Hybrid Gaussian–$B$-spline basis for the electronic continuum: Photoionization of atomic hydrogen

As a first step towards meeting the recent demand for new computational tools capable of reproducing molecular-ionization continua in a wide energy range, we introduce a hybrid Gaussian–$B$-spline basis (GABS) that combines short-range Gaussian functions, compatible with standard quantum-chemistry computational codes, with $B$ splines, a basis appropriate to represent electronic continua. We illustrate the performance of the GABS hybrid basis for the hydrogen atom by solving both the time-independent and the time-dependent Schrödinger equation for a few representative cases. The results are in excellent agreement with those obtained with a purely $B$-spline basis, with analytical results, when available, and with recent above-threshold ionization spectra from the literature. In the latter case, we report fully differential photoelectron distributions which offer further insight into the process of above-threshold ionization at different wavelengths.

Figure 1

Radial part of Gaussian (solid lines) and $B$-spline (dashed lines) representatives of the monocentric GABS basis as a function of radius. The first $B$-spline node is located at $R_0=10$ a.u. The basis defines three characteristic regions: (i) $r\in [0,R_0]$, with only Gaussian functions; (ii) $r\in [R_0,R_1]$, Gaussian and $B$-spline functions overlap; (iii) $r\in [R_1,R_{\mathrm{box}}]$, the Gaussian functions are negligible.

Figure 2

Radial part of the $7s$ (top panel) and $15s$ (bottom panel) hydrogen Rydberg states as a function of $\sqrt{r}$, computed with $B$ splines (dots) and GABS (solid line). For the latter case, the Gaussian component (dashed line) and the $B$-spline component (dash-dotted line) are separately shown.

Figure 3

(Top panel) Radial part of the ${\psi }_{sE}$ hydrogen scattering state with $E=0.1$ a.u., computed analytically (dots) and numerically using GABS (solid line) $R_0=10$ a.u., ${\ell }_{\max }=20$. The Gaussian (dashed line) and the $B$ spline (dash-dotted line) components of the numerical function are also shown. (Bottom panel) Absolute local error of the scattering states ${\psi }_{sE}$ computed with GABS for $E=0.1$ a.u. (dots), $E=1$ a.u (solid line), and $E=4$ a.u. (dotted line). See text for more details.

Figure 4

(Top panel) Absolute value of selected reduced dipole transition matrix elements in velocity gauge computed using the GABS basis and compared with the corresponding analytical results (solid squares). (Bottom panel) Relative percent deviation between numerical and analytical results. The following transitions are shown: $1s-{\epsilon }_p$ (solid line), $2s-{\epsilon }_p$ (dashed line), $2p-{\epsilon }_s$ (dash-dotted line), $2p-{\epsilon }_d$ (dotted line).

Figure 5

Relative percent deviation of $|(E-E_{1s})\langle {\psi }_{E_p}\parallel {\mathcal{O}}_1^{\left(l\right)}\parallel {\psi }_{1s}\rangle |$ and $|{(E-E_{1s})}^{-1}\langle {\psi }_{E_p}\parallel {\mathcal{O}}_1^{\left(a\right)}\parallel {\psi }_{1s}\rangle |$ from $|\langle {\psi }_{E_p}\parallel {\mathcal{O}}_1^{\left(v\right)}\parallel {\psi }_{1s}\rangle |$ (solid line and dashed line, respectively), computed with a GABS basis using $R_0=10$ a.u., ${\ell }_{\max }=20$.

Figure 6

Absolute value of the reduced transition matrix elements $(E_p-E_s)\langle {\psi }_{E_p}\parallel {\mathcal{O}}_1^l\parallel {\psi }_{E_s}\rangle$ (black dots), $\langle {\psi }_{E_p}\parallel {\mathcal{O}}_1^v\parallel {\psi }_{E_s}\rangle$ (red circles), ${(E_p-E_s)}^{-1}\langle {\psi }_{E_p}\parallel {\mathcal{O}}_1^a\parallel {\psi }_{E_s}\rangle$ (green solid squares), computed with GABS for several scattering states by truncating the radial integral at $R_{\mathrm{box}}$. Three representative energies for the $s$ state are shown: 1 a.u. (first peak), 2 a.u. (second peak), and 3 a.u. (third peak). The numerical results are compared with the exact analytical result in velocity gauge (gray solid line).

Figure 7

Absolute value of the reduced velocity-gauge hydrogen dipole matrix element $\langle {\psi }_{E_p}\parallel {\mathcal{O}}_1^v\parallel {\psi }_{E_s}\rangle$ from three $s$ scattering states ($E=1,2,3$ a.u.) to several $p$ scattering states evaluated as the sum of two contributions (thin solid line). The first contribution is the numerical radial integral shown in Fig. 6, which is computed with the GABS basis and truncated at $R_{\mathrm{box}}$, and the second contribution is the CBC term derived in Appendix pp1. The agreement between the numerical result and the exact analytical result (thick solid line) is excellent.

Figure 8

ATI photoelectron spectrum from hydrogen ground state due to a 30-cycle ${cos}^2$ laser pulse with $\omega =0.35$ a.u. and $I=14$ ${\mathrm{TW}/\mathrm{cm}}^2$. The present results, computed by solving the TDSE using either a purely $B$-spline basis (dots) or the GABS basis (solid line), are compared with the spectrum obtained under similar conditions by Rodríguez et al. [80]. See text for more details.

Figure 9

Normalized photoelectron angular distribution corresponding to the first and second ATI peaks in Fig. 8, computed here by solving the TDSE in either a $B$-spline or a GABS basis, and compared to the results obtained by Rodríguez et al. [80] in similar conditions.

Figure 10

Doubly differential photoelectron distribution $d^{2}P/dE\mathrm{d}cos\theta$ for the process described in Fig. 8.

Figure 11

Normalized photoelectron angular distribution for several selected energies in Fig. 10. The energies 0.225, 0.294, and 0.319 a.u. correspond to the $2p$, $3p$, and $4p$ resonances, respectively.

Figure 12

ATI photoelectron spectrum from hydrogen ground state due to a 20-cycle ${cos}^2$ laser pulse with $\omega =0.114$ a.u. and $I=0.1$ ${\mathrm{PW}/\mathrm{cm}}^2$. The present results, computed by solving the TDSE using either a purely $B$-spline basis (black dots) or the GABS basis (solid line), are compared with the spectrum obtained under similar conditions by Grum-Grzhimailo et al. [81] (light solid squares). See text for more details.

Figure 13

Schematic illustration of the dynamic-interference mechanism underlying the appearance of the multipeak substructure in the nonresonant multiphoton photoelectron signals when an atom is ionized with intense isolated ultrashort pulses. The external pulse induces an ac-Stark shift of the energy levels of the atom. Depending on the value of the time-dependent intensity of the laser, the pulse promotes at different times transitions to laser-dressed states that are adiabatically connected to field-free states with different energies, thus resulting in a global shift of the photoelectron signal. Two temporally separated amplitudes contribute to each final energy. The phase difference acquired by the first amplitude with respect to the second one in the intermediate time lapse can give rise to either constructive or destructive interference, thus resulting in a peak or a zero in the photoelectron spectrum, respectively. See text for more details.

Figure 14

Doubly differential photoelectron distribution $d^{2}P/dE\mathrm{d}cos\theta$ for the process described in Fig. 12.

Figure 15

Absolute value of the reduced velocity-gauge hydrogen dipole matrix element $\langle {\psi }_{E_p}\parallel {\mathcal{O}}_1^{\left(v\right)}\parallel {\psi }_{E_s}\rangle$ from the $s$ scattering state with $E_s\simeq 2$ a.u. to several $p$ scattering states evaluated as the sum of two contributions. The first contribution is the numerical integral shown in Fig. 6, which is computed with the GABS basis and truncated at $R_{\mathrm{box}}$, and the second contribution is either the AOI (solid circles) or the CBC (thin solid line). The thick gray solid line on the background shows the exact analytical results. See text for more details.