Double ionization of a three-electron atom: Spin correlation effects
Dmitry K. Efimov, Jakub S. Prauzner-Bechcicki, Jan H. Thiede, Bruno Eckhardt, Jakub Zakrzewski
DDouble ionization of a three-electron atom: Spin correlation effects
Dmitry K. Efimov, ∗ Jakub S. Prauzner-Bechcicki, Jan H. Thiede, Bruno Eckhardt, † and Jakub Zakrzewski
1, 3 Instytut Fizyki imienia Mariana Smoluchowskiego,Uniwersytet Jagiello´nski, (cid:32)Lojasiewicza 11, 30-348 Krak´ow, Poland Philipps-University Marburg, Biegenstraße 10, 35037 Marburg, Germany Mark Kac Complex Systems Research Center, Jagiellonian University, (cid:32)Lojasiewicza 11, 30-348 Krak´ow, Poland (Dated: December 9, 2019)We study the effects of spin degrees of freedom and wave function symmetries on double ion-ization in three-electron systems. Each electron is assigned one spatial degree of freedom. Theresulting three-dimensional Schr¨odinger equation is integrated numerically using grid-based Fouriertransforms. We reveal three-electron effects on the double ionization yield by comparing signals fordifferent ionization channels. We explain our findings by the existence of fundamental differencesbetween three-electronic and truly two-electronic spin-resolved ionization schemes. We find, for in-stance, that double ionization from a three-electron system is dominated by electrons that have theopposite spin.
I. INTRODUCTION
Approaching the attosecond time scale in experimentsallows one to directly probe internal atomic dynamics.The improvement in experimental techniques encouragesone to develop new and refine existing theoretical toolsin a way that will advance our understanding of pro-cesses and correlations inside atoms and molecules thattake place under strong electromagnetic field irradiation.That task remains at the focus of current strong-fieldphysics.Correlations between electrons in atoms and moleculesand their interaction with femtosecond pulses have beenstudied extensively [1–3]. The range and methods ofthese studies depended on how an electronic correlationwas understood by researchers. For example, a theoreti-cal definition of an electronic correlation was introducedin [4, 5], but its correspondence with experimentally mea-surable quantities stays unclear. On the other hand,there are many experimentally accessible signatures ofcorrelations. In the context of multiple ionization theseare a characteristic “knee” in the field amplitude depen-dent double ionization yield [3, 6] and the shape of two-electron momentum distributions [7–12]. The former wasexplained by the presence of two double ionization chan-nels: sequential double ionization (SDI), when electronsare released without experiencing a dynamical interac-tion, and non-sequential double ionization (NSDI), forwhich electronic interactions play a decisive role, for ex-ample by means of an electron–parent-ion recollision pro-cess [13–16]. In the electron momenta distribution, thepronounced “fingerlike” structure originates in Coulombcorrelations [17–22]. Additionally, angular distributionsof electronic momenta possess a correlation fingerprintsas well [23]. All these observations are nicely reviewedin [24, 25]. Momentum distributions also allow one to ∗ dmitry.efi[email protected] † † Deceased 7 August 2019. test how electronic correlations affect the delay betweentwo electron ionization times [26, 27]. Finally, multielec-tronic correlations can manifest themselves in High Har-monic Generation (HHG) [28–31] and affect sub-barrierstrong-field tunneling times [32]. For instance, Coloumbcorrelations prevent simultaneous rescattering of ionizingelectrons from the ion they left behind, thus shifting thesecond plateau cutoff position of high harmonic spectra[33].From a theoretical point of view, correlations of elec-trons may be revealed by the examination of subsys-tem dynamics. For example, in [32, 34] multielectroncorrelations are considered to affect the single-electronionization and HHG. The electron can also experiencesimultaneous rescattering on mixed neutral and ion-ized states and thus influence generation of harmonics[35, 36]. Chattopadhyay and Madsen [37] consider atwo-electron correlation effect on a single ionization ofdiatomic molecules.A minimal system where possible effects due to thirdactive electron on double ionization may be examined hasjust three electrons, such as Li. An early seminal studyof Li atom ionization demonstrated also the necessity tocarefully take into account the electron spin [38]. This isto be contrasted with standard two-active electrons treat-ments that implicitly assume a spatially symmetric wavefunction (corresponding to antisymmetric configurationof spins). The effects resulting from three-electron spinconfiguration we shall call spin correlation effects. While[38] considered the ionization of lithium with few largefrequency photons, the progress in numerical methods al-lows us nowadays to treat three active electrons at opticalfrequencies [39], though within a reduced dimensionalitymodel.The spin correlation effects are to be distinguishedfrom the Coulomb electron-electron correlation and inparticular from Coulomb interaction of the third electronwith the remaining pair. The latter is not an easy objectto study as any arbitrary changes in the internal atomicpotential yield the considerable shifts of ionization po-tential values and thus of Keldysh parameter values [40]. a r X i v : . [ phy s i c s . a t o m - ph ] D ec In the present paper we employ our three-active elec-trons model [39] to study effects of three-electron cor-relations on the double ionization process. In the pure3-electron atom, Li, the three electrons are not symmet-ric, since there is a single weakly bound p electron andtwo strongly bound 1s electrons. While Li ionizationpotentials are unfavorable for studying multiphoton cor-relation effects [38], to identify the effects of correlations,it seems better to consider atoms like boron or aluminum(with ns np electrons forming the outer shell). Even thisis highly difficult as a real multiphoton regime would re-quire going to very low frequencies. To stay with stan-dard optical frequencies, say ω = 0 .
06 a.u. which corre-sponds to about 760 nm wavelength of laser irradiation,we consider instead an artificial 3-electron atom with sin-gle, double and triple ionization thresholds correspondingto Ne. To be closer to experiments, we simulate ion-ization dynamics under the influence of experimentallyachievable 5-cycles-long laser pulses, rather than the veryshort 2-cycle pulse we considered previously [39].The correlation effects are revealed via comparing re-sults of strong-field ionization simulation for 3-electronand 2-electron models of the same atom. The latter isspecially defined to be in a good correspondence with theformer, preserving space geometry and double ionizationpotential values. The spin correlation effects study im-plies keeping interaction terms in two- and three- poten-tials as close to each other as possible.The paper is organized as follows. First we describein Section II models of two- and tree-electron atoms weuse, underlining physical differences between them thatcan not be eliminated by simple changes of parameters.We then discuss in Section III the different ionization pro-cesses in the presence of spin degree of freedom. In Sec-tion IV we show how such differences manifest themselvesin the output of atomic ionization simulations. We closewith a summary and conclusions in Section V . Atomicunits are used throughout this paper unless stated other-wise. For the sake of clarity, we note that 1 a.u. of energyis equal to 27.2 eV; at the same time 0.1 a.u. of electricfield corresponds to 3 . · W/cm of laser intensity. II. SIMPLIFIED ATOMIC MODELS
A numerical calculation of the dynamics of three elec-trons in the full space is still beyond the reach of currentnumerical capabilities. We introduce, therefore, mod-els where each electron is restricted to move along one-dimensional track so that the dimensionality of positionspace does not exceed three. We first construct the three-active electrons model and then define the two activeelectrons restricted model. a. Three-active electrons model.
In this model, themotion of each electron is restricted to a one-dimensionalline, forming (i) an angle of π/ γ (tan γ = (cid:112) /
3) with the electric fieldpolarization direction. We should point out here that the vector of electric field is not lying in any of the two-electronic planes, but forms equal angles with them. Thisunusual configuration is identified on the basis of an adi-abatic analysis that assumes that the ionization processis most effective along the lines defined by the saddles ofthe potential energy in the presence of the instantaneousstatic electric field [41]. These saddles can be consid-ered as transition states leading to efficient channels forionization. As the field amplitude changes during thepulse, these saddles move along lines inclined at constantangles with respect to the field polarization axis and toeach other in a fixed configuration, independent of fieldstrength [41]. Those lines are then taken as tracks towhich each electron’s motion is confined. The Hamilto-nian of three-electron system in the discussed geometrythen reads H = (cid:88) i =1 p i V ( r , r , r ) (1)with V ( r , r , r ) = − (cid:88) i =1 (cid:32) (cid:112) r i + (cid:15) + (cid:114) F ( t ) r i (cid:33) + (cid:88) i,j =1 i 83 a.u. so that I p = 4 . 63 a.u. i.e. 126 eV. Time-dependent Schr¨oodinger equation is solved on a spatialgrid with the use of split operator technique and FastFourier transform with algorithms described in detailselsewhere [39, 42]. The largest grid used, having 2048points in each direction covering 400 a.u. of physical co-ordinate space, required about a week of 192 cores for5-cycles pulse evaluation. The initial state was found byimaginary time propagation in an appropriate symme-try subspace for much smaller grid involving, typically,512 points in each direction corresponding to 100 a.u.The ionization yields are calculated by integrating theelectronic fluxes through different ionization regions bor-ders, which in turn are expressed as a surface integralsof probability currents calculated directly from the wave-function and its gradient. The procedure is described indetail in [39]. We use absorbing boundary conditions atedges of the integration box [42]. UU U UUD U DUD U DUD antisymmetric symmetric FIG. 1. (Color online) Level schemes for different ionizationscenarios (marked with numbers in green squares). Ioniza-tion events are visualized from top to the bottom. Scenarios1 and 2 correspond to an ionization in the three-active elec-trons model with potential (1), softening parameter (cid:15) = √ . (cid:15) antisym = √ . 68 a.u. is considered, whereas scenario 4corresponds to a symmetric configuration (U) vs (D) or (D) vs(U) with (cid:15) sym = √ . 18 a.u. Letters in circles denote the spinof the electron ionized in a particular event. Numbers reflectthe ionization potential values: the bold blue ones are resultsof numerical simulations while the black ones are obtained assimple differences. b. Two-active electrons model. The two-active elec-trons model is built consistently from the three electronsmodel discussed above. We restrict the electronic motionto one-dimensional tracks that form a plane and cross atangle π/ π/ 6) = √ / (cid:112) / 3, as introducedearlier in (1). The two-electron Hamiltonian then reads H = (cid:88) i =1 (cid:32) p i − Z (cid:112) r i + (cid:15) + (cid:114) F ( t ) r i (cid:33) +1 (cid:112) ( r − r ) + r r + (cid:15) , (4)where Z is a nuclear charge, set either as Z = 2 (Ne atom)or Z = 3 (Ne + ion). Two dimensional calculations havebeen performed on 2048 points grid corresponding to 400a.u. per direction with initial state found on a smallergrid of 512 points. c. Spin configurations. In the case of three electronsthe whole system evolution can be described by a singlethree-electronic spatial function (see Appendix A for thediscussion). Without loss of generality we may considera case with two spin-up (U) and one spin-down (D) elec-trons. The spatial wave function is antisymmetric in theUU plane and symmetric in all other planes.During double ionization (U) and (D) or (U) and (U)electrons can be ionized. To mimic this picture withtwo-active electrons model, one needs to consider twoproblems with different symmetries: spatially symmet-ric, corresponding to (U) and (D) ionized electrons, andspatially antisymmetric, corresponding to (U) and (U)ionized electrons. d. Calculation of ionization potentials. The defini-tion of atomic models immediately implies the wayfor calculating a set of ionization potentials. To findthe ground state wave function and the correspond-ing eigenenergy we employ the imaginary-time evolutionstarting from a function with the required symmetry.Such a procedure works for one-, two- or three-electronfunctions.The three-electron model allows one to resolve ioniza-tion channels in respect to number, sequence and spin ofionized electrons (see the details in [39]), and thus intro-duce different scenarios for discussed processes. Scenariosanalyzed in the present study are presented in Sec. IV.For the sake of clarity and completeness of methodol-ogy section, we shall expose procedure of computation ofionization potential discussing a particular example of anionization channel in the three-electron model. Let the(U) electron be ionized first, and followed then by (D)and (U) electrons (the scenario 1 in Fig. 1). The tripleionization potential 4.63 a.u. is defined by the groundstate of the three-electron wavefunction. One can easilycalculate the single ionization potential of a double ion( Z = 3), i.e. when two electrons are already ionized.The double ion has just one electron, so the potentialvalue 2.53 a.u. is spin-independent. The difference of itwith the triple ionization potential gives the double ion-ization potential value of 2.10 a.u. The single ionizationpotential is calculated as the difference of triple ioniza-tion potential and double ionization potential 4.25 a.u.of a single ion ( Z = 3) possessing (U) and (D) electrons(and thus spatially symmetric) yielding 0.39 a.u.In the two-electron model getting ionization potentialsis even simpler. First, one finds ground state of a two-electron system ( Z = 2), i.e. a double ionization poten-tial. Then, the single ionization potential (ground state)of a single ion ( Z = 2) is calculated in correspondingone-electron system. Difference between these two ion-ization potentials gives the single ionization potential ofthe two-electron system. III. SPINS AND IONIZATION POTENTIALS In the system considered several ionization scenariosare possible, accounting for the different channels of sin-gle, double and triple ionization. In the following we willfocus primarily on double ionization events, thus tripledirect ionization is not discussed at all. Discussion ofthe triple ionization as described within our three-activeelectrons model can be found elsewhere [39]. Also, forthe parameters used in our simulations, triple ionizationyields are about 3 orders smaller in magnitude than thosefor double ionization, thus we can assume that the lossof electrons to triple ionization channel is negligible.Generalizing the already mentioned spin independenceof the Ne ++ ionization potential, one comes to an equal-ity of double ionization (DI) potentials of 2 . 10 a.u. forall possible spin configurations of first two electrons, i.e.(DU), (UD) and (UU) (see Fig. 1). Here (DU) denotesthat the first of the escaping electrons has spin down (D)and the second has spin up (U), and similar for the othercases. These two electrons can escape either sequentiallyor simultaneously. Of course, for the simultaneous escape(DU) and (UD) configurations are indistinguishable.The value of the single electron ionization (SI) poten-tial of an atom does depend, however, on the ionizationchannel. If the first ionizing electron is (U) – like in theexample from the previous section – then its release leadsto the formation of a singly charged ion with a (UD) pairthat is described by a spatially symmetric wave function.We shall call this path scenario 1 (see Fig. 1). However,if the first ionizing electron is (D) then a (UU) pair isformed, with a spatially antisymmetric wave function.We shall call this path scenario 2 (see Fig. 1).One should note that the two (U) electrons are in-distinguishable in the analyzed model. Therefore thereare only two single ionization channels, i.e. SI (U) andSI (D). Consequently both (U) electrons contribute toSI (U) channel, whereas only one electron contributes toSI (D) channel.Now, our objective is to explore to what extent onemay mimic the double ionization of a three-electron atomwith a two-active electrons model. To that end, we con-sider the two scenarios introduced above. The presenceof a third electron is taken into account by two factors:by the nuclear charge Z = 2 and by considering spin de-grees of freedom of the two-active electrons. The latterare deduced from the underlying ionization scenarios inthe three-active electrons model. Namely, in scenario 1, a (U) electron is ionized first, followed by (U) or (D).Thus there are two possibilities, where either a (UU) ora (UD) pair is extracted in the double ionization event.In scenario 2, by contrast, the first electron is the (D)electron, which leaves two equivalent U electrons for thesecond ionization step, so that the two-electron ioniza-tion always results in a (DU) pair. Note, that (UD)and (DU) pairs are indistinguishable in a two-electronsystem. Each spin pair induces a wave function with aproper spatial symmetry — (UU) indicates a problemwith an antisymmetric wave function, whereas (UD) and(DU) refer to a spatially symmetric wave function. Thisis why the two different two-active electrons models, de-noted by scenarios 3 and 4 in Fig. 1, corresponding to(UU) and (UD) pairs are to be considered. We will re-fer to them as the antisymmetric and symmetric models,respectively.For consistency, the DI potentials for the two electronmodels are imposed to be equal to the one from the three-electron model. The decision of keeping the DI potentialvalues the same for both scenarios 3 and 4 is dictatedby the fact that the DI potential has a strong effect onthe ionization yield. At the same time ground state ener-gies for symmetric and antisymmetric models are knownto be quite different – recall those of para- and ortho-helium [44, 47], for instance. The softening parametersfor antisymmetric and symmetric models used to achieveour goal are (cid:15) antisym = √ . 68 a.u. and (cid:15) sym = √ . 18 a.u.correspondingly. The corresponding values of ionizationpotentials for both models are shown in the scenarios 3and 4 in Fig. 1.It is important to note that SI potentials are differ-ent for different scenarios. In the three-active electronsmodel SI potential is either 0.39 a.u. (scenario 1, when(U) electron is first ionized) and 1.21 a.u. (scenario 2,when (D) electron is first ionized). This should not be asurprise, as the remaining single ion has either two elec-trons with opposite spins or two electrons with the samespins. Thus corresponding wave functions are spatiallysymmetric or antisymmetric, and may be related to theground and the exited state of the single ion, respectively.There is a good correspondence between three-active andtwo-active electrons models, with respect to SI potentials,when it is assumed that the first two escaping electronsboth have spin up – see scenarios 1 and 3 in Fig. 1. Inthose two cases SI potentials are very close to each other(i.e. 0.39 a.u. and 0.35 a.u. in scenario 1 and 3, respec-tively). That correspondence suggests that the ionizationoccurs in the similar way. And this appears to be thecase, as we will discuss in Sec. IV. Note, however, thatthe antisymmetric model fits to one path realized in sce-nario 1, i.e. to (UU) only. The difficult part comes whenone considers the first two escaping electrons having op-posite spins. In such a case, two-electron model cannotdistinguish between (UD) and (DU) channels, whereasthree-active electrons model allows for the separation ofthem ((UD) is included in scenario 1, and (DU) – in sce-nario 2). Consequently, the SI potential in scenario 4is different from the SI potential in the first and secondscenario. Fortunately, its value lies approximately in themiddle between those of scenario 1 and scenario 2, andis equal to 0.72 a.u. Let us note that scenarios 1 and 2include quantum trajectories that end in the same state,namely, double ion with (U) electron. One could expectan interference between the (UD) and (DU) paths. Suchan interference is an intrinsic element of the symmetricmodel.Any improvements of potential (4) by splitting, for ex-ample, the softening parameter into two, one correspond-ing to nuclear-electronic and the other to electronic-electronic interaction terms (like discussed in [33]), fortuning SI potential values are not helpful. Such a manip-ulation cannot change symmetry properties of scenario 3and scenario 4 that follow solely from their two-electronorigin. IV. RESULTS AND DISCUSSIONA. Preliminaries In Fig. 2 and Fig. 3 we show the results of simu-lations obtained within the three-active and two-activeelectrons models, respectively – single and double ioniza-tion yields are plotted as functions of the field amplitude.Note that respective ionization yields are split into vari-ous channels. To discern these channels we use a spatialcriterion based on a division of space into regions cor-responding to atom, single and double ions (and tripleions for the three-active electrons case), and calculatingprobability fluxes through boundaries between these re-gions [39, 42, 43].The spatial criterion in our model allows one to dis-tinguish between the two types of double ionization: thesimultaneous, instantaneous escape of electrons and timedelayed processes. The former is called a recollision-impact ionization (RII), known also as an electron-impact ionization (EII) or a recollision induced directionization (REDI) [48–50]. The latter includes a vari-ety of mechanisms such as the sequential double ioniza-tion (SDI) and recollision excitation with a subsequentionization (RESI) [24, 51–53]; another recently proposedtime-delayed mechanism to be noted is “slingshot non-sequantial double ionization” [54]. The channels distin-guished by our space criteria are denoted RII and timedelayed ionization (TDI), respectively. TDI in our modelputs together SDI and RESI yields despite RESI beingphysically a non-sequential process. This is a drawbackof our approach. As it is well known the importance ofRESI manifests itself by the existence of the “knee”. It isclearly visible in our TDI channels in Fig. 3, while for atruly sequential channel such a “knee” should not appear.For the purpose of expressing the electrons ionizationorder in time delayed processes we use notation (0-D-U) introduced previously in [39]: from neutral atom “0”the first (D) electron escapes, then does the second (U). For RII processes the notation without accounting forionization order is used: (0-DU).In the following we will compare results obtained withthe use of three-active and two-active electrons mod-els. Unfortunately, a direct comparison of ionizationyields obtained within different models is not fruitful asmodels act in different restricted geometries. In such acase we may analyze instead slopes, trends and overallshape of the curves rather than the numerical values ob-tained. Whenever we compare different ionization chan-nels within the same model we can additionally comparetheir magnitudes. We organize the section as follows:first we discuss the three-electron model results, thenproceed with describing the two-electron model resultscomparing them with the three-electron model results. B. Three-active electrons model results a. Single ionization. First we analyze single ioniza-tion yields. In the three-active electrons model it isclearly seen that the magnitude of the SI signal dependson the spin of ionized electron (compare SI (U) and SI(D) curves in Fig. 2). The SI (U) signal evidently domi-nates over SI (D). While two (U) electrons contribute toSI (U) yield (as compared to a single one for (D) yield)the origin of the observed large difference in yields canbe traced to vastly different single ionization potentialsin both cases. As seen from Fig. 1, for the U electron I U = 0 . 39 a.u. while for the D electron I D = 1 . 21 a.u.Using perturbative ideas one estimates that, for ω = 0 . F = 0 . − in Fig. 2. b. Double ionization – TDI channel. An interestingobservation within the discussed models is the fact thatthe three-active electrons model allows to distinguish be-tween the process in which a (U) electron is ionized firstfollowed by (D) electron from the process in which theyionize in a reverse order. The two processes are inherentlyindistinguishable in the two-active electrons model andare embedded in the symmetric model only. The remain-ing channel, i.e. (0-U-U), can be analyzed in both three-active and two-active electrons (antisymmetric) models.As evident from Fig. 2, ionization yields for TDI (0-U-D) and (0-D-U) show quite a different behavior. Tosome degree, this again reflects the different SI poten-tials for (U) and (D) electrons, while DI potentials arethe same. For time delayed ionization one needs to con-sider both SDI and RESI. While for higher fields, above F = 0 . -5 -4 -3 -2 -1 Ionization Yields Field Amplitude [a.u.]SI (U)SI (D)TDI (0-U-D)TDI (0-U-U)TDI (0-D-U)RII (0-DU)RII (0-UU) FIG. 2. (Color online) Ionization yields as a function of elec-tric field amplitude resolved for different ionization channelsof the three-active electrons model (1). The channels are de-noted as single ionization (SI), time delayed ionization (TDI)and recollision-impact ionization (RII) with the correspondingspin sequence of ionizing electrons placed in parentheses. 5-cycles long sin -shaped pulse of frequency 0.06 a.u. (Eq. (3))has been used for simulations. -5 -4 -3 -2 -1 Ionization Yields Field Amplitude [a.u.]SI [A]TDI (0-U-U) [A]RII (0-UU) [A]SI [S]TDI (0-U-D/0-D-U) [S]RII (0-DU) [S] FIG. 3. (Color online) Ionization yields vs electric field ampli-tude resolved for different ionization channels of two-electronatom (4) in both symmetric and antisymmetric configura-tions. Notation follows Fig. 2. Data corresponding to thesymmetric and antisymmetric models are labeled [S] and [A],respectively. U-D) signal clearly dominates over TDI (0-D-U) as thefield amplitude increases – see Fig. 2. On the other hand,one can argue that more efficient ionization of the first(U) electron should lead to its more efficient rescatteringand a higher excitation of the parent ion and thus shouldfinally result in higher RESI when compared to the chan-nel with the (D) electron being ionized first. However,this reasoning fails to describe trends observed for bothsignals in the low field amplitude region F < . F ≈ . + bound eigenstates depend on whether theion contains same-spin or different spin electrons.Within the discussion above it is assumed that the rec-olliding electron is at the same time the electron that ion-izes first. This assumption is not obvious, since the sce-nario when the striking electron can be captured by theion while transferring energy for a release of a previouslybound electron is possible. Still we suppose the latteroption to be less probable because the recollision crosssection is proportional to the ion’s ionization cross sec-tion [56] which drops with increasing of energy the boundelectron gets during ionization. In the extreme case whenthe rescattering electron has large energy and transfersmost of it to the bound electron, the rescattering crosssection is reduced to the classical Rutherford formula [56]and thus is proportional to 1 /E , where E is the energygained by the bound electron. So, it is more likely forthe rescattering electron to remain free rather than to becaptured by the ion. The direct proof of the suggestionin our quantum simulation is not possible though. Ourassumption may be partially validated by the classicaltrajectory method that allows one to track swapping ofelectrons during recollisions; Haan et al [57] estimated thenumber of double ionization events with electron swap-ping at 30% within their classical model using similarfield parameters (780 nm of wavelength, F = 0 . c. Double ionization – RII channel. Let us now con-sider direct electron escapes. RII signal for electronsescaping with opposite spins, i.e. the channel (0-DU),is always significantly larger than for electrons escapingwith the same spin, i.e. the channel (0-UU) – compareFig. 2. The ponderomotive energy needed for RII (UU)is U p = 0 . 54 a.u. that corresponds to F = 0 . 045 a.u.,while for RII (DU) U p = 0 . 28 a.u. with F = 0 . 03 a.u.Comparison of RII and TDI signals shows that the sig-nal in the RII (0-DU) dominates other double ionizationsignals for amplitudes up to F ≈ . 15 a.u. Just below F = 0 . F > . ψ ( r i , r j ) = − ψ ( r j , r i ))corresponding to (0-UU) channel has a nodal line along r i = r j that is close to the direction of the correlatedelectronic escape during RII. This causes low probabilityof RII (0-UU) in comparison to that of RESI and RII(0-DU). Another factor explaining high efficiency of (0-DU) channel is the unusually small value of 0.89 a.u. forthe second electron ionization potential versus the sin-gle ionization potential I = 1 . 21 a.u. in scenario 2: therescattering electron has a high chance to directly ion-ize the parent ion rather than excite it. Such a propertycan be attributed to 3-electron configuration, because ofspin degrees of freedom (see the discussion on ionizationscenarios in Sec. III), as the second electron ionizationpotential can never be smaller than the single ionizationpotential in any 2-electron model. d. Double ionization – final remarks. Besides in-volving the dynamics of different spin-resolved channels,an overall enhancement of the double ionization signal forlow field amplitudes in the three-active electrons modelmight be attributed to electron-electron Coulomb inter-actions that make rescattering cross section of the re-turning electron larger. For double ionization/excitationthe third electron could serve like a catalyst in the smallfield regime. Such mechanism would be analogous to thatresponsible to a formation of “knee” as discussed by Szy-manowski et al. [40]. However, there is no easy way tosupport such a conclusion in a present study. -5 -4 -3 -2 -1 Ionization Yields Field Amplitude [a.u.]TDI (0-U-U) [A,2E]RII (0-UU) [A,2E]TDI (0-U-U) [3E]RII (0-UU) [3E] FIG. 4. (Color online) Ionization yields vs electric field am-plitude: selected curves from Figs. 2 and 3. C. Comparison with two-active electrons modelresults Comparison of curves corresponding to the same ion-ization channels in three-electron model (Fig. 2) andtwo-electron models (Fig. 3) implies that the symmetricmodel data differ in shape considerably from their three-electron counterparts. On the contrary, the antisymmet-ric model data, essentially, follow their analogues (seeFig. 5). Nevertheless, we briefly review data obtainedwith both models contrasting them to proper channelsin the three-electron model. a. Antisymmetric model discussion. In the antisym-metric two-active electrons model (see Fig. 3) there areonly (U) electrons, therefore its SI signal correspondssolely to the SI(U) channel in the three-active electronsmodel. So it is of no wonder that both the SI(U) sig-nal of the three-active electrons model and the SI signalof the antisymmetric model exhibit an almost completesaturation of the yield in the range of field amplitudesconsidered.TDI signal obtained within antisymmetric model maybe compared to TDI (0-U-U) signal of the three-electronmodel only -see Fig. 4. Generally, we conclude thatboth signals have similar shapes. For lower field am-plitudes, however, the TDI curve in the antisymmetrictwo-electron model is slightly steeper than its counter-part in the other model.The results of simulations for RII channel (0-UU) inthree- and two- active electron models, are also comparedin Fig. 4. A very good agreement between the models isfound for large field amplitudes F > . 15. For loweramplitudes, while both curves are still parallel to eachother, the RII from the three-electron case starts to growearlier. Such a feature might suggest the importance ofthe three-electron correlations. b. Symmetric model discussion. The SI channel inthe symmetric model corresponds partially to the SI(U) -5 -4 -3 -2 -1 Ionization Yields Field Amplitude [a.u.]full DI (2E [S])full DI (2E [A])full DI (3E) FIG. 5. (Color online) Ionization yields vs electric field am-plitude: selected curves from Figs. 2 and 3. The “full DI”curves were obtained by summing all possible double ioniza-tion channels. and partially to the SI(D) channels of the three-electronmodel, but the SI[S] signal does not match neither SI(U)nor SI(D). Rather, it shows the shape known from earlierstudies of two-electron models (see for example [43, 58,59]). Clearly even a single electron physics of the three-electron atom cannot be simulated within the restrictedtwo-electron symmetric model.In general, double ionization signals for the symmetricmodel grow more rapidly with a field amplitude than thecorresponding signals for the three-electron model, re-gardless the channel. Let us compare TDI (0-U-D) and(0-D-U) from Fig. 2 with TDI[S] from Fig. 3. For fieldamplitudes larger than F = 0 . F = 0 . V. CONCLUSIONS We have studied double ionization of three-electronatom within a simplified three-active electrons model.We have compared single and double ionization yieldswith those obtained with the use of judiciously chosentwo-active electrons models. The two-active electronsmodels have been designed to include spin degrees of free-dom by a proper choice of symmetry of initial wave func-tions. In each of the analyzed models we investigated sig-nals of different spin ionization channels. It is currentlynot possible to experimentally resolve these channels, butas our results show they have effects on (i) absolute val-ues of double ionization (normalized to single ionization yield, for example) and (ii) of knee slope position.Electronic correlations can affect double ionization be-cause of Coulomb interactions and spin configurations.The former might be partially responsible for the shift ofthe double ionization knee slope to lower fields. The lat-ter defines peculiarities of each ionization channel yieldsdependencies on field amplitude and, thus, total dou-ble ionization yield. The influence of Coulomb electron-electron interactions on the double ionization while beingintriguing appeared to elude the research method used.Comparison of the different cases shows that doubleionization in the case of a correlated three-electron sys-tem can not, in principle, be properly represented by aset of two-electron subsystems. This results in a con-siderable limitation of precision and applicability of anytwo-electron model used for simulating three- and pre-sumably other multi-electron atoms double ionization.Taking into account wavefunction symmetry propertiesresulting from spin considerations allows to define dif-ferent ionization channels. The differences in shapes ofionization curves for those channels can be used to differ-entiate between them. This is in contrast to the situationin the two-electron atom case, where such differences cannot be resolved.From our analysis it follows that two-electron antisym-metric model is in better qualitative agreement with dou-ble ionization of correlated three-electron atoms than thesymmetric one. Curiously, antisymmetric model shares aproperty with the celebrated Rochester model [60]: bothprevent simultaneous escape of electrons, however, dueto different reasons. In the first case, the simultaneousdouble ionization suppression is caused by the wave func-tion symmetry (spin) depleting the area around r = r ,while in the second case it is the overestimated Coulombrepulsion between electrons that restricts electrons fromapproaching each other when they are away from nucleus.Despite application of restricted dimensionality mod-els in this work, the results obtained can be generalizedto real three-dimensional atoms. The spin structure re-mains the same, so that the qualitative effects discussedhere should be independent of space dimensions. Ourobservations extend previous studies of triple ionization[38, 39] and show that spin electron correlations can havesignificant effects on double ionization as well. VI. ACKNOWLEDGEMENTS A support by PL-Grid Infrastructure is acknowledged.This work was realized under National Science Centre(Poland) project Symfonia No. 2016/20/W/ST4/00314. Appendix A: Three-electron ground state wavefunction. In a model atom with 3 active electrons one has to takecare about spin degrees of freedom [38]. For two-activeelectrons models it is typically assumed that electronshave opposite spins, and the spatial part of the groundstate wave function is symmetric with respect to an ex-change of particles. Now, when the third electron comesinto play it is impossible to construct a wave function ina way that leaves its spatial part symmetric with respectto exchange of each pair of particles. For the sake ofsatisfying Pauli’s exclusion principle the general form ofelectronic wave function can be written as [38]:Ψ ααβ ( r , r , r , t ) ∝ α (1) α (2) β (3) ψ ( r , r , r , t )+ β (1) α (2) α (3) ψ ( r , r , r , t )+ α (1) β (2) α (3) ψ ( r , r , r , t ) , (A1) where α ( i ) and β ( i ) denote spin functions correspondingto spin-up and spin-down states, respectively. 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