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Imaging molecular geometry with electron momentum spectroscopy

Electron momentum spectroscopy is a unique tool for imaging orbital-specific electron density of molecule in momentum space. However, the molecular geometry information is usually veiled due to the single-centered character of momentum space wavefunction of molecular orbital (MO). Here we demonstrate the retrieval of interatomic distances from the multicenter interference effect revealed in the ratios of electron momentum profiles between two MOs with symmetric and anti-symmetric characters. A very sensitive dependence of the oscillation period on interatomic distance is observed, which is used to determine F-F distance in CF4 and O-O distance in CO2 with sub-Ångström precision. Thus, using one spectrometer, and in one measurement, the electron density distributions of MOs and the molecular geometry information can be obtained simultaneously. Our approach provides a new robust tool for imaging molecules with high precision and has potential to apply to ultrafast imaging of molecular dynamics if combined with ultrashort electron pulses in the future.

The physical and chemical properties of molecules directly depend on their geometries and electronic structures that both have always been the central issues in molecular physics. The geometry of a molecule is conventionally obtained by the methods of X-ray 1,2 or electron diffraction [3][4][5][6] , from which the atomic positions are determined with sub-Ångström spatial resolution. An alternative imaging approach emerged in the past decade, which is referred to as the laser induced electron diffraction [7][8][9][10][11] , has also been demonstrated to image molecular structures with sub-Ångström precision. In this technique, an intense laser field is employed to extract electron from a molecule itself, and within one laser period a fraction of the tunneled electron wave packet will be forced back to re-collide and diffract from the parent molecular ion. The well-established method in the conventional electron diffraction is then applicable to retrieve the bond lengths of molecule.
On the other hand, the tunneled electron wave packet that directly emerges into the vacuum retains information about the orbital from which the electron is ionized 9 . By measuring the momentum distribution for these direct electrons, the fingerprint of the highest occupied molecular orbital can be observed through the filter of the suppressed binding potential through which the electron tunnels 9 . Thus one set of measurements simultaneously identifies the orbital wavefunction of molecule and the position of the atoms in the molecule in this laser induced electron tunneling and diffraction technique. Information about the ionizing orbital of neutral molecule is also imprinted on the high-harmonic radiation produced by the recombination of the re-collision electron with the parent ion in the laser field and allows the three-dimensional shape of the highest electronic orbital to be measured 12 .
Electron momentum spectroscopy (EMS), which is based on the electron-impact single ionization or (e, 2e) experiment near the Bethe ridge, is a well-established technique that can obtain the spherically averaged electron momentum distributions, or electron momentum profiles (see Supplementary Information Note 1), for any individual molecular orbitals (MOs) in principle [13][14][15] . This unique ability of imaging MOs makes the EMS a robust technique for exploring the electronic structures of molecules in gas phase 16 . However, the geometry information of molecule is usually veiled due to the single-centered character of the momentum space wavefunction for MO. In momentum space, for a MO which can be approximated by a linear combination of atomic orbitals (LCAOs), the information about the equilibrium nuclear positions R J is only present in the phase factors exp(− ip ⋅ R J ) introduced by Fourier transform of the wavefunction from position space to momentum space (see Methods for details). Therefore the electron momentum distribution of a MO will be modulated by a cosine or sine function b is the distance between atoms J a and J b . This oscillation phenomenon is usually referred to as bond oscillation 17 , which can also be regarded as a result of the Cohen-Fano type 18 or the Young-type interference effect originated from the coherent superposition of the (e, 2e) amplitudes from the atoms in the molecule. This type of molecular scale interference was first proposed by Cohen and Fano 18 in photoionization and was successively demonstrated in the ionization of molecules induced by heavy ions [19][20][21][22][23][24][25] , photons [26][27][28][29][30][31][32][33][34][35] , as well as electrons [36][37][38] .
In the EMS experiments, the interference effect was first discussed in the 1980 s 17 and clearly observed only recently in the experiments of CF 4 37 , H 2 38,39 , and SF 6 40,41 . Direct observation of the interference pattern in electron momentum distribution is usually very difficult due to the weak modulation on the rapidly decreasing intensity at large momentum. The feasible way is to compare the experimental cross section of a molecule with the theoretical or experimental one-center atomic cross section 37,39,40 or to compare the cross sections between two different vibrational states 38 . Kushawaha et al. 33 in their photoionization work suggested a more obvious way to observe the interferences by measuring the ratio of two cross sections corresponding to the MOs with symmetrical and anti-symmetrical characters, which are expected to give oscillations in antiphase, thus magnifying the interference pattern.
In the present work, the similar scheme has effectively been applied in EMS experiments to uncover the multi-center interferences in CF 4 and CO 2 . The scheme is pictorially illustrated in Fig. 1a. With CF 4 as an example, the three outermost MOs (1t 1 , 4t 2 , 1e) of this molecule are essentially due to lone-pair electrons or 2p atomic orbitals (AOs) on the F atoms. Figure 1b shows the calculated electron momentum profiles (see Supplementary Information Note 2) for 4t 2 and 1e orbitals at equilibrium geometry. In the logarithmic coordinate both of the momentum profiles show weak oscillations extending to large momentum region due to the multi-center interferences from the ionization of the four F atoms. Different orientations of the constituent 2p AOs in 4t 2 and 1e orbitals lead to the oscillations almost completely in antiphase (Fig. 1b) 37 . The interference pattern can be significantly magnified by plotting the ratio of the momentum profiles for these two MOs, as illustrated in Fig. 1a σ(1 )/σ(4 ) In this study, the accurate measurements are carried out for CF 4 and CO 2 by using a high-sensitivity angle and energy dispersive multichannel electron momentum spectrometer with simultaneous detection in 2π angle range 42 . Two-dimensional (2D) electron density map of binding energy and relative azimuthal angle for the outer-valence MOs for these two molecules have been obtained. The experimental electron momentum profiles for the relevant MOs are extracted. A strong dependence of the oscillation period on the interatomic distance is observed in the ratios of electron momentum profiles between two MOs with oscillations in antiphase, which is used to determine F-F distance in CF 4 and O-O distance in CO 2 with sub-Ångström precision. Thus, in our new approach, we can simultaneously obtain the electron density distributions of MOs and the molecular geometry information in one set of measurements. Benefited from the wide momentum range (from 0 to 8 a.u.) of this new version EMS spectrometer 42 , more than two periods of oscillations are included in the interference fringes. Besides, the present observation of interference effect totally depends on the experimental measurements and does not rely on the comparison with the one-center atomic cross section. These features make our approach a robust tool for imaging molecules with high precision and has the potential to apply to ultrafast imaging of molecular dynamics if combined with the ultrashort electron pulses 43 in the future.

Results
2D electron density maps. Figure 2a and b show the 2D electron density maps for CF 4 and CO 2 measured at impact energy of 1.2 keV plus binding energy (see Methods). These 2D maps are the (e, 2e) TDCSs as functions of binding energy and relative azimuthal angle φ (i.e. the momentum of the orbital electron) and contain all the information on binding energy spectra (BES), electron momentum distributions, and symmetries for various ionization states. Figure 2c and d show the total BES summed over all the measured φ for CF 4 and CO 2 respectively. Gaussian functions as shown by the solid curves, which correspond to the ionizations from different MOs, are invoked to fit the BES. The MO specific electron momentum profiles can be extracted by deconvoluting the Multicenter interference effect. The orbital images for 1t 1 , 4t 2 , 1e MOs of CF 4 and for 3σ u , 4σ g MOs of CO 2 are shown at the top right of Fig. 3. For CF 4 molecule, the three outer most MOs, 1t 1 , 4t 2 and 1e, are composed of 2p lone-pair electrons on the F atoms. As we have mentioned, both the momentum profiles for 4t 2 and 1e orbitals show weak oscillations due to the multi-center interferences from the ionization of the four F atoms. The phase of the interference factor depends on the different orientations of the constituent 2p AOs in the MOs 37 . In 4t 2 orbital the 2p AOs of the four F atoms orient parallel to each other, while in 1e orbital the 2p AOs of each two pairs of F atoms are in opposite orientations. The different orientations lead to the interference oscillations of momentum profiles almost completely in antiphase (Fig. 1b). Besides 4t 2 , 1e orbital pair, the momentum profiles of 1t 1 , 4t 2 orbital pair of CF 4 and 3σ u , 4σ g orbital pair of CO 2 are also modulated by the interference factors in antiphase (see Supplementary Information Note 3 and Fig. S1 for detail).
The interference pattern will significantly be magnified by plotting the ratio of the momentum profiles as indicated in Fig. 1c. Figure 3a and b show the ratios of the measured momentum profiles σ(1t 1 )/σ(4t 2 ) and σ(1e)/σ(4t 2 ) for CF 4 by solid circles. Both ratios exhibit significant oscillations around constant values with more than two periods, which is the distinct evidence of the multi-center interference effect. The constant is the product of the ratio of the electron occupation numbers of MOs (6 for 1t 1 , 4t 2 and 4 for 1e) and the ratio of the pole strengths of the corresponding ionization peaks. The pole strengths of the main ionizations peaks for the outer valence orbitals of molecules are usually approximately equal to unity. Therefore, the constant is roughly dependent on the ratio of the electron occupation numbers, which is about 1 for σ(1t 1 )/σ(4t 2 ) and 0.67 for σ(1e)/σ(4t 2 ) as is the case shown in Fig. 3a and b. We also illustrate in the figures the theoretical ratios for σ(1t 1 )/σ(4t 2 ) and σ(1e)/σ(4t 2 ) of CF 4 calculated at the equilibrium interatomic F-F distance R FF = 2.1551 Å 44 as well as at the distances changing − 0.2 Å, − 0.1 Å,+ 0.1 Å,+ 0.2 Å. The theoretical momentum profiles are calculated by B3LYP density functional method adopting aug-cc-pVTZ basis sets (see Supplementary Information Note 2). A very sensitive dependence of the oscillation interference pattern on the interatomic F-F distance can be observed. The theoretical results at equilibrium geometry give the best agreement with the experiments.
For CO 2 molecule, 3σ u and 4σ g MOs, which are hybrid orbitals of the oxygen (O) lone-pairs, are anti-symmetrical (u) and symmetrical (g) that are expected to give oscillations in antiphase. The experimental and theoretical momentum profile ratios of 3σ u and 4σ g MOs are shown in Fig. 3c. As is expected, the experimental ratio presents regular oscillation around a constant of about 0.85 that corresponds to the pole strength ratio of 4σ g and 3σ u (0.72/0.85) 45 . Similar to the situation of CF 4 , a very sensitive dependence of the interference pattern on the interatomic O-O distance is observed and the theoretical result at equilibrium geometry (R OO = 2.3267 Å 44 ) give approximately the best agreement with the experiment.
It should also be noted that the experimental data obviously deviate from the theoretical predictions at large momentum. These derivations should be ascribed to the distorted wave effect which is a common phenomena in EMS 14 at large momentum region and such effect may be different for different MOs. However, it still remains an unresolved problem to include the distorted wave effect in the calculations for the molecular system. Determining interatomic distance. As is discussed above, the oscillation period of the interference pattern is very sensitive to the change of interatomic distance, which provides a possible way to determine the interatomic distances with high precision. This is the well-known benefit in precision of any interferometric measurements like the Young's double-slit experiment. In order to determine the exact values of the equilibrium interatomic distances from the present experimental data, a series of theoretical momentum profile ratios are calculated at various interatomic distances R and a least-square fitting procedure is performed (see Supplementary Information 4). The χ 2 values, which is defined as the sum of the squared differences between experimental and theoretical momentum profile ratios, are shown as open circles in Fig. 4 as functions of relative interatomic distances (R − R eq )/R eq , where R eq are the equilibrium interatomic distances of CF 4 and CO 2 reported in ref. 44. Three-order polynomials (solid line) are used to fit the χ 2 distributions. As can be seen in Fig. 4a-c, the minimum points of χ 2 values are (R − R eq )/R eq = 0.033, 0.018 and − 0.059 for the momentum profile ratios of 1t 1 /4t 2 , 1e/2t 2 of CF 4 and 4σ g /3σ u of CO 2 . Therefore the exact values of the equilibrium interatomic distances of the present work can thus be determined to be R FF = 2.23 Å or 2.19 Å (2.21 Å on average) for CF 4 and R OO = 2.19 Å for CO 2 . On the other hand, the uncertainty of χ 2 value, shown as error bar in Fig. 4, can be deduced from that of the experimental data, which includes the statistical and deconvolution uncertainties. The corresponding error bars show that the minimum points of χ 2 distributions can just be resolved from the points of (R − R eq )/R eq = 0.00, 0.07 for 1t 1 /4t 2 of CF 4 , (R − R eq )/R eq = − 0.01, 0.05 for 1e/4t 2 of CF 4 and (R − R eq )/R eq = − 0.09, − 0.03 for 4σ g /3σ u of CO 2 , as indicated by the dashed lines in Fig. 4a-c. The uncertainties of the determined values of equilibrium interatomic distances are thereby ± 0.08 Å or ± 0.06 Å (± 0.07 Å on average) for CF 4 and ± 0.07 Å for CO 2 , which are about 3-4% of interatomic distances. By further improving the momentum resolution and reducing the statistical uncertainty, it would not be difficult to reach 1% or better in geometry determination.

Discussion
We demonstrate a robust method for the retrieval of the interatomic distances from the multicenter interference effect of molecules with EMS. A sensitive dependence of the oscillation period on the interatomic distance is observed in the ratios of electron momentum profiles between two MOs with oscillations in antiphase. A least-square fitting procedure is used to precisely determine the equilibrium F-F distance in CF 4 and O-O distance in CO 2 with sub-Ångström precision. The result for F-F distance is R FF = 2.21 Å ± 0.07 Å, which is consistent with the value reported by electron diffraction 44 within the experimental uncertainty. As for O-O distance in CO 2 , the result is determined to be R OO = 2.19 Å ± 0.07 Å. It is slightly smaller than the value from the electron diffraction experiments 44 . EMS is readily a well-established technique to obtain the spherically averaged electron momentum distributions for individual MOs. Therefore, by unveiling its new ability of determination of molecular bond lengths, EMS is now able to obtain the electron density distributions of MOs and the molecular geometry information simultaneously in one set of measurements. On the other hand, the recent advances in ultrashort electron pulses allowing one to achieve 4D electron diffraction 3-6 as well as 4D electron microscopy 46,47 . The most recent work 48,49 also demonstrated the feasibility of time-resolved EMS measurements of short-lived transient species, where an ultrashort photon pulse is used for exciting the dynamics of interest and an ultrashort electron pulse is applied to probe the system as a function of the delay time between them. Therefore, by employing the new approach of the present work as well as ultrashort electron pulses, EMS has the potential to apply to ultrafast imaging of the molecular dynamics by exploring not only the change of electron densities but also the change of molecular structures for transient species.

Methods
Experiment. The experiment is carried out using a high-sensitivity angle and energy dispersive multichannel electron momentum spectrometer with nearly 2π azimuthal angle range (2π-EMS). The details of the 2π-EMS can be seen in ref. 42. Briefly, the experiment involves coincidence detection of two outgoing electrons produced by electron impact ionization of the target molecule. The electron beam generated from a thermal cathode electron gun is accelerated to the energy of 1200 eV plus the binding energy to collide with the gas-phase target in the gas cell. The symmetric non-coplanar kinematics is employed. The scattered and ejected electrons with equal polar angles (θ 1 = θ 2 = 45°) and energies are analyzed by a spherical electrostatic analyzer with 90° sector and 2π azimuthal angle range. The two outgoing electrons are detected in coincidence by a position sensitive detector placed at the exit plane of the analyzer. The passing energies of energy analyzer are 600 eV for CF 4 and 200 eV for CO 2 , respectively. The performances of EMS-2π are calibrated by electron impact ionization of Argon before experiment. The energy resolution, polar angle resolution and azimuthal angle resolution are determined to be Δ E = 2.2 eV, Δ θ = 1.0° and Δ φ = 2.4° for CF 4 experiment and Δ E = 1.4 eV, Δ θ = 1.0° and Δ φ = 2.9° for CO 2 experiment, respectively.