Anomalous polarization dependence in vibrationally resolved resonant inelastic x-ray scattering of ${\mathrm{H}}_2\mathrm{O}$

It is well established that different electronic channels, in resonant inelastic x-ray scattering (RIXS), display different polarization dependences due to different orientations of their corresponding transition dipole moments in the molecular frame. However, this effect does not influence the vibrational progression in the Franck-Condon approximation. We have found that the transition dipole moments of core excitation and deexcitation experience ultrafast rotation during dissociation in the intermediate core-excited state. This rotation makes the vibrational progression in RIXS sensitive to the polarization of the x-ray photons. We study the water molecule, in which the effect is expressed in RIXS through the dissociative core-excited state where the vibrational scattering anisotropy is accompanied also by violation of parity selection rules for the vibrations.

Figure 1

Surfaces of the components of the transition dipole moment ${\mathbf{d}}_{c0}$ as functions of the bond lengths $R_1$ and $R_2$. Panel (a) displays the antisymmetric $y$ component while panel (b) shows the symmetric $z$ component.

Figure 2

Schematic picture of opposite rotation of the transition dipole moment in the course of coherent dissociation along different pathways. The green color map displays the potential-energy surface of the dissociative $|\text{O}1s^{-1}4a_1^1\rangle$ core-excited state as a function of the bond lengths $R_1$ and $R_2$. The transition dipole moment ${\mathbf{d}}_{c0}$ is shown by the orange arrows.

Figure 3

Natural orbitals involved in the dissociation of the OH bond. The orbitals are extracted from the multielectronic RASSCF wave functions of the ground state and the $|\text{O}1s^{-1}4a_1\rangle$ core-excited state. The upper row displays the $3a_1$ and $4a_1$ orbitals at equilibrium geometry $R_1=R_2=R_{\text{eq}}=1.81$ a.u. while the bottom row shows the same in the region of dissociation $R_1=R_{\text{eq}}=1.81$ a.u., $R_2=6$ a.u.

Figure 4

Panels (a) and (b) show how the $z$ and $y$ components of the transition dipole moment effectively change the symmetry of the initial vibrational state. Panels (c)–(f) show that contrary to ${\mathrm{\Psi }}_{zz}\left(0\right)$ and ${\mathrm{\Psi }}_{yy}\left(0\right)$ the $y$ component of ${\mathbf{d}}_{c0}$ makes antisymmetric the wave packets ${\mathrm{\Psi }}_{zy}\left(0\right)$ and ${\mathrm{\Psi }}_{yz}\left(0\right)$.

Figure 5

Total cross section $\sigma$ (solid green) and partial cross sections ${\sigma }_{\text{odd}}$ (dotted red) and ${\sigma }_{\text{even}}$ (dashed blue) [Eq. (17)] for $\chi =0^{\circ }$ (upper panel) and $\chi ={90}^{\circ }$ (lower panel). The insets in both panels compare the strict $\sigma$ (thin green) and $\sigma$ computed using the FC approximation (thick black). The spectra are normalized so that the peak $n=1$ in $\sigma$ equals 1.

Figure 6

Polarization dependence of the ratio of the areas of the $n\mathrm{th}$ and the first peaks in the RIXS spectrum [see Eq. (18)].

Figure 7

(a) Polarization dependence of the ratio ${\rho }^{\left(1\right)}$ of intensities of the symmetry allowed (10) and forbidden (01) RIXS channels [see Eq. (19)]. Panel (b) displays $\sigma$ for the peak $n=1$ in the FC approximation. Panels (c) and (d) show the same taking into account the rotation of the transition dipole moment for $\chi =0$ and ${90}^{\circ }$, respectively. The total profile shown by the thick green line is calculated using a Gaussian with half width at half maximum equal to 0.02 eV. The RIXS cross section bars for levels ${\epsilon }_{1,0}$ and ${\epsilon }_{0,1}$ are shown by thick blue and magenta bars, respectively. The vertical thin green lines show weak dependence of the peak position of unresolved peaks on the scattering angle $\chi$.