Fullerene photoemission time delay explores molecular cavity in attoseconds
Maia Magrakvelidze, Dylan M. Anstine, Gopal Dixit, Mohamed El-Amine Madjet, Himadri S. Chakraborty
aa r X i v : . [ phy s i c s . a t m - c l u s ] A p r Fullerene photoemission time delay explores molecular cavity in attoseconds
Maia Magrakvelidze, Dylan M. Anstine, Gopal Dixit, Mohamed El-Amine Madjet, and Himadri S. Chakraborty Department of Natural Sciences, D. L. Hubbard Center for Innovation and Entrepreneurship,Northwest Missouri State University, Maryville, Missouri 64468, USA Max Born Institute, Max-Born-Strasse 2A, 12489 Berlin, Germany Qatar Environment and Energy Research Institute (QEERI), P.O. Box 5825, Doha, Qatar (Dated: September 28, 2018)Photoelectron spectroscopy earlier probed oscillations in C valence emissions, producing series ofminima whose energy separation depends on the molecular cavity. We show here that the quantumphase at these cavity minima exhibits variations from strong electron correlations in C , causingrich structures in the emission time delay. Hence, these minima offer unique spectral zones todirectly explore multielectron forces via attosecond RABITT interferometry not only in fullerenes,but also in clusters and nanostructures for which such minima are likely abundant. PACS numbers: 32.80.Fb, 61.48.-c, 31.15.E-
I. INTRODUCTION
Resolving electron dynamics in real-time offers the ac-cess into a plethora of electron-correlation driven pro-cesses in atomic, molecular and more complex systems.Advent in technology in producing isolated ultrashortlaser pulses and pulse trains dovetails a new landscapeof active and precision research of light-matter interac-tions on ultrafast time scale [1–4]. For instance, in pump-probe laser spectroscopy, a pump pulse initiates an elec-tronic process while a subsequent probe pulse exploresthe electron’s motion with a temporal resolution of a fewfemtoseconds to several attoseconds. This serves as a mi-crocosm of a fundamental mechanism that a laser-drivenprocess can be viewed as dynamical electronic wavepack-ets with evolving amplitudes, phases and group delays.Relative delay between 2 s and 2 p photoemission inneon was measured in a pilot experiment by attosecondstreaking metrology [5]. Also, for argon, the relative de-lay between 3 s and 3 p photoemissions at energies belowthe 3 s Cooper minimum [6, 7] and the group delay in 3 p photorecombination across the 3 p Cooper minimum [8]were accessed using attosecond interferometry, known asRABITT. For simple molecules like diatomic nitrogen,two-color photoionization, resolved in attoseconds, wasthe subject of a recent study [9]. Moving to the otherextreme in the structure scale, the condensed-phase sys-tems, recent activities include measurements of the rela-tive delay between the emission from conduction and va-lence band states of monocrystalline magnesium [10] andtungsten [11]. Further, theoretical studies to explore de-lays in photoelectrons from metal surfaces brought aboutimportant insights [12].Straddling the line between atoms and condensed mat-ters are clusters and nanostructures that not only havehybrid properties of the two extremes, but also exhibitspecial behaviors with fundamental effects and technolog-ical applications. Time-resolved access into the photoe-mission processes in fullerenes can be singularly attrac-tive due to their eminent symmetry and stability. Recentefforts were made to predict the time delay in photoe- missions from atoms endohedrally confined in C [13–15]. However, these studies did not address the direct re-sponse of C electrons, but instead focused at the effectsof confinement. Only recently, an electron momentumimaging measurement is performed to study the photo-electron angular distribution of C , establishing an in-direct connection to the emission time delay at the plas-mon resonance [16]. Evidently, hardly anything has beendone to temporally explore cluster systems. In this Let-ter, we report an investigation of the time delay in pho-toemission from two highest occupied molecular orbitals,HOMO and HOMO-1, of C which uncovers dramaticattosecond response at characteristic emission minima.Results carry signatures of C cavity, opening a newapproach for molecular imaging applications, and mostimportantly establish an attosecond route to probe a re-markable aspect of electron correlations. II. ESSENTIAL DETAILS OF THE METHOD
Time-dependent local density approximation(TDLDA) is employed to simulate the dynamicalresponse of C to incident photons [17]. The dipoleinteraction, z , with the light that is linearly polarizedin z -direction induces a frequency-dependent complexchange in the electron density arising from dynamicalelectron correlations. This can be written, using theindependent particle (IP) susceptibility χ , as δρ ( r ; ω ) = Z χ ( r , r ′ ; ω )[ z ′ + δV ( r ′ ; ω )] d r ′ , (1)in which δV ( r ; ω ) = Z δρ ( r ′ ; ω ) | r − r ′ | d r ′ + (cid:20) ∂V xc ∂ρ (cid:21) ρ = ρ δρ ( r ; ω ) , (2)where the first and second term on the right hand sideare, respectively, the induced change of the Coulomb andthe exchange-correlation potentials. Obviously, δV in-cludes the dynamical field produced by important elec-tron correlations within the linear response regime.The gradient-corrected Leeuwen and Baerendsexchange-correlation functional [LB94] [18] is used forthe accurate asymptotic behavior of the ground statepotential. The C molecule is modeled by smearingsixty C ions into a spherical jellium shell, fixed inspace, with an experimentally known C mean radius( R = 3.54 ˚A) and a width (∆ = 1.3 ˚A) determined ab initio [17]. Inclusion of molecular orientations willhave minimal effect on the result due to the C symmetry [19]. The delocalized system of total 240valence electrons from sixty carbon atoms constructsthe ground state in the Kohn-Sham frame [17] usingLB94. This produced HOMO and HOMO-1 to be of 2 h ( l = 5) and 2 g ( l = 4) character respectively with eachhaving a radial node – a result known from the quantumchemical calculation [20] supported by direct and inversephotoemission spectra [21], and from energy-resolvedelectron-momentum density measurements [22]. TDLDApredicted oscillatory photoemission cross sections ofHOMO and HOMO-1 in C which agreed well withthe experiment [23] and with quantum chemical calcu-lations [23, 25]. Fig. 1 shows a very good agreementbetween measurements and TDLDA ratio of HOMOand HOMO-1 cross sections for the four low energyoscillations. An extra peak at 175 eV for TDLDA, anda slight offset between the theory-experiment positionsof two high-energy peaks, plus some mismatch betweentheir widths, are likely limitations of the jellium core.These oscillations are due to the interference betweenemissions from C shell-edges as was shown by Fouriertransforming the above ratio [23, 26] and evident fromthe fact in Fig. 1 that the reciprocal, 2 π/ ∆ k , of the av-erage peak separation (∆ k ∼ . k ) roughly equals the fullerene diameter.The comparison gives confidence on the use of LB94.Similar geometry-based oscillations in high-harmonicspectra of icosahedral fullerenes were predicted [19]. Thispoints to a common spectral implication between pho-toionization and recombination matrix elements. III. CAVITY MINIMA
Studies of ionization time delay at resonances and min-ima (anti-resonances) are attractive, since electron cor-relations can directly influence the result. Of particularinterest is a Cooper minimum which arises at the zeroof the wave function overlap in the matrix element whenthe bound wave contains at least one radial node [27].Around this minimum, the ionization probability is di-minished which allows couplings with other electrons todominate, offering a unique spectral zone to probe thecorrelation. We show that the minima in the oscillationof C valence emissions also appear from zeros in thematrix element, and thus can be of great value in cap-turing time-resolved many-electron dynamics.Choosing the photon polarization along z -axis, thephotoionization dipole amplitude in the IP picture,
50 100 150 200 250 3000.00.51.01.5 k=1.6 H O M O / H O M O - r a t i o Photon energy (eV)
TDLDA Expt. Ruedell et al.2.2 2.8 3.3 3.5 3.9 4.4
FIG. 1. (Color online) Oscillations in the ratio of HOMO toHOMO-1 photoemission cross sections of C calculated us-ing TDLDA/LB94 and compared with the experimental ra-tio [23]. Similar in [23], a smooth background, that roughlyfits the total photoionization cross section of atomic car-bon [24], is added to TDLDA cross sections to approximatelyaccount for local scatterings from carbon atoms. Photoelec-tron momenta at the peaks are indicated to illustrate that theconjugate of the oscillation period relates to the C diameter. that omits the electron correlation dynamics, is d = h Ψ kl ′ | z | Φ nl i in which Φ nl = φ nl ( r ) Y lm ( Ω r ) is HOMO orHOMO-1 wave function, and the continuum wave func-tion with l ′ = l ± kl ′ ( r ) = (8 π ) X m ′ e iη l ′ ψ kl ′ ( r ) Y l ′ m ′ ( Ω r ) Y ∗ l ′ m ′ ( Ω k ) , (3)where the phase η l ′ ( k ) includes contributions from theshort range and Coulomb potentials, besides a constant l ′ π/
2. Using Eq. (3), the radial matrix element (in lengthgauge) embedded in d is h r i = h ψ kl ′ | r | φ nl i . This matrixelement can also be expressed in an equivalent accelera-tion gauge as h ψ kl ′ | dV /dr | φ nl i , which embodies the no-tion that an ionizing (recoil) force dV /dr is available toan electron in a potential V ( r ). Both C radial groundstate potential and its derivative are shown in Fig. 2(a).The potential exhibits rapid variations at the inner ( R in )and outer ( R out ) radii but has a flatter bottom. Conse-quently, the derivative peaks (or anti-peaks) at the shell-edges, allowing two dominant contributions in the inte-gral so one can approximate the matrix element as [26] h r i ≈ A ( k )[ a in ψ kl ′ ( R in ) + a out ψ kl ′ ( R out )] , (4)where a in and a out are the values of φ nl at R in and R out , and A ( k ) is a decaying function of k similar to theone calculated semi-classically for metal clusters [28]. Inessence, this means a strong cancellation effect in the ma-trix elements at the interior region of the potential due tooverlaps between oscillating ψ kl ′ and radially symmetric φ nl . This symmetry, not present in atoms (where elec-trons are localized toward the nucleus), is a character ofnanosystems with delocalized electrons; see the HOMOand HOMO-1 wave functions in Fig. 2(a). In any case,each term in Eq. (4) oscillates in k and vanishes whena node of ψ kl ′ moves through R in or R out or, equiva-lently, when an integer number of half-periods of contin-uum oscillation fits within R in or R out ; decreasing periodof continuum waves with increasing energy is illustratedin Fig. 2(a). For each term, the effect is analogous toa single spherical-slit diffraction. Since the combination(interference) of two oscillations is itself an oscillation, h r i must also contain zeros, as shown in Fig. 2(b) for HOMO → k ( l + 1). Evidently, unlike the zero of a Cooper mini-mum, that depends on the node in the bound wave func-tion, these zeros arise from nodes in the continuum wavefunction and can be termed as the cavity minima . R out HOMO HOMO-1 Cont. 30 eV Cont. 150 eV P o t en t i a l and w a v e f un c t i on ( a . u . ) Radial coordinate (a.u.) dVdrV(r) R in (a)50 100 150 200 250 300-0.4-0.20.0 x (b) R ad i a l m a t r i x e l e m en t ( a . u . ) Photon energy (eV)
FIG. 2. (Color online). (a) Ground state radial potential andits gradient, radial wave functions of HOMO and HOMO-1,and the continuum wave of ( l +1) angular momentum for a lowand a high energy are shown. (b) The real IP radial matrixelement compares to the real and imaginary components ofcomplex TDLDA matrix element. Besides two scaling regions,the imaginary part is multiplied by an overall factor of 5. IV. RESULTS AND DISCUSSION
Wigner-Smith time delay, the energy differential ofthe phase of the photoemission amplitude [29], is ac-cessible by “two-color” XUV-IR schemes like attosecondstreak camera and RABITT. This is because the extradelay introduced by the IR probe pulse, the Coulomb-laser coupling delay, can be independently calculated anddeducted from the data [15, 30, 31]. Our results [32]of Wigner-Smith delay using the current TDLDA/LB94scheme showed excellent agreements with RABITT mea-surements [6–8] for argon. This standard techniquesto extract the IR-induced delay information from theCoulomb and the short-range potentials are well de-scribed within the IP frame [30]. Ref. [31] derivesthis coulomb-laser coupling delay from a universal phasebrought by the absorption of the IR photon in the pres-ence of the Coulomb potential with charge Z. Whethermultielectron effects like configuration interactions fromlevel compactness could modify this delay is only a ques-tion for future research. In fact, experimental efforts tomeasure the current predictions or those from Ref. [16]can only verify the validity of this question.The IP radial matrix element h r i is real, implyingthat the IP phase is directly η in Eq. (3) and, hence,insensitive to the zeros in the matrix element. How-ever, the phase becomes sensitive to the cavity min-ima when TDLDA includes correlations via an energy-dependent complex induced potential δV in the ampli-tude: D = h Ψ kl ′ | z + δV ( r ) | Φ nl i ; see Ref. [17] for detailsof the formalism. Hence, the many-body effects could bedirectly probed by the phase and group delay measure-ments at these minima. The TDLDA phase γ = η +arctan (cid:20) Im h r + δV i Re h r + δV i (cid:21) = η +arctan (cid:20) Im h δV ih r i +Re h δV i (cid:21) , (5)since h r i is real. In Eq. (5), the new radial matrix element h r + δV ( r ) i being complex suffers a π phase-shift as itsreal part moves through a zero at a cavity minimum.TDLDA quantum phases for two dipole channels fromeach of HOMO and HOMO-1 are presented in Figs. 3.Phase-shifts of about π at all cavity minima are noted;for HOMO-1 the shifts are roughly synchronized betweenthe two channels [Fig 3(b)]. The direction of a phase-shift, upwards or downwards, depends on the details ofthe TDLDA matrix element. Eq. (5) suggests that h r i iscorrelation-corrected by Re h δV i , but this correction di-minishes at higher energies as seen in Fig. 2(b). WhenRe h r + δV i sloshes through a zero, a π -shift occurs. Butthe direction of the shift depends on the sign of Im h r + δV i – a quantity entirely correlation-induced. The oscilla-tions in the imaginary part [Fig. 2(b)] arise from a multi-channel coupling with a large number of C inner chan-nels which are open at these energies. The amplitudes ofthese inner channels do not oscillate in-phase and havediverse phase offsets in relation to HOMO and HOMO-1oscillations [26]. Consequently, the position of zeros in
50 100 150 200 250 3002468 (b) P ha s e ( un i t s o f ) (a)
50 100 150 200 250 30012345 P ha s e ( un i t s o f ) Photon energy (eV)
FIG. 3. (Color online). TDLDA phases as a function of thephoton energy for ionization through dipole allowed channelsfor (a) HOMO and (b) HOMO-1 electrons. Calculated ion-ization total phases from these levels are also shown. Im h r + δV i is a function of correlations via this multi-channel process. Indeed, while the real and imaginarycomponents are seen to oscillate roughly out-of-phase,the zeros of one do not occur systematically on a definiteside of the zeros of the other, causing the phase changeto follow a pattern that directly maps the correlation thevalence emission experiences at a cavity minimum.In the RABITT experiment, one measures the delayassociated with a phase Γ which is not resolved in thephotoemission direction Ω k :Γ = arg[ ¯ D l +1 exp( iγ l +1 ) + ¯ D l − exp( iγ l − )] , (6)where ¯ D l ′ = R d Ω k | D l ′ ( Ω k ) | . Since for a channel σ l ′ ∼ R d Ω k | D l ′ ( Ω k ) | , we approximate Eq. (6) by replacing ¯ D by the square root of respective channel cross sections.Fig. 3 also presents these calculated total phases.TDLDA Wigner-Smith time delays, energy-differentials of total phases, are shown in Fig. 4.To obtain the delay from the phases, one can usearbitrarily small energy steps for the differential.Measurements based on RABITT metrology typicallyuses 800 nm ( ω = 1.55 eV) IR probe pulse that leadsto the extraction of the delay from measured Γ by
50 100 150 200 250 300-2000-1000010002000 (a) W i gne r- S m i t h t i m e de l a y ( a s ) HOMO Finite difference50 100 150 200 250 300-400-2000200400600800 (b) W i gne r- S m i t h t i m e de l a y ( a s ) Photon energy (eV) HOMO-1 Finite difference
FIG. 4. (Color online). Wigner-Smith time delays for (a)HOMO and (b) HOMO-1 calculated within TDLDA frame-work and its comparison with the delays determined by afinite difference approach, where 800 nm IR pulse is used forenergy differential (see the text). [Γ( E + ω ) − Γ( E − ω )] / ω . Resulting “finite difference”TDLDA delays for HOMO and HOMO-1 are also shownin Figs. 4. Structures, corresponding to negative orpositive delays, at the cavity minima indicate strikingvariations in the photoelectron speed. The fast (slow)emissions are effects of dynamical anti-screening (screen-ing) from the multichannel coupling based on the Fanoscheme [33]. In this, the correlation h δV ( r ) i for theemission from nl (HOMO or HOMO-1) reads as [17] h δV i nl = X λ lim δ → Z dE ′ h ˜ ψ λ ( E ′ ) | | r nl − ~r λ | | ˜ ψ nl ( E ) i E − E ′ + iδ d λ ( E ′ ) , (7)where the sum is over all other open channels λ andtwo-body wave functions ˜ ψ involve both bound and con-tinuum states in an IP channel. h δV i can be large,since bound wave functions of delocalized electrons oc-cupy similar regions in space enabling large overlap inEq. (7). We note that the details of the correlation hereis pretty complex, as all the open channels (about 30 ina jellium frame), constituting 240 delocalized electrons,are coupled. A simple interpretation of the results maystill be outlined. At an XUV energy of current inter-est, each molecular level can ionize in its uncoupled IPchannel. However, the interchannel coupling in Eq. (7)may include another possibility: An inner electron caninitially absorb the XUV photon and then transfer theenergy via Coulomb interactions to HOMO or HOMO-1to cause an outer emission. Thus, since this repulsive1 /r underpins the coupling landscape [Eq. (7)], andsince the correlation must dominate near a minimum ofa channel, either of the valence electrons feels a strongoutward force, via interchannel couplings, from the hostof inner electrons and hence ionize faster. This is seenin Fig. 4 in predominant negative-delay structures. Theexception at 190 eV needs further investigation. TheHOMO-1 level, being below HOMO, feels some blockadefrom the inward Coulomb push via its coupling with theouter HOMO, and therefore gets relatively slower overalland, in particular, shows a second positive delay at 150eV.Probing correlation forces by the attosecond spec-troscopy is the main focus of this work. Even thoughthe separation between HOMO and HOMO-1 is 1.3 eV,our results can be experimentally accessed, since the res-olution of RABITT measurements is not limited to thespectral-width of the attosecond pulse but to that ofthe individual harmonics ( ∼
100 meV) of the resultingfrequency comb. Further, by approximating ψ by theasymptotic form cos( kr − l ′ π/
2) of the spherical Besselfunction [34], Eq. (4) becomes sinusoidal in k . This re-sults oscillations in the momentum space with radii beingthe frequencies. Hence, the reciprocals ( π/ ∆ k ) of the sep-arations (periods) ∆ k between the minima, or betweenthe delay extrema, connect to C radii. Obviously, forlarger (smaller) fullerenes the structures will compact-ify (spread out). Furthermore, this technique may applyto access time information in a spheroidal fullerene, acarbon nanotube, or nanostructures of partial symmetryby properly orienting the polarization of XUV photon tominimize non-dipole effects from deformity [36]. The utilization of plane waves, instead of the contin-uum solutions as Eq. (3), should produce cavity minimain the cross section, since, as discussed above, the originof these minima is the nodes in the photoelectron wavethat plane waves have. But in this case, the minima willappear at spectral positions different from the presentresult. Futhermore, as plane waves omit the Coulomband short range phases of Eq. (3), the phase and time-delay profiles will differ from the current prediction. Theplane waves routinely form the basis of the strong fieldapproximation. But since the correlation effects dimin-ish in a strong field environment, the delay structure mayconsiderably weaken or be altered directly by the field. V. CONCLUSION
In summary, photoemission quantum phases andWigner-Smith time delays for HOMO and HOMO-1 elec-trons of a C molecule are investigated. Results showstructures at the cavity minima in the energy range abovethe plasmon resonances and below the carbon K -edgewhich carry the direct imprint of the dynamical corre-lation and the molecular size. Even though a jelliumdescription of the ion core omits the scattering from lo-cal carbon ions [17], the structures should still be ob-served, but may soften in strength. We also calculatedthe results with a different, but less accurate than thecurrent (LB94), XC functional. In specific, using a func-tional as in Ref. [35] has shown similar qualitative re-sults. We plan to include the comparison in a futurepaper. Besides fullerenes, the detection of photoemissionminima in metal clusters [37] suggests a possible uni-versality of the phenomenon in cluster systems, or evenquantum dots [36], that confine finite-sized electron gas.The work predicts a new research direction to apply at-tosecond RABITT metrology in the world of gas-phasenanosystems. ACKNOWLEDGMENTS
The research is supported by the NSF, USA. [1] M. Hentschel, R. Kienberger, C. Spielmann, G. A. Rei-der, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann,M. Drescher, and F. Krausz, Nature , 509 (2001).[2] P. B. Corkum and F. Krausz, Nature Physics , 381(2007).[3] F. Krausz and M. Ivanov, Rev. Mod. Phys. , 163(2009).[4] G. Sansone, F. Calegari, and M. Nisoli, Journal of Se-lected Topics in Quantum Electronics, , 507 (2012).[5] M. Schultze, M. Fie, N. Karpowicz, J. Gagnon, M. Korb-man, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komni-nos, T. Mercouris, C. A. Nicolaides, R. Pazourek,S. Nagele, J. Feist, J. Burgdrfer, A. M. Azzeer, R. Ern-storfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, and V. S. Yakovlev, Science , 1658 (2010).[6] K. Kl¨under, J. M. Dahlstr¨om, M. Gisselbrecht,T. Fordell, M. Swoboda, D. Gunot, P. Johnsson,J. Caillat, J. Mauritsson, A. Maquet, R. Taeb, andA. L’Huillier, Physical Review Letters , 143002(2011).[7] D. Gu´enot, K. Kl¨under, C. L. Arnold, D. Kroon, J. M.Dahlstr¨om, M. Miranda, T. Fordell, M. Gisselbrecht,P. Johnsson, J. Mauritsson, E. Lindroth, A. Maquet,R. Ta¨ıeb, A. L‘Huillier, and A. S. Kheifets, PhysicalReview A , 053424 (2012).[8] S. B. Schoun, R. Chirla, J. Wheeler, C. Roedig, P. Agos-tini, L. F. DiMauro, K. J. Schafer, and M. Gaarde, Phys-ical Review Letters , 153001 (2014). [9] J. Caillat, A. Maquet, S. Haessler, B. Fabre, T. Ruchon,P. Sali`eres, Y. Mairesse, and R. Ta¨ıeb, Physical ReviewLetters , 093002 (2011).[10] S. Neppl, R. Ernstorfer, E. M. Bothschafter, A. L. Cava-lieri, D. Menzel, J. V. Barth, F. Krausz, R. Kienberger,and P. Feulner, Physical Review Letters , 87401(2012).[11] A. L. Cavalieri, N. Mller, T. Uphues, V. S. Yakovlev,A. Baltuska, B. Horvath, B. Schmidt, L. Blmel,R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg,P. M. Echenique, R. Kienberger, F. Krausz, andU. Heinzmann, Nature , 1029 (2007).[12] C. -H Zhang and U. Thumm, Physical Review Letters , 123601 (2009).[13] G. Dixit, H. S. Chakraborty, and M. E. Madjet, PhysicalReview Letters , 203003 (2013).[14] P. C. Deshmukh, A. Mandal, S. Saha, A. S. Khaifets,V. K. Dolmatov, and S. T. Manson, Physical Review A , 053424 (2014).[15] R. Pazourek, S. Nagele, and J. Burgd¨orfer, Faraday Dis-cuss. , 353 (2013).[16] T. Barillot, C. Cauchy, P -A. Hervieux, M. Gisselbrecht,S. E. Canton, P. Johnsson, J. Laksman, E. P. Mans-son, J. M. Dahlstr¨om, M. Magrakvelidze6, G. Dixit,M. E. Madjet, H. S. Chakraborty, E. Suraud, P. M.Dinh, P. Wopperer, K. Hansen, V. Loriot, C. Bordas,S. Sorensen and F. L´epine, Physical Review A , 033413(2015).[17] M. E. Madjet, H. S. Chakraborty, J. M. Rost, and S. T.Manson, Journal of Physics B , 105101 (2008).[18] R. Van Leeuwen and E. J. Baerends, Physical Review A , 2421 (1994).[19] M. F. Ciappina, A. Becker, and A. Jaro´n-Becker, Physi-cal Review A , 063406 (2007); Physical Review A ,063405 (2008).[20] N. Troullier and J. L. Martins, Physical Review B ,1754 (1992).[21] J. H. Weaver, J. L. Martins, T. Komeda, Y. Chen, T. R.Ohno, G. H. Kroll, and N. Troullier, Physical Review Letters , 1741 (1991).[22] M. Vos, S. A. Canney, I. E. McCarthy, S. Utteridge, M. T.Michalewicz, and E. Weigold, Physical Review B ,1309 (1997).[23] A. R¨udel, R. Hentges, U. Becker, H. S. Chakraborty,M. E. Madjet, and J. M. Rost, Physical Review Letters , 125503 (2002).[24] K. T. Taylor and P. G. Burke, Journal of Physics B ,L353 (1976).[25] S. Korica, A. Reink¨oster, M. Braune, J. Viefhaus,D. Rolles, B. Langer, G. Fronzoni, D. Toffoli, M. Stener,P. Decleva, O. M. Al-Dossary, U. Becker, Surface Science , 1940 (2010).[26] M. A. McCune, M. E. Madjet, and H. S. Chakraborty,Journal of Physics B , 201003 (2008).[27] J. W. Cooper, Physical Review , 681 (1962).[28] O. Frank and J. -M. Rost, Zeitschrift f¨ur Physik D ,59 (1996).[29] E. P. Wigner, Physical Review , 145 (1955); F. T.Smith, Physical Review , 349 (1960).[30] M. Ivanov and O. Smirnova, Physical Review Letters , 213605 (2011).[31] J. M. Dahlstr¨om, D. Gu´enot, K. Kl¨under, M. Gissel-brecht, J. Mauritsson, A. L‘Huillier, A. Maquet, andR. Ta¨ıeb, Chemical Physics , 53 (2012).[32] M. Magrakvelidze, M. E. Madjet, G. Dixit, M. Ivanov,and H. S. Chakraborty, submitted .[33] U. Fano, Physical Review , 1866 (1961).[34] L. D. Landau, and E. M. Lifshitz, Quantum Mechanics:Nonrelativistic Theory 3rd ed., Pergamon Press Oxford,1977 (p. 136).[35] O. Gunnarsson and B. Lundqvist, Physical Review B ,4274 (1976).[36] H. S. Chakraborty, R. G. Nazmitdinov, M. E. Madjet,and J. -M. Rost, arXiv:cond-mat/0111383.[37] K. J¨ank¨al¨a, M. Tchaplyguine, M. -H. Mikkel¨a,O. Bj¨omeholm, and H. Huttula, Physical Review Let-ters107