Stability of the time-dependent configuration-interaction-singles method in the attosecond and strong-field regimes: A study of basis sets and absorption methods

We investigate the behavior of several spatial grid methods and complex absorbers for strong-field and attosecond scenarios when using the time-dependent configuration-interaction singles method to solve the multi-electron time-dependent Schrödinger equation for atoms. We compare the pseudospectral grid, finite-element, and finite-element-discrete-variable-representation (DVR) methods with each other and discuss their advantages and disadvantages. Additionally, we study the performances of complex absorbing potential (CAP) and smooth exterior complex scaling (SES) to absorb the outgoing electron. We find that SES performs generally better than CAP for calculating high-harmonic generation spectra and XUV photoelectron spectra. In both of these cases, the DVR and even more the FEM grid representations show more reliable results—especially when using SES. Both absorbers show drawbacks when calculating photoelectron spectra in the strong-field regime.

Figure 1

(a) FEM functions (purple solid line) ${\kappa }_0^{\left(i\right)}\left(r\right)$, (green dashed line) ${\kappa }_1^{\left(i\right)}\left(r\right)$, and (blue dotted line) ${\kappa }_2^{\left(i\right)}\left(r\right)$ for $r_{i-1}=0,r_i=1$, and $r_{i+1}=3$. (b) First and (c) second derivatives of the FE functions.

Figure 2

The expectation value $〈\dot{z}〉\left(t\right)$ for different $l_{\text{max}}$ and $e_c$ calculated (a), (c) in the length and (b), (d) in the velocity gauges. All other parameters are kept fixed at their reference values (see Table 1).

Figure 3

HHG spectra corresponding to the results shown in Fig. 2. The HHG spectra are calculated with $〈\dot{z}〉\left(t\right)$.

Figure 4

HHG spectra in the velocity gauge for (a), (d) PSG, (b), (e) DVR, and (c), (f) FEM radial basis sets. Results in the upper panels (a)–(c) are based on a CAP absorber, and an SES absorber is used in the lower panels (d)–(f). The radius of the numerical box is 50 and the absorber starts around 40 (red dashed line) and 45 (blue solid line). The black dotted lines indicate the converged reference HHG spectrum.

Figure 5

HHG spectra for different $r_{\text{max}}$ keeping the width of the absorbing region fixed at 10. The black dotted line indicates the converged reference HHG spectrum. All calculations are done in the FEM representation by using the SES absorber.

Figure 6

HHG spectra for (a), (d) PSG, (b), (e) DVR, and (c), (f) FEM representation are shown for different grid-point distributions. A single Möbius-type grid (solid blue lines) and two Möbius-type grids (dashed red lines) concatenated at $r=35a_0$ are used. For the PSG grid method, only the single Möbius-type grid is used. Results are shown for (a)–(c) 151 and (d)–(f) 301 radial basis functions. The black dotted lines indicate the reference HHG spectrum. In all cases SES is used with $\lambda =0.1$.

Figure 7

HHG spectra for the SES (solid blue line) and CAP (dotted red line) absorbers for a $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ pulse. The box radius is 50. The reference spectrum (dashed black line) is calculated with a box size of 125.

Figure 8

The Gabor transformation associated with the HHG spectra in Fig. 7 calculated with (a) SES and (b) CAP absorbers. The Gabor window function has a Gaussian profile with a width of 6 a.u. (194 as).

Figure 9

Photoelectron spectrum (PES) of argon ionized by a 105 eV pulse with a duration of 1.21 fs (50 a.u.) and a peak intensity of $87.5\mathrm{TW}/{\mathrm{cm}}^2$. The PES are shown for all three grid methods (PSG, DVR, FEM) and both absorbers (CAP, SES), where the CAP strength $\eta$ and the SES smoothing parameter $\lambda$ are varied, respectively. The t-SURFF radius is $r_{\text{tsurff}}=100$ and the absorber starts at $r_{\text{absorb}}=120$.

Figure 10

Photoelectron spectrum (PES) of the ionized $3p$ electron of argon for all three grids using the SES absorber. The t-SURFF radius $r_{\text{tsurff}}$ and the SES absorber radius $r_{\text{absorb}}$ are varied, indicated by $(r_{\text{tsurff}},r_{\text{absorb}})$. The SES smoothing parameter is $\lambda =0.1$ and the SES angle is $\theta ={25}^{\circ }$. The pulse is the same as in Fig. 9.

Figure 11

ATI photoelectron spectrum of hydrogen ionized by a 20-cycle ${cos}^2$ pulse with a wavelength of 400 nm and ${10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ peak intensity for a box size of (solid red line) $r_{\text{max}}=70$ with $r_{\text{tsurff}}=30$ and (dotted blue line) $r_{\text{max}}=150$ with $r_{\text{tsurff}}=125$. The FEM representation is used in combination with the SES absorber. The low-energy reference spectrum (dashed-dotted black line) is taken from Ref. [70].