Basic studies toward ultrafast soft x-ray photoelectron diffraction; its application to probing local structure in iodobenzene molecules

Ultrafast x-ray photoelectron diffraction (UXPD) for free molecules has a promising potential to probe the local structures of the molecules in an element-specific fashion. Our UXPD scheme consists of three steps: (1) near-infrared laser (NIR) with ns pulse duration aligns sample molecules, (2) ultra-violet laser with fs pulse duration pumps the aligned molecules, and (3) soft x-ray free-electron laser (SXFEL) with fs pulse duration probes the molecules by measuring x-ray photoelectron diffraction (XPD) profiles. Employing steps of (1) and (3), we have measured I 3d XPD profiles from ground state iodobenzene aligned by the NIR laser with the SXFEL. Then, we have intensively calculated I 3d XPD profiles with density functional theory, taking degrees of alignments of the molecules into account, to extract a distance between C and I atoms in iodobenzene from the experimental I 3d XPD profiles. Although we have failed to determine the distance from the comparison between the experimental and theoretical results, we have succeeded in concluding that the degeneracies of the initial state eliminate the sensitivity on molecular structure in the I 3d XPD profiles. Thus, the observation of fine structures in the XPD profiles could be expected, if a nondegenerate molecular orbital is selected for a probe of UXPD. Finally, we have summarized our criteria to perform UXPD successfully: (1) to use SXFEL, (2) to prepare sample molecules with the degree of alignment higher than 0.8, and (3) to select a photoemission process from a nondegenerate inner-shell orbital of sample molecules.


I. INTRODUCTION
Photoelectron angular distributions (PADs) from isolated molecules have been measured for an increasing variety of purposes, including examining details of electron correlation and photoemission dynamics, demonstrating the electron diffraction as a structural probe of single molecules and probing time-resolved photochemical reactions (Refs. 1-5 and references therein). For such PAD studies, we found that molecular frame photoelectron angular distributions (MFPADs) for the core-shell photoelectrons, with an energy of >100 eV, are less influenced by details of molecular potentials; thus, the MFPADs are well described using an x-ray photoelectron diffraction (XPD) picture. 6 Theoretically, photoelectron diffraction is a consequence of interference between a directly emitted photoelectron wave and elastically scattered waves by surrounding atoms. Thus, the richly structured MFPAD or XPD profiles might provide a means of determining molecular structure in an element-specific fashion. Surface structural analysis based on the XPD theory has been widely developed. [7][8][9][10][11] Moreover, several methodologies for the structural analysis have been applied, such as the trial-and-error method, 12 the real-space triangular method, 13 and the holography 14 in the surface science field. In the latter two methods, an XPD profile in momentum space is inversely transformed into an atomic position map in real space via Fourier transform with some corrections and modifications. Nevertheless, these ambitious methods are not applicable to the XPD profiles from the isolated molecules. This inapplicability is because the scattering amplitudes are not isotropic, and phase shifts of the amplitudes are strongly dependent on the scattering processes. Hence, we adopted the trial-and-error method to determine structures of molecules in the electronic ground state from our experimental XPD profiles. 15 Then, for linear CO 2 , bent NO 2 , planar BF 3 , and prolate symmetric top CH 3 F molecules, our method confirmed that bond lengths and angles can be determined with a resolution of less than 0.1 Å and 10 , respectively. Therefore, on the basis of these achievements, we propose an ultrafast XPD (UXPD) scheme using soft x-ray free-electron lasers (SXFELs) to observe experimentally ultrafast rather simple photochemical reactions, e.g., dissociations, eliminations, and isomerization. Meanwhile, with the advent of femtosecond lasers and x-ray free-electron lasers (XFELs), monitoring ultrafast molecular transformations, such as chemical reactions and phase transitions, has been possible for a decade. 16,17 Electron and x-ray diffraction are the two principal experimental probes for molecular structure determination. Moreover, UXPD has been suggested as a promising probe by a few teams, including ours, for having the element-specific fashion. Therefore, several proposals 15,[18][19][20][21] and test experiments [22][23][24][25][26] for the UXPD have been published. However, results on the transient structure of molecules during photochemical reactions have not yet been reported, except for the recent paper 27 on the breakup process in O 2 molecules, applying SXFEL-pump and SXFEL-probe measurements. Meanwhile, we have extracted structure of the I 2 molecules in the electronic ground state using the SXFEL probe. 26 The ultrafast dynamics of O 2 molecules have been obtained through both triple coincidence and highrepetition SXFEL pulse capabilities. 27 However, triple coincidence rates are extremely low compared with single photoelectron detection rates, and the processing of triple coincidence data is cumbersome compared with direct measurement of a single photoelectron momentum image. Therefore, the outline of our UXPD scheme is as follows: (1) a near-infrared (NIR) laser pulse with ns duration aligns sample molecules, (2) an optical pulse with fs duration excites aligned sample molecules, and (3) SXFEL with fs pulse width probes the molecules by measuring 2D x-ray photoelectron image with velocity map imaging (VMI) spectrometer. 25,26 In this paper, as a feasibility study toward the UXPD, we report I 3d XPD profiles from iodobenzene molecules (IPh) aligned with a Nd:YAG laser. These molecules were obtained using SXFEL pulses from PAL-XFEL. 28,29 The benefits of using SXFEL are as follows: (1) the photoionization cross-sections for soft x-rays <1 keV are two orders of magnitude larger than those for hard x-rays >4 keV (Ref. 30) and (2) the bandpass of SXFEL is one order of magnitude smaller than that of XFEL. The measured I 3d XPD profiles for the aligned IPh molecules were compared with those via density functional theory (DFT) calculations. On the basis of the intensive comparison between these experimental and theoretical results, we propose the most favorable experimental conditions to realize the UXPD, which will be able to capture the transient structure of molecules during photochemical reactions. We also discuss specific applicability of UXPD for tracking photodissociation under the currently achievable experimental conditions.

II. EXPERIMENT A. Experimental setup and procedure
The experiments were conducted at the soft x-ray scattering and spectroscopy (SSS) beamline of PAL-XFEL. 31 SXFEL pulses, with the energy of $100 lJ/pulse and duration of $50 fs, were fired into the interaction region of our diffractometer comprising the VMI spectrometer, which has been previously introduced in Refs. 25 and 26 and is schematically shown in Fig. 1. The typical bandwidth of an SXFEL is $0.5%, i.e., $4 eV at the photon energy of 750 eV. The injectionseeded Nd:YAG laser pulses (Spectra Physics, PRO 230-50), with the energy of $700 mJ/pulse and duration of $10 ns, were focused by a plano-convex lens placed outside the vacuum chamber. SXFEL and Nd:YAG laser beams were merged collinearly by a holey mirror inside the vacuum chamber. The spatial overlap of the SXFEL and Nd:YAG laser beams was first examined by monitoring images on a Ce:YAG phosphor screen in the interaction region, and then, the spatial overlap was confirmed by monitoring the degree of alignment of the sample molecules. The typical spot size of SXFEL was $60 and $80 lm in the y-and z-directions for its 1/e 2 values, respectively, as shown in Fig. 1. The spot size of the Nd:YAG laser is $60 lm in the full width at half maximum with the Gaussian distribution. The temporal overlap between the SXFEL and Nd:YAG laser pulses was monitored using a fast photodiode. The polarization vectors of both the SXFEL and Nd:YAG laser beams were parallel to the z-direction.
Transistor-transistor logic (TTL) signals delivered from the PAL-XFEL facility, which fires XFEL pulses with the repetition rate of 30 Hz, were used as a master trigger for the present timing measurements. The master trigger was guided to the pulse generator (Quantum Composers, Model 9250) for the SXFEL pulses to synchronize with the Nd:YAG laser pulses, the pulse valve for molecular beam, and the shutter of the sCMOS cameras. The repetition rates were 30 Hz for both the Nd:YAG laser and sCMOS cameras and 15 Hz for the pulse valve.
A pulsed supersonic molecular beam was formed by expanding a gas mixture of the sample IPh molecules diluted in 40 bar helium through a pulsed valve, which is developed by Even and Lavie, 32 into the vacuum chamber. The pulsed valve was heated to 60 C to provide a partial pressure of $1 kPa for the sample IPh molecules. The molecular beam passed through a 3-mm-diameter skimmer, and it was introduced into the interaction region, where the Nd:YAG laser and SXFEL pulses overlapped. The source and main chambers were differentially pumped by turbo-molecular pumps, and their typical pressures during the experiments were 3 Â 10 À4 and 2.5 Â 10 À6 Pa, respectively. The pulse duration of the valve was settled to 22 ls by monitoring the pressure of the source chamber. The timing of the valve was optimized by observing the ion signals of the VMI.
The photoions and photoelectrons produced by the SXFEL pulses were measured using the faced VMIs. 25,26 The velocity-focused photoelectrons were detected by a chevron-stacked dual microchannel plate Structural Dynamics ARTICLE scitation.org/journal/sdy (MCP) backed by a phosphor screen. The image on the phosphor screen was recorded shot by shot using the sCMOS camera, and it is stored on the PC. The photoelectron 2D momentum images obtained with and without the sample gas were alternately measured, and the objective images that originated from the residual-free sample gas were then obtained by subtracting the latter from the former. Both the photoelectron detection and the fragment photoions were accelerated simultaneously toward the other VMI, and they were detected by the same system. We acquired a total momentum image data of 50 000 and 20 000 shots for photoelectrons and fragment ions, respectively.

Degree of alignment of iodobenzene molecules
By absorbing an x-ray photon of 750 eV, IPh molecules are multiply ionized via Auger decays, and then, fragment photoions are produced by the Coulomb explosion. Among various fragment photoions, we selected I þ ions by applying the pulse-gate voltage to the MCP of the VMI because they dissociate along the C-I axis of the IPh molecules. Hence, on the basis of the measured angular distributions of I þ ions, we evaluated the degree of alignment for the polarization vector of the Nd:YAG laser. Figures 2(a) and 2(b) show the I þ ion images of IPh generated by SXFEL pulses without the alignment Nd:YAG laser and with the Nd:YAG laser, respectively. The images are presented in the laboratory frame of reference, whose z-and x-axis are the polarization vector of the Nd:YAG laser and its propagation direction, respectively, as shown in Fig. 1. In the IPh molecular ensembles, which are aligned adiabatically by the electric fields of the Nd:YAG laser, the "head vs tail" distinction is lost, and the distribution is axially symmetric for the C-I axis. That is, the angular distributions of I þ ions are described by only the mutual angle, H, between the C-I axis and the polarization vector of the Nd:YAG laser. Hence, the raw data in the quad screen of the image of upper and lower sides and those of left and right sides were averaged in the figures. With this average procedure, the position-dependent detection efficiencies of the MCP can be removed.
Without the alignment Nd:YAG laser, the I þ ions exhibit an isotropic distribution because the C-I axis of IPh is randomly oriented. Indeed, the degree of alignment, cos 2 H h i ¼ 0:33, has been evaluated from the angular distribution of I þ ions shown in Fig. 2(c). With the alignment Nd:YAG laser, the angular distribution is confined along the z-axis, i.e., the C-I axis of IPh is aligned preferentially along the zaxis. To determine the degree of alignment of the IPh molecular ensembles, we simulated the angular distributions using the rotational temperature of molecules and the peak intensity of the Nd:YAG laser pulses, following Evaluation of degree of molecular alignment in the Methods section of Ref. 26. In the present case, as IPh is an asymmetry top rotor, the polarizability anisotropy of it is approximated by subtracting the average of the two lower components from the largest one, which is parallel to the C-I axis of IPh. Then, the simulation with the rotational temperature of 8 K and peak intensity of 7 Â 10 11 W/cm 2 resulted in the angular distribution of f ðHÞ ¼ 0:0197 þ 0:0617P 2 þ 0:0457P 4 þ 0:0219P 6 þ 0:009P 8 þ 0:004P 10 þ 0:002P 12 ; (1) where P n is the nth Legendre polynomial. Equation (1)    show 2D I 3d photoelectron images of IPh without and with the Nd:YAG laser, respectively, in the laboratory frame. These images were measured with SXFEL pulses with a photon energy of 750 eV. In the figures, the raw data in the quad screen of the image of upper and lower sides and those of left and right sides were averaged because of the symmetry restriction. The inner and outer rings around the radius of 2.5 a.u. correspond to I 3d 3/2 and I 3d 5/2 photoelectrons, respectively. Although the degrees of alignment of the sample IPh molecular ensembles are quite different, as shown in Fig. 2, the differences between the I 3d photoelectron images from the randomly oriented ensembles and that from the aligned molecular ensembles are barely discernible. To discuss more details about the differences, the polar plots relevant to Figs. 3(a) and 3(b) with the current theoretical results are shown in Figs. 3(c) and 3(d), respectively. The polar plots were obtained from the outer ring image since we confirmed that there are no detectable differences between the polar plots from the outer and inner ring images. Compared with I 3d laboratory frame photoelectron angular distribution (LFPAD) from the randomly oriented IPh (hereafter LFPAD 3d random ), I 3d laboratory frame photoelectron angular distribution from the aligned IPh (hereafter LFPAD 3d align ) exhibits a slightly preferential direction along the polarization vectors of both the alignment Nd:YAG laser and XFEL, as shown in Figs. 3(c) and 3(d). Both the comparison between the experiment and theory and the reason why the LFPAD 3d align has such a structureless simple profile are explained in Sec. IV B.
The photoelectron spectra obtained from Figs. 3(a) and 3(b) are shown in Fig. 4, where the spin-orbit splitting of I 3d sub-shells is Structural Dynamics ARTICLE scitation.org/journal/sdy resolved. The 3d photoelectron spectrum of Xe, which was reported in Ref. 33, is also depicted in Fig. 4 as a reference since the photoelectrons of IPh were measured under the same VMI conditions and photon energy as those for Xe. Therefore, referring to the ionization potentials (IPs) of Xe 3d 5/2,3/2 in Ref. 34, we determined the IPs of IPh: 632.64 6 0.8 and 643.41 6 0.8 eV for the I 3d 5/2 and I 3d 3/2 , respectively. On the one hand, the relativistic calculations with Grasp92 þ RATIP 35 gave the values of 628.540 and 639.938 eV for j ¼ 2 I 3d 5/2 5p 3/2 and j ¼ 2 I 3d 3/2 5p 3/2 , respectively. As shown in Fig. 4, the spectra exhibit the different spectral widths. The bandwidth of 8.31 eV for IPh is broader than that of 6.90 eV for Xe due to the vibrational broadening in the IPh molecules. Furthermore, there is a slight change of spectral widths, i.e., from 8.31 to 10.09 eV for I 3d 5/2 without and with the Nd:YAG lasers. This broadening in the IPh molecules is due to the above-threshold ionization (ATI) induced by the Nd:YAG laser fields. We have estimated less than three sidebands due to ATI under the present experimental conditions, referring to Ref. 36. On the other hand, the ponderomotive energy is estimated as 0.074 eV from the Nd:YAG laser intensity of 0.7 TW/cm 2 . 36 On the basis of these considerations, we think that the angular distributions of the I 3d photoelectrons are unaffected by the electric fields of the Nd:YAG laser.

III. FUNCTIONAL FORM OF PHOTOELECTRON ANGULAR DISTRIBUTION FROM ALIGNED MOLECULES A. Molecular frame photoelectron angular distribution
The ground state electronic density of IPh and the continuum wave functions have been calculated with the nonrelativistic DFT, thereby employing the LB94 exchange correlation functional. 37 On the basis of the electron diffraction and microwave spectroscopy data, the equilibrium molecular structure of IPh was determined. 38 IPh belongs to the C 2v point group; hence, the nomenclature of the molecular orbitals is given by the conventional definition in the molecular frame (MF) of reference, where the Z-axis represents the principal symmetry axis of the C-I and the Y-axis is orthogonal to the Z-axis and on the plane of the phenyl ring. In the nomenclature for C 2v , the iodine 3d atomic orbitals (3d z 2 , 3d xz , 3d yz , 3d x 2 Ày 2 , and 3d xy ) of IPh are related to the 6a 1 , 3b 1 , 3b 2 , 1a 2 , and 7a 1 lone-pair molecular orbitals, respectively, as shown in Fig. 5. The initial electron density is taken from a conventional bound-state LCAO-DFT calculation, with the program Amsterdam Density Functional (ADF). 39 The computations have then been performed, applying a basis set of multicenter B-spline functions that are centered on all the nuclei. 40,41 The origin is placed on the C1 nucleus of IPh. The maximum angular momentum of the spherical wave expansion of the continuum is chosen as l max ¼ 18. The order of where u i is the Kohn-Sham orbital that is currently ionized and u plÀ klh is the continuum orbitals normalized to incoming wave S-matrix boundary conditions. In u plÀ klh , p is the irreducible representation, l is its subspecies in the case of degeneracy, l is the angular momentum, and h is an index that identifies independent contribution when p and l are the same. These objects are related to the unitary transformation of spherical harmonics (Y lm ) into real symmetry adapted spherical harmonics (X The functional form of MFPAD is expressed by the general treatment of Refs. 43 and 44: where X ¼ (U, H, X) represents the Euler angles that define the lab frame (LF) with respect to the molecular frame (MF),k ¼ðh; /Þ expresses the direction of the photoelectron momentum in the MF, a is the fine structure constant, x is the incident photon energy, n i is the occupation number of the ionized orbital, and m p is the polarization quantum number: 0 and þ1 or À1 for linear and left or right circular polarization, respectively. The A LM 0 coefficients, which appear in Eq. (5), are calculated from dipole matrix elements and phase shifts as follows:

B. Laboratory frame photoelectron angular distribution from aligned molecules
The spherical harmonics Y LM 0 ðkÞ in the MF are expressed by the relevant function Y LM ðk 0 Þ in the LF: By substituting Eq. (7) into Eq. (5) and then the following formulas of the product of rotation matrices, we have The MFPAD of Eq. (5) is rewritten in the LF as follows: Here, we consider the molecular-axis distributions of the aligned molecular ensemble, which have been mentioned in Sec. II B 1. Such distributions are expressed generally as follows: 45 f ðX À1 Þ ¼ X K¼even F K P K ðcos ðÀHÞÞ: Therefore, once Eq. (10) has been multiplied by the distribution function of Eq. (11) and then integrated over X, applying the relation between the rotational matrix and the Legendre polynomial P K ðcos ðÀHÞÞ ¼ R KÃ 00 ðX À1 Þ and the orthogonality of the rotational matrices, LFPAD 3d align considering the distribution is written as follows: where the expansion coefficient hA L i is given by and h 0 stands for the polar angle of the photoelectron momentum in the LF. 46,47 For the aligned molecular ensembles, X ¼ (U, H, X) reduces to X ¼ (0, H, 0); thus, LFPAD 3d align depends on the two parameters of ðH ; h 0 Þ. When the polar angle H is fixed at a certain value H 0 , the distribution function is expressed by the delta function of and LFPAD 3d H0 ð h 0 Þ is written by LFPAD 3d H0 ðh 0 Þ ¼ ð LFPAD 3d align ðH; h 0 Þdðcos H À cos H 0 Þ sin HdH: The functional form LFPAD 3d H0 ðh 0 Þ can be obtained by inserting

IV. COMPUTED RESULTS A. MFPAD
As mentioned in the introduction, the MFPAD for the innershell photoelectrons having energies of >100 eV can be interpreted as an XPD profile. The XPD profile is formed by the direct photoemission wave, the singly scattered wave, and the interference between them. The molecular structure is reflected in the XPD profile mainly through the interference terms, which are inversely proportional to the internuclear distances. 48 Therefore, the MFPAD reflects the molecular geometry, when the x-ray is polarized in the direction from the electron emitter to the scattering site, and the molecular orbital, which is ionized, has large distribution in this direction.
In Fig. 6, the MFPADs from the nearly degenerate 6a 1 , 3b 1 , 3b 2 , 1a 2 , and 7a 1 orbitals of IPh are shown on the YZ plane. On this plane, the polarization vectors of both the SXFEL and Nd:YAG laser are parallel to the C-I axis. However, on this plane, the MFPADs from the 1a 2 and 3b 1 orbitals are 0 because both the orbitals have nodes on the plane, as shown in Fig. 5. The scales of the plots reflect the relative values of the differential cross-sections from the 6a 1 , 7a 1 , and 3b 2 orbitals and their sum. The shape of the MFPAD from the 6a 1 orbital is Structural Dynamics ARTICLE scitation.org/journal/sdy basically like f Z 3 wave, although the MFPAD exhibits the backwardscattering effect by the phenyl ring. The MFPAD from the 7a 1 orbital barely exhibits a scattering effect because the 7a 1 orbital is strongly distributed in the direction perpendicular to the C-I axis, as shown in Fig. 5. The MFPAD from the 3b 2 orbital exhibits the forward-focusing effect 49 by the phenyl ring because the 3b 2 orbital has a relatively large distribution in the direction toward the C2 and C6 atoms. Moreover, the sum of MFPADs is characterized by the backward-scattering and forward-focusing effects by the phenyl ring.

B. LFPAD from aligned iodobenzene
The calculated total ionization cross sections r from the 6a 1 , 3b 1 , 3b 2 , 1a 2 , and 7a 1 orbitals are 0.5206, 0.5264, 0.5247, 0.5221, and 0.5224 MB, respectively. The calculated asymmetry parameter b for the relevant five orbitals is 0.2665, 0.5995, 0.4234, 0.5419, and 0.5217, respectively. Hence, the calculated total b parameter for the I 3d orbitals of IPh is 0.47, which expresses the profile of LFPAD 3d random . Although the experimental b parameter is 0.38 6 0.014, the experimental and theoretical profiles of LFPAD 3d random are similar to each other, as shown in Fig. 3(c). Figure 7 shows the calculated LFPAD 3d align on the zx plane from the nearly degenerate 6a 1 , 3b 1 , 3b 2 , 1a 2 , and 7a 1 orbitals of IPh (hereafter LFPAD p align , where p ¼ 6a 1 , 3b 1 , …) and their sum (hereafter LFPAD sum align ). In contrast to MFPAD, ionizations from both the 3b 1 and 1a 2 orbitals contribute to the LFPAD 3d align . In the calculations, we used the experimentally determined value of the degree of alignment for the IPh molecules, cos 2 H h i ¼ 0:78, where the values of F K substituted in Eq. (13) are the coefficients in Eq. (1). The measured LFPAD 3d align was compared with the calculated one, as shown in Fig. 3. Furthermore, in Fig. 3, the calculated result agrees with the measured one within the experimental uncertainties. The above-mentioned agreements of the b parameter and LFPAD 3d align between the theory and the experiment guarantee reliability of the present calculations.
To see the details of LFPAD sum align , each contribution from the nearly degenerate I 3d orbitals is examined by referring to Fig. 7. The backward-scattering effect in LFPAD 6a1 align is unclear, and the scattering effects in LFPAD 1a2 align and LFPAD 7a1 align are barely discernible because both the molecular orbitals do not have large density distributions in directions toward the phenyl ring (see Fig. 5). By contrast, the forward-focusing effects by the phenyl ring can still be observed in LFPAD 3b1 align and LFPAD 3b2 align . Once the phenyl ring has tilted from the zx plane, the 3b 1 and 3b 2 orbitals (Fig. 5) have similar density distributions on the zx plane, and scattering effects by the phenyl ring on electrons emitted from these orbitals lead to similar LFPAD patterns. However, once the LFPAD p align from the nearly degenerate five orbitals has been summed (see LFPAD sum align ), the fine structures in the MFPAD, such as the backward-scattering and forward-focusing effects, are FIG. 6. MFPADs from the 6a 1 , 3b 1 , 3b 2 , 1a 2 , and 7a 1 orbitals of IPh. SXFEL polarization vector is parallel to the C-I axis, and the photoelectron energy is set to 120 eV. Panel below shows the sum of the five MFPADs with the geometry depicted by boll and stick molecular model of IPh.
FIG. 7. LFPAD p align from the 6a 1 , 3b 1 , 7a 1 , 3b 2 , 1a 2 , and 7a 1 orbitals of the aligned molecular ensemble of IPh. The photoelectron energy was set to 120 eV. In the lower panel, left: the sum of the five LFPAD p align , middle: the polarization geometry, and right: the polar plot of the molecular axis distribution determined experimentally, the same as Fig. 2(d), which was used in the LFPAD calculations.

ARTICLE
scitation.org/journal/sdy that the molecular structure is reflected in the MFPADs mainly through the interference terms between the directly emitted and singly scattered waves. Thus, sensitivity of the MFPADs to the molecular structure depends on the density distribution of the molecular orbitals from which the photoelectrons are emitted into the scattering sites. From this, a nondegenerate molecular orbital that meets this condition should be selected for the UXPD.

V. OUTLOOK
As we discussed in Ref. 21, the interference Coulombic nature in the MFPAD makes it possible to track photodissociation or photoelimination over longer internuclear distances by UXPD, where timeresolved experiments that detect electronic structures are inapplicable. Such an advantageous aspect of UXPD is expected in tracking the elimination of the iodine atom from IPh molecule excited to the 3B 1diabatic state with a 266 nm laser pulse. 50 To know whether the LFPAD sum align maintains the advantage or not, we calculated both the MFPAD and LFPAD sum align for the C-I bond lengths of 2.7 and 3.2 Å , which are shown together with those calculated at the equilibrium bond length 2.1 Å , as shown in Fig. 10. The degree of alignment has been set to the same value as the present experiment cos 2 H h i ¼ 0:78. To examine only the effects concerning the molecular geometry, we calculated both the MFPADs and LFPAD sum align for the ground-state potential. The computed results demonstrate that both the MFPADs and LFPAD sum align are insensitive to change in the C-I bond length near the equilibrium geometry (from 2.1 to 2.7 Å ). This insensitivity is due to the contributions from photoelectrons emitted from the 1a 2 and 7a 1 orbitals through the pseudo degeneracy of the molecular orbitals to be ionized. However, when the C-I bond has been elongated by more than 1 Å than the equilibrium, the change of forward-focusing and backward-scattering effects is observed in the MFPAD, and they appeared in the LFPAD sum align as the change of the XPD profiles as FIG. 8. LFPADs depending on both the ionized molecular orbitals and the molecular alignment. The first, second, and third row show LFPAD p H0 ðh 0 Þ at H 0 ¼ p/2, p/4, and 0, respectively, and the fourth row shows LFPAD p align with the degree of alignment of h cos 2 Hi ¼ 0:78 that is the same as upper of Fig. 7. Both the molecular axis distribution and the delta function given by Eq. (14) are normalized with respect to integral over H. Note that the LFPAD 6a1 H0¼0 ðh 0 Þ is plotted in a different scale.
Structural Dynamics ARTICLE scitation.org/journal/sdy well, as shown in Fig. 10. Based on these theoretical predictions, we can summarize that the advantage of the MFPADs, which can track the temporal molecular geometries during photodissociation or photoelimination, is kept slightly in the LFPAD sum align on the sum over the contributions from the degenerated orbitals under the currently achieved degree of alignment. Namely, for the larger change of molecular geometry of more than 1 Å , one can detect it through the change of the LFPAD sum align profiles by UXPD.
FIG. 9. LFPAD p align from the 6a 1 , 3b 1 , 7a 1 , 3b 2 , 1a 2 , and 7a 1 orbitals of the aligned molecular ensemble of IPh. The photoelectron energy was set to 120 eV, and the degree of alignment, h cos 2 Hi ¼ 0:93, was assumed. In the lower panel, left: the sum of the five LFPAD p align , middle: the polarization geometry, and right: the polar plot of the molecular axis distribution.

VI. CONCLUSION
In our previous work, 25,26,47 we reported that a higher degree of alignment of sample molecules, h cos 2 Hi > 0:8, is demanded to determine the molecular structures from XPD profiles for the aligned sample molecular ensembles. According to this criterion, we measured the I 3d XPD profile of IPh with h cos 2 Hi ¼ 0:78. However, we could not determine the local structure, i.e., the C-I bond length in the static ground-state IPh. This unexpected result is due to photoemission from the nearly degenerate molecular orbitals with respect to energy eigenvalues in the current experiment. The XPD profiles from the nearly degenerate five molecular orbitals contribute to the experimental LFPAD 3d align , as discussed in Sec. IV B. Thus, intrinsic features of interference effects, which are observed in each XPD profile, are smeared out in the LFPAD 3d align . If one selects a photoemission process from a nondegenerate molecular orbital, one can observe the intrinsic features of interference effects in the XPD profile, as demonstrated in the intensive calculations in Sec. IV B.
Finally, we summarize the criteria to perform the UXPD successfully as follows: (1) to use SXFEL, (2) to prepare sample molecules with the degree of alignments higher than 0.8, and (3) to select a photoemission process from a nondegenerate inner-shell orbital of the sample molecules, e.g., 1s orbitals of the second-row elements in the periodic table.