Fano meets nuclei: Configuration interaction theory explicitely including nuclear degrees of freedom
aa r X i v : . [ phy s i c s . c h e m - ph ] F e b Fano meets nuclei:Configuration interaction theory explicitely including nuclear degrees of freedom
Elke Fasshauer ∗ Department of Physics and Astronomy, Aarhus UniversityNy Munkegade 120, 8000 Aarhus, Denmark (Dated: February 12, 2021)Electronic decay processes like the Auger-Meitner process and the Interparticle Coulombic Decay(ICD) are omnipresent and occur in areas like metallurgy, analysis of surfaces, semi-conductors,water, solvents, noble gas clusters and even in the active centers of proteins. These processes areinitated by removal or excitation of a sub-outervalence electron, which can be achieved by, e.g., lightin the XUV to x-ray range. These energies allow the creation of ultrashort laser pulses and hencetime-resolved measurements with high time resolutions.So far, electronic decay processes were described by Fano’s theory [1]. However, this theoryconsiders only the involved electronic states while neglecting the nuclear parts of the wavefunction.Especially in the case of ICD, though, the nuclear dynamics are required for an accurate description.We therefore present an analytical ansatz for the explicit inclusion of the nuclear wavefunction withinthe Born-Oppenheimer approximation into Fano theory.We thereby gain a model, which can serve for illustrative interpretations of electronic decayprocesses, in which nuclear motion is relevant.
I. INTRODUCTION
Since the advent of ultrashort laser pulses in the at-tosecond and femtosecond range [2, 3], it has in principlebecome possible to study the time evolution of chemicalreactions in detail.However, the duraction of a laser pulse is inverselyproportional to its mean photon energy. Hence, ultra-short laser pulses typically have mean photon energies inthe extreme ultraviolet (XUV) or even x-ray range. Wetherefore aim to understand the processes initiated bythese ultrashort laser pulses and which information wecan extract from their respective time-resolved spectra.The interaction of ultrashort laser pulses with matterprompts ionizations and excitations from the inner shellsof atoms and molecules. The resulting excited states canthen decay under electronic rearrangement and emissionof an electron, which carries the excess energy. Such elec-tronic decay processes have been known for many years.The Auger-Meitner process was discovered almost 100years ago [4, 5] and also the first theoretical predictionof the Interatomic Coulombic Decay (ICD) is turning 25next year [6].The different electronic decay processes can be dis-tinguished by the number of involved entities (atoms,molecules or strongly bound conglomerates) and the kindof electronic rearrangement during the process. In theAuger-Meitner process, only one entity A is involved. Af-ter primary ionization from an inner shell, the vacancyis filled by an electron from an outer shell and anotherelectron is emitted: A hν −→ A ∗∗ + + e − photo → A + e − photo + e − Auger-Meitner . ∗ [email protected] Here, we indicate an inner-shell vacancy by two stars ∗∗ .The Auger-Meitner process is found in a variety of dif-ferent systems like atoms, molecules and solids. As aconsequence and because it is element specific, Auger-Meitner electron spectroscopy is used for surface analysisin metallurgy, quality analysis of microelectronics as wellas for basic studies of chemical reaction mechanisms [7].It has also been a test process for time-resolved measure-ments in general [8, 9] and is often observed as a sideproduct of modern x-ray spectroscopies [10].In the ICD process, the energy released during the va-cancy filling is transferred through space to a differententity, which consequently emits the ICD electron: AB hν −→ A ∗∗ + B + e − photo → A + + B + + e − photo + e − ICD . After the process, the two involved entities are both pos-itively charged and therefore repell each other.ICD and ICD-like processes occur in many differentsystems with weakly interacting units such as noble gasclusters [11–14], proteins [15], solvents [12, 13, 16–21] aswell as semi-conductors [22–24] and are discussed as akey mechanism of radiative DNA damage [25–29].For time-resolved spectroscopy of electronic decay pro-cesses, most of the developed theory considers atomicsystems [30–32], for whose description the purely elec-tronic Schr¨odinger equation suffices. However, for Auger-Meitner processes of molecules, molecules bound to a sur-face, like in heterogeneous catalysis, or an ICD process,the nuclear degrees of freedom will affect the spectra ofthe emitted electrons.Recent technical improvements now allow to resolvethe features caused by the nuclear degrees of freedom[33, 34] and therefore, can be compared to previous the-oretical simulations [35–38]. The numerical results, how-ever, don’t allow for simple physical interpretations ofthe results. We therefore aim for an intuitively analyt-ical description of electronic decay processes of systemsconsisting of more than one atom.By careful observation of the assumptions we makein our derivations, it will be possible to assign furtherproperties observed in experiment to those parts, whichwe neglect.The analytical description of time-independent spectraof electronic decay processes dates back to the work byFano [1] and the generalization of his theory by Mies [39].Fanos basic theory focusses on purely electronic wave-functions, whereas Mies did not specify the nature of theinvolved wavefunctions and had a strong focus on cou-pling resonances. The generality of Ref [39] provides thebasis for numerical simulations of any case, but leaves thescientist bare of qualitative insight on how different ef-fects influence the spectrum of the emitted electrons. Wewill therefore, within the Born-Oppenheimer approxima-tion, provide an ansatz for the wavefunction of a decayingstate including both electronic and nuclear eigenstates ofthe system for non-interacting resonant states and dis-cuss the limits of our approach based on the approxima-tions we introduce.Moreover, we will show how our ansatz can explainthe long dispute between theory and experiment over theICD lifetime of the neon dimer [40].The paper is structured as follows: In section ?? , wechoose an ansatz for the Fano wavefunction including nu-clear degrees of freedom for one electronic resonant andone electronic final state and derive the corresponding co-efficients. We discuss the general results and the limita-tions and consequences of our introduced approximationsin section II and summarize in section III. A. Derivation of the Fano coefficients includingnuclear wavefunctions
Fano’s purely electronic and time-independent wave-function for one resonant and one final state is [1] | Ψ E i = a ( E ) | r i + Z d E ′ b E ′ ( E ) | E ′ i . (1)Here, | r i is the electronic wavefunction of the bound reso-nant state and and | E ′ i is the electronic continuum wave-function including both the bound final state and theemitted free electron. a and b are coefficients, which aredetermined for the different cases.We extend this ansatz to include nuclear degrees offreedom by adding a summation over the difference nu-clear eigenstates of the two respective potential energycurves. We note, that we in this step have assumed theBorn-Oppenheimer approximation. | Ψ E i = X λ a λ ( E ) | r i | χ λ i + X µ ′ Z d E ′ b E ′ ,µ ′ ( E ) | E ′ i | χ µ ′ i . (2)In order to determine the coefficients, we follow thesame strategy as for the pure electronic Fano ansatz inRef. [1] and solve the following set of equations. h r | ˆ H | Ψ E i = X λ h r | ˆ Ha λ ( E ) | r i | χ λ i + X µ ′ Z d E ′ h r | ˆ Hb E ′ ,µ ′ ( E ) | E ′ i | χ µ ′ i (3)Here, the Hamilton operator consists of the unper-turbed Hamiltonian of the system H , which involvesboth electronic and nuclear operators and the configu-ration interaction operator V : H = H + V (4)We assume that the coefficients are not explicitely de-pendent on neither the electronic nor the nuclear coordi-nates to obtain: h r | ˆ H | Ψ E i = X λ a λ ( E )( E r + E λ ) | χ λ i + X µ ′ Z d E ′ b E ′ ,µ ′ ( E ) V rE ′ | χ µ ′ i . (5)We project out the contribution of a single nucleareigenfunction h χ λ ′ | and hereby assume that the electronicconfiguration interaction term V rE ′ does not directly de-pend on the nuclear coordinates. This approximation isanalog to the Condon approximation for the transitiondipole moments. h χ λ | h r | ˆ H | Ψ E i = a λ ( E )( E r + E λ )+ X µ ′ Z d E ′ b E ′ ,µ ′ ( E ) V rE ′ h χ λ | χ µ ′ i (6)= E a λ ( E ) (7)Analagously, we obtain h χ µ | h E ′′ | ˆ H | Ψ E i = X λ a λ ( E ) V E ′ r h χ µ | χ λ i + b E ′ ,µ ( E )( E ′ + E µ ′ ) (8)= E b E ′ ,µ ′ ( E ) . (9)By solving Eq. (9) for b E ′ ,µ ′ ( E ) we arrive at b E ′ ,µ ′ ( E ) = X λ a λ ( E ) V E ′ r h χ µ ′ | χ λ i E − E ′ − E µ ′ + X λ a λ ( E ) V E ′ r h χ µ ′ | χ λ i z ( E ) δ ( E − E ′ − E µ ′ )(10)Here, we implicitely assume, that the first part of thesum becomes the principal part upon integration over oneof the energies. For only one resonant and one final state,we would insert this expression into Eq. 7 and solve for z ( E ). However, when including nuclear wavefunctions,we arrive at( E − E r − E λ ) a λ ( E ) = P X µ ′ ,λ ′ Z d E ′ a λ ′ ( E ) | V E ′ r | h χ λ | χ µ ′ i h χ µ ′ | χ λ ′ i E − E ′ − E µ ′ + X µ ′ ,λ ′ a λ ′ ( E ) | V ( E − E µ ′ ) r | z ( E ) h χ λ | χ µ ′ i h χ µ ′ | χ λ ′ i . (11)A full solution of this equation is not possible becauseonly the decay of several resonant states to the samefinal state or the decay of one resonant state into sev-eral final states can be properly normalized in the waydiscussed by Fano [1]. In our case, the sum over µ ′ or λ ′ prevents this normalization. We therefore make theassumption that the nuclear wavefunctions of the reso-nant state are not coupled via the continuum and henceintroduce δ λ,λ ′ . This approximation is valid, when thecoupling of different nuclear resonant states is small, orwhen the Franck-Condon factor for one value of λ ′ andone value of µ ′ is significantly larger than for all othervalues of λ ′ . Then, all other contributions than the onefor λ = λ ′ are small and a large part of the sum is coveredby the main contribution.With this in mind, we, analog to Fano, use1( E − E ′ − E µ ′ )( E − E ′ − E µ ′ )= 1 E − E (cid:18) E − E ′ − E µ ′ − E − E ′ − E µ ′ (cid:19) + π δ ( E − E ) δ ( E ′ − ( E − E µ ′ )) (12) δ ( E − E ′ − E µ ′ ) δ ( E − E ′ − E µ ′ )= δ ( E − E ) δ ( E ′ −
12 ( E + E − E µ ′ )) (13)to deduct z ( E ) = E − E r − E λ − P µ ′ F µ ′ ( E ) W λ . (14)Here, W λ is the weighted sum of absolute squares ofthe configuration interation terms of the different nuclear final states for a single nuclear resonant state W λ = X µ ′ | V ( E − E µ ′ ) r | | h χ λ | χ µ ′ i | (15)and F µ ′ ( E ) = R d E ′ | V E ′ r | |h χ λ | χ µ ′ i| E − E ′ − E µ ′ .Let us put this expression in Eq. (14) into perspec-tive by comparing it to its purely electronic counterpartfor the decay of one electronic resonant state to severalelectronic final states: z el ( E ) = E − E r − P f F f ( E ) P f | V rf | (16)The similarities between them are obvious. The onlydifference is the weighting of the different decay widthsby the absolute square of Franck-Condon overlaps.By using δ λ,λ ′ consistently in the normalization of thewave function by calculating h Ψ E | Ψ E i and by assumingthat F ( E ) is real, we obtain X λ | a λ ( E ) | W λ ( π + | z ( E ) | ) = δ ( E − E ) . (17)To solve this equation for the coefficient of only onevibrational eigenstate of the resonant state λ we assumethat the contributions of the different vibrational eigen-states are equally distributed amongst all N λ bound vi-brational states of the resonant state to give | a λ ( E ) | W λ ( π + | z ( E ) | ) = 1 /N λ . (18)We finally determine the coefficients within the appliedapproximations to be a λ ( E, h R i ) = − √ W λ √ N λ r ( E − E r − E λ − P µ ′′ F µ ′′ ( E )) + π W λ (19) b E ′ ,µ ′ ( E, h R i ) = X λ ′ V E ′ r a λ ′ ( E ) h χ µ ′ | χ λ ′ i E − E ′ − E µ ′ − X λ ( E − E r − E λ − P µ ′′ F µ ′′ ( E )) V E ′ r a λ √ N λ W λ δ ( E − E ′ − E µ ′ ) h χ µ ′ | χ λ ′ i . (20)Because the F µ ( E ) are small shifts in the energy position of the resonant state | r i | χ λ i , which are further dampedby Franck-Condon factors and because they describe thecoupling between the nuclear resonant states via the con-tinuum, we neglect them in the following. II. DISCUSSION
The main benefit of the above derivation is the expres-sion given in Eq. (15), where the decay width is given byΓ λ = 2 πW λ . It comprises several key properties.Firstly, it analytically shows that the decay widths andhence the lifetimes are different for each nuclear eigen-state of the resonant state. This property in in agreementwith previous results [33, 41–43].Secondly, if we assume that the dependence of the con-figuration interaction elements do not depend on the en-ergy of the final state, the decay width of the individualnuclear state of the resonant state becomes the purelyelectronic lifetime weighted by the sum over the absolutesquares of its nuclear wavefunction with all nuclear finalstates: Γ λ = 2 π X µ ′ | V r | | h χ λ | χ µ ′ i | (21)This allows interpretations analog to those of transi-tion dipole moments using the Franck-Condon factors[44, 45]. If a specific nuclear wavefunction of the reso-nant hardly overlaps with any of the nuclear final states,then the corresponding decay width will be small and thelifetime will be large. The larger the total overlap withthe nuclear final states is, the larger is the decay widthand the smaller is the lifetime.Moreover, within this approximation, it becomes ev-ident that the effective decay width for either of thenuclear resonant states can not become larger than thepurely electronic decay width, because the sum over theabsolute square of the overlap integrals can not exceed 1.Hence, the lifetime of these states is longer or equal tothe purely electronic decay width.This behaviour has already been observed in connec-tion to the ICD lifetime of the neon dimer after primaryionization from the Ne2 s . The purely electronic lifetimeunderestimates the experimentally determined lifetimeand only numerical simulations including nuclear dynam-ics were able to reproduce the experimental value withinthe errors of the respective methods and measurements.[40, 46] This effect should be taken into account in futurecomparisons between theory and experiment. A. Introduced approximations and theirconsequences
In this paper, we have derived expressions for the time-resolved description of electronic decay processes and thebuild-up of their signals including both several resonantand several final states. In a rigorous treatment, it is only possible to include either several resonant or severalfinal states, but not several states of both kinds. In orderto still treat their combination, we have introduced a fewapproximations, whose impact we would like discuss.1. We use the Born-Oppenheimer approximationthroughout the paper and therefore, the derived ex-pressions will not be applicable, when strong non-adiabatic couplings exist, which involve the reso-nant or the final states.2. In the derivation of the coefficients of the Fanowave function, we assume that the different reso-nant states do not couple via the continuum. Inthis case, the only meaningful contribution in Eq.(11) is the one for λ ′ = λ . It also means that theprincipal part of the integral over E ′ must be zero,which we use later in the derivation. It moreovermeans that | V E ′ r | must be independent of the en-ergy E ′ . While neglecting F ( E ) in the purely elec-tronic solution for one resonant and one final stateresults in neglecting a small shift in the resonantenergy, while the overall shape of the correspond-ing peak remains the same, it is more vital, whenseveral resonant states are involved. By neglecting F µ ′ ( E ), we neglect shifts of all resonant energies,which can change the energy differences betweenthem, but we also neglect changes in the respectivedecay widths and hence lifetimes. Strong couplingsvia the continuum will therefore lead to qualita-tively different spectra than shown in this paper.In extreme cases for two resonant states, one reso-nant state can combine the entire decay capabilityof the system into itself, while the other state isstabilized.It would be beneficial to go beyond the current lim-itations in future work.3. The approximation of negligible couplings of dif-ferent resonant states via the continuum leads toan under defined expression in Eq. (18). We there-fore assume equal contributions of | a λ ( E ) | W λ ( π + | z ( E ) | ) for all vibrational resonant states. This ap-proximation is very likely incorrect. If the contribu-tions for different λ are not equal, it would changethe relative size of the different contributions andintroduce additional prefactors for each resonantor indirect contribution. The overall shape of theequations would not change though.4. We assume in the derivation that the coupling el-ements V Er are independent of the nuclear coordi-nates. Analog to the transition dipole moments,this approximation is only reasonable, when theBorn-Oppenheimer approximation is valid. We canfurthermore expect this approximation to be rea-sonable, when the excited nuclear bound states donot differ too much from the solutions of a harmonicoscillator and hence are almost symmetric with re-spect to the equilibrium distance, because eventualdeviations from the electronic decay widths at theequilibrium geometry are averaged out. III. SUMMARY
We have derived Fano wavefunction within the Born-Oppenheimer approximation, which explicitely includenuclear degrees of freedom. We analytically show that,within our approximations, each nuclear eigenstate of the electronic resonant state decays with a different life-time. Moreover, this lifetime can not be shorter, thanthe purely electronic lifetime.The model presented in this paper allows illustrativeinterpretations of the lifetimes of different nuclear reso-nant states analog to the Franck-Condon factors for thetransition dipole moments.We have carefully chosen the used approximations andqualitatively discuss their consequences for spectra sim-ulated with this approach and how the different effects,which have not been taken into account in this paper willmanifest itself in a spectrum. [1] U. Fano, Phys. Rev. , 1866 (1961).[2] F. Krausz and M. Ivanov,Rev. Mod. Phys. , 163 (2009).[3] M. Galli, V. Wanie, D. P. Lopes, E. P. M˚ansson,A. Trabattoni, L. Colaizzi, K. Saraswathula, A. Cartella,F. Frassetto, L. Poletto, F. L´egar´e, S. Stagira, M. Nisoli,R. M. V´azquez, R. Osellame, and F. Calegari,Opt. Lett. , 1308 (2019).[4] L. Meitner, Z. Phys. , 131 (1922).[5] P. Auger, C. R. Acad. Sci. , 169 (1923).[6] L. S. Cederbaum, J. Zobeley, and F. Tarantelli,Phys. Rev. Lett. , 4778 (1997).[7] M. P. Seah, “Microscopic methods in metals,” (Springer,Berlin, Heidelberg, 1986) Chap. Auger Electron Spec-troscopy.[8] M. Drescher, M. Hentschel, R. Kienberger, M. Uib-eracker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh,U. Kleineberg, U. Heinzmann, and F. Krausz,Nature , 803 (2002).[9] O. Smirnova, V. S. Yakovlev, and A. Scrinzi,Phys. Rev. Lett. , 253001 (2003).[10] G. Greczynski and L. Hultman, Mat. Sci. , 100591(2020).[11] R. Santra, J. Zobeley, L. S. Cederbaum, and N. Moi-seyev, Phys. Rev. Lett. , 4490 (2000).[12] U. Hergenhahn, J. Electron Spectrosc. Relat. Phenom. , 78 (2011).[13] T. Jahnke, J. Phys. B , 082001 (2015).[14] E. Fasshauer, New J. Phys. , 043028 (2016).[15] P. H. P. Harbach, M. Schneider, S. Faraji, and A. Dreuw,J. Phys. Chem. Lett. , 943 (2013).[16] I. B. M¨uller and L. S. Cederbaum,J. Chem. Phys. , 204305 (2006).[17] C. Richter, D. Hollas, C.-M. Saak, M. F¨orstel, T. Miteva,M. Mucke, O. Bj¨orneholm, N. Sisourat, P. Slav´ıˇcek, andU. Hergenhahn, Nature Comm. , 4988 (2018).[18] N. V. Kryzhevoi and L. S. Cederbaum,Angew. Chem. Int. Ed. , 1306 (2011).[19] S. D. Stoychev, A. I. Kuleff, and L. S. Cederbaum,J. Am. Chem. Soc. , 6817 (2011).[20] N. V. Kryzhevoi and L. S. Cederbaum,J. Phys. Chem. B , 5441 (2011).[21] B. Oostenrijk, N. Walsh, J. Laksman, E. P. Mansson,C. Grunewald, S. L. Sorensen, and M. Gisselbrecht,Phys. Chem. Chem. Phys. , 932 (2018).[22] A. Bande, K. Gokhberg, and L. S. Cederbaum, J. Chem.Phys. , 144112 (2011). [23] A. Bande, J. Chem. Phys. , 214104 (2013).[24] P. Dolbundalchok, D. Pel´aez, E. F. Aziz, and A. Bande,J. Comp. Chem. , 2249 (2016).[25] E. Alizadeh, T. M. Orlando, and L. Sanche,Annu. Rev. Phys. Chem. , 379 (2015).[26] B. Bouda¨ıffa, P. Cloutier, D. Hunting, M. A. Huels, andL. Sanche, Science , 1658 (2000).[27] F. Martin, P. D. Burrow, Z. Cai, P. Cloutier, D. Hunting,and L. Sanche, Phys. Rev. Lett. , 068101 (2004).[28] E. Brun, P. Cloutier, C. Sicard-Roselli, M. Fromm, andL. Sanche, J. Phys. Chem. B , 10008 (2009).[29] E. Surdutovich and A. V. Solov’yov,Eur. Phys. J. D , 206 (2012).[30] M. Wickenhauser, J. Burgd¨orfer, F. Krausz, andM. Drescher, Phys. Rev. Lett. , 023002 (2005).[31] M. Wickenhauser, J. Burgd¨orfer, F. Krausz, andM. Drescher, J. Mod. Opt. , 247 (2006).[32] E. Fasshauer and L. B. Madsen,Phys. Rev. A , 043414 (2020).[33] Y.-C. Lin, A. Fidler, A. Sandhu, R. R. Lucchese,B. McCurdy, S. R. Leone, and D. M. Neumark,Faraday Discuss. , (2020).[34] A. Wituschek, L. Bruder, E. Allaria, U. Bangert,M. Binz, R. Borghes, C. Callegari, G. Cerullo, P. Cin-quegrana, L. Giannessi, M. Danailov, A. Demidovich,M. Di Fraia, M. Drabbels, R. Feifel, T. Laarmann,R. Michiels, N. S. Mirian, M. Mudrich, I. Nikolov, F. H.O’Shea, G. Penco, P. Piseri, O. Plekan, K. C. Prince,A. Przystawik, P. R. Ribiˇc, G. Sansone, P. Sigalotti,S. Spampinati, C. Spezzani, R. J. Squibb, S. Stranges,D. Uhl, and F. Stienkemeier, Nat. Comm. , 883 (2020).[35] A. Palacios, J. Feist, A. Gonz´alez-Castrillo, J. L. Sanz-Vicario, andF. Mart´ın, ChemPhysChem , 1456 (2013),https://chemistry-europe.onlinelibrary.wiley.com/doi/pdf/10.1002/cphc.201200974.[36] A. Palacios, H. Bachau, and F. Mart´ın,Phys. Rev. Lett. , 143001 (2006).[37] A. Palacios, H. Bachau, and F. Mart´ın,Phys. Rev. A , 031402 (2006).[38] A. Palacios, H. Bachau, and F. Mart´ın,Phys. Rev. A , 013408 (2007).[39] F. H. Mies, Phys. Rev. , 164 (1968).[40] K. Schnorr, A. Senftleben, G. Schmid, S. Augustin,M. Kurka, A. Rudenko, L. Foucar, A. Broska, K. Meyer,D. Anielski, D. Anielski, R. Boll, D. Rolles, M. K¨ubel,M. F. Kling, Y. H. Jiang, S. Mondal, T. Tachibana, K. Ueda, T. Marchenko, M. Simon, G. Brenner,R. Treusch, S. Scheit, V. Averbukh, J. Ullrich,T. Pfeifer, C. D. Schr¨oter, and R. Moshammer,J. Electron Spectrosc. Relat. Phenom. , 245 (2015).[41] M. N. Piancastelli, R. F. Fink, R. Feifel,M. B¨assler, S. L. Sorensen, C. Miron, H. Wang,I. Hjelte, O. Bjørneholm, A. Ausmees, S. Svens-son, P. Salek, F. K. Gelmukhanov, and H. ˚Agren,Journal of Physics B: Atomic, Molecular and Optical Physics , 1819 (2000).[42] Q. Bian, Y. Wu, J. G. Wang, and S. B. Zhang,Phys. Rev. A , 033404 (2019).[43] H. J. B. Marroux, A. P. Fidler, A. Ghosh, Y. Kobayashi, K. Gokhberg, A. I. Kuleff, S. R. Leone, and D. M. Neu-mark, Nat. Chem. , 5810 (2020).[44] J. Franck and E. G. Dymond,Trans. Faraday Soc. , 536 (1926).[45] E. U. Condon, Phys. Rev. , 858 (1928).[46] K. Schnorr, A. Senftleben, M. Kurka, A. Rudenko,L. Foucar, G. Schmid, A. Broska, T. Pfeifer, K. Meyer,D. Anielski, R. Boll, D. Rolles, M. K¨ubel, M. F.Kling, Y. H. Jiang, S. Mondal, T. Tachibana, K. Ueda,T. Marchenko, M. Simon, G. Brenner, R. Treusch,S. Scheit, V. Averbukh, J. Ullrich, C. D. Schr¨oter, andR. Moshammer, Phys. Rev. Lett.111