Time-dependent renormalized-natural-orbital theory applied to laser-driven ${{\mathrm{H}}_2}^{+}$

Recently introduced time-dependent renormalized-natural-orbital theory (TDRNOT) is extended towards a multicomponent approach in order to describe ${{\mathrm{H}}_2}^{+}$ beyond the Born-Oppenheimer approximation. Two kinds of natural orbitals, describing the electronic and the nuclear degrees of freedom are introduced, and the exact equations of motion for them are derived. The theory is benchmarked by comparing numerically exact results of the time-dependent Schrödinger equation for an ${{\mathrm{H}}_2}^{+}$ model system with the corresponding TDRNOT predictions. Ground-state properties, linear-response spectra, fragmentation, and high-order harmonic generation are investigated.

Figure 1

Plot of the correlated ground-state probability density ${\left|\mathrm{\Psi }(x,R)\right|}^2$ for (a) $N_{\mathrm{o}}=1$, (b) $N_{\mathrm{o}}=2$, (c) $N_{\mathrm{o}}=4$, and (d) $N_{\mathrm{o}}=8$ for negative values of $x$. For $x>0$ the absolute difference to the exact probability density is plotted.

Figure 2

Linear-response spectra obtained from TDRNOT calculations with different numbers of RNOs $N_{\mathrm{o}}$ compared to the exact TDSE result. Each spectrum is normalized to one, and the spectra are shifted vertically by factors of ${10}^{-5}$ in order to make them distinguishable.

Figure 3

Time-dependent nuclear probability density upon the interaction with an 800-nm four-cycle pulse with $I_0={10}^{14}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$. Again different numbers of orbitals were used: (a) $N_{\mathrm{o}}=1$, (b) $N_{\mathrm{o}}=2$, (c) $N_{\mathrm{o}}=4$, (d) $N_{\mathrm{o}}=8$, and (e) $N_{\mathrm{o}}=10$. With more RNOs included, the agreement with the exact result of the TDSE, given in (f), is considerably improved.

Figure 4

Energy spectra for nuclei and photoelectrons, calculated using the (extended) virtual-detector method. The laser parameters are the same as in Fig. 3. (a) Kinetic-energy spectra of the nuclei for dissociation (blue) and DI (red). The vertical gray lines denote $2\omega$, $3\omega$, and $4\omega$ absorption from the vibrational ground state. (b) TDRNOT photoelectron spectrum for 10 RNOs per degree of freedom (red, dashed) compared to the exact result from the TDSE (orange, solid).

Figure 5

HHG spectra for an 800-nm 10-cycle flat-top pulse with $I_0=3.0\times {10}^{14}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ calculated with TDRNOT using 1, 2, 4, and 8 RNOs per degree of freedom compared to the exact spectrum obtained from the TDSE. The insets show a section of each spectrum plotted on a linear scale.