Violation of centrosymmetry in time-resolved coherent x-ray diffraction from rovibrational states of diatomic molecules

Owing to increasing applications of time-resolved coherent x-ray scattering for the investigation of molecular reaction dynamics, we develop a theoretical model for time-dependent x-ray diffraction from molecular and/or electronic motion in molecules. Our model shows that the violation of centrosymmetry (VOC) is a general phenomenon in time-resolved diffraction patterns. We employ our theoretical model to illustrate the VOC in time-resolved coherent x-ray diffraction from two oriented diatomic molecules undergoing rovibrational motion: lithium hydride (LiD) and hydrogen (HD). Our simulations show asymmetric x-ray diffraction images that reflect the directions of the molecular motions.

Figure 1

Molecular potential energy curves for the $X{}^1{\mathrm{\Sigma }}^{+}$ and $A{}^1{\mathrm{\Sigma }}^{+}$ states of lithium hydride. The molecule undergoes a vertical transition caused by some pump procedure, and the ensuing rovibrational motion of the excited molecular wave packet is imaged by short x-ray pulses as a function of pump-probe delay. The inset shows the Franck-Condon factors as a function of the vibrational number $v$ of the $A{}^1{\mathrm{\Sigma }}^{+}$ state for the rovibrational transition $J^{″}=0\to J^{\prime }=1$.

Figure 2

Right column: Density distribution of the molecular wave packet of LiD in the $A{}^1{\mathrm{\Sigma }}^{+}$ state in the $yz$ plane as a function of time after the pump pulse. The molecular axis (from Li to D atoms) is oriented such that the axes of the majority of molecules point toward positive $z$. Left column: Time-resolved molecular scattering intensities for 1-fs (FWHM) x-ray pulses from the oriented rovibrational motion of LiD as a function of pump-probe delay $t_d$. The azimuthal scattering angle $\phi =0^{\circ }$ corresponds to the positive $z$ direction. Owing to symmetry, the diffraction images and molecular densities are only presented in the upper half of the $yz$ plane.

Figure 3

Comparison of the molecular scattering intensities at azimuthal scattering angles $\phi =0^{\circ }$ and ${180}^{\circ }$ as a function of pump-probe delay. Note that the time-varying scattering intensities of the left and right panels oscillate out of phase for $q\ge 3.0$ a.u.

Figure 4

Asymmetry of the EADP for the LiD molecule as a function of momentum transfer $q$ for six different time delays. The asymmetry is defined in Eq. (43).

Figure 5

(a) Molecular potential curves of the $X{}^1{\mathrm{\Sigma }}_g^{+}$ and $B{}^1{\mathrm{\Sigma }}_u^{+}$ states of the hydrogen molecule. The inset shows the Franck-Condon factors for the rovibrational transitions $(v^{″}=0,J^{″}=0)\to (v^{\prime }=v,J^{\prime }=1)$. (b) Radial density distribution of the oriented rovibrational motion of HD in the $B{}^1{\mathrm{\Sigma }}_u^{+}$ state as a function of time.

Figure 6

Molecular scattering intensities for 1-fs (fwhm) x-ray pulses from the molecular rovibrational motion of HD as a function of pump-probe delay time $t_d$ at azimuthal scattering angles $\phi =0^{\circ }$ and ${180}^{\circ }$.