Time-dependent generalized-active-space configuration-interaction approach to photoionization dynamics of atoms and molecules

We present a wave-function-based method to solve the time-dependent many-electron Schrödinger equation with special emphasis on strong-field ionization phenomena. The theory builds on the configuration-interaction (CI) approach supplemented by the generalized-active-space concept from quantum chemistry. The latter allows for a controllable reduction in the number of configurations in the CI expansion by imposing restrictions on the active orbital space. The method is similar to the recently formulated time-dependent restricted-active-space CI method [D. Hochstuhl and M. Bonitz, Phys. Rev. A 86, 053424 (2012)]. We present details of our implementation and address convergence properties with respect to the active spaces and the associated account of electron correlation in both ground-state and excitation scenarios. We apply the time-dependent generalized-active-space CI theory to strong-field ionization of polar diatomic molecules and illustrate how the method allows us to uncover a strong correlation-induced shift of the preferred direction of emission of photoelectrons.

Figure 1

Schematic of the generalized-active-space (GAS) method with $G$ subspaces GAS-1 to GAS-G. The spin-orbital partition is given by ${\mathbf{N}}_b=[1,n_b^2,\cdots ]$ and the allowed number of electrons in each subspace by ${\mathbf{N}}_{\mathrm{el}}=[(n_1^1,\cdots ),\cdots ,(n_1^G,\cdots )]$. The energy eigenvalues of the single-particle orbitals are labeled by $E_1^{\alpha ,\beta }$, where $\alpha$ and $\beta$ denote the spin coordinate. For the nonrelativistic studies in this work, these are degenerate, $E_i^{\alpha }=E_i^{\beta }$.

Figure 2

Schematic of the GAS partitioning mainly used in this work. The scheme is denoted by ${\mathrm{CAS}}^{*}(N_{\mathrm{el}}^C,K)$ and consists of a fixed core with $N_{\mathrm{el}}-N_{\mathrm{el}}^C$ electrons in the same number of spin orbitals (this space is empty if $N_{\mathrm{el}}^C=N_{\mathrm{el}}$) and an active space with $N_{\mathrm{el}}^C$ electrons in $2K$ spin orbitals, from which one electron can be removed and excited into GAS-G describing the one-electron continuum.

Figure 3

Schematic view of the partially rotated basis set with pseudo-orbitals ${\phi }_i$ in the central region $[-x_c,x_c]$ (solid blue lines) close to the minimum of the binding potential (dashed black line). In addition, a correlated single-particle density, cf. Eq. (34), for the ground state of a model for beryllium (see Sec. 4b) calculated for the complete computational grid $[-x_s,x_s]$ is given by the dashed red line (logarithmic scale to the right). The asymptotics is indicated by the thin black lines labeled “$n\left(x\right)$”; cf. Eq. (26). The underlying FE-DVR grid (all functions are included in the calculation) is sketched in gray. Parameters in the figure are $x_s=30$ and $x_c=10$. We used 30 elements with eight DVR functions per element. Quantities on the abscissa are given in atomic units (a.u.).

Figure 4

Single-particle orbitals in the central region $[-x_c,x_c]$ for 1D beryllium ($N_{\mathrm{el}}=4$, $x_c=10$). Comparison of pseudo-orbitals ${\phi }_i^{p_1}$, ${\phi }_i^{p_2}$ [cf. Eqs. (27) and (28)] and natural orbitals ${\phi }_i^{\mathrm{n}}$ [Eq. (32)]. The HF virtuals are additionally plotted in gray for comparison. The parameters are the same as in Fig. 3.

Figure 5

GAS partitions for the two-electron model. The acronyms of the different approximations are SAE, single-active electron; CIS, configuration-interaction singles; CAS*(2,$K$), complete active space with a CAS including $K$ spatial orbitals and single excitations outside. See Fig. 1 and Sec. 2b for notations.

Figure 6

Ionization probability $\mathcal{P}(t_f=4000,\omega$), cf. Eq. (40), of the heliumlike model for different GAS approximations as a function of the photon energy for a fixed pulse duration. The results for pseudo-orbitals ${\phi }^{p_1}$, Eq. (27), and natural orbitals ${\phi }^n$, Eq. (32), in the rotated basis are compared. The left panel (a) shows the whole range of frequencies on a logarithmic scale and the right panels magnifications of the one-electron excitations ($1e\mathrm{n}o$, b) and the first two-electron resonances ($2o\mathrm{n}e$, c) on a linear scale. The dipole-excitation spectrum $S\left(\omega \right)$ [cf. Eq. (45)] for an infinitesimally short pulse from a fully correlated TDSE simulation is drawn in gray to help identify the positions of the excited states. The field parameters are $F_0=0.001,\sigma =100,t_0=400$, ${\phi }_{\mathrm{CEP}}=0$ in Eq. (44).

Figure 7

Photoelectron spectra of the 1D heliumlike model for a short pulse [Eq. (44)] with $\sigma =5$, $F_0=0.01$, ${\phi }_{\mathrm{CEP}}=0$ and a photon energy of $\omega =2.1$ using (a) pseudo-orbitals ${\phi }_i^{p_1}\left(x\right)$ and (b) natural orbitals ${\phi }_i^n\left(x\right)$. The ${\mathrm{CAS}}^{*}(2,2)$ results are scaled by a factor of 1/10 in the insets.

Figure 8

Schematics of the GASs for the four-electron berylliumlike model. The label v (c) refers to an active valence (core) orbital. ${\mathrm{CAS}}^{*}(2,K)$ and ${\mathrm{CAS}}^{*}(4,K)$ are active spaces with two and four electrons, respectively, with single excitations out of the CAS. The case CIS-v equals ${\mathrm{CAS}}^{*}(2,1)$. See also Fig. 2 and Sec. 2b.

Figure 9

Dipole excitation spectrum $S\left(\omega \right)$ [cf. Eq. (45)] of the 1D four-electron berylliumlike model for an excitation of $\sigma =0.1$, $t_0=1$, $F_0=0.001$ [Eq. (43)] in different GAS approximations; shown is the total energy range from the GSE to full fourfold ionization ($I_p^4=-E_0$). The first ionization potentials for ionization from the valence orbital, $I_p^v$, and core orbital, $I_p^c$, are indicated by dashed vertical lines. The labels “v” and “c” refer to the valence and core orbitals, respectively. Red lines (higher energy, limited by $I_p^c$) correspond to core electrons only; blue is for the valence shell (lower energy, $I_p^v$).

Figure 10

Parts of the dipole excitation spectrum $S\left(\omega \right)$ of the four-electron berylliumlike model for different GAS approximations and orbital basis sets in the central region. (a) Fixed core; (b) all electrons are active. Dashed vertical lines are guides to the eye for reference to the best [${\mathrm{CAS}}^{*}$(2,11) for (a) and ${\mathrm{CAS}}^{*}$(4,7) for (b)] approximation (bottom line). The individual lines are vertically shifted for better visibility. The full spectrum and parameters are given in Fig. 9.

Figure 11

Total energy of the 1D four-electron LiH-like model. The dissociation limit is indicated by dashed lines for fully relaxed Li ($E_d$) and for Li with fixed core ($E_d^{\mathrm{v}}$). See text for details. Dot-dashed and dot-long-dashed red lines converging to $E_d^{\mathrm{v}}$ are for two active [${\mathrm{CAS}}^{*}$(2,.)] electrons, blue lines converging to $E_d$ for all four being active [${\mathrm{CAS}}^{*}$(4,.)]. The TDSE result (full black line) is for a smaller FE-DVR basis set [11].

Figure 12

Sketch of the scenarios considered with the 1D four-electron LiH-like model molecule under strong-field ionization. The electric field $F\left(t\right)$ is shown in the left part for (a) ${\phi }_{\mathrm{CEP}}=0$ and (b) ${\phi }_{\mathrm{CEP}}=\pi$. The two doubly occupied core and valence HF orbitals are plotted along with the single-particle initial density and potential. For better visibility of the densities, the dipole potential (diagonal dashed lines) at the time of maximum electrical field is magnified in the figure.

Figure 13

Single-particle density $n\left(x\right)$ [cf. Eq. (34)] of the 1D four-electron LiH-like molecule exposed to single-cycle pulses (i) after a propagation of $t=500$.