Comment on “Origin of symmetry-forbidden high-order harmonic generation in the time-dependent Kohn-Sham formulation”

In their recent paper [Phys. Rev. A 103, 043106 (2021)], Zang et al. theoretically investigated high harmonic generation (HHG) in benchmark two-electron systems that are inversion symmetric with time-dependent density functional theory (TDDFT) in the Kohn-Sham formulation. They found that the theory wrongly predicted the emission of symmetry-forbidden even harmonics and concluded that this error originates from an inherent problem of TDDFT that unphysically populates one- and two-electron excited states. They further claimed that this effect results in an incorrect HHG cutoff energy. We reproduced their main results, but found that the unphysical even harmonics that they observed originated from numerical errors introduced by the boundary conditions. We show that contrary to their claims, the HHG cutoff energy calculated within TDDFT agrees perfectly with the standard and well-established models of HHG.

Figure 1

Origin of spurious even harmonics in HHG calculations from ${\mathrm{H}}_2$. (a) HHG spectra from ${\mathrm{H}}_2$ calculated with KS-TDDFT with a 15 bohr wide CAP. (b) Same as (a) but with a 30 bohr wide CAP. (c) Same as (a) but for a SAE model from the TDSE. (d) Same as (b) but for a SAE model from the TDSE. Arrows highlight positions of unphysical even harmonics for the narrow absorber, and their absence in the wide absorber case. HHG yield is presented in logarithmic scale (with basis 10).

Figure 2

HHG cutoff energy scaling with respect to laser power in TDDFT. Plots show HHG spectra from ${\mathrm{H}}_2$ calculated with TDDFT in the same approach as in Fig. 1, but for various laser powers. Red dashed lines denote the $I_p$, red solid lines denote the Lewenstein model cutoff prediction, and dashed blue lines signify the numerical cutoff. Odd harmonics are highlighted by gray lines. HHG yield is presented in logarithmic scale (with basis 10).