Single-Molecule X-Ray Interferometry: Controlling Coupled Electron-Nuclear Quantum Dynamics and Imaging Molecular Potentials by Ultrahigh-Resolution Resonant Photoemission and Ab Initio Calculations

This paper reports an advanced study of the excited ionic states of the gas-phase nitrogen molecule in the binding-energy region of 22–34 eV, combining ultrahigh-resolution resonant photoemission (RPE) and ab initio configuration-interaction calculations. The RPE spectra are recorded for nine photon energies within the N $1s\to {\pi }^{*}$ absorption resonance of ${\mathrm{N}}_2$ by using a photon bandwidth that is considerably smaller than lifetime broadening, and the dependence on excitation energy of the decay spectra is analyzed and used for the first assignment of 12 highly overlapped molecular states. The effect on the RPE profile of avoided curve crossings between the final ${\mathrm{N}}_2^{+}$ ionic states is discussed, based on theoretical simulations that account for vibronic coupling, and compared with the experimental data. By use of synchrotron radiation with high spectral brightness, it is possible to selectively promote the molecule to highly excited vibrational sublevels of a core-excited electronic state, thereby controlling the spatial distribution of the vibrational wave packets, and to accurately image the ionic molecular potentials. In addition, the mapping of the vibrational wave functions of the core-excited states using the bound final states with far-from-equilibrium bond lengths has been achieved experimentally for the first time. Theoretical analysis has revealed the rich femtosecond nuclear dynamics underlying the mapping phenomenon.

Popular Summary

Absorption of an x-ray photon of proper wavelength by a molecule can take the molecule to a highly excited electronic state accompanied by molecular vibration. Molecules in these states are very reactive chemically. Such highly excited and reactive molecules are ubiquitous and important in diverse fields such as plasma physics, astrochemistry, radiation physics, and photochemistry. But, how much do we actually understand the quantum dynamics of these unstable species? The answer is: not yet much, even in the case of one of the simplest molecules, the nitrogen molecule. In this work, we apply state-of-the-art resonant photoemission spectroscopy to the study of the electron and nuclear dynamics of highly excited nitrogen molecules and map out, with an unprecedented resolution and degree of comprehensiveness, the vibrational wave functions and their ultrafast evolution with time.

A highly electronically excited nitrogen molecule is very unstable and lives over an extremely short time scale of a few femtoseconds. Its decay yields a molecular ion through the so-called Auger process, where a high-energy electron is ejected when another electron falls into the hole left by the initial excited electron. The Auger-electron emission spectra naturally contain fundamental information about the extremely short-lived intermediate states, and not least, also on what the vibrational motion of the resulting molecular ion is like—knowledge that has not been available so far. It is such spectra that we measure experimentally and analyze.

Indeed, taking advantage of the high spectral brightness of the synchrotron radiation at the PLEIADES beam line at the SOLEIL Synchrotron, Paris-Saclay campus, we are able to selectively excite nitrogen molecules, control the spatial forms of the vibrational wave functions of the excited molecules, and follow the time evolution of the vibrational wave functions of the final molecular ions. Adding to our toolbox first-principles calculations of the evolution of the vibrational wave functions, that correctly account for the coupling between the nuclear and electronic motion in highly excited, and thus deformed, molecules, we have succeeded in not only mapping out for the first time the actual shapes of the vibrational wave functions of highly excited nitrogen molecules, but also identifying a few ionic states that have not been known before.

This approach is not only limited to the nitrogen molecule: It can be extended to studies of excited ionic states of even larger molecules, and can also be easily transposed to neutral molecular states by detecting the radiative decay instead of the electron emission by Auger decay.

Figure 1

Experimental x-ray absorption spectrum. The labels show photon energies used for the RPE spectra discussed in this paper: 400.88 eV ($\nu =0$), 401.00 eV ($\nu =0$ and 1), 401.11 eV ($\nu =1$), 401.24 eV ($\nu =1$ and 2), 401.33 eV ($\nu =2$), 401.54 eV ($\nu =3$), 401.75 eV ($\nu =4$), 401.96 eV ($\nu =5$), and 402.15 eV ($\nu =6$).

Figure 2

Ab initio potential-energy curves of the ground (lower panel), intermediate (upper panel), and relevant final states (middle panel). The centers of the Franck-Condon regions for the transition from the ground and core-excited states are shown by the vertical dashed arrows.

Figure 3

Experimental resonant Auger spectra with x-ray photon energy tuned to the $\nu =0$ vibrational resonance of the N $1s\to {\pi }^{*}$ core-excited electronic state (see Fig. 1). The assignment is made using the original ab initio calculations and some previous photoelectron studies; see Refs. 10, 16, 18, 20.

Figure 4

Experimental (red lines) vs theoretical (black lines) RPE cross sections for several excitation energies (see Fig. 1). The main contributions of various final states for the excitation of $\nu =0$ and $\nu =3$ are singled out and presented in detail in Figs. 5, 6, 7. All 12 final-state contributions for the excitation of $\nu =1$ and $\nu =2$ are shown in Supplemental Fig. 1 [48].

Figure 5

The details of the experimental and theoretical RPE spectra (see Fig. 4) in the binding-energy range of 22–24.6 eV. The spectral structure for the resonant excitation to $\nu =0$ (upper plot) and $\nu =3$ (lower plot) sublevels of the core-excited state are shown. The profiles of the individual states of different symmetries are shifted horizontally for reasons of clarity (see the labels in the upper panel). The total theoretical spectrum is shown by a solid black line, while blue dots represent the experimental spectrum. Theoretical contributions for only the most important states are shown. The colors and line types of the individual electronic states are the same as those used for the PECs in Fig. 2 (as in Figs. 6, 7).

Figure 6

The details of the experimental and theoretical RPE spectra (Fig. 4) in the binding-energy range of 24–27 eV. The excitation energies and line notations are the same as in Fig. 5.

Figure 7

The details of the experimental and theoretical RPE spectra (see Fig. 4) in the binding-energy range of 27–30 eV. The excitation energies and line notations are the same as in Fig. 5.

Figure 8

The PECs of the (a) ${}^2\Pi _g$ and (b) ${}^2\Pi _u$ states around the avoided crossing points. The marked points (see the legends) show the ab initio adiabatic energies, and dashed lines propose the $R$ dependence of the diabatic states. The equilibrium position of the core-excited state (FC region) is shown with the vertical dashed line.

Figure 9

(a) RPE spectra related to the isolated $1^2{\Pi }_g$ and $2^2{\Pi }_g$ states compared to the calculations that take into account the vibronic coupling between them. (b) Total theoretical RPE profiles related to the ${}^2\Pi _u$ states computed using the diabatic (shaded areas) and adiabatic (dashed lines) PECs (see Fig. 8).

Figure 10

Illustration of the PEC mapping in the framework of the high-resolution RPE spectroscopy. (a) The seven lowest stationary vibrational wave functions in the core-excited state. (b) Comparison between the reconstructed molecular potentials of the $1^2{\Pi }_g$ and $1^2{\Delta }_g$ final ionic states (solid lines) based on the ultrahigh-resolution RPE data and ab initio calculated potentials (open circles). The uncertainty in the reconstructed potential curves is represented by the thickness of the lines. The right turning point of the core-excited wave packet at $\nu =6$ is shown by the dashed line. (c) The experimental RPE spectra are presented in relation to the reconstructed PECs.

Figure 11

(a) Geometric interpretation of the phenomenon of mapping of the core-excited vibrational wave functions onto a bound final state based on the reflection principle (see the text), using the $1^2{\Pi }_g$ state as an example. The squared moduli of the core-excited vibrational wave functions ($\nu =0,\dots ,3$) are shown in the inset. The FC region is marked by the vertical dashed lines. (b) Theoretical RPE spectra of the $1^2{\Pi }_g$, $2^2{\Sigma }_g^{+}$, and $3^2{\Sigma }_g^{+}$ final bound states resulting from excitation of the $\nu =0,\dots ,3$ core-excited vibrational wave functions. The spectral profile reflects the nodal structure of the corresponding vibrational wave function [see the inset in (a)].

Figure 12

The time-dependent dynamics of the vibrational wave packets in the final $1^2{\Pi }_g$ state $||\Psi (\tau )⟩|$ (2) following resonant excitations to the vibrational sublevels $\nu =0,\dots ,3$ of the core-excited state.