ARTICLE Corrected: Publisher correction DOI: 10.1038/s41467-017-02437-9 OPEN Imaging the square of the correlated two-electron wave function of a hydrogen molecule 1 2 1 1 1 1 1 1 3 M. Waitz , R.Y. Bello , D. Metz , J. Lower , F. Trinter , C. Schober , M. Keiling , U. Lenz , M. Pitzer , 4 4 5 6 7 1 8 1 K. Mertens , M. Martins , J. Viefhaus , S. Klumpp , T. Weber , L.Ph.H. Schmidt , J.B. Williams , M.S. Schöfﬂer , 9 10 2,13 2 2,11,12 1 1 V.V. Serov , A.S. Kheifets , L. Argenti , A. Palacios , F. Martín , T. Jahnke & R. Dörner The toolbox for imaging molecules is well-equipped today. Some techniques visualize the geometrical structure, others the electron density or electron orbitals. Molecules are many- body systems for which the correlation between the constituents is decisive and the spatial and the momentum distribution of one electron depends on those of the other electrons and the nuclei. Such correlations have escaped direct observation by imaging techniques so far. Here, we implement an imaging scheme which visualizes correlations between electrons by coincident detection of the reaction fragments after high energy photofragmentation. With this technique, we examine the H two-electron wave function in which electron–electron correlation beyond the mean-ﬁeld level is prominent. We visualize the dependence of the wave function on the internuclear distance. High energy photoelectrons are shown to be a powerful tool for molecular imaging. Our study paves the way for future time resolved correlation imaging at FELs and laser based X-ray sources. 1 2 Institut für Kernphysik, J. W. Goethe Universität, Max-von-Laue-Str. 1, 60438 Frankfurt, Germany. Departamento de Química, Universidad Autónoma 3 4 de Madrid, 28049 Madrid, Spain. Universität Kassel, Heinr.-Plett-Strasse 40, 34132 Kassel, Germany. Institut für Experimentalphysik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany. FS-PE, Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, 6 7 Germany. FS-FLASH-D, Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany. Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. Department of Physics, University of Nevada Reno, 1664 N. Virginia Street, Reno, NV 89557, USA. 9 10 Department of Theoretical Physics, Saratov State University, 83 Astrakhanskaya, Saratov, 410012, Russia. Research School of Physical Sciences, The Australian National University, Canberra, ACT 0200, Australia. Instituto Madrileo de Estudios Avanzados en Nanociencia, 28049 Madrid, Spain. Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049 Madrid, Spain. Present address: Department of Physics and CREOL College of Optics & Photonics, University of Central Florida, Orlando, FL 32816, USA. Correspondence and requests for materials should be addressed to F.M. (email: email@example.com) or to R.D. (email: firstname.lastname@example.org) NATURE COMMUNICATIONS 8: 2266 DOI: 10.1038/s41467-017-02437-9 www.nature.com/naturecommunications 1 | | | 1234567890 Intensity (arb. units) Intensity (arb. units) Intensity (arb. units) ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-02437-9 maging the wave function of electrons yields detailed infor- on the one hand and coincident detection of reaction fragments on mation on the properties of matter, accordingly, experiments the other hand. Our novel experimental approach allows us to Ihave pursued this goal since decades. For example, in solid visualize the square of the H correlated two-electron wave func- state physics photoionization is routinely used as a powerful tool tion. In the ionization step, one of the electrons is mapped onto a for single-electron density imaging. Photon-based techniques detector and simultaneously the quantum state of the second have the particular strength that they can be in principle imple- electron is determined by coincident detection of the fragments. mented in pump-probe experiments opening additionally the perspective to go from still images to movies. For atoms and Results molecules photoionization has also been proposed as a promising Concept of correlation imaging. The properties of a photo- technique to image orbitals , but no positive outcomes were ionization event, given by the ionization amplitude D, are reported so far. The reverse process of photoionization, namely determined (within the commonly used dipole approximation) by high harmonic generation, has succeeded in accomplishing this only three ingredients: the initial state of the system ϕ , which we goal of orbital imaging . Further techniques for imaging mole- want to image, the properties of the dipole operator μ ^ (respon- cular orbitals are electron momentum spectroscopy or strong sible for the photoionization) and the ﬁnal state representing the ﬁeld tunnel ionization . remaining cation and a photoelectron with momentum k, χ : While the toolbox to image single electrons is well equipped, endeavors to directly examine an entangled two-electron wave D ¼ ϕ ðrÞμ ^ðrÞχ ðrÞdr; ð1Þ 0 k function have, so far, not been successful and corresponding techniques are lacking. This is particularly unfortunate, as elec- where r represents the coordinates of target electrons. The initial tron correlation which shapes two-electron wavefunctions is of wave function is directly accessible provided that the other two major importance across physics and chemistry. It is electron constituents do not introduce signiﬁcant distortions. This is the correlation which is at the heart of fascinating quantum effects case when utilizing circularly polarized light and examining high 6 7 such as superconductivity or giant magnetoresistance . Even in energy electrons (Born limit) within the polarization plane. As an single atoms or molecules, electron correlation plays a vital role illustration, let us consider the one-electron H molecular ion. At and continues to challenge theory. For example, the single- a high enough energy, the continuum electron can be described photon double ionization, i.e., the simultaneous emission of two by a plane wave. In this case, the photoionization differential electrons after photoabsorption, is only possible due to cross section in the electron emission direction (θ, φ) (the so- electron–electron correlation effects, as the photon cannot called molecular frame photoelectron angular distribution, interact with two electrons simultaneously. Instead, the second MFPAD) is simply proportional to the square of the Fourier electron is emitted either after an interaction with the ﬁrst elec- transform (FT) of the initial state, ϕ (k) (see methods section): tron (which is typically described as a “knock-off process”)or because of the initial entanglement of the two electrons due to dP 1 3=2 2 ikr ð2Þ ¼ k ð2πÞ ϕ ðrÞe dr electron correlation prior to the absorption of the photon (in a 0 3=2 dðcos θÞdk 2π process termed “shake off”) . Although the importance of electron correlation is intuitively understandable in processes which 2 3=2 2 ð3Þ ¼ k ð2πÞ jj ϕ ðkÞ : obviously involve two electrons, it turns out, that even for bound stationary states of atoms and molecules, electron correlation contributions are crucial: within the commonly used Here θ denotes the polar angle with respect to the molecular axis Hartree–Fock approximation, the calculated values of binding and φ the corresponding azimuthal angle. Thus, by choosing energies are often in no satisfying accordance to those actually high-photon energies and restricting the measurement of the occurring in nature. Here the basic cause is that the Hartree–Fock MFPAD to the polarization plane (φ = 90° and 270°) of the cir- approximation is a mean-ﬁeld theory, which considers only an cularly polarized light, the initial electronic wave function is overall mean potential generated by the ensemble of electrons, directly mapped onto the emitted photoelectron. Figure 1 illus- and as such neglects electron–electron correlation by deﬁnition. trates this mapping procedure for the ground state of H (Fig. 1a: In this manuscript, we show that the correlated molecular wave electronic wave function in coordinate space; Fig. 1b the square of function can be visualized by the simultaneous use of two well- the Fourier transform of Fig. 1a; Fig. 1c the same in logarithmic established and well-understood methods: photoelectron emission color scale). As can be seen from Fig. 1d, the MFPAD for an 90° 60° 120° a –3 b c 15 10 10 10 0.6 150° 30° 5 5 –5 0 0 0 0° –5 –5 –5 –10 1 210° 330° –10 0.1 –14 –10 –10 10 –15 0 240° 300° –15 –10 –5 0 5 10 15 –10 –5 0 5 10 –10 –5 0 5 10 270° x (a.u.) Momentum x (a.u.) Momentum x (a.u.) þ þ Fig. 1 Imaging of the H one-electron wave function. a The electronic wave function of H in the polarization plane for an internuclear distance R = 1.4 a.u. 2 2 The positions of the two nuclei are indicated by black dots. b The square of the Fourier transform of a in the (k , k ) plane. c The same as (b), but in x y logarithmic color scale. Notice the appearance of nearly vertical fringes, whenjj k is signiﬁcantly different from zero. The approximate periodicity of these fringes is Δk 2π=R. The dashed line indicates the region of momentum space associated with an electron kinetic energy of 380 eV (i.e., a radius of jj k ¼ 5:3 a.u.) and θ is the angle with respect to the molecular axis. d Polar plot of the intensity distribution in c along the dashed line (red) and the corresponding MFPAD in the plane of polarization of the ionizing radiation obtained from nearly exact calculations (green) 2 NATURE COMMUNICATIONS 8: 2266 DOI: 10.1038/s41467-017-02437-9 www.nature.com/naturecommunications | | | y (a.u.) Momentum y (a.u.) Momentum y (a.u.) Intensity (arb. units) Intensity (arb. units) Intensity (arb. units) NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-02437-9 ARTICLE gj Fully correlated Photofragmentation ad Square of wavefunction 90° 90° Hartree-fock fully correlated 120° 60° 120° 60° 150° 30° 150° 30° 5 5 –5 0 0 180° 0° 180° 0° –10 –5 –5 210° 330° 210° 330° –15 –10 –10 –10 –5 05 10 –10 –5 0 510 300° 240° 240° 300° 270° 270° Momentum x (a.u.) Momentum x (a.u.) hk 90° 90° 120° 60° 120° 60° be 150° 30° 150° 30° 5 5 –5 0 0 180° 0° 180° 0° –10 –5 –5 210° 330° 210° 330° –15 –10 –10 –10 –5 0 5 10 –10 –5 0 5 10 240° 300° 240° 300° 270° 270° Momentum x (a.u.) Momentum x (a.u.) il 90° 90° 120° 120° 60° 60° c f 150° 150° 30° 30° 5 5 –5 0 0 0° 180° 0° 180° –10 –5 –5 330° 210° 330° 210° –15 –10 –10 300° –10 –5 05 10 –10 –5 05 10 300° 240° 240° 270° 270° Momentum x (a.u.) Momentum x (a.u.) Fig. 2 Correlation imaging of the H two-electron wave function. a–f Momentum distributions of electron A resulting from the projection of the two- electron wave function of H onto different H states of electron B; a, c uncorrelated Hartree-Fock wave function; d, f fully correlated wave function. The different quantum states of electron B are 2pσ (a, d), 2sσ (b, e) and 1sσ (c, f). Circular lines showjj k ¼ 5:3 a.u. (c, d, f) andjj k ¼ 5:2 a.u. b, e which u g g correspond to ionization by a photon of 400 eV energy. g–i ground state wave function (intensity distributions along the circular lines shown in (d, f). j–l Experimental and theoretical MFPADs (symbols and green line, respectively) obtained after photoionization with circularly polarized photons of an energy of 400 eV for the same ﬁnal states of electron B measured in coincidence. Ions and electrons are selected to be in the plane of polarization of the ionizing photon and data for left and right circularly polarized light are added. Molecular orientation as indicated. The error bars indicate the standard deviation of the mean value electron of 380 eV is very similar tojj ϕ ðkÞ for the chosen Application on H . Figure 2 illustrates this concept and high- momentum k (the square of the FT along the dashed line shown lights the differences between the uncorrelated Hartree–Fock in Fig. 1c). Notice that, due to the smallness of the cross section at wave function and the highly correlated nearly exact wave func- such high electron momentum, the main features of the FT are tion. The corresponding one-electron momentum distributions only apparent in the logarithmic plot shown in Fig. 1c. resulting from the projection of the corresponding ground state This tool of high energy photoelectron imaging can now be wave functions onto different states of the bound electron B, n , combined with coincident detection of the quantum state of a are depicted in Fig. 2a–c (Hartree–Fock) and Fig. 2d–f (exact) as second electron to visualize electron correlation in momentum functions of the momentum components parallel (k ) and x,A space. We dissect the entangled two-electron wave function by perpendicular (k ) to the molecular axis. The different rows y,A analyzing a set of conditional angularly resolved cross sections correspond to the different states in which the second electron B corresponding to a high energy continuum electron (A) and a is left after photoionization, i.e., they correspond, from bottom to bound electron (B) detected in a different region of the two- top, to projections of the ground state wave function onto the electron phase space. Quantum mechanically, this is equivalent to n ¼ 1sσ ,2sσ , and 2pσ states of H . Thus, as in our one- λ g g u projecting the initial two-electron wave function onto products of electron example shown in Fig. 1, the different panels in Fig. 1 different H (bound) molecular orbitals (B) and a plane wave (A) contain direct images of different pieces of the ground state of H (see Methods section). In doing so, one can thus determine if and through the square of the corresponding FTs. The role of electron how the density distribution of one electron changes upon correlation is quite apparent in this presentation: Fig. 1a is empty changing the region of phase space in which one detects the other, for the uncorrelated Hartree–Fock wave function, since projec- correlated, electron. tion of the latter wave function onto the 2pσ orbital is exactly NATURE COMMUNICATIONS 8: 2266 DOI: 10.1038/s41467-017-02437-9 www.nature.com/naturecommunications 3 | | | 1sσ 2sσ σ 2pσ g g u Momentum y (a.u.) Momentum y (a.u.) Momentum y (a.u.) ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-02437-9 zero, while this is not the case for the fully correlated wave function (Fig. 1d); also, Fig. 1b, c for the uncorrelated description are identical, while Fig. 2e and f for the correlated case are signiﬁcantly different. As in the example of Fig. 1c, a ﬁxed energy corresponds to points around the circumference of a circle. The density distributions pertaining to points around the circles of Fig. 2a–f are shown in Fig. 2g–i. 2sσ Experimentally, these conditional probabilities are obtained by H + H (n =2) measuring in coincidence the momentum of the ejected electron 25 and the proton resulting from the dissociative ionization reaction 2pσ þ H + H (n =1) γð400 eVÞþ H ! H ðn Þþ e ð4Þ 2 λ 1sσ 1 + X Σ HðnÞþ H ; ð5Þ 02468 10 Internuclear distance (a.u.) which, as explained below, allows us to determine the ﬁnal ionic state characterized by the quantum number n .Fig. 2j–l depicts the experimental results of the measured angular distributions of 0.3 electron A together with numerical data resulting from a nearly exact theoretical calculation of the photoionization process. As can be seen, the measured and calculated MFPADs shown in Fig. 2j–l Total are very similar to the calculated projections in momentum space 2sσ 0.2 of the fully correlated ground state wave function shown in Fig. 2g–i. In other words, the momentum of the ejected photoelectron 2pσ faithfully reﬂects and maps the momentum of a bound state 4B/2E Experiment 2F electron in the molecular ground state when the momentum of the 4C 0.1 4A second bound electron is constrained by projection of the H wave 2D function onto different molecular-ion states; this represents the correlation between the two electrons. Note in particular Fig. 2gis not empty and Fig. 2h, i are not identical, as it would be for an uncorrelated H ground state (compare with Fig. 2a–c). 2 0 5 10 15 20 25 Kinetic energy release (eV) Identifying the quantum state of the second electron. In more Fig. 3 Correlation diagram and kinetic energy distribution for dissociation of detail, the angular emission distributions and the ﬁnal quantum H a Potential energy curves for the ground state of H (lower curve) and state of electron B are obtained in our experiment by measuring the 1sσ ,2sσ , and 2pσ ionization thresholds (upper curves). The latter g g u the momenta of the charged particles generated by the photo- correspond to electronic states of H . The violet shaded area represents ionization process in coincidence. As the singly charged molecule the Franck-Condon region associated to the ground vibrational state of H . dissociates in the cases presented here into a neutral H atom and Notice the break in the energy scale for a better visualization. The dashed a proton, we can obtain the spatial orientation of the molecular violet line shows how the initial internuclear distance of the molecule is axis by measuring the vector momentum of the proton (i.e., its mapped onto the kinetic energy release (KER) of the reaction applying emission direction after the dissociation). The electron emission the”reﬂection approximation” . b KER distribution obtained after single- direction in the molecular frame is then deduced from the relative photon ionization of H employing photons of hν = 400 eV. Symbols: emission direction of the proton and the vector momentum of the experiment, lines: theory. The calculation depicted by the black curve electron. Additionally, the magnitude of the measured ion includes the twelve states with the highest photo ionization cross sections momentum provides the kinetic energy release (KER) of the (up to n = 4). The main contributions (besides 1sσ at low KER) are shown reaction. The latter enables an identiﬁcation of the quantum state in blue (2sσ ) and red (2pσ ), others are not visible on that scale. The þ g u of electron B (i.e., the H electronic state), which is demonstrated shaded areas indicate the regions of KER selected in Figs. 2d–f and 4a, c in Fig. 3. Fig. 3a shows the relevant potential energy curves of H and Fig. 3b the measured (and theoretically predicted) KER spectra. From the measured sum of the kinetic energies of the of electron B, the maxima in the momentum distribution of electron and the proton we furthermore identify the asymptotic electron A become minima and vice versa. This can be intuitively electronic state of the neutral H fragment (not detected in the understood in coordinate space. The maxima in the k-space dis- experiment), mostly H(n = 1) and H(n = 2). tribution correspond to the constructive interference of the part of the electron density close to one or the other nucleus spaced by R. Nodal structure of the wave function. Our experimentally Thus, inverting maxima to minima in k-space corresponds to a obtained spectra not only show the imprint of correlation, but also phase shift of π between the wave function at one or the other nucleus in coordinate space. For H , the two-electron wave allow us to separate the contribution of different pieces of the electronic wave function to this correlation. Indeed, the momen- function is gerade, i.e., it has the same sign of the overall phase at both centers. For a large part of the two-electron wave function, tum distribution of electron A depends strongly on the properties of electron B. The most dramatic example can be seen by com- this symmetry consideration is also valid for each individual paring the upper and middle rows in Fig. 2, which show electron electron (it reﬂects the fact that both electrons occupy the 1sσ A under the condition that electron B is detected in the 2pσ and orbital most of the time). Therefore, both electrons have the same 2sσ states of H , respectively. Upon this change in the selection phase at both nuclei, which, in turn, is directly reﬂected in the 4 NATURE COMMUNICATIONS 8: 2266 DOI: 10.1038/s41467-017-02437-9 www.nature.com/naturecommunications | | | Intensity (arb. units) Energy (eV) KER (2pσ ) KER (2sσ ) g NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-02437-9 ARTICLE ad Square of 2sσ σ wave function Image by photofragmentation R = 1.6 a.u. KER 9–11 eV 90° 90° 120° 120° 60° 60° 150° 150° 30° 30° 180° 0° 180° 0° 210° 330° 210° 330° 240° 300° 240° 300° 270° 270° be R = 1.4 a.u. KER 11–13 eV 90° 90° 120° 60° 120° 60° 30° 30° 150° 150° 180° 0° 180° 0° 210° 330° 210° 330° 300° 240° 240° 300° 270° 270° cf R = 1.2 a.u. KER 13–15 eV 90° 90° 120° 60° 120° 60° 30° 30° 150° 150° 180° 0° 180° 0° 210° 330° 210° 330° 240° 300° 240° 300° 270° 270° Fig. 4 Dependence of the momentum distribution on the internuclear distance R a–c and KER d–f of the molecule at the instant of photoionization. Molecular orientation as indicated. a to c: Square of the correlated wave function, as shown in Fig. 2h, but for internuclear distances as stated in the legends. Electron B is projected onto the 2sσ state while electron A is depicted. d–f: Experimental and theoretical MFPADs (symbols and black line, respectively) for the KER ranges corresponding to the internuclear distances in a, c resulting from applying the reﬂection approximation through the 2sσ potential energy curve. The error bars indicate the standard deviation of the mean value maximum at k = 0 and the corresponding maximum in the In addition to identifying the ﬁnal state of electron B, direction perpendicular to the molecular axis in Fig. 2e, f. Due to the measured KER provides further insights into the ionized electron correlation, however, this is not strictly true for all parts H molecule. As soon as the potential energy curve relevant for of the wave function: Projecting electron B onto the 2pσ state the process is known, one can infer the internuclear distance R highlights this small fraction of the wave function where electron of the two atoms of the molecule at the instant of photoabsorp- A has the opposite phase at the two nuclei. As explained before, tion by using the reﬂection approximation (see Fig. 3). This this part of the wave function does not exist for a Hartree–Fock allows us to investigate more details of the two-electron wave wave function and Fig. 2a is therefore empty. This phase change of function: The distributions in Fig. 2d–f shows nodal lines the wave function between the nuclei leads to the nodal line that lead to corresponding nodes in the angular distributions in through the center in Fig. 2d and the nodes in Fig. 2g, j in the Fig. 2g–i. As mentioned above, these nodes in k-space are direction perpendicular to the molecular axis. separated by Δk ¼ 2π=R. Within the range of R covered by the NATURE COMMUNICATIONS 8: 2266 DOI: 10.1038/s41467-017-02437-9 www.nature.com/naturecommunications 5 | | | Internuclear distance ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-02437-9 −10 62.5 MHz. A residual gas pressure of 2 × 10 mbar in the reaction chamber led to about 200 Hz of ions detected without the gas jet in operation. The count rates during the experiment were ~ 350 Hz on the ion detector and about 5 kHz on the electron detector. Dissociative ionization events (reaction (5)) were selected by gating on the ion and electron time of ﬂight and on the ion kinetic energy. After all conditions applied to the data, we end up with ~ 200,000 events which we analyze in the MFPADs. Correlation imaging.To ﬁrst order of perturbation theory, the ionization ampli- tude of a one-electron molecular system is given by (within the dipole approx- imation) D ¼ ϕ ðrÞ μ ^ðrÞ χ ðrÞ ; ð6Þ hj ji n k where ϕ is the initial state, μ ^ is the dipole operator, and χ is the ﬁnal state n k y representing a photoelectron with momentum k. At high photoelectron energies, ikr one can approximate the ﬁnal state by a plane wave, χ ðrÞ¼ e . Thus, if we k consider circularly polarized light propagating along the z-axis and a molecule ﬁxed along the x-axis (see Fig. 5), the transition amplitude can be written, in the velocity gauge: ikr D ’ ϕ ðrÞjj μ ^ðrÞ e ∂ ∂ ikr ¼ hj ϕ ðrÞ þ i e ð7Þ ∂x ∂y ikr ¼k þ ik ϕ ðrÞje : y x The corresponding photoionization probability (or equivalently the photoionization cross section), differential in the electron emission angles and momentum (or MFPAD), is proportional to the square of the transition amplitude (see Fig. 5 for notations): Fig. 5 Geometrical deﬁnitions: polar angle θ and azimuthal angle φ deﬁning the direction of the electron momentum k with respect to the plane deﬁned dP 2 2 2 ikr ’ k þ k ϕ ðrÞje : ð8Þ x y n by the internuclear axis (x) and the propagation direction k dðcos θÞdφdk Restricting the detection of the electrons to the plane containing the molecule and perpendicular to the light propagation direction, the above expression reduces Franck-Condon region, the nodal structure of the electronic wave to: function changes signiﬁcantly and Fig. 4 demonstrates how the k- dP 2 space distribution of the two-electron wave function changes 2 ikr ¼ k ϕ ðrÞje : ð9Þ dðcos θÞdk accordingly as a function of R (or KER respectively). The This expression is only applicable to differential probabilities in the (x, y) plane. corresponding experimental and theoretical MFPADs resulting It can be seen that the integral over r is proportional to the Fourier transform ϕ (k) from high energy photoionization follow a similar pattern. of the ϕ (r): In conclusion, high energy angular resolved photoionization is Z 2 a promising route to access molecular wave functions in dP 1 2 3=2 ikr ¼ k ð2πÞ drϕ ðrÞe ; ð10Þ 3=2 momentum space. The process of molecular dissociation in dðcos θÞdk ð2πÞ combination with shake up of the bound electron is universal by 3/2 its nature. Shake up of an electron into a continuum state instead where we have introduced a factor of (2π) to make this relationship clearer, i.e., of a bound state, i.e., double ionization of the molecule, might dP 2 3=2 2 ¼ k ð2πÞ jj ϕ ðkÞ : ð11Þ also come into play. Therefore, this approach can in principle be dðcos θÞdk extended to molecules with more than two electrons. In detail, it Thus, at high photon energies, the MFPADs measured in the polarization plane depends on the shape of the potential energy surfaces which of the circularly polarized light directly map the initial electronic wave function. Let us now generalize this concept to the case of a correlated initial state as that determines to which extend different ionic states can be separated of the H molecule. The amplitude describing photoionization from the ground by the kinetic energy of the fragments. Combined with state, Ψ (r , r ), can now be written as: 0 1 2 coincidence detection, this technique opens the door to image D ¼hΨ ðr ; r ÞjOðr ; r ÞjΦ ðr ; r Þi ð12Þ 0 1 2 1 2 f 1 2 correlations in electronic wave functions. Similar approaches have also been proposed for imaging correlations in superconduc- where Oðr ; r Þ¼ μ ^ðr Þþ μ ^ðr Þ and Φ ðr ; r Þ is the ﬁnal continuum state. At 10 1 2 1 2 f 1 2 tors . With the advent of X-ray free electron lasers and the high photoelectron energies, the latter can be approximately written as a product of extension of higher harmonic sources to high photon energies, an H continuum wave function χ ðr Þ that describes a photoelectron with linear k 2 such correlation imaging bares the promise to make movies of the momentum k and an H bound wave function ϕ ðr Þ that describes the electron 2 n remaining in the ion: time evolution of electron correlations in molecules and solid materials. ^ ð13Þ D ¼hj Ψ ðr ; r Þ Oðr ; r Þji ϕ ðr Þχ ðr Þ : 0 1 2 1 2 n 1 k 2 We now write the fully correlated ground state wave function of H as a linear combination of two-electron conﬁgurations expressed as antisymmetrized products Methods of Hartree–Fock (HF) orbitals Experiment. The experiment has been performed at beamline P04 of the PETRA HF HF HF HF Ψ ¼ 1sσ ðr Þ1sσ ðr Þþ c 2sσ ðr Þ2sσ ðr Þ 0 1 2 1 1 2 g g g g III facility at DESY in Hamburg. The circularly polarized photon beam (400 eV ð14Þ HF HF photon energy, about 1.3 × 10 photons per second, 100 μm focus diameter) was þ c 2pσ ðr Þ2pσ ðr Þþ ::: 2 1 2 u u crossed with a supersonic H gas jet (diameter 1.1 mm, local target density 5 × 10 −2 12–14 cm ) at right angle in the center of a COLTRIMS spectrometer . A homo- where we have factored out the antisymmetric spin wave function corresponding to −1 geneous electric ﬁeld of 92 V cm guided electrons and ions towards position- a singlet multiplicity and c ; c 1. The ﬁrst term in this expansion represents the 1 2 sensitive micro-channel plate detectors (active area 80 mm diameter) with hex- ground state of H in the HF approximation, agonal delayline readout . In the ion arm of the spectrometer a 55 mm accel- HF HF HF Ψ ðH Þ¼ 1sσ ðr Þ1sσ ðr Þ; ð15Þ 2 1 2 eration region was followed by a 110 mm drift region. The electron arm of the 0 g g spectrometer was formed by a 37 mm long acceleration region. A magnetic ﬁeld of 35.5 G parallel to the electric ﬁeld guided the electrons on cyclotron trajectories. which includes screening and exchange but neglects electron correlation. The data was taken in 480-bunch operation mode, equaling a repetition rate of Substituting Eq. (14) in Eq. (13), retaining the lowest-order non-zero terms, and 6 NATURE COMMUNICATIONS 8: 2266 DOI: 10.1038/s41467-017-02437-9 www.nature.com/naturecommunications | | | NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-02437-9 ARTICLE using Eq. (11), the partial differential photoionization cross sections (or partial Ab initio calculations. The ab initio method used to obtain the dissociative MFPADs) associated with the lowest three ionization channels, 1sσ ,2sσ , and g g ionization spectra and the corresponding angular distributions has been described 3=2 2pσ , can be written (up to a trivial factor of k ð2πÞ ): 22,23 elsewhere . It has been successfully applied to evaluate photoionization cross DE sections and MFPADs of the H molecule in both time-dependent and time- 2 2 2 HF ^ 22–25 ð16Þ hj Ψ O 1sσ χ ’ 1sσ j1sσ ϕ HFðkÞ ; independent scenarios . Due to the high photoelectron energies produced in 0 g k g 1sσ the experiment, we have made use of the Born-Oppenheimer approximation, which allows us to describe the initial and ﬁnal continuum wave functions as DE 2 2 2 products of an electronic wave function and a nuclear wave function. The ground HF Ψ O 2sσ χ ’ 1sσ j2sσ ϕ HFðkÞ ; ð17Þ 0 g g k g 1sσ g state electronic wave function has been obtained by performing a conﬁguration interaction calculation in a basis of antisymmetrized products of one-electron H orbitals, and the ﬁnal electronic continuum states by solving the multichannel 2 2 HF scattering equations in a basis of uncoupled continuum states that are written as ð18Þ Ψ O 2pσ χ ’ c 2pσ ji 2pσ ϕ HFðkÞ ; 0 u k 2 u 2pσ products of a one-electron wave function for the bound electron and an expansion on spherical harmonics and B-spline functions for the continuum electron. The where the dependence on r and r is now implicit in all equations. Hence, the multichannel expansion includes the six lowest ionic states (1sσ ,2pσ ,2pπ ,2sσ , 1 2 g u u g partial differential cross sections are proportional to the representation of the 3dσ , and 3pσ ) and partial waves for the emitted electron up to a maximum g u ground state HF orbitals in momentum space and to the overlap between these HF angular momentum l = 7 enclosed in a box of 60 a.u., which amounts to around max orbitals and the H orbitals that deﬁne the different ionization thresholds. As can 61,000 discretized continuum states. The one-electron orbitals for the bound be seen, in the absence of electron correlation, i.e., when the initial state is simply electron are consistently computed in the same radial box using single-center HF described by Ψ and therefore the c coefﬁcients are zero, ionization can be direct expansions with corresponding angular momenta up to l = 16. The electronic 0 max (i.e., an electron is ejected into the continuum and the other remains in the 1sσ g wave functions have been calculated in a dense grid of internuclear distances orbital, Eq. (16) or can be accompanied by excitation of the remaining electron into comprised in the interval R = [0, 12] a.u. The nuclear wave functions have been the 2sσ state (shake-up mechanism, Eq. (17)). Ionization and excitation into the g obtained by diagonalizing the corresponding nuclear Hamiltonians in a basis of B- 2pσ state is only possible when c is different from zero (Eq. (18)), i.e., when splines within a box of 12 a.u. We have thus computed the photoionization u 2 electron correlation is not negligible. amplitudes and cross sections for circularly polarized light for the dissociative To get additional information about the relative magnitude of the partial cross ionization process from ﬁrst order perturbation theory sections, we write the HF orbitals as linear combinations of H orbitals. To the ﬁrst D ¼ Φ ðr ; r Þξ ðRÞ Oðr ; r Þ Ψ ðr ; r Þξ ðRÞ : ð24Þ f 1 2 1 2 0 1 2 order of perturbation theory, f 0 HF 1sσ ¼ 1sσ þ λ 2sσ þ ::: g 1 g ð19Þ At variance with Eq. (12), the previous equation includes the initial ξ and ﬁnal ξ g 0 f vibrational wave functions and integration is performed over both electronic and nuclear coordinates. HF The present methodology does not account for the double ionization channel, 2pσ ¼ 2pσ þ λ 3pσ þ ::: ð20Þ u 2 u which is open at the photon energies used in the present work. However, this channel is expected to have a marginal inﬂuence in the reported results since the and so on, where λ 1. Substituting Eqs. (19) and (20)in (16), (17), and (18), and corresponding cross section is at least an order of magnitude smaller than that for retaining the lowest-order non-zero terms in λ , one obtains the following the single ionization channel. In addition, according to the Franck–Condon simpliﬁed expressions for the three ionization channels above: picture, double ionization could only contribute to the KER spectrum in the region 2 around 19–20 eV, i.e., outside the region of interest discussed in the present work. ð21Þ Ψ O 1sσ χ ’ ϕ ðkÞ 0 g k 1sσ Data availability. The data that support the ﬁndings of this study are available from the authors on reasonable request. Ψ O 2sσ χ ’ λ ϕ ðkÞ ; ð22Þ 0 g 1 k 1sσ Received: 17 July 2017 Accepted: 30 November 2017 ð23Þ Ψ O 2pσ χ ’ c ϕ ðkÞ ; 0 u k 2 2pσ where we have used the fact that the H orbitals form an orthonormal basis. As can be seen, the dominant mechanism is direct ionization from the 1sσ orbital (Eq. (21)). Ionization with simultaneous excitation of the remaining electron (Eqs. References (22) and (23)) is much less likely, since both λ and c are small. Ionization through 1 2 1. Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission studies other channels only contribute to second or higher order, thus explaining why they of the cuprate superconductors. Rev. Mod. Phys. 75, 473–541 (2003). barely contribute to the ionization cross section. According to this simple 2. Santra, R. Imaging molecular orbitals using photoionization. Chem. Phys. 329, formalism, for both the 1sσ and 2sσ channels, the MFPADs map the 1sσ orbitals g g g 357–364 (2006). in momentum space. 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NATURE COMMUNICATIONS 8: 2266 DOI: 10.1038/s41467-017-02437-9 www.nature.com/naturecommunications 7 | | | ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-02437-9 European Union’s Horizon 2020 research and innovation programme under grant 14. Jahnke, T. et al. Carbon k-shell photo ionization of CO: molecular frame agreement No 654220. T.W. was supported by the U.S. Department of Energy Basic angular distributions of normal and conjugate shakeup satellites. J. Electron Energy Sciences under Contract No. DE-AC02-05CH11231. A.P. acknowledges a Ramón Spectrosc. Relat. Phenom. 183,48–52 (2011). y Cajal contract from the Ministerio de Economa y Competitividad. We thank M. Lara- 15. Jagutzki, O. et al. Multiple hit readout of a microchannel plate detector Astiaso for providing us with the Hartree–Fock wave functions. with a three-layer delay-line anode. IEEE Trans. Nucl. Sci. 49, 2477–2483 (2002). 16. Smirnov, Yu. F., Pavlitchenkov, A. V., Levin, V. G. & Neudatschin, V. G. 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Weber, T. et al. Complete photo-fragmentation of the deuterium molecule. Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- 017-02437-9. Nature 431, 437–440 (2004). 20. Knapp, A. et al. Mechanisms of photo double ionization of helium by 530 eV photons. Phys. Rev. Lett. 89, 033004 (2002). Competing interests: The authors declare no competing ﬁnancial interests. 21. Waitz, M. et al. Two-particle interference of electron pairs on a molecular level. Reprints and permission information is available online at http://npg.nature.com/ Phys. Rev. Lett. 117, 083002 (2016). reprintsandpermissions/ 22. Martín, F. Excitation of atomic hydrogen by protons and multicharged ions. J. Phys. B 32, 501–511 (1999). Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in 23. Palacios, A., Sanz-Vicario, J. L. & Martín, F. Theoretical methods for attosecond published maps and institutional afﬁliations. electron and nuclear dynamics: applications to the H molecule. J. Phys. B 48, 242001 (2015). 24. Martín, F. Single photon-induced symmetry breaking of H dissociation. Science 315, 629–633 (2007). Open Access This article is licensed under a Creative Commons 25. Dowek, D. Circular dichroism in photoionization of H . Phys. Rev. Lett. 104, 233003 (2010). Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Acknowledgements Commons license, and indicate if changes were made. The images or other third party This work was funded by the Deutsche Forschungsgemeinschaft, the BMBF, the Eur- material in this article are included in the article’s Creative Commons license, unless opean Research Council under the European Union Seventh Framework Programme indicated otherwise in a credit line to the material. If material is not included in the (FP7/2007-2013)/ERC grant agreement 290853 XCHEM, the MINECO projects article’s Creative Commons license and your intended use is not permitted by statutory FIS2013-42002-R and FIS2016-77889-R, and the European COST Action XLIC CM1204. regulation or exceeds the permitted use, you will need to obtain permission directly from All calculations were performed at the CCC-UAM and Mare Nostrum Supercomputer the copyright holder. To view a copy of this license, visit http://creativecommons.org/ Centers. We are grateful to the staff of PETRA III for excellent support during the beam licenses/by/4.0/. time. K.M. and M.M. would like to thank the DFG for support via SFB925/A3. A.K. and V.S. thank the Wilhelm und Else Heraeus-Foundation for support. J.L. would like to thank the DFG for support. S.K. acknowledges support from the European Cluster of © The Author(s) 2017 Advanced Laser Light Sources (EUCALL) project which has received funding from the 8 NATURE COMMUNICATIONS 8: 2266 DOI: 10.1038/s41467-017-02437-9 www.nature.com/naturecommunications | | |
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