Resonant Photoelectron Confinement in the SF6 Molecule - The

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A: Spectroscopy, Molecular Structure, and Quantum Chemistry 6

Resonant Photoelectron Confinement in the SF Molecule Etienne Plésiat, Sophie E. Canton, John D. Bozek, Piero Decleva, and Fernando Martin J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b12237 • Publication Date (Web): 04 Jan 2019 Downloaded from http://pubs.acs.org on January 8, 2019

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The Journal of Physical Chemistry

Resonant Photoelectron Confinement in the SF6 Molecule Etienne Plésiat,† Sophie E. Canton,‡ John D. Bozek,¶ Piero Decleva,§ and Fernando Mart´ın∗,†,k,⊥ †Departamento de Qu´ımica, Módulo 13, Universidad Autónoma de Madrid, 28049 Madrid, Spain, EU ‡ELI-ALPS, ELI-HU Non-Profit Ltd., Dugonics ter 13, Szeged 6720, Hungary, EU ¶Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, BP 48, 91192 Gif-sur-Yvette Cedex, France §Dipartimento di Scienze Chimiche e Farmaceutiche, Universit´ a di Trieste and IOM-CNR, 34127 Trieste, Italy, EU kInstituto Madrile˜ no de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), Cantoblanco, 28049 Madrid, Spain, EU ⊥Condensed Matter Physics Center (IFIMAC), Universidad Aut´ onoma de Madrid, 28049 Madrid, Spain, EU E-mail: [email protected]

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Abstract Recent thorough experimental activity aiming to generate high harmonics in the SF6 molecules requires the knowledge of, on the one hand, accurate valence-shell photoionization cross sections and phases, from threshold up to a few tens of eV, where resonances are likely to appear, and, on the other hand, the effect of the nuclear vibrational dynamics on the process. In this work, we have experimentally determined and theoretically evaluated vibrationally resolved photoionization cross sections of SF6 up to 80 eV photon energies, with emphasis on the E2 T1u channel, for which vibrational progressions are fully resolved in the experiment. Our results reveal the presence of shape resonances due to excitation to SF6 virtual states lying just above the ionization threshold, in agreement with previous synchrotron radiation work and theoretical calculations. More interestingly, our calculations also disclose resonance features at photoelectron energies as high as 40-50 eV, which are due to the transient confinement of the ejected electron in the octahedral cage formed by the peripheral F atoms. In the vicinity of all resonances, including those due to confinement, the calculated ionization phases experience an excursion of about π and significantly depend on the final vibrational state of the remaining cation. Both effects should be taken into account to correctly interprete ongoing high-harmonic generation work in SF6 . A similar behavior is expected for other symmetric molecules containing a central atom, such as BF3 , CF4 , and the like.

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Introduction For more than four decades, core, inner-valence and valence shell photoionization of SF6 has amply been investigated by using synchrotron radiation 1–14 and theoretical methods, 15–22 covering a wide range of photon energies, from the threshold up to a few hundreds of eV. At the lower photon energies, these investigations have revealed a variety of resonant structures that show up both in total and partial photoionization cross sections (either angle integrated or differential), and are specific for each ionization channel. In the last few years, there has been a renewed interest in the problem, spurred by significant experimental and theoretical efforts pursuing high harmonic (HH) generation from SF6 molecules irradiated with strong IR fields. 23–29 That HH emission from SF6 is technically feasible was demonstrated as early as in 1996. 30 However, it has not been until very recently that experiments have shown that such an emission has very special characteristics: (i) the resulting harmonic spectra exhibit pronounced amplitude modulations 25,27,29 and (ii) the emitted light is elliptically polarized in selected harmonics. 27 The first property, in addition to providing information on the electron dynamics occurring in the molecule, allows one to design attosecond pulses with interesting spectral characteristics. The second one offers the possibility to control the degree of ellipticity of such attosecond pulses. Both features are difficult to achieve when using atomic or simpler molecular targets. High harmonic emission is the result of a three-step process: 31,32 (i) tunnel ionization induced by the strong IR field, (ii) round trip of the ejected electron following the applied field for approximately half a laser cycle, and (iii) recombination of the ejected electron with the parent ion accompanied by emission of the excess of kinetic energy in the form of XUV light. For the IR fields used in typical HH experiments, the first step mainly involves the valence shells of molecules. The third step is the inverse of photoionization. Thus, knowledge acquired in valence-shell photoionization experiments is extremely valuable to identify the features observed in the HH spectrum. However, as HH emission is a coherent process, the information provided by the photoionization cross sections is not enough to fully characterize 3



the process: the ionization phases are also needed. As the latter cannot be obtained from the analysis of the photoelectron spectra, their theoretical evaluation is the only practical alternative, at least for complex molecular targets such as SF6 . In most experiments on SF6 , harmonics are emitted up to photon energies close to 40 eV (i.e., equivalent to a kinetic energy of the recombining electron of 40 eV), but the rapid experimental progress in this field will soon allow one to overcome this limit; for example, by combining the usual strong IR field with pulses of a shorter wavelength (VUV and XUV assisted HH emission 33–35 ). Most unusual features observed in the HH spectrum mentioned above have been explained by the trapping of the recombining electron in shape resonances, 25,27,29 since this can lead to strong variations of the electron phase in a narrow interval of electron energies, hence to modulations in the HH spectrum. In general, shape resonances are interpreted as arising from either trapping of continuum wave functions in potential energy barriers created by electronegative atoms 36 or excitations into valence virtual (antibonding) orbitals above threshold. 37 In SF6 , resonances of both kinds have been identified (see, e.g., Ref. 19). Most of these resonances are expected to appear at low electron kinetic energies. However, earlier valence-shell photoelectron spectra obtained with synchrotrons and theoretical calculations suggest that resonance features might also exist at rather high energies, up to several tens of eV above threshold. 5,17,19 What is the origin of resonances appearing at such high energies? This is a relevant question because any resonance should have an important effect in the phase of the ejected electron and, by extension, in the HH spectra. In this work we provide unambiguous experimental and theoretical evidence for the existence of resonances in the electronic continuum of SF6 at around 40 eV. Our conclusion is based on the analysis of vibrationally resolved photoelectron spectra of SF6 , which are much more sensitive to the different features appearing in the electronic continuum (e.g., resonances, 38,39 double- and multi-slit interferences, 38–42 photoelectron diffraction, 43,44,44–47 nuclear recoil effects, 44–46,48,49 etc) than total cross sections. In particular, we show that the

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origin of the first of these high-energy resonances appearing at ∼ 40 eV photoelectron energy in the E2 T1u electronic band is the transient confinement of the ejected electron in the octahedral cage formed by the peripheral F atoms. As a consequence, the calculated ionization phase experiences an excursion of about π in this energy region, which should lead to strong modulations of the HH spectra. Furthermore, we also show that the ionization phase significantly varies with the vibrational state of the remaining cation, which must therefore be also taken into account when interpreting HH spectra from SF6 . Similar effects are expected to show up in symmetric molecules containing a central atom such as BF3 (triangular cage), CF4 (tetrahedral cage), and the like.

Methods The experiments were performed at the high-resolution AMO beamline 10.0.1 of the Advanced Light Source (ALS). An effusive molecular beam of gaseous SF6 was ionized by linearly polarized synchrotron radiation. The photoelectron spectra were measured using a Scienta SES-200 hemispherical analyzer 50 placed at the so-called magic angle 54.7 with respect to the light polarization axis, thereby eliminating angular effects on the measured cross sections. All the spectra were corrected afterwards for spectrometer transmission and background subtraction. Owing to the high performance of the analyzer, which has a resolution of ∼120 meV (see 51,52 for details), the six lowest photoelectron bands were kept resolved throughout the photon energy range covered. With negligible stray background, reliable areas proportional to the cross sections were extracted by summing the counts in a given kinetic energy window, avoiding systematic uncertainties attached to baselines coming from overlapping peaks and bypassing recourse to fitting with analytical line shapes. Theoretical calculations were performed within the Born-Oppenheimer approximation as described in Ref. 39. Briefly, the electronic wave functions were calculated by using the static-exchange DFT method with an LB94 exchange-correlation functional and a basis of 150

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B-splines expanded in a box of 20 a.u.. The maximum angular momentum was fixed to 15 for the one-center expansion and to 2 for the off-center expansions around each atomic center (the radius of the latter was limited to 1.6 a.u.). The electronic wave functions were evaluated for 38 different molecular geometries associated with the symmetric stretching mode, including the equilibrium geometry. These geometries were built by varying synchronously the six S-F distances between 2 and 6 a.u. with a denser grid around the equilibrium geometries of the ground and excited states. The potential energy curves along the symmetric stretching coordinate were built by making use of (i) the accurate harmonic potential reported in Ref. 18 (ωe = 833 cm−1 ) for the ground state and (ii) the accurate Morse potential of Ref. 5 (ωe = 563 cm−1 , ωe χe = 2.26 cm−1 ) for the E2 T1u excited state. The ionization potential has been set to match the current experimental one, i.e. IP=28.1eV. The nuclear wave functions were obtained by diagonalizing the corresponding nuclear Hamiltonian in a basis of 1000 B-splines and a box of 10 a.u.. The vibrationally resolved photoionization cross sections were evaluated by using first-order perturbation theory and the dipole approximation.

Results Fig. 1 (top panel) shows the measured photoelectron spectrum of SF6 at a photon energy of 50 eV. The spectrum shows the peaks associated to the lowest six electronic bands, X2 T1g , A2 T1u , B2 T2u , C2 Eg , D2 T2g , and E2 T1u , which correspond to removing the electron from, respectively, the 1t1g , (HOMO), 5t1u (HOMO-1), 1t2u (HOMO-2), 3eg (HOMO-3), 1t2g (HOMO-4), and 4t1u (HOMO-5) molecular orbitals of SF6 . As can be seen, the B2 T2u and A2 T1u bands overlap and show no vibrational structure. The C2 Eg band exhibits a complex vibrational structure whose progression (flat on the top) seems to indicate a splitting of the doubly degenerate state by Jahn-Teller interactions. The vibrational structure of the D2 T2g and E2 T1u bands is clearly resolved, although it is only apparent in the low energy part of the former. In both cases, the totally symmetric stretching mode is responsible for

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Figure 1: Photoelectron spectrum of SF6 at a photon energy of 50 eV. Upper panel: Measured photoelectron spectrum revealing the contribution from different ionization channels accessible at this photon energy. Middle panel: Blow up of the peak corresponding to ionization from the 4t1u orbital. A wide progression of final vibrational states for the symmetric stretching mode of the SF+ 6 cation is clearly resolved. The vertical lines show the results of our theoretical calculations. Lower panel: One-dimensional cut of the potential energy surfaces of the ground state of SF6 and the E2 T1u state of SF+ 6 along the symmetric stretching mode coordinate. Superimposed to the latter curve, the vibrational levels and corresponding wave functions for the symmetric stretching mode are shown.

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the vibrational progression. The peaks of the E2 T1u band are possibly broadened by predissociation, since the threshold for SF+ 3 formation is approximately located at the position of this state. Fig. 1 (middle panel) shows a blow up of the E2 T1u band. As the orbital from which the electron is extracted is bonding (4t1u ) , the S-F distance of the remaining cation is ∼ 0.044˚ A larger. As a consequence, the potential energy surfaces of the neutral and ionized molecules are shifted with respect to each other (see bottom panel in Fig. 1). In combination with the large value of the reduced mass of SF6 , the existence of this shift leads to the very broad vibrational progression shown in Fig. 1 (middle panel), which peaks at v 0 = 4, as one could infer by assuming a vertical transition from the ground state. The energy difference between the v 0 = 0 and v 0 = 1 peaks is about 67 meV, i.e. ∼540 cm−1 , in good agreement with the harmonic frequency of 562.5 cm−1 given by Holland et al. 5 The theoretically predicted intensities agree well with the experimental ones. Fig. 2 shows the vibrationally resolved cross sections as a function of photoelectron energy. For a better visualization, all cross sections have been normalized to the dominant v 0 = 4 one, leading to the so-called v-ratios. In doing so, the rapidly decreasing background of the cross sections is automatically removed and the resonant features are clearly displayed. We can see two of these resonances appearing at around 28 and 39 eV photon energies (∼ 5 and 16 eV photoelectron energies, respectively) in all v-ratios, which as described in previous work 19 correspond to excitations from the 4t1u orbital to the lowest t2g and eg virtual ones. The theoretical peaks are sharper than the experimental ones due to the lower energy resolution of the latter. In the vicinity of these resonances, the v-ratios clearly deviate from the value of the corresponding Franck-Condon factors. There is a third, broader peak at ∼ 69 eV (∼ 40 eV photoelectron energy), which is also clearly seen in both experiment and theory, especially for the largest v-ratios (v 0 = 2 − 6). Interestingly, as discussed in Ref. 44, the peak profiles in the v-ratios reverse when going from v 0 < 4 to v 0 > 4, as it is quite apparent in both the experimental and theoretical results. The agreement between the calculated and

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Figure 2: Vibrationally resolved photoelectron spectra of SF6 as a function of photon energy. All spectra have been normalized to the v = 4 one. Continuous lines: theoretical results. Circles: experimental results. Horizontal dashed lines: values of the Franck-Condon factors. measured ratios worsens as v 0 departs from the reference value (v 0 = 4), especially for v 0 = 0. This is the logical consequence of the ratios becoming progressively smaller, which makes calculated and measured cross sections more sensitive to small errors. For completeness, we compare in Fig. 3 the calculated 4t−1 1u photoionization cross section, obtained by summing over all vibrationally resolved partial-wave cross sections, with that reported by Holland et al 5 in earlier measurements. The latter were obtained in absolute scale, so the comparison is direct (no rescaling). As can be seen, the general agreement is 9



Figure 3: Vibrationally unresolved total and partial 4t−1 1u photoionization cross sections of SF6 as functions of photon energy. Continuous lines: theoretical results convoluted with the experimental energy resolution. Circles: experimental results of Holland et al. 5 quite good, especially below a photon energy of 60 eV. In particular, the two resonances at around 28 and 39 eV are clearly seen both in the calculations and the measurements. In contrast, the resonance at 69 eV, which is quite apparent in the calculations, is rather weak in the experimental cross section. This is the region where the cross section is smallest and, therefore, where it can be more affected by the presence of an unphysical background. By measuring v-ratios, one avoids potential problems of this kind and resonance features can be more clearly seen.

Discussion To understand the origin of the third resonance we have first adopted the usual strategies proposed in the literature. Fig. 3 shows the contributions of continuum states of different symmetries to the 4t−1 1u cross section. Only the dominant contributions to the three res-

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30 0.4

l=4

Cross section (Mb)

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40

50

60

70

80

v'=2

SF6 (4t1u → εlt1g)

0.2 0

v'=4

0.4

l=6

l=4

0.2

l=8 0

v'=6

0.4 0.2 0

30

40

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60

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Photon energy (eV) Figure 4: Partial wave contributions to the t1g photoionization cross sections of SF6 leading −1 to SF+ 6 (4t1u ). Continuous lines: partial wave contributions. Dashed lines: total t1g cross section. onances discussed in the text are shown. The indicated symmetry is that of the ejected electron. In agreement with existing literature 19 and the above discussion, the lowest two resonances appear in the t2g and eg continua. The resonance at around 70 eV mainly appears in the t1g continuum. Therefore, we have concentrated our analysis in the continuum of this symmetry. Fig. 4 shows the contribution of the different partial waves to this channel for three different final vibrational states of the SF+ 6 cation. As can be seen, for all vibrational states, there are three partial waves that contribute significantly to the resonance: l = 4, 6 and 8. None of them prevails over the others in the energy range covered by the resonance, so that it is difficult to attribute the existence of such resonance to a single centrifugal barrier

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in the potential seen by the escaping electron. An alternative analysis carried out in ref. 19 consists in looking at the so called dipole prepared states. We have evaluated these states in the t1g continuum in the whole energy range of photon energies investigated in the present work (in ref. 19 , this analysis was restricted to energies lying in the vicinity of the resonance). We have found that, from the threshold up to 100 eV, the qualitative shape of the dipole prepared states (number, sign and shape of the lobes, number of nodal planes, etc) is the same as for the dipole prepared state reported in ref. 19 in the resonance region. This shape is qualitatively similar to that of the 2t2g virtual orbital of SF6 . Therefore, there is no special feature in the dipole prepared state that can explain the presence of this resonance in the photon energy range where it is observed. As none of these interpretations is completely satisfactory, here we have adopted a different approach. We have solved the Schrödinger equation that describes the quantum mechanical motion of an electron inside an octahedral cage of infinite walls with a side length identical to that of two neighboring F atoms in the actual SF6 molecule at the equilibrium geometry (3.122 ˚ A). The energies corresponding to the lowest eigenfunctions are 37.0 eV, 74.5 eV, 109.5 eV, etc. We propose that the peak observed at ∼ 40 eV in the v-ratios corresponds to a resonance due to the trapping of the ejected electron in the ground state of this octahedral cage. To prove it, Fig. 5 shows the different terms contributing to the dipole matrix element describing such a resonant transition. The bra contains one of the three-fold degenerate 4t1u Kohn-Sham orbitals of SF6 , from which the electron is removed, as obtained from our DFT calculations. This electron is coupled to the ionization continuum through the dipole operator (represented by x in the figure). The ket contains the ground-state wave function of the electron moving inside the above-mentioned octahedral cage. Multiplying the latter state by the dipole operator leads to the function shown in the ket of the right-hand side of the equation. As can be seen, the latter ”orbital” is very similar to the actual 4t1u orbital inside the cage (which contains most of the electron density). Therefore, the resonance appearing at 40 eV photoelectron energy must be due to the transient confinement

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Figure 5: Origin of the resonance at ∼ 40 eV photoelectron energy. The bra shows one of the three-fold degenerate 4t1u Kohn-Sham orbitals of SF6 , from which the electron is removed. This electron is coupled to the ionization continuum through the dipole operator (represented by x in the figure). Instead of the actual continuum orbital, the ket in the left-hand side of the equation has been replaced by the ground-state wave function of an electron moving inside an octahedral cage of infinite walls and side distance equal to that of two neighboring F atoms in the actual SF6 molecule (3.122 ˚ A). The eigen-energy corresponding to this state is 37 eV (corresponding to 60 eV in photon energy), in qualitative agreement with the position of the highest resonance shown in Fig. 2. Multiplying the latter state by the dipole operator leads to the function shown in the ket of the right-hand side of the equation. of the ejected electron inside the octahedral cage defined by the F atoms. A similar picture explains the appearance of a resonant peak at around the same energy in the 5t1u cross section (not shown), since the corresponding molecular orbital resembles the 4t1u one. In comparison with the other two analyses described above, this confinement model has the advantage that, in addition to be very simple, it provides a clear explanation about the origin of the resonance by only making use of observable quantities. Therefore, it might be experimentally verified by measuring, e.g., the photo-absorption spectrum of an electron in a quantum well mimicking an octahedral potential box with the appropriate size. Obviously, such an experiment is not necessary because the spectrum can be exactly calculated by solving the Schrdinger equation, but the spirit is the same. The model predicts that the resonance can only appear at a well-defined energy, which is not the case of the other two models, and explains why the dipole transition element should be large at this particular energy. The resonance is indeed due to the particular shape of the potential, but not the shape that corresponds to a simple centrifugal barrier or to a specific virtual molecular orbital appearing at a given energy. We now analyze the effect of these confinement resonances on the ionization phases. Fig. 6 13



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v 0 xk

Figure 6: Phase φ4t−1 (ω) of the E2 T1u (4t−1 1u.1 ) dipole transition matrix element correspond1u.1 ing to an electron ejected parallel to the polarization direction (aligned with x, as in Fig. 5) for the different vibrational states of the remaining SF+ 6 cation. Due to the symmetry of the orbital 4t1u.1 (see Fig. 5), the phases are zero when the polarization vector is aligned v 0 yk v 0 zk with y or z (φ4t−1 (ω) = φ4t−1 (ω) = 0). For the other degenerate subspecies, we have: v 0 zk

1u.1

v 0 yk

v 0 xk

1u.1

v 0 yk

v 0 xk

v 0 xk

v 0 zk

φ4t−1 (ω) = φ4t−1 (ω) = φ4t−1 (ω) and φ4t−1 (ω) = φ4t−1 (ω) = φ4t−1 (ω) = φ4t−1 (ω) = 0. 1u.2

1u.3

1u.1

1u.2

1u.2

1u.3

1u.3

shows the phase of the E2 T1u dipole transition matrix element corresponding to an electron ejected at zero degrees with respect to the polarization direction x (the same direction as in Fig. 5) for the different vibrational states of the remaining cation. We have chosen this ejection direction because in HH generation experiments the electron mostly follows the direction of the applied intense IR field, so that recombination of that electron with the remaining molecule also occurs in this direction. As can be seen, the phases vary abruptly with photoelectron energy in the vicinity of the confinement resonance. The variation can be as large as π/2. We do not see the typical phase jump of π that is expected in the vicinity of an isolated resonance because, as one can see in Fig. 4, the resonance peak is due to the contribution of resonance structures appearing in the l = 4, 6 and 8 partial waves. Most

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remarkably, the phase significantly depends on v 0 , e.g., at ∼ 65 eV photon energy the phase difference between the v = 0 and v = 10 channels is as large as 0.4 rad. These differences should be taken into account to correctly assign phases to different harmonics observed in experiments. Indeed, although in these experiments the molecule is initially in the ground v=0 state, for not-too-short IR pulses, a nuclear wave packet is formed after a few laser cycles due to Raman transitions involving the ground and the lowest electronically excited states. As a consequence, after this, ionization can take place from excited vibrational states (not only from v = 0), and consequently recombination with the remaining molecular cation can also occur in excited vibrational states (see, e.g., recent work on the H+ 2 molecular cation 53–55 ). Hence, vibrationally resolved phases are necessary in order to interpret HH generation experiments performed in molecules.

Conclusions In conclusion, we have measured and calculated vibrationally resolved photoionization cross sections of SF6 , which allow us to identify resonances previously assigned to excitation to virtual molecular states, as well as a new kind of resonances whose origin is the confinement of the ejected electron in the cage formed by the peripheral atoms. The latter resonances appear at rather high photoelectron energies (40-50 eV) and have a strong influence on the ionization phases that are necessary to analyze high harmonic spectra resulting from this and other molecules. Furthermore, we have shown that the calculated phases significantly depend on the vibrational state in which the molecular cation is left, which again must be taken into account to interpret high harmonic spectra.

Acknowledgments This work has been supported by the ERC advanced grant 290853 - XCHEM - within the seventh framework programme of the European Union, the MINECO projects FIS201315



42002-R and FIS2016-77889-R, and the European COST Action XLIC CM1204. We also acknowledge computer time from CCC-UAM and Marenostrum Supercomputer Center. FM acknowledges support from the ’Severo Ochoa’ Programme for Centres of Excellence in R&D (MINECO, Grant SEV-2016-0686) and the ’Mar´ıa de Maeztu’ Programme for Units of Excellence in R&D (MDM-2014-0377). EP acknowledges a Juan de la Cierva contract from the Ministerio de Econom´ıa y Competitividad (Spain). The ELI-ALPS project (GINOP2.3.6-15-2015-00001) is supported by the European Union and co-financed by the European Regional Development Fund.

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harmonic generation in isotropic media of dissociating homonuclear molecules. Sci. Rep. 2016, 6, 32653.

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Resonant Photoelectron Confinement in the SF6 Molecule - The

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