelli, C. Marante, method: a highly efficient algorithm for propagating 11

wavepackets,” Phys. Rep. 324, 1–105 (2000). II,” Phys. Scr. 3, 133–155 (1971). [39] O. Koch, W. Kreuzer, and A. Scrinzi, “Approximation [48] A. E. Kramida and G. Nave, “The Ne II spectrum,” Eur. of the time-dependent electronic Schrödingerequation by Phys. J. D - At. Mol. Opt. Plasma Phys. 39, 331–350 MCTDHF,” Appl. Math. Comput. 173, 960–976 (2006). (2006). [40] Jeppe Olsen, BjörnO Roos, Poul Jørgensen, and Hans [49] G.V. Marr and J.B. West, “Absolute photoionization Jørgen Aa. Jensen, “Determinant based configuration in- cross-section tables for helium, neon, argon, and kryp- teraction algorithms for complete and restricted configu- ton in the VUV spectral regions,” At. Data Nucl. Data ration interaction spaces,” J. Chem. Phys. 89, 2185–2192 Tables 18, 497–508 (1976). (1988). [50] J. A. R. Samson and W. C. Stolte, “Precision measure- [41] Michael Klene, Michael A Robb, Llus Blancafort, and ments of the total photoionization cross-sections of He, Michael J Frisch, “A new efficient approach to the direct Ne, Ar, Kr, and Xe,” J. Electron Spectros. Relat. Phe- restricted active space self-consistent field method,” J. nomena 123, 265–276 (2002). Chem. Phys. 119, 713–728 (2003). [51] E. Foumouo, G. L. Kamta, G. Edah, and B. Piraux, [42] Tsuyoshi Kato and Hirohiko Kono, “Time-dependent “Theory of multiphoton single and double ionization of multiconfiguration theory for electronic dynamics of two-electron atomic systems driven by short-wavelength molecules in intense laser fields: A description in terms of electric fields: An ab initio treatment,” Phys. Rev. A 74, numerical orbital functions,” J. Chem. Phys. 128, 184102 063409 (2006). (2008). [52] L. B. Madsen, L. A.A. Nikolopoulos, and P. Lam- [43] D. J. Haxton, K. V. Lawler, and C. W. McCurdy, “Single bropoulos, “Extracting generalized multiphoton ioniza- photoionization of Be and HF using the multiconfiguration cross-sections from nonperturbative time-dependent tion time-dependent Hartree-Fock method,” Phys. Rev. calculations: An application in positronium,” Eur. Phys. A 86, 013406 (2012). J. D 10, 67–79 (2000). [44] T. Sato, K. L. Ishikawa, I. Bˇrezinová, F. Lackner, [53] D. Hochstuhl and M. Bonitz, “Time-dependent restricted S. Nagele, and Joachim Burgdörfer,“Time-dependent active space Configuration Interaction theory applied to complete-active-space self-consistent-field method for the photoionization of neon,” J. Phys. Conf. Ser. 427, atoms: Application to high-order harmonic generation,” 012007 (2013). Phys. Rev. A 94, 023405 (2016). [54] L. B. Madsen, L. A. A. Nikolopoulos, T. K. Kjeldsen, and [45] Carlos F. Bunge, JoséA. Barrientos, and Annik Vivier J. Fernández,“Extracting continuum information from Bunge, “Roothaan-Hartree-Fock Ground-State Atomic Ψ(t) in time-dependent wave-packet calculations,” Phys. Wave Functions: Slater-Type Orbital Expansions and Rev. A 76, 063407 (2007). Expectation Values for Z = 2-54,” At. Data Nucl. Data [55] Charles W. Clark, “Coulomb phase shift,” Am. J. Phys. Tables 53, 113–162 (1993). 47, 683–684 (1979). [46] E. B. Saloman and C. J. Sansonetti, “Wavelengths, En- [56] J. M. Dahlström,T. Carette, and E. Lindroth, “Dia- ergy Level Classifications, and Energy Levels for the grammatic approach to attosecond delays in photoion- Spectrum of Neutral Neon,” J. Phys. Chem. Ref. Data ization,” Phys. Rev. A 86, 061402 (2012). 33, 1113–1158 (2004). [47] W. Persson, “The Spectrum of Singly Ionized Neon, Ne