Electron Correlation in the Ionization Continuum of Molecules

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Letter

Electron Correlation in the Ionization Continuum of Molecules: Photoionization of N in the Vicinity of the Hopfield Series of Autoionizing States 2

Markus Klinker, Carlos Marante, Luca Argenti, Jesús González-Vázquez, and Fernando Martin J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.7b03220 • Publication Date (Web): 24 Jan 2018 Downloaded from http://pubs.acs.org on January 30, 2018

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The Journal of Physical Chemistry Letters

Electron Correlation in the Ionization Continuum of Molecules: Photoionization of N2 in the Vicinity of the Hopfield Series of Autoionizing States Markus Klinker,∗,† Carlos Marante,† Luca Argenti,†,§ Jesús González-Vázquez,∗,† and Fernando Martín∗,†,‡,¶ Departamento de Química, Módulo 13, Universidad Autónoma de Madrid, 28049 Madrid, Spain, EU, Instituto Madrileño de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), Cantoblanco, 28049 Madrid, Spain, EU, and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049 Madrid, Spain, EU E-mail: [email protected]; [email protected]; [email protected]

∗ To

whom correspondence should be addressed de Química, Módulo 13, Universidad Autónoma de Madrid, 28049 Madrid, Spain, EU ‡ Instituto Madrileño de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), Cantoblanco, 28049 Madrid, Spain, EU ¶ Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049 Madrid, Spain, EU § Current address: Department of Physics and CREOL College of Optics & Photonics, University of Central Florida, Orlando, Florida 32816, USA † Departamento

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The Journal of Physical Chemistry Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Direct measurement of autoionization lifetimes by using time-resolved experimental techniques is a promising approach when energy-resolved spectroscopic methods do not work. Attosecond time-resolved experiments have recently provided the first quantitative determination of autoionization lifetimes of the lowest members of the well-known Hopfield series of resonances in N2 . In this work, we have used the recently developed XCHEM approach to study photoionization of the N2 molecule in the vicinity of these resonances. The XCHEM approach allows us to describe electron correlation in the molecular electronic continuum at a level similar to that provided by multi-reference configuration interaction methods in bound state calculations, a necessary condition to accurately describe autoionization, shake-up and inter-channel couplings occurring in this range of photon energies. Our results show that, electron correlation leading to interchannel mixing is the main factor that determines the magnitude and shape of the N2 photoionization cross sections, as well as the lifetimes of the Hopfield resonances. At variance with recent speculations, non-adiabatic effects do not seem to play a significant role. These conclusions are supported by the very good agreement between the calculated cross sections and those determined in synchrotron radiation and attosecond experiments.

N2+ + e N2 + 2p 2s 1s

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The advent of attosecond light pulses has opened the way to perform time resolved measurements of electron dynamics in atoms and molecules, thus allowing one to reach the ultimate frontier responsible for chemical bonding. 1–5 The attosecond time resolution provided by these novel light sources has permitted, among other accomplishments, to determine the lifetimes of atomic autoionizing states from the direct observation of their exponential decay with time. 1,6 For isolated resonances, the lifetimes resulting from such measurements are in perfect agreement with those obtained from photoelectron spectra obtained in synchrotrons. However, when resonances overlap in the spectral domain, the usual situation in molecules when several ionization threshold are accessible, extraction of the corresponding lifetimes from such spectra is not so straightforward, and a direct time resolved measurement could be a better alternative, especially when the lifetimes are substantially different. 5 A common characteristic of all attosecond measurements, which is also shared with synchrotron radiation measurements, is that ionization is produced by absorption of a single XUV or X-ray photon. In addition to direct ionization, where a single electron takes the excess of photon energy, a variety of processes in which this energy is shared between two or more electrons can also occur: inner-shell ionization followed by Auger decay, autoionization from Rydberg or multiply excited states, ionization accompanied by excitation of the remaining ion (shake up), Auger decay in combination with shake up, inter-channel couplings between different ionization continua, etc. 1 All these processes are mainly governed by electron correlation. Recently, attosecond XUV-pump / IR-probe experiments have been performed to determine, for the first time, the lifetimes of molecular autoionizing states, 7 namely of the lowest members of the Hopfield 8,9 series of autoionizing states in the N2 molecule. These states lie above the second ionization threshold of N2 and, therefore, can autoionize by emitting an electron and leaving N+ 2 in either the ground or the lowest excited state (shake up). Furthermore, the lowest members of the series lie only a few hundreds of meV above the threshold, so that they can autoionize by emitting a rather slow electron, which can therefore strongly interact with the electrons remaining in the molecular cation (inter-channel coupling). A conspicuous result of these measurements is the large

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differences in the autoionizing lifetimes of two of these series (one with much longer lifetimes than the other). This was tentatively attributed in Ref. 10 to interference stabilization between nearly degenerate Rydberg states, and more specifically in Ref. 11 to non-adiabatic rotational couplings between these states. However, there is no direct experimental or theoretical proof that this is indeed the reason for the differences in the observed lifetimes. The Hopfield series has also been investigated by using synchrotron radiation. 12–16 Although the total photoelectron spectra resulting from these earlier experiments are qualitatively similar to the most recent time-resolved ones, the shape and assignment of the resonance peaks are significantly different. Furthermore, the lack of experimental information on the partial photoionization cross sections and on molecular orientation, as well as the fact that many of these resonances overlap in the same spectral region, has prevented synchrotron radiation measurements from providing the lifetimes for the lowest Hopfield resonances. Therefore, the accuracy of the lifetimes reported in such attosecond measurements, which are benchmarks for similar applications of attosecond technology to more complicated molecules, remains to be checked. An accurate theoretical description of N2 photoionization in this energy region is only possible by appropriately describing electron correlation in the molecular continuum. This is a challenge for most existing theoretical methods, which have thus mainly focused on the description of the direct ionization process at different levels of approximation. 17–19 To our knowledge, the only existing theoretical calculations of N2 photoionization that account for the lowest members of the Hopfield series were performed as early as in 1983 20 and 1991 21 by using multichannel quantum defect theory (MCQDT) and multichannel frozen-core Hartree-Fock approximation (MCFCHF), respectively. At that time, only an approximate description of the various matrix elements that enter either the MCQDT or the MCFCHF equations was possible (frozen-core, static exchange and singlecentre continuum approximations, among others), so that they cannot be used as reference for the measured autoionization lifetimes. Thus, in spite of the apparent simplicity of the N2 molecule, a precise understanding of its correlated ionization dynamics is still to be achieved. Getting an accurate description of multichannel resonant ionization in this fundamental system is a prerequisite

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to understand correlation-induced ionization processes in more complicated molecules. In particular, autoionizing Auger decay has been identified as an important cause of molecular damage in biological systems. 22 In this work, we have evaluated the photoionization spectrum of N2 between the second and the third ionization thresholds within the fixed-nuclei approximation by using the recently proposed XCHEM 23 code, which includes electron correlation in the electronic continuum at the same level as the most advanced quantum chemistry methods do for bound states. Except very close to threshold, the calculated total photoionization spectrum is in excellent agreement with the most recent spectra obtained with synchrotron radiation, and the calculated resonance lifetimes are very close to those determined in synchrotron radiation experiments and of the order of those obtained from the attosecond pump-probe experiments mentioned above. From these calculations, we conclude that the large differences in the lifetimes of the different states that compose the Hopfield series are almost entirely due to differences in electron correlation and not to non-adiabatic rotational effects as claimed in earlier work. From the analysis of the partial photoionization cross sections, 2 we show that the dominant ionization channel leaves the N+ 2 ion in the Πu excited state, not in the 2 Σ+ g

ground state, and that the overall shape of the resonances observed in the total photoionization

spectrum is mainly due to molecules oriented parallel to the polarization direction of the electric field. We start by briefly describing the methodology used in the present work. In the XCHEM approach, the Ne -electron continuum wave function Ψ− αE is written as the close-coupling (CC) expansion ˆ ˆ Ne )φi (rNe )cβ i,αE , (1) Ψ− αE (x1 , · · · , xNe ) = ∑ ℵi (x1 , · · · , xNe )ci,αE + ∑ Nβ i A ϒβ (x1 , · · · , xNe −1 , x i

βi

where the first term includes the so-called short range states ℵi , in which all electrons reside within a radius R0 , and the second term contains anti-symmetrized products between one-electron radial continuum functions φi (rN ) and (Ne − 1)-electron channel functions ϒβ i . The latter represent an

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(Ne − 1)-electron parent ion state Φb multiplied by the angular part of the Ne -th (continuum) electron (Xlm ) and coupled to the spin (χ) of the latter electron, so as to leave the Ne -electron system in a state of well defined total spin S and z-component Σ. Hence, every ϒβ can be unambiguously identified by the channel index β = {b, l, m, S, Σ}. The utility of this approach becomes apparent upon observing that, among the terms appearing in equation 1, only φ (rNe ) extends beyond R0 , which is essential to impose the appropriate incoming-wave boundary conditions of continuum wave functions for the photoionization problem. All other terms can thus be computed by using the tools of quantum chemistry packages (QCP) by expressing both short-range ℵi and parent-ion Φb states in terms of orbitals represented by a basis of polycentric Gaussian functions (GPi (r)) centered at the atomic sites of the molecular system. In this work we have carried out state-average Restricted Active Space SCF (SA-RASSCF) calculations by using the cc-pVQZ 24 basis set, which contains 110 functions, allowing for all configurations in which the 1σg/u orbitals are doubly occupied, the 2σg/u , 3σg/u , 1πg/u orbitals are occupied by any physically admissible number of electrons, and the 4σg/u , 5σg/u , 6σg/u , 2πg/u , 3πg/u and 1δg/u orbitals contain at most two electrons (see Fig. 1a 1 for notations). The state average was performed by optimizing the neutral states X 1 Σ+ g , A Πu , 1 + 25 B1 Σ+ u and C Σu . The RASSCF orbitals were computed with MOLPRO, exploiting its capability

to perform a state average calculation over states of different symmetry. All subsequent RASSCF calculations were performed with MOLCAS. 26 To describe φi , we use a monocentric hybrid Gaussian/B-spline (GABS) 27 basis placed at the center of mass of the molecule, with the B-Splines (Bi (r)) being non-zero for radii > R0 and the monocentric Gaussians (GM i (r)) being nonzero for radii < R1 , such that R1 > R0 . The Gaussian 2

2k+l e−αi r , with part of the GABS basis contains a set of 22 even-tempered functions GM i (r) ∝ r

αi = α0 β i (α0 = 0.001, β = 1.46, i = 0, ..., 21), k ≤ 2 and l ≤ 3, which amounts to 1056 functions. The B-Spline part is composed of a set of 390 B-Splines of order 7 extending from R0 = 7.0 a.u. up to Rmax = 200 a.u.. The products in the second term of equation 1 were obtained by augmenting the parent ions Φb with orbitals expanded in the GPi (r), GM i (r) and Bi (r) functions. The radius R0 was chosen so that the polycentric Gaussian functions GPj effectively vanish in the region where

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the B-spline functions Bi (r) are defined (r > R0 ). This implies orthogonality between polycentric Gaussians and B-splines, i.e., hBi |GPj i = 0 (no effective overlap) for all i and j, thus drastically simplifying the calculation of operator matrix elements, 23 in particular, the Hamiltonian matrix H. Subsequently, physically relevant scattering states are constructed by fitting linear combinations of the eigenstates of H to the asymptotically correct expression of Ψ− αE enforcing photoionization boundary conditions. 23,27,28 In the present application, our close-coupling expansion includes all those channels that stem from the parent ions associated with the first three ionization thresholds (see Fig. 1). As a whole, the XCHEM approach allows us to exploit the flexibility and efficiency of QCPs by using polycentric Gaussians, and to use B-Splines for the long-range behavior of the continuum electron, all the while sidestepping many of the problems normally associated with the use of only Gaussians and/or B-Splines in molecular photoionization. Another interesting property is that, as the number of monocentric Gaussian functions is much larger than the number of polycentric Gaussian functions, and the monocentric basis used for N2 is also good enough to describe the wave function of the continuum electron in larger molecules, increasing the size of the system up to around ten atoms does not lead to a significant increase in computational effort. For more details on the GABs basis, mathematical intricacies of the XCHEM approach as well as benchmark results, we refer to Refs. 23, 27, 29 and references therein. Fig. 1 shows the energy positions and the ionization channels of N2 in the region of interest. The Hopfield series of autoionizing states lie above the A2 Πu and below the B2 Σ+ u ionization thresholds, and is reached by photons with energy between 17.1 and 18.7 eV (the ionization po1 tential of N2 is 15.6 eV). Due to the dipole selection rule, only states of 1 Σ+ u and Πu symmetries

can be populated. For each symmetry and ionization threshold, an infinite number of channels is open, but their relative importance decreases with the value of the angular momentum of the ejected electron. Fig. 1 only shows the most relevant channels. As can be seen, the Hopfield −1 −1 states of 1 Σ+ u symmetry have a dominant 2σu nsσg or 2σu ndσg character, and can decay to the 2 + ground state of N+ 2 , X Σg , by ejecting an electron to the ε pσu or the ε f σu continua accompanied

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Figure 1: (a) Schematic representation of processes yielding different cationic states by removal of an electron from different molecular orbitals. The number preceding the symmetry notation for a particular molecular orbital indicates ordering within that symmetry. (b) Channels included in the CC expansion: five channels leave the systems in states of 1 Σ+ u symmetry (red), and six 1 in states of Πu symmetry (blue). The series of autoionizing states of interest to this work lie between the A2 Πu and B2 Σ+ u ionic states (corresponding to removing an electron from the 1πu and 2σu molecular orbitals, respectively). These autoionizing states are indicated by two series of red (nsσg and ndσg ) and one series of blue (ndπg ) horizontal lines. Depending on the final symmetry of the system, these states may decay into either three or five open channels. In this panel, we have used the common united-atom limit notation introduced in Ref. 20 to label the excited electron. by de-excitation of one of the 3σg electrons to the 2σu orbital, or decay to the first excited state 2 of N+ 2 , A Πu , by ejecting an electron to the εdπg continuum accompanied by de-excitation of one

of the 1πu electrons to the 2σu orbital. Similarly, the states of 1 Πu symmetry have a dominant + 2σu−1 ndπg character and can decay to the X2 Σ+ g state of N2 , by ejecting an electron to the ε pπu

or the ε f πu continua accompanied by de-excitation of one of the 3σg electrons to the 2σu orbital, or decay to the A2 Πu state of N+ 2 , by ejecting an electron to the εsσg , εdσg , or εdδg continua accompanied by de-excitation of one of the 1πu electrons to the 2σu orbital. For completeness, the energy diagram also shows the expected positions of the resonances lying between the first and the second ionization thresholds (i.e., below 17.1 eV). These resonances will not be investigated in the present work.

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Figure 2: Total and partial photoionization cross sections of N2 . In both panels the black line corresponds to the total cross section and the colored lines to partial cross sections. Continuous lines: present results in length gauge. The dash-dotted and dotted lines are the theoretical results of Refs. 20 and 21, respectively. Panel (a) discriminates the cross section according to the symmetry of the Ne -electron system after ionization and clearly shows the relatively minor impact of the ndπg series on resonances. Panel (b) discriminates according to the (Ne − 1)-electron parent ion left behind upon ejection of an electron, showing that after ionization it is more likely to find the ion in the excited 2 Πu state than in the ionic ground state. This panel indicates the position of the lowest 3sσg , 3dσg and 3dπg resonances. For simplicity in the notation, in the text we have used the same index n for all resonances in the same group. Fig. 2 shows the calculated total and partial photoionization cross sections obtained in length gauge, together with the available theoretical results 20, 21. In Fig. 3, the calculated total cross 9



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Figure 3: Comparison of the total photoionization cross section obtained using the XCHEM approach with (a) the attosecond measurements of Reduzzi et al 7 and (b) the synchrotron radiation measurements of Gürtler et al, 13 Peatman et al, 30 Dehmer et al, 15 and Huber et al. 16 Each blue box "contains" three resonance features according to the three series of autoionizing states diagrammatically shown in Fig.1b, collectively labelled by n. sections are compared with the existing experimental results: the time-resolved measurements of Ref. 7 (panel a), the earlier synchrotron radiation measurements of Refs. 13, 15, 16, 30 (panel b). For a meaningful comparison with experiment, in all figures we have used the experimental ionization potential of N2 to define the energy scale and in Fig. 3 the theory curves have been convoluted with a Gaussian function of width 0.03 eV (panel a) and 0.015 eV (panel b) to account for the 10


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limited energy resolution reported in the corresponding experiments. It is important to emphasize that the comparison between the theoretical results and the synchrotron radiation data is performed on the absolute scale, i.e., no rescaling of the calculated cross section has been performed. For both total and partial cross sections, the gauge invariance is very good (as an illustration, Fig. 3a shows results obtained with both length and velocity gauges), thus suggesting that the basis set used in the calculations is almost complete. As can be seen, the agreement between our calculated total cross sections and the synchrotron radiation data is very good in the whole range of photon energies, both in magnitude and shape, except for the lowest 2σu−1 3dσg resonance lying just above the A2 Πu threshold at ∼ 17.17 eV, which is slightly broader and is shifted to higher energies in the theoretical results. This might be due to an incorrect description of the electronic continuum or to the breakdown of the adiabatic approximation just above the A2 Πu threshold, where the electron ejected into the 2σu−1 εdπg continuum is very slow and, therefore, has a velocity comparable to that of the nuclei. Interestingly, for this specific resonance, the agreement is slightly better when comparing with the time-resolved experimental results of Ref. 7 (see Fig. 3a). For all other resonances, including the other (n = 3) ones appearing at ∼ 17.39 eV (2σu−1 3sσg ) and ∼ 17.35 eV (2σu−1 3dπg ), the agreement between calculated and experimental synchrotron radiation data is remarkable. As can be seen in Fig. 2, the shapes of these resonance peaks significantly differ from those reported in previous theoretical works. In any case, the general good agreement between theory and experiment suggests that non-adiabatic effects should not play an important role here, in line with the absence of isotopic effects in the N2 photoelectron spectrum reported in the most recent experimental work. 31 1 The 1 Σ+ u and Πu contributions to the total cross sections shown in Fig. 3a correspond, up

to a factor, to the ionization cross sections of molecules oriented parallel and perpendicular to the polarization direction, respectively. 32,33 The analysis of these contributions reveals that almost all the features observed in the experimental spectra are exclusively due to resonances of 1 Σ+ u symmetry, namely the 2σu−1 nsσg and 2σu−1 ndσg ones. These resonances show up as pronounced asymmetric peaks and dips, respectively. The peaks associated to the 2pπ −1 ndπg resonances,

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which show up in the 1 Πu contribution, are very weak and have a characteristic asymmetric Fano profile. They are barely visible in the total photoionization cross sections and are well separated from each other. The partial cross sections of Fig. 2b show that, at these low photon energies, 2 photoionization of N2 leaves preferentially the N+ 2 cation in the A Πu excited state, instead of in

the X2 Σ+ g ground state. In other words, direct and resonant ionization from the HOMO-1 dominates over direct and resonant ionization from the HOMO, with a possibly significant contribution from shake-up processes. Indeed, the rather large width of the Hopfield states is indicative of a strong coupling between different ionization channels in N2 . From the calculated photoionization spectra, we have determined the energy positions and autoionization widths of the three series of Hopfield resonances. This was done by fitting the computationally obtained data to the analytical expression 34 that describes the behavior of the scattering phase in the vicinity of an autoionizing state, δ (E) = δb + tan−1

Γn , 2(E − En )

(2)

where En is the resonance position, Γn the resonance width and δb is a slowly varying background. The results thus obtained are shown in Fig. 4. The figure also includes the experimental values of the widths (i.e., the inverse of the lifetimes) determined in Refs 7, 10, 13, 16 for the 2σu−1 3sσg and 2σu−1 3dσg resonances. As in the atomic case, the energies and widths of the resonances of the three Hopfield series roughly scale as 1/n2 and 1/n3 , respectively. Our calculated widths for n ≥ 4 are in good agreement with those determined in the synchrotron radiation experiments of Refs 13, 16. For n = 3 the only available experimental data are those obtained from the timeresolved measurements of Refs. 7, 10. In this case, the calculated widths are significantly larger than the measured ones; however, their relative value (∼1.5 for the 3dσg and 3sσg resonances) is in good agreement with the experimentally measured ratio. Therefore, the main finding of the time-resolved experiment of Ref 7, which is the significant difference in the lifetimes of these two resonances, is confirmed by the present calculations. Furthermore, Fig. 4 shows that a similar

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Figure 4: (a) Energies of the three relevant series of autoionizing states, extracted from the XCHEM results as outlined in the text (nsσg in blue, ndσg in red, ndπg in green). Clearly visible is the approximate n−2 scaling behavior. (b) The widths of the autoionizing states of panel (a), displaying an approximate n−3 scaling behavior. The XCHEM results are compared to reference data from synchrotron radiation measurements by Gürtler et al (1977) 13 and Huber et al (1993), 16 as well as to the 2016 time-resolved results from Reduzzi et al 7 and Eckstein et al, 10 the only ones available for the 3sσg and 3dσg resonances.

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difference in magnitude is observed for all the members of the 2σu−1 nsσg and 2σu−1 ndσg series of resonances. To gain a deeper insight into the role of electron correlation, we have performed two sets of additional calculations for the 1 Σ+ u electronic continuum, in which various kinds of electron correlation are artificially removed, namely, (i) The interaction between the 2σu−1 nsσg and 2σu−1 ndσg resonances due to electron correlation is switched off. This has been done by performing calculations in which, all 2σu−1 nsσg configurations and all 2σu−1 ndσg configurations are alternatively removed, so that they do not mix with each other. In both cases, the multi-reference description of the cationic states as well as all the other inter-channel couplings are preserved as in the full calculation. (ii) The cationic states are described by using the smallest possible active space, which only includes the 1s, 2s and 2p atomic orbitals of nitrogen. Although small, this is still much better than adopting a Hartree-Fock or a single Slater determinant description of the cationic states, since each state is still described by about 150 configurations. Couplings between all open channels are preserved as in the full calculation. The resonance positions and widths resulting from these two sets of calculations are given in Table 1: Relative energy positions Ensσg − Endσg and autoionization widths Γnsσg and Γndσg for the 2σu−1 nsσg and 2σu−1 ndσg resonances of N2 resulting from three different sets of calculations: fully correlated (full), partly uncorrelated (i), and partly uncorrelated (ii). See text for details of the latter two calculations. n=3

full approx. (i) approx. (ii) n=4 full approx. (i) approx. (ii) n=5 full approx. (i) approx. (ii)

Ensσg − Endσg (eV) Γnsσg (meV) Γndσg (meV) 0.2175 98.3215 63.3144 0.0794 164.9940 43.4736 − 132.8978 − 0.1062 59.8804 27.2729 0.0701 78.3084 12.3347 0.1044 50.3598 28.5583 0.0567 28.4093 11.1197 0.0253 33.9670 5.2365 0.0539 22.7393 10.8079

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Table 1. Calculation (ii) is not able to reproduce the 2σu−1 3dσg resonance appearing just above threshold (at ∼ 17.17 eV), however the relative energies and widths of all remaining resonances are rather similar to those obtained from the full calculation. In contrast, resonance positions and widths resulting from calculation (i) are significantly different, in particular for the lowest states. As can be seen, for a given n, removing the coupling between the 2σu−1 nsσg and 2σu−1 ndσg autoionizing states leads to significantly smaller energy differences and to substantially different autoionization widths: one of the resonance widths becomes much smaller and the other one much larger. Hence, we conclude that accounting for the coupling between the 2σu−1 nsσg and 2σu−1 ndσg resonances due to electron correlation is crucial for an accurate evaluation of energy positions and widths. We have also found that the photoelectron spectra resulting from both (i) and (ii) are completely different from the actual one and from each other. The non resonant background is off by more than 100% and the resonance profiles bare no similarities with the observed ones. Also, gauge invariance is very poor: differences between gauges are as large as 100% or more. This shows that the tiniest imperfection in the description of electron correlation, either in the electronic continuum or in the states of the remaining cation, has a dramatic effect on the photoelectron spectrum, which is the consequence of the extreme sensitivity of dipole matrix elements to the quality of the wave function. The present analysis confirms once again that the main reason for the observed features is electron correlation, rather than non-adiabatic effects. In conclusion, we have used the recently developed XCHEM approach to study photoionization of the N2 molecule between the second and the third ionization thresholds, where the Hopfield series of autoionizing resonances show up. The XCHEM approach allows us to describe electron correlation for the electronic continuum at the same level of accuracy as multi-reference CI methods do for bound states. Our results show that non-adiabatic couplings play a minor role in this particular problem and that electron correlation is the main factor that determines the magnitude and shape of the N2 photoionization cross sections, as well as the lifetimes of the Hopfield resonances. These conclusions are supported by the good agreement between the calculated cross sections and those determined in early synchrotron radiation experiments and more recent attosec-

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ond measurements.

Acknowledgments This work has been supported by the ERC advanced grant 290853 - XCHEM - within the seventh framework programme of the European Union, the ERC proof-of-concept grant 780284 - ImagingXChem - within the Horizon2020 Framework Programme, and the MINECO projects FIS201342002-R and FIS2016-77889-R (AEI/FEDER, UE). We also acknowledge computer time from CCC-UAM and Marenostrum Supercomputer Centers. L.A. acknowledges support from the TAMOP NSF Grant No. 1607588, as well as UCF funding.

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(16) Huber, K. P.; Stark, G.; Ito, K. Rotational Structure in the Hopfield Series of N2 . J. Chem. Phys. 1993, 98, 4471–4477. (17) Bachau, H.; Cormier, E.; Decleva, P.; Hansen, J. E.; Martín, F. Applications of B-Splines in Atomic and Molecular Physics. Rep. Prog. Phys. 2001, 64, 1815–1943. (18) Toffoli, D.; Stener, M.; Fronzoni, G.; Decleva, P. Convergence of the Multicenter B-Spline DFT Approach for the Continuum. Chem. Phys. 2002, 276, 25–43. (19) Gozem, S.; Gunina, A. O.; Ichino, T.; Osborn, D. L.; Stanton, J. F.; Krylov, A. I. Photoelectron Wave Function in Photoionization: Plane Wave or Coulomb Wave? J. Phys. Chem. Lett. 2015, 6, 4532–4540. (20) Raoult, M.; Le Rouzo, H.; Raseev, G.; Brion, L. H. Ab Initio Approach to the Multichannel Quantum defect Calculation of the Electronic Autoionisation in the Hopfield Series of N2 . J. Phys. B-At Mol. Opt. 1983, 16, 4601. (21) Lucchese, R. R.; Zurales, R. W. Comparison of the Random-Phase Approximation with the Multichannel Frozen-Core Hartree-Fock Approximation for the Photoionization of N2 . Phys. Rev. A 1991, 44, 291–303. (22) Howell, R. W. Auger Processes in the 21st Century. Int. J. Rad. Biol. 2008, 84, 959. (23) Marante, C.; Klinker, M.; Corral, I.; González-Vázquez, J.; Argenti, L.; Martín, F. HybridBasis Close-Coupling Interface to Quantum Chemistry Packages for the Treatment of Ionization Problems. J. Chem. Theory Comput. 2017, 13, 499–514. (24) Dunning Jr, T. H. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007–1023. (25) Werner, H.-J.; Knowles, P. J.; Knizia, G.; Manby, F. R.; Schütz, M. Molpro: a GeneralPurpose Quantum Chemistry Program Package. Wires. Comput. Mol. Sci. 2012, 2, 242–253.

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100The Journal of Physical ChemistryPage Letters 20 of 23 LG Lucchese et al. (1991) n=6n=7 Raoult et. al. (1983) n=5 1 80 n=4 n=3 2 3 4 60 5 6 7 40 Total 1 u 8 9 10 20 11 12 1 u 13 0 b 14 100 15 LG Lucchese et al. (1991) 16

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80 21 Page TheofJournal 23n=3 of Physical Chemistry Letters (convoluted - length gauge) (convoluted - velocity gauge) 70 Reduzzi et al. (2016) 1 2 60 n=4 n=5 n=6 3 4 50 n=7 5 6 40 7 8 9 30 10 11 20 12 13 10 b 14 Huber et al. (1993) 15 80 Dehmer et al. (1984) 16 Gürtler et al. (1977) 17 70 Peatman et al. (1978) (convoluted) 18 19 60 20 21 50 22 23 40 24 25 30 26 27 28 20 29 ACS Paragon Plus Environment 30 10 17.5 18.0 18.5 31 Photon Energy (eV) 32

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nd g ns g np g Reduzzi et. al. (2016) Gürtler et al (1977) Eckstein et al (2016) Huber et. al. (1993) Reduzzi et. al. (2016) Gürtler et al (1977)

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