Time-resolved pump-probe spectroscopy to follow valence electronic motion in molecules: Application

Numerical experiments are performed using a recently published formalism [Phys. Rev. A 88, 013419 (2013)] for computing the transient absorption of an attosecond x-ray probe by a molecule in a nonstationary valence-excited state. This study makes use of an all-electron correlated electronic-structure model to compute electronic dynamics ensuing after a localized excitation on a model chromophore group. Simulated absorption of a delayed attosecond pulse is then used to investigate the presence of an extra valence electron or vacancy around atoms of a given element as a function of time, by tuning the carrier frequency to the associated core-valence energy gap. We show correlations between the predicted absorption of such pulses and visualizations of the particle and hole locations in test molecules. Given the strong role played by the relative orientation of the molecules and the probe polarization, results are presented for a few different alignment schemes. For the molecules studied, effective pump-induced alignment is sufficient to recover easily interpreted information.

Figure 1

Cartoon of the attosecond probe process. A dynamic superposition of valence-excited states (a) is promoted to a state (b) in the space of core-valence excitations by an x-ray probe. The attosecond duration of this probe, and the associated bandwidth, renders the excitation process atomically local, and the transient absorption therefore provides information about the time-dependent valence occupancy around atoms of a targeted element type. Electronic occupancy is represented here by shading in a site-local basis, consisting of one (nominally) occupied and virtual orbital per atom, shown as circles for simplicity. The lower cartoon illustrates a state with an excited electron (particle) spatially removed from a valence vacancy (hole), with both in motion. The completely occupied core orbitals are shown with different sizes, indicative of the extremely large difference in their element-specific binding energies, which allows for elemental specificity of the probe excitation; in this case, the third atom from the left (where the valence hole happened to be) has been targeted.

Figure 2

Molecules simulated in this work. The left-hand system contains a neutral organic acid group $\left(–{\mathrm{CO}}_2\mathrm{H}\right)$, which contains a strongly positively charged C atom and negatively charged O atoms, and the right-hand system contains an overall positive ammonium group $\left(–\mathrm{NH}{}_3^{+}\right)$. Unlabeled atoms are carbon (black) or hydrogen (white). The principal inertial axes of each molecule are aligned with the Cartesian frame shown, and the center of mass is located at the coordinate origin (molecules have mirror symmetry across the plane of the page). The angle between the long axis of the molecule and the C=C bond is illustrated for each molecule $({\alpha }^{\prime }=21.{05}^{\circ },{\alpha }^{\prime \prime }=22.{26}^{\circ })$. A tic is placed one atomic unit $\left(a_0\right)$ from the origin on the $x$ axis for scale.

Figure 3

System dipole moments, starting from nonstationary $\pi \to {\pi }^{*}$ excitations localized over the C=C bond. (a) Trajectory of the molecular dipole vector in the $xz$ plane, relative to the dipole of the ground state (in atomic units, $e{a}_0$). This is clear evidence of spontaneous charge separation along the long $\left(z\right)$ axis of the molecule, equivalent to unit charges displaced by a few Bohr radii (see Fig. 2 for orientation and scale); the $y$ component is always negligible, due to symmetry. The marked points occur at evenly spaced time intervals, corresponding to the marked points on the thick lines of panel (b). (b) The $z$ component of the difference in the dipole as a function of time. The thick lines are the $z$ components of the trajectories in panel (a) (same vertical axis). The thin lines are decompositions of these into particle and hole contributions. This presents numerical evidence of the result (also visualized in Figs. 5 and 6) that, depending on the system, the overall dipole trajectory is dominated by either the travel of the particle toward the $–{\mathrm{CO}}_2\mathrm{H}$ group or the travel of the hole away from the $–\mathrm{NH}{}_3^{+}$ group. The hatched points on the thin lines correspond the first three visualization frames in Figs. 5 and 6 for the respective molecules.

Figure 4

Color and brightness scale for visualizations in Figs. 5 and 6. The difference in the total electron density, relative to the ground state, will be decomposed into particle and hole contributions, whose magnitude is mapped to a color. These densities will be integrated in the out-of-page $\left(y\right)$ direction, giving units of charge per area ($e/a_0^2$, in atomic units). The particle contribution represents an increase in total density, due to the excited electron, and the hole contribution represents a depletion, due to the vacancy left behind, the sum of which gives the observable change in density that integrates to zero overall. Since the scale used is logarithmic, in order to visualize both high- and low-density regions, small changes in color can represent a significant redistribution of charge.

Figure 5

Delay-dependent absorptivity of individual atoms in the $–{\mathrm{CO}}_2\mathrm{H}$ system for probe polarization orthogonal to the molecular symmetry plane. In this simulation, the molecule is fixed at a single orientation in space. The change in absorptivity $\Delta {\sigma }_{\mathrm{eff}}$ for each atom at delay $T$ is directly proportional to the increase or decrease (relative to the ground state) in the probability of the probe promoting an electron from a core orbital at that site to the valence. Since only the combined absorption from all atoms of the same type is meaningful in the formalism used here, the lower panel gives this sum. The relevant dynamics is visualized at key times in terms of the particle and hole contributions to the electron density difference (molecule rotated relative to Fig. 2; see Fig. 4 for color scale). A clear correlation can be seen between the probability of excitation at the F atom and the extent to which the hole is at that atom. Similarly, the probability of excitation at the acidic C atom decreases as the particle moves there. For the atom-summed signals, the F and C curves are seen to be countercorrelated in time.

Figure 6

Delay-dependent absorptivity of individual atoms in the $–\mathrm{NH}{}_3^{+}$ system for probe polarization orthogonal to the molecular symmetry plane. The lower panel gives the atom-summed signal, and the relevant particle and hole dynamics are visualized at key points in time. A clear correlation is again seen between the probability of excitation at the F atom and the extent to which the hole is at that atom. A couple of brief, weak fluctuations of the particle towards the $–\mathrm{NH}{}_3^{+}$ group can be resolved as dips in the absorptivity of the N atom. For the atom-summed signals, the F and C curves are again counter-correlated in time.

Figure 7

Time-resolved absorptivity curves for various alignment cases of the $–{\mathrm{CO}}_2\mathrm{H}$ system. In the left-hand panels, the angle $\beta$ between the pump and probe polarization is varied, whereby the pump causes an effective alignment of the excited molecules. The lower two panels on the right-hand side are for molecules with an overall pre-pump alignment (in addition to effective alignment by the pump), and the top right panel is the isotropic “control” case of no alignment at all. In the isotropic case, and when $\beta ={45}^{\circ }$, the most easily interpreted structure is washed out. When the pump and probe polarizations are orthogonal ($\beta ={90}^{\circ }$, and for the prealigned molecules), there is increased probability that the probe is orthogonal to the mirror plane of the molecule, enhancing the signal, especially when the molecules are prealigned. When $\beta =0^{\circ }$, similar structure is apparent, but inverted; this is attributed to this pulse geometry being more sensitive to electronic relaxations than to the primary particle-hole structure of the excitation.

Figure 8

Time-resolved absorptivity curves for various alignment cases of the $–\mathrm{NH}{}_3^{+}$ system. In the left-hand panels, the angle $\beta$ between the pump and probe polarization is varied. The lower two panels on the right-hand side are for molecules with an additional overall alignment, and the completely unaligned case is at the top right. As for the $–{\mathrm{CO}}_2\mathrm{H}$ system, the dynamic signal is enhanced when the pump and probe polarizations are orthogonal, and when the molecules are prealigned. The inverted structure in the $\beta =0^{\circ }$ case is attributed to sensitivity to electronic relaxations.

Figure 9

Individual elements of the tensors $\Delta {\mathit{a}}_Z^{\left[1\right]\mathrm{FN}}$ for each atom type $Z$ in the $–{\mathrm{CO}}_2\mathrm{H}$ system (on the left) and in the $–\mathrm{NH}{}_3^{+}$ system (on the right). This information is sufficient to compute, and therefore interpret, all of the foregoing data, except the signals contributions of individual atoms. Since the inertial axes have no special significance regarding the electronic structure, these tensor elements are given for a rotated orientation, in which the C=C bond is aligned with the $z$ axis (the $y$ axis still traverses the mirror plane). These plots are useful for understanding the sensitivity of certain pulse geometries to electronic relaxations, as opposed to the primary excitation (see text for discussion).