Time-dependent quantum description of molecular double-core-hole-state formation: Impact of the nuclear dynamics on sequential two-photon processes

We present a theoretical model to investigate double-core-hole-state formation in molecules through sequential absorption of two x-ray photons from a femtosecond laser pulse. A complete time-dependent quantum description taking into account x-ray absorption and nuclear dynamics explicitly and Auger decay phenomenologically is established within the local approximation. Using this model, we assess the impact of the nuclear dynamics on double-core photoionization processes in the case of the carbon monoxide molecule. We show that sequential absorption of two x-ray photons modifies significantly the vibrational distribution of the photoelectron spectra of the double-core-hole states compared to direct single x-ray photon absorption. Depending on the shape of the potential energy curves involved in the sequential absorption processes, lower or higher vibrational levels may be favored. Furthermore, in the case where the final state is dissociative, the electron spectrum is further broaden and blue-shifted in the two-photon process.

Figure 1

Ab initio PECs for the ground, SCH, single-site and two-site DCH states of CO, calculated at the CISD level of theory. The equilibrium internuclear distances are indicated for each electronic state.

Figure 2

Left panel: Photoelectron-photoelectron coincidence spectrum for ${\left[\mathrm{C}\left({\mathrm{K}}^{-2}\right)\mathrm{O}\right]}^{2+}$ formation (logarithmic scale in arbitrary units). The labels 0–3 indicate the vibrational level of the SCH state. The labels a–e indicate the vibrational level of the DCH state. See text for more details. Right panel: Comparison of the spectra of the total photoelectron energy ${\varepsilon }_T$. Black solid line: Two-photon process including nuclear dynamics. Green dotted line: Two-photon process simulated from vertical transitions at the equilibrium internuclear distance of the ground electronic state. Brown dashed line: One-photon process including nuclear dynamics. The three spectra are scaled with respect to their most intense peak. The red vertical bars correspond to the Franck-Condon factors for the production of each (a–e) vibrational level of the DCH state.

Figure 3

Left panel: Photoelectron-photoelectron coincidence spectrum of ${\left[\mathrm{CO}\left({\mathrm{K}}^{-2}\right)\right]}^{2+}$ state (logarithmic scale in arbitrary units). Right panel: Comparison of the spectra of the total photoelectron energy ${\varepsilon }_T$. For further details on the different spectra, see the caption of Fig. 2.

Figure 4

Left panel: Photoelectron-photoelectron coincidence spectrum of the ${\left[\mathrm{C}\left({\mathrm{K}}^{-1}\right)\mathrm{O}\left({\mathrm{K}}^{-1}\right)\right]}^{2+}$ state, with ${\left[\mathrm{C}\left({\mathrm{K}}^{-1}\right)\mathrm{O}\right]}^{+}$ as the intermediate SCH state (logarithmic scale in arbitrary units). Right panel: Comparison of the spectra of the total photoelectron energy ${\varepsilon }_T$. Red dashed and blue solid vertical bars: Contributions of the singlet and triplet cationic states, respectively. For further details on the different spectra, see the caption of Fig. 2.

Figure 5

Left panel: Photoelectron-photoelectron coincidence spectrum of the ${\left[\mathrm{C}\left({\mathrm{K}}^{-1}\right)\mathrm{O}\left({\mathrm{K}}^{-1}\right)\right]}^{2+}$ state, with ${\left[\mathrm{CO}\left({\mathrm{K}}^{-1}\right)\right]}^{+}$ as the intermediate SCH state (logarithmic scale in arbitrary units). Right panel: Comparison of the spectra of the total photoelectron energy ${\varepsilon }_T$. The difference between the $(2\omega ,{\varepsilon }_T)$ spectra with the carbon and oxygen as intermediate SCH states is given as gray shaded area, together with the scaling factor. Red dashed and blue solid vertical bars: Contributions of the singlet and triplet cationic states, respectively. For further details on the different spectra, see the caption of Fig. 2.