Full-dimensional treatment of short-time vibronic dynamics in a molecular high-order-harmonic-generation process in methane

We present derivation and implementation of the multiconfigurational strong-field approximation with Gaussian nuclear wave packets (MC-SFA-GWP)—a version of the molecular strong-field approximation which treats all electronic and nuclear degrees of freedom, including their correlations, quantum mechanically. The technique allows realistic simulation of high-order-harmonic emission in polyatomic molecules without invoking reduced-dimensionality models for the nuclear motion or the electronic structure. We use MC-SFA-GWP to model isotope effects in high-order-harmonic-generation (HHG) spectroscopy of methane. The HHG emission in this molecule transiently involves the strongly vibronically coupled ${}^{2}F_2$ electronic state of the ${\mathrm{CH}}_4{}^{+}$ cation. We show that the isotopic HHG ratio in methane contains signatures of (a) field-free vibronic dynamics at the conical intersection (CI); (b) resonant features in the recombination cross sections; (c) laser-driven bound-state dynamics; as well as (d) the well-known short-time Gaussian decay of the emission. We assign the intrinsic vibronic feature (a) to a relatively long-lived ($\ge 4$ fs) vibronic wave packet of the singly excited ${\nu }_4$ ($t_2$) and ${\nu }_2$ ($e$) vibrational modes, strongly coupled to the components of the ${}^{2}F_2$ electronic state. We demonstrate that these physical effects differ in their dependence on the wavelength, intensity, and duration of the driving pulse, allowing them to be disentangled. We thus show that HHG spectroscopy provides a versatile tool for exploring both conical intersections and resonant features in photorecombination matrix elements in the regime not easily accessible with other techniques.

Figure 1

MR-CIS Dyson orbitals for the three degenerate components of the lowest (${}^{2}F_2$) ionization channel in ${\mathrm{CH}}_4$. The orbitals are shown at the high-symmetry $T_d$ point. Isosurface levels are $0.02a_0^{-3/2}$.

Figure 2

Magnitude (a) and phase (b) of the dipole photoionization matrix elements [Eq. (37)] for the $D_3$ component of the ${}^{2}F_2$ ionization channel at the high-symmetry point (Fig. 1). The $\vec{k}$ vector is taken along the positive $X$ Cartesian direction. The cradle contributions [Eq. (19)] are significant beyond $\approx 150$ eV. Omitting the cradle terms leads to the “plane-wave” (PW) curve.

Figure 3

Nuclear autocorrelation function in methane cation. Initial nuclear wave packet $\left|c\right(0)\rangle$ is the ground vibrational state of the neutral species. Solid red line: ${\mathrm{CH}}_4$, this work. Solid blue line: ${\mathrm{CD}}_4$, this work. Dotted orange line: ${\mathrm{CH}}_4$ using field-diabatized cationic energy surface (Ref. [52]). Dotted purple line: ${\mathrm{CD}}_4$ from Ref. [52]. Panel (a) shows the squared absolute magnitude of the autocorrelation function. Panel (b) gives the phase of the autocorrelation function. Linear phase due to the zero-point energy has been subtracted. Dotted, gray vertical lines indicate time delays for the cutoff trajectories for 800- and 1600-nm driving fields.

Figure 4

Transient population of dominant singly excited vibronic basis functions in ${\mathrm{CH}}_4{}^{+}$. The initially prepared wave packet is ground vibrational state of the neutral molecule, placed on the $D_1$ diabatic surface ($D_1|\mathrm{gs}\rangle$, black squares). This vibronic basis function is directly coupled to the singly excited ${\nu }_2$ mode on the same electronic surface ($D_1|{\nu }_2\rangle$, red diamonds), singly exited ${\nu }_4$ mode on the $D_2$ and $D_3$ surfaces ($D_2/D_3|{\nu }_4\rangle$, purple triangles), and singly excited ${\nu }_3$ mode on the same surfaces ($D_2/D_3|{\nu }_3\rangle$, green inverted triangles).

Figure 5

Nuclear autocorrelation function in vibrationally excited methane cation. Initial nuclear wave function $\left|c\right(0)\rangle$ has a single vibrational quantum in the asymmetric bend (${\nu }_4, 1356/1028{\mathrm{cm}}^{-1}$ in ${\mathrm{CH}}_4/{\mathrm{CD}}_4$) or asymmetric stretch (${\nu }_3, 3204/2379{\mathrm{cm}}^{-1}$) vibrational modes. Panels and the scale of the plots is the same as in Fig. 3; see Fig. 3 caption for further details.

Figure 6

MC-SFA-GWP high-order-harmonic spectrum of methane for a single-sided recollision in 800-nm driving field. Only short trajectories are included. The ionization potential ($I_{\mathrm{p}}$) and the harmonic cutoff ($3.17U_{\mathrm{p}}+1.3I_{\mathrm{p}}$) are indicated by vertical grey lines. The labels are as follows (see text for details): Nuclear geometry frozen, no nuclear factors included: FS (${\mathrm{CD}}_4$: solid black line; ${\mathrm{CH}}_4$: dotted black line). Nuclear geometry frozen, autocorrelation factor included: AC (${\mathrm{CD}}_4$: blue; ${\mathrm{CH}}_4$: red). MC-SFA-GWP, vibronic terms due to laser coupling omitted: ND (${\mathrm{CD}}_4$: purple; ${\mathrm{CH}}_4$: orange). MC-SFA-GWP, all terms included: DS (${\mathrm{CD}}_4$: teal; ${\mathrm{CH}}_4$: magenta). Panel (a) 300 TW ${\mathrm{cm}}^{-2}$. The ${\mathrm{CD}}_4$ FS and ${\mathrm{CH}}_4$ FS (not shown) curves are visually indistinguishable. (b) 1 PW ${\mathrm{cm}}^{-2}$.

Figure 7

MC-SFA-GWP high-order-harmonic spectrum of methane in 1.2-$\mu \mathrm{m}$ driving field. (a) 300 TW ${\mathrm{cm}}^{-2}$. (b) 1 PW ${\mathrm{cm}}^{-2}$. Also see Fig. 6 caption. The HHG spectra for the frozen nuclear configuration are off the scale, and are not shown.

Figure 8

MC-SFA-GWP high-order-harmonic spectrum of methane in 1.6-$\mu \mathrm{m}$ driving field. (a) 300 TW ${\mathrm{cm}}^{-2}$. (b) 1 PW ${\mathrm{cm}}^{-2}$. Also see Fig. 6 caption. The HHG spectra for the frozen nuclear configuration are off the scale, and are not shown. Parts of the ND and DS spectra between the dotted vertical grey line and the cutoff may show increased computational errors; see Sec. 3d.

Figure 9

Isotope effects in methane for the 800-nm driving field. Vertical axis: The ratio of the calculated HHG yields for ${\mathrm{CD}}_4$ and ${\mathrm{CH}}_4$ (Fig. 6). Horizontal axis: Time delay between ionization and recollision events, calculated from the harmonic energy using classical simple man's model (see text). AC (blue line with diamonds): direct product of the autocorrelation function and frozen-nuclei electron dynamics; ND (green line with triangles): correlated treatment, field-induced bound-state dynamics neglected; DS (red line with squares): the full treatment, including field-induced vibronic transitions. Dashed grey line gives the ratio of ground-state autocorrelation factors (Fig. 3). Dash-dotted and dotted grey lines in (a) represent the ratio of autocorrelation factors for ${\nu }_3$ and ${\nu }_4$ vibrationally excited initial wave packets (Fig. 5), respectively. Panels (a) and (b) correspond to laser intensity of 300 and 1000 TW ${\mathrm{cm}}^{-2}$, respectively. Experimental points in (a) are from Ref. [41], measured at the estimated intensity of 200 TW ${\mathrm{cm}}^{-2}$.

Figure 10

Isotope effects in methane for 1.2-$\mu \mathrm{m}$ driving field, from HHG data in Fig. 7. Also see Fig. 9 caption.

Figure 11

Isotope effects in methane for 1.6-$\mu \mathrm{m}$ driving field, from HHG data in Fig. 8. Time delays beyond the grey vertical dotted line at 2.6 fs may suffer from increased numerical errors; see Sec. 3d of the text. Also see Fig. 9 caption.