Concept of correlation imaging

The properties of a photoionization event, given by the ionization amplitude D, are determined (within the commonly used dipole approximation) by only three ingredients: the initial state of the system ϕ 0 , which we want to image, the properties of the dipole operator \(\hat \mu \) (responsible for the photoionization) and the final state representing the remaining cation and a photoelectron with momentum k, χ k :

$$D = {\int} \phi _0({\mathbf{r}})\hat \mu ({\mathbf{r}})\chi _{\mathbf{k}}({\mathbf{r}}){\mathrm d}{\mathbf{r}},$$ (1)

where r represents the coordinates of target electrons. The initial wave function is directly accessible provided that the other two constituents do not introduce significant distortions. This is the case when utilizing circularly polarized light and examining high energy electrons (Born limit) within the polarization plane. As an illustration, let us consider the one-electron \({\mathrm{H}}_2^ + \) molecular ion. At a high enough energy, the continuum electron can be described by a plane wave. In this case, the photoionization differential cross section in the electron emission direction (θ, φ) (the so-called molecular frame photoelectron angular distribution, MFPAD) is simply proportional to the square of the Fourier transform (FT) of the initial state, ϕ 0 (k) (see methods section):

$$\frac{{{\mathrm d}P}}{{{\mathrm d}({\mathrm{cos}}\,\theta ){\mathrm d}k}} = k^2(2\pi )^{3/2}\left| {\frac{1}{{2\pi ^{3/2}}}{\int} \phi _0({\mathbf{r}})e^{i{\mathbf{kr}}}{\mathrm d}{\mathbf{r}}} \right|^2$$ (2)

$$ = k^2(2\pi )^{3/2}\left| {\phi _0({\mathbf{k}})} \right|^2.$$ (3)

Here θ denotes the polar angle with respect to the molecular axis and φ the corresponding azimuthal angle. Thus, by choosing high-photon energies and restricting the measurement of the MFPAD to the polarization plane (φ = 90° and 270°) of the circularly polarized light, the initial electronic wave function is directly mapped onto the emitted photoelectron. Figure 1 illustrates this mapping procedure for the ground state of \({\mathrm{H}}_2^ + \) (Fig. 1a: electronic wave function in coordinate space; Fig. 1b the square of the Fourier transform of Fig. 1a; Fig. 1c the same in logarithmic color scale). As can be seen from Fig. 1d, the MFPAD for an electron of 380 eV is very similar to \(\left| {\phi _0({\mathbf{k}})} \right|^2\) for the chosen momentum k (the square of the FT along the dashed line shown in Fig. 1c). Notice that, due to the smallness of the cross section at such high electron momentum, the main features of the FT are only apparent in the logarithmic plot shown in Fig. 1c.

Fig. 1 Imaging of the \({\mathrm{H}}_2^ + \) one-electron wave function. a The electronic wave function of \({\mathrm{H}}_2^ + \) in the polarization plane for an internuclear distance R = 1.4 a.u. The positions of the two nuclei are indicated by black dots. b The square of the Fourier transform of a in the (k x , k y ) plane. c The same as (b), but in logarithmic color scale. Notice the appearance of nearly vertical fringes, when \(\left| {\bf{k}} \right|\) is significantly different from zero. The approximate periodicity of these fringes is \({\mathrm{\Delta }}k_x\sim 2\pi {\mathrm{/}}R\). The dashed line indicates the region of momentum space associated with an electron kinetic energy of 380 eV (i.e., a radius of \(\left| {\bf{k}} \right| = 5.3\) 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) Full size image

This tool of high energy photoelectron imaging can now be combined with coincident detection of the quantum state of a second electron to visualize electron correlation in momentum space. We dissect the entangled two-electron wave function by analyzing a set of conditional angularly resolved cross sections corresponding to a high energy continuum electron (A) and a bound electron (B) detected in a different region of the two-electron phase space. Quantum mechanically, this is equivalent to projecting the initial two-electron wave function onto products of different \({\mathrm{H}}_2^ + \) (bound) molecular orbitals (B) and a plane wave (A) (see Methods section). In doing so, one can thus determine if and how the density distribution of one electron changes upon changing the region of phase space in which one detects the other, correlated, electron.

Application on H 2

Figure 2 illustrates this concept and highlights the differences between the uncorrelated Hartree–Fock wave function and the highly correlated nearly exact wave function. The corresponding one-electron momentum distributions resulting from the projection of the corresponding ground state wave functions onto different states of the bound electron B, \(n_\lambda \), are depicted in Fig. 2a–c (Hartree–Fock) and Fig. 2d–f (exact) as functions of the momentum components parallel (k x,A ) and perpendicular (k y,A ) to the molecular axis. The different rows correspond to the different states in which the second electron B is left after photoionization, i.e., they correspond, from bottom to top, to projections of the ground state wave function onto the \(n_\lambda = 1s\sigma _{\mathrm{g}}\), \(2s\sigma _{\mathrm{g}}\), and \(2p\sigma _{\mathrm{u}}\) states of \({\mathrm{H}}_2^ + \). Thus, as in our one-electron example shown in Fig. 1, the different panels in Fig. 1 contain direct images of different pieces of the ground state of H 2 through the square of the corresponding FTs. The role of electron correlation is quite apparent in this presentation: Fig. 1a is empty for the uncorrelated Hartree–Fock wave function, since projection of the latter wave function onto the \(2p\sigma _{\mathrm{u}}\) orbital is exactly 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.

Fig. 2 Correlation imaging of the H 2 two-electron wave function. a–f Momentum distributions of electron A resulting from the projection of the two-electron wave function of H 2 onto different \({\mathrm{H}}_2^ + \) 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 \(2p\sigma _{\mathrm{u}}\) (a, d), \(2s\sigma _{\mathrm{g}}\) (b, e) and \(1s\sigma _{\mathrm{g}}\) (c, f). Circular lines show \(\left| {\bf{k}} \right| = 5.3\) a.u. (c, d, f) and \(\left| {\bf{k}} \right| = 5.2\) a.u. b, e which 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 Full size image

Experimentally, these conditional probabilities are obtained by measuring in coincidence the momentum of the ejected electron and the proton resulting from the dissociative ionization reaction

$$\gamma (400\,{\mathrm{eV}}) + {\mathrm{H}}_2 \to {\mathrm{H}}_2^ + (n_\lambda ) + e^ - $$ (4)

$$ \searrow $$

$${\mathrm{H}}(n) + {\mathrm{H}}^ + ,$$ (5)

which, as explained below, allows us to determine the final ionic state characterized by the quantum number \(n_\lambda \). Fig. 2j–l depicts the experimental results of the measured angular distributions of 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 are very similar to the calculated projections in momentum space of the fully correlated ground state wave function shown in Fig. 2g–i. In other words, the momentum of the ejected photoelectron faithfully reflects and maps the momentum of a bound state electron in the molecular ground state when the momentum of the second bound electron is constrained by projection of the H 2 wave function onto different molecular-ion states; this represents the correlation between the two electrons. Note in particular Fig. 2g is not empty and Fig. 2h, i are not identical, as it would be for an uncorrelated H 2 ground state (compare with Fig. 2a–c).

Identifying the quantum state of the second electron

In more detail, the angular emission distributions and the final quantum state of electron B are obtained in our experiment by measuring the momenta of the charged particles generated by the photoionization process in coincidence. As the singly charged molecule dissociates in the cases presented here into a neutral H atom and a proton, we can obtain the spatial orientation of the molecular axis by measuring the vector momentum of the proton (i.e., its emission direction after the dissociation). The electron emission direction in the molecular frame is then deduced from the relative emission direction of the proton and the vector momentum of the electron. Additionally, the magnitude of the measured ion momentum provides the kinetic energy release (KER) of the reaction. The latter enables an identification of the quantum state of electron B (i.e., the \({\mathrm{H}}_2^ + \) electronic state), which is demonstrated in Fig. 3. Fig. 3a shows the relevant potential energy curves of \({\mathrm{H}}_2^ + \) and Fig. 3b the measured (and theoretically predicted) KER spectra. From the measured sum of the kinetic energies of the electron and the proton we furthermore identify the asymptotic electronic state of the neutral H fragment (not detected in the experiment), mostly H(n = 1) and H(n = 2).

Fig. 3 Correlation diagram and kinetic energy distribution for dissociation of \({\mathrm{H}}_2^ + \) a Potential energy curves for the ground state of H 2 (lower curve) and the \(1s\sigma _{\mathrm{g}}\), \(2s\sigma _{\mathrm{g}}\), and \(2p\sigma _{\mathrm{u}}\) ionization thresholds (upper curves). The latter correspond to electronic states of \({\mathrm{H}}_2^ + \). The violet shaded area represents the Franck-Condon region associated to the ground vibrational state of H 2 . Notice the break in the energy scale for a better visualization. The dashed violet line shows how the initial internuclear distance of the molecule is mapped onto the kinetic energy release (KER) of the reaction applying the”reflection approximation”9. b KER distribution obtained after single-photon ionization of H 2 employing photons of hν = 400 eV. Symbols: experiment, lines: theory. The calculation depicted by the black curve includes the twelve states with the highest photo ionization cross sections (up to n = 4). The main contributions (besides \(1s\sigma _{\mathrm{g}}\) at low KER) are shown in blue (\(2s\sigma _{\mathrm{g}}\)) and red (\(2p\sigma _{\mathrm{u}}\)), others are not visible on that scale. The shaded areas indicate the regions of KER selected in Figs. 2d–f and 4a, c Full size image

Nodal structure of the wave function

Our experimentally obtained spectra not only show the imprint of correlation, but also allow us to separate the contribution of different pieces of the electronic wave function to this correlation. Indeed, the momentum distribution of electron A depends strongly on the properties of electron B. The most dramatic example can be seen by comparing the upper and middle rows in Fig. 2, which show electron A under the condition that electron B is detected in the \(2p\sigma _{\mathrm{u}}\) and \(2s\sigma _{\mathrm{g}}\) states of \({\mathrm{H}}_2^ + \), respectively. Upon this change in the selection of electron B, the maxima in the momentum distribution of electron A become minima and vice versa. This can be intuitively understood in coordinate space. The maxima in the k-space distribution correspond to the constructive interference of the part of the electron density close to one or the other nucleus spaced by R. Thus, inverting maxima to minima in k-space corresponds to a phase shift of π between the wave function at one or the other nucleus in coordinate space. For H 2 , the two-electron wave 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, this symmetry consideration is also valid for each individual electron (it reflects the fact that both electrons occupy the \(1s\sigma _{\mathrm{g}}\) orbital most of the time). Therefore, both electrons have the same phase at both nuclei, which, in turn, is directly reflected in the maximum at k x = 0 and the corresponding maximum in the direction perpendicular to the molecular axis in Fig. 2e, f. Due to electron correlation, however, this is not strictly true for all parts of the wave function: Projecting electron B onto the \(2p\sigma _{\mathrm{u}}\) state highlights this small fraction of the wave function where electron A has the opposite phase at the two nuclei. As explained before, this part of the wave function does not exist for a Hartree–Fock wave function and Fig. 2a is therefore empty. This phase change of the wave function between the nuclei leads to the nodal line through the center in Fig. 2d and the nodes in Fig. 2g, j in the direction perpendicular to the molecular axis.

In addition to identifying the final state of electron B, the measured KER provides further insights into the ionized H 2 molecule. As soon as the potential energy curve relevant for the process is known, one can infer the internuclear distance R of the two atoms of the molecule at the instant of photoabsorption by using the reflection approximation9 (see Fig. 3). This allows us to investigate more details of the two-electron wave function: The distributions in Fig. 2d–f shows nodal lines that lead to corresponding nodes in the angular distributions in Fig. 2g–i. As mentioned above, these nodes in k-space are separated by \({\mathrm{\Delta }}k_x = 2\pi {\mathrm{/}}R\). Within the range of R covered by the Franck-Condon region, the nodal structure of the electronic wave function changes significantly and Fig. 4 demonstrates how the k-space distribution of the two-electron wave function changes accordingly as a function of R (or KER respectively). The corresponding experimental and theoretical MFPADs resulting from high energy photoionization follow a similar pattern.

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 \(2s\sigma _g\) 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 \(2s\sigma _{\mathrm{g}}\) potential energy curve. The error bars indicate the standard deviation of the mean value Full size image

In conclusion, high energy angular resolved photoionization is a promising route to access molecular wave functions in momentum space. The process of molecular dissociation in combination with shake up of the bound electron is universal by its nature. Shake up of an electron into a continuum state instead of a bound state, i.e., double ionization of the molecule, might also come into play. Therefore, this approach can in principle be extended to molecules with more than two electrons. In detail, it depends on the shape of the potential energy surfaces which determines to which extend different ionic states can be separated by the kinetic energy of the fragments. Combined with coincidence detection, this technique opens the door to image correlations in electronic wave functions. Similar approaches have also been proposed for imaging correlations in superconductors10. With the advent of X-ray free electron lasers and the extension of higher harmonic sources to high photon energies, such correlation imaging bares the promise to make movies of the time evolution of electron correlations in molecules and solid materials.