Influence of shape resonances on the angular dependence of molecular photoionization delays

Characterizing time delays in molecular photoionization as a function of the ejected electron emission direction relative to the orientation of the molecule and the light polarization axis provides unprecedented insights into the attosecond dynamics induced by extreme ultraviolet or X-ray one-photon absorption, including the role of electronic correlation and continuum resonant states. Here, we report completely resolved experimental and computational angular dependence of single-photon ionization delays in NO molecules across a shape resonance, relying on synchrotron radiation and time-independent ab initio calculations. The angle-dependent time delay variations of few hundreds of attoseconds, resulting from the interference of the resonant and non-resonant contributions to the dynamics of the ejected electron, are well described using a multichannel Fano model where the time delay of the resonant component is angle-independent. Comparing these results with the same resonance computed in e-NO+ scattering highlights the connection of photoionization delays with Wigner scattering time delays.


Supplementary Note 1: Experimental setup
The monochromatized synchrotron radiation has a bandwidth of around 2.5 meV in the employed energy range and is focused at the center of a COLTRIMS-type 3D electron-ion coincidence momentum spectrometer. This set-up, derived from earlier versions 1,2 , is optimized here to investigate dissociative photoionization of molecules which occurs at the intersection of the light beam and a supersonic molecular beam generated through a 70 µm nozzle and two skimmers (500 and 700 µm diameter). Ions and electrons, guided towards time and position sensitive delay line detectors (RoentDek DLD-80) by a set of variable electrostatic and magnetic fields, combined with an electrostatic lens focusing the trajectories of energetic ion fragments 3 , are collected over a 4π solid angle. In the study of dissociative photoionization of NO, the velocity vectors vN+ and ve of N + ions and photoelectrons are derived from the impact positions and time of flight at the detectors employing a multichannel time to digital converter for positions and ion time of flight and a synchronized time to amplitude converter for the electron time of flight 3 .

Supplementary Note 2: Extraction of experimental MFPADs
In this work, we investigate photoionization of NO( 2 Π) to the NO + ( 3 Π, v = 0) state, with the ionization potential IP=21.73 eV corresponding to ionization from the 4 inner valence molecular orbital, dissociating to the N + ( 3 P) + O( 3 P) limit at 21.03 eV relative to the NO( 2 Π, v = 0) origin. In brief, the coincident events (N + ,e) corresponding to this DPI process were selected from the electron-ion kinetic-energy correlation histogram, and the MFPAD (̂, Ω) = ( , , , ) was constructed from the (vN+, ve) ion and photoelectron velocity vectors, as described in 4 and references therein, within the axial-recoil approximation 5 . ̂≡ ( , ) defines the emission direction in the molecular frame with and being the polar and azimuthal angles of ve relative to vN+. The Euler angles Ω ≡ ( , , ) with the polar and azimuthal angle of vN+ relative to the light quantization axis ê and , define the orientation of the molecular axis, i.e., the rotation of the molecular frame into the field frame (FF). The MFPADs are expanded using the FLN-function formalism developed previously 6,7 . For dissociative photoionization of linear molecules induced by circularly polarized light, within the dipole approximation, the ( , , ) MFPAD is determined by five one-dimensional ( ) functions according to the general expression 7 : ± ( , , ) = 00 ( ) − where are the associated Legendre polynomials. The functions and their statistical uncertainties extracted from the fit of the measured ( , , ) distribution according to Eq. (S1) enable us to reconstruct the MF emission diagram ( , ) for any orientation of the molecular axis relative to the quantization axis of circularly or linearly polarized light. In particular, for a molecular orientation parallel to the polarization of linearly polarized light considered to characterize the role of the 4 → k* shape resonance, the MFPAD is fully described with =0°( ) = 00 ( ) + 20 ( ). (S2) and, e.g., for a molecular perpendicular orientation it writes as The photoionization amplitude can be written as where Ψ̂( −) is the wave function of the final state for the emitted photoelectron in the MF with a kinetic energy , Ψ 0 is the initial state, ⃗ the dipole operator, → is the electric field. In the dipole approximation, with circularly polarized light propagating in the ̂F F direction and helicity ℎ = ±1, or with light linearly polarized in the ̂F F ≡ ( , ) direction, for which we take ℎ = 0, the corresponding PDA, can be written as where ,ℎ (1) (Ω) is a rotation matrix and = √ 4 3 1, ( , ) is the dipole operator with being the projected angular momentum of the photon on the molecular axis. The N electron final scattering state including the ( − 1) ion state and the photoelectron is currently expanded in partial waves as where the are the Coulomb scattering phases, = arg Γ ( + 1 − ), and Δ = − 0 . Note that 0 is singular as → 0 and behaves as where k is the photoelectron momentum = ( 1 2 ) 2 , but the Δ are all finite at k=0. The dipole matrix element for each partial wave (DMEs) is defined as where , , is the real-valued magnitude of the DME and , , is its phase. Combining Eq. (S6-S9) leads to the expression of the PDA as given in Eq.
(2) of the main paper: This provides then the one-photon ionization time delay (̂, Ω, ) as the energy derivative of its argument (̂, Ω, ):

Supplementary Note 4: Extraction of experimental DMEs from measured MFPADs
The determination of the partial-wave dipole matrix elements , , relies on the expansion of the five FLN functions in Legendre polynomials: where the ′ coefficients are expressed in terms of the DMEs as Note that in the studied reaction, to a very good approximation, only = DMEs contribute to the PDA, so we drop the  label in the following.
For each photon energy, a non-linear least-squares fit of the experimental data, as described in 6,7 , is performed giving access to the magnitudes , and phases  , of the one-photon ionization dipole matrix elements. The experimental ( ) functions (Eq. (S12)) were used to generate a set of MFPAD data for a range of angles, with the light quantization axis being parallel, perpendicular, and at the magic angle with respect to the molecular axis, and with both linear and circular polarization, for a total of 113 data points that were fit at each photon energy. The parameter space of possible dipole matrix elements was sampled, usually with 10,000 random initial parameter sets, with the only constraint being that the total cross section was the same as that of the data being fit. The parameters at the nearest local minimum of the sum of the squares of the deviations, starting from each initial parameter set, were then found using the Levenberg-Marquardt algorithm. For the studied energies, a single local minimum, or two local minima corresponding to a similar quality of the fits, were found, which were significantly better than all the other fits 7 . In the presence of two solutions, the selection was based on a criterion of closest resemblance with the calculation. The statistical uncertainties of the DMEs were deduced from the root-mean-square (rms) deviation of the fit.
The MFPADs measured at different excitation energies were normalized relative to the total cross section for PI into the NO + ( 3 Π) ionic state 8,9 : here we chose to calibrate the measured C000 Legendre polynomial coefficients to those obtained in the reported calculations. On the other hand, we have used the phase of a selected partial-wave DME, here the  2,1 phase of the (d) partial wave, which is not affected by the shape resonance (sec. III.), as a phase reference for all DMEs at the different excitation energies, such that ̃= − 21 . Choosing such a common reference establishes the desired consistency between the different energies. It also removes the low energy divergence of the Coulomb phase from the phase of the PDAs. The effect of using such relative phases is discussed further in Supplementary Note 5. We note that, if desired, the part of the Coulomb phase that was subtracted with the reference phase could be reintroduced by multiplying all DMEs by exp( =2 ) (see Eq. (S9)). Fig. S1 shows the experimental and computed DME magnitudes and phases as a function of the electron/photon energy. Relying on the smooth dependence of the DMEs across a broad shape resonance (see also Fig. S1 b,d for the theory), a spline fit of the experimental data was used to obtain the PDAs on a finer energy grid. To improve the fit at the boundaries of the eleven main data points, additional data points recorded at 22.5 eV and 48.4 eV were used.  1)). Error bars (two standard deviations) feature the statistical uncertainties in extracting the DME magnitudes and phases. The full and dashed lines in a and c are spline fits to the experimental data. The phases ̃ are relative to the phase 21 . 6 The magnitudes of the ( , 0) ≡ ( ) DMEs, describing the parallel transition, dominate the explored energies, with a characteristic strong contribution of the ( ) partial wave from the 4 → * shape resonance clearly identified both in the measured and computed DMEs. The ( ) and ( ) partial waves are also found to take part in the resonance for the measured MFPADs. All extracted phases are referenced to the phase of the ( ) partial wave, which is not affected by the resonance and displays a slow variation with the energy (see Fig. S2): ̃= − 21 . The ̃ phases do not include the low energy singularity which comes from the long-range Coulomb potential.

Supplementary Note 5: Effect of using relative phases
In Fig. S2, we give the computed absolute DME phases including the Coulomb phases for the photoionization of NO to the Π 3 state of NO + . The effect of the Coulomb phase is most noticeable at low energies. The non-Coulomb phase of the (l,m)=(2,1) DME, whose magnitude dominates the perpendicular transition unaffected by the * shape resonance (cf. Fig. S1), displays a weak variation in the studied energy range. It is chosen as a reference for the phases of all (l,m) DMEs providing the ̃= − 21 phases reported in Fig. S1c and d. This subtraction removes the divergence in the Coulomb phase at low energy from the phase of each partial-wave DME.  Taking such a reference thus removes the rapid rise of the Coulomb phase at low electron energies and the resulting positive contribution to the time delays, highlighting specifically the influence of the scattering from the short-range molecular potential and the dipole coupling from which the electron is removed. As stated in Note 4, the standard contribution of the Coulomb phase can be added back in analytically if desired (cf. Eq. (S9)). The DMEs reported in Fig. S1 providing the theoretical PDAs and photoionization time delays presented in Fig. 2 and Fig. S4 for photoionization of NO( 2 Π) to the 3 Π state of NO + , corresponding to ionization from the 4 inner valence molecular orbital, were computed using the multichannel Schwinger configuration interaction method as described in 4 . Briefly, the calculations utilized a singlecenter expansion, with max = 60, and the ion and initial states were given by complete active space configuration interaction wave functions with 8 active orbitals (four orbitals and four orbitals) obtained from a valence complete active space self-consistent field calculation on the ground state of the molecule. They were performed at a fixed inter-nuclear distance of 1.15077 Å 10 (experimental re) with an augmented correlation-consistent polarized valence triple-zeta basis 11,12 to represent the bound orbitals. The photoionization calculation 8,13 used a close-coupling expansion of the photoionized state, including ten open channels with ion states of vertical ionization potentials near or below the ionization potential of the 3 Π state. The states included were the 1 Σ + , 3 Σ + , 3 Π, 3 Δ, ′ 3 Σ − , ′ 1 Σ − , 1 Π, 1 Δ, 3 Π, and 1 Π states. These calculations successfully describe the role of electronic correlations in photoionization of NO leading to the 3 Π state, as well as the 4 → k* shape resonance which affects the parallel component of the transition 7 . In order to extract the resonant scattering time delay coming only from the shape resonance in the c 3  channel, we used a one-channel scattering calculation restricted to the c 3  state of NO + and to the  symmetry. The boundary conditions in the scattering calculation were for the (+) scattering state (i.e. normal scattering state) which has an incoming Coulomb wave with asymptotic momentum ⃗⃗ and outgoing spherical scattered Coulomb waves. The number of partial waves was the same as in the photoionization calculations but the amplitudes were given by the scattering S-matrix and not by the photoionization transition amplitudes. Indeed, when using the same ten-state wave function expansion for the e-ion scattering calculation as was used in the photoionization calculation, there were contributions from a number of overlapping broad resonances coming from scattering from the different ion states included. This led to an eigenphase sum from which it was difficult to extract the time delay just from the resonance from scattering from c 3  state of NO +. The eigenphase sums from the single channel calculations as a function of scattering energy and half of the corresponding Wigner time delay are presented in Fig. 5. Fig. S4 is a complementary representation of the 2D polar plot in Fig. 2 showing the magnitude, phase, and time delay along different emission directions for experiment and theory, respectively. represent the PDA magnitudes, panels c and d the PDA phases using the 2,1 phase as a reference for the DME phases, and panels e and f the corresponding photoionization time delays. In (a, c, e) the thickness of the lines features the statistical uncertainties of the PDA magnitudes and phases, and that of the photoionization time-delays, deduced from those characterizing the magnitudes and phases of the DMEs in Fig. S1.

Supplementary Note 8: Comportments of the PDA curves in the complex plane in the Fano model
To illustrate some typical behaviors of the PDA and resulting time delays in the multichannel Fano model, we combine the expressions of the PDA summed over partial waves in Eq. (3) and that of each DME given in Eq. (5) with the additional assumption that the non-resonant DMEs, where Δ = − tan −1 ( 1 ) is the scattering phase shift due to the short-range, non-Coulomb part of the electron-molecule interaction potential in the pure Breit-Wigner form for resonant scattering. Eq. (S15) shows that the phase change in the PDA comes from both the exponential term, Δ , and the term (− sin Δ + cos Δ) which is due to the interference between the resonant and non-resonant parts of the PDA. The exponential term by itself forms a curve that is a half-circle as goes from −∞ → +∞ so that its phase increases by . Multiplying the exponential term by the interference term closes this curve to become a circle. The PDA then always starts and ends with the same phase if the infinite range of is considered, so that its phase will change by 0 or 2 depending on the radius of the circle and the location of its center. Examples for different parameters are displayed in Fig. S5  The value of the phase of the PDA with the same set of 0 values. c The energy derivative of the phases expressed as time delays. Note that in this example, since the energies are unitless quantities, the corresponding time delays are also unitless.
For the studied shape resonance in photoionization of NO, with res 8.5eV and Γ7eV, the range of energies considered here is limited to about two times the width of the resonance. This leads to PDA phase changes ranging from 0 to 2 . The major photoemission process around ( = 180°) is a clear illustration of the model, with a time delay displaying both positive and negative values, as reported in Fig. 2c and Fig. S4e and f, and a net PDA phase change close to zero. The phase change in the PDA for 60° is closer to . This behavior results from the nearly zero non-resonant PDA at 60°, as shown in Fig.  4b. The peak in the computed photoionization time delay at 60° (202 as), seen in Fig. 2c and Fig. S4f, is also very close to the time delay of the resonant PDA (237 as) reported in Fig. 4e.