Time-dependent density-functional-theory calculation of high-order-harmonic generation of H${}_2$

The observation of the isotope effect in the high-order-harmonic generation (HHG) of H${}_2$ presents a challenge for time-dependent density-functional-theory (TDDFT) methods, since this effect is related to the dynamics of the ion created in the tunneling ionization step of HHG and it depends on the harmonic order. As an initial step toward describing this effect within current computational capacity, we benchmark a method in which the nuclear and electronic degrees of freedom are separated and both treated quantum mechanically. For the electrons two TDDFT formalisms are adopted. Although the ion-dynamics effect is not described in our method, it reproduces the measured D${}_2$-to-H${}_2$ HHG ratios up to the 25th harmonic when the 35th is the classical cutoff. Beyond the 25th harmonic, however, our results show substantial deviation and are sensitive to the laser intensity. A higher intensity reproduces the experimental results. Analysis reveals an $R$-dependent phase factor as the cause of the isotope effect in our calculation. We isolate this phase factor and propose a strong-field-approximation-phase model, which reproduces experimental data, including those for which the ion-dynamics model has to be further modified. We show that the model that we propose is intrinsically related to the ion-dynamics model. Our model provides a correction to the TDDFT approach when the ion-dynamics effect becomes significant. It also indicates that the isotope effect is not only a probe for the ion created by the external field but is ultimately a more useful probe for the ground-state nuclear wave function. For all molecules whose vertical ionization potential strongly depends on the nuclear geometry, HHG may serve as a sensitive ultrafast probe of nuclear dynamics.

Figure 1

The vertical $I_p$ calculated by full CI, LB${}_{\alpha }$, and SIC methods together with the ground vibrational wave functions of H${}_2$ and D${}_2$.

Figure 2

Squared dipole acceleration $|a(R_{\mathrm{eq}},\omega ){|}^2$ and ${\left|\langle a\left(\omega \right)\rangle \right|}^2$ of H${}_2$ calculated with the TDLB${}_{\alpha }$ (black solid line and green $\times$) and TDSIC (red dashed line and blue circles) methods.

Figure 3

The amplitudes in atomic units and phases in units of $2\pi$ of $a(R,\omega )$ for $\omega =3{\omega }_0,5{\omega }_0,...,35{\omega }_0$, as a function of $R$, calculated with the TDLB${}_{\alpha }$ method.

Figure 4

The D${}_2$-to-H${}_2$ spectral density ratios calculated with the TDLB${}_{\alpha }$ method at two intensities, compared to experiment [4]. The black solid line and the dashed red line are calculated for 800 nm lasers. The blue dash-dotted line is for 1064 nm 2$\times {10}^{14}$ W/cm${}^2$ lasers. The magenta line with triangles is the result of the SFA-phase model, and the green dashed line with circles is the result of Lein's ion-dynamics model [15]. The unit of intensity $I$ is ${10}^{14}\text{W}/{\text{cm}}^2$.

Figure 5

Comparison of the D${}_2$-to-H${}_2$ ratio calculated with the SFA-phase model with experimental data [5]. Note that there are two sets of experimental data for each intensity presented with the filled and unfilled symbols respectively. Results from the ion-dynamics calculations are also shown. The unit of intensity $I$ is 10${}^{14}$ W/cm${}^2$.

Figure 6

The dipole acceleration squared, ${\left|a(t,\omega )\right|}^2$, for harmonics $\omega =3{\omega }_0,5{\omega }_0,...,35{\omega }_0$, as a function of the delay time $t$ between the preparation of the vibrational wave packet and the HH generating pulse. The three panels on the left correspond to a wave packet which consists of 10$\%$ $v=1$ ($|c_1{|}^2=0.1$) and the panels on the right correspond to a wave packet with equal contributions for $v=0$ and $v=1$ vibrational states.