Time-dependent multiconfiguration self-consistent-field study on resonantly enhanced high-order harmonic generation from transition-metal elements

We theoretically study high-harmonic generation (HHG) from transition-metal elements Mn and ${\mathrm{Mn}}^{+}$ using full-dimensional, all-electron, first-principles simulations. The HHG spectra calculated with the time-dependent complete-active-space self-consistent-field (TD-CASSCF) and occupation-restricted multiple-active-space (TD-ORMAS) methods exhibit a prominent peak at $\sim 50$ eV, successfully reproducing resonant enhancement observed in previous experiments [Opt. Express 20, 25239 (2012)]. Artificially freezing $3p$ orbitals in simulations results in its disappearance, which shows the essential role played by $3p$ electrons in the resonant harmonics (RH). Further transition-resolved analysis unambiguously identifies constructively interfering $3p\text{−}3d$ ($m=0,\pm 1$) giant resonance transitions as the origin of the RH, as also implied by its position in the spectra. Time-frequency analysis indicates that the recolliding electron combines with the parent ion to form the upper state of the transitions. In addition, this study shows that the TD-CASSCF and TD-ORMAS methods can be applied to open-shell atoms with many unpaired inner electrons.

Figure 1

Orbital diagram used for TD-CASSCF with 15 orbitals for (a) Mn and (b) ${\mathrm{Mn}}^{+}$. All the electron arrangements within Active1 are allowed. The various frozen spaces are indicated by the dashed lines. For TD-ORMAS simulations with 24 orbitals, the schemes in (c) and (d) are used for Mn and ${\mathrm{Mn}}^{+}$, respectively, restricting up to two electrons in Active2.

Figure 2

HHG spectra calculated with the TD-CASSCF method for several frozen-core settings shown in Figs. 1 and 1 for (a) Mn and (b) ${\mathrm{Mn}}^{+}$. The results of the MCTDHF simulations, in which all 15 orbitals in Figs. 1 and 1 are treated as active, are also shown.

Figure 3

HHG spectra calculated with the TD-ORMAS method for (a) Mn and (b) ${\mathrm{Mn}}^{+}$ (dark green). The vertical arrows mark the cutoff positions expected from the three-step model [50, 51]. The HHG spectrum from ${\mathrm{Mn}}^{2+}$ obtained using TD-CASSCF with 14 orbitals (the same orbital diagram as Fig. 1 except that $4s$ orbital is absent) is also plotted (dashed blue).

Figure 4

HHG spectra from (a) Mn and (b) ${\mathrm{Mn}}^{+}$ calculated with the TD-ORMAS method for 1333-nm wavelength and ${10}^{14}{\mathrm{W}/\mathrm{cm}}^2$ peak intensity.

Figure 5

Photoexcitation spectra calculated with (a) TD-CASSCF and (b) TD-ORMAS. The same orbital subspace decomposition used for the ${\mathrm{Mn}}^{2+}$ HH spectrum in Fig. 3 is used for its photoexcitation spectrum in (a). The spectra of ${\mathrm{Mn}}^{+}$ and ${\mathrm{Mn}}^{2+}$ are multiplied by ${10}^{-3}$ and ${10}^{-6}$, respectively, for better visibility. The excitation spectrum of ${\mathrm{Mn}}^{2+}$ could not be stably calculated with the TD-ORMAS method.

Figure 6

Temporal evolution of the survival probabilities of Mn, ${\mathrm{Mn}}^{+}$, and ${\mathrm{Mn}}^{2+}$ during the TD-ORMAS simulations starting from (a) Mn and (b) ${\mathrm{Mn}}^{+}$ under the conditions used in Fig. 3.

Figure 7

The time-frequency spectrograms of HHG from (a) Mn and (b) ${\mathrm{Mn}}^{+}$ obtained by Gabor transforming the TD-ORMAS results (Fig. 3) with a time window size of 7.5 a.u. The lower and upper groups of recombination energy (sum of the kinetic energy of the returning electron and the ionization potential) curves are for ${\mathrm{Mn}}^{+}$ and ${\mathrm{Mn}}^{2+}$, respectively.

Figure 8

Power spectra of each $3p\leftrightarrow 3d$ transition, $3p\leftrightarrow 4s$, the sum of the three $3p\leftrightarrow 3d$ transition components, as well as the sum of all the 18 transitions listed in Eq. (11) for (a) Mn and (b) ${\mathrm{Mn}}^{+}$.