Time-dependent restricted-active-space configuration-interaction method for the photoionization of many-electron atoms

We introduce the time-dependent restricted-active-space configuration-interaction method to solve the time-dependent Schrödinger equation for many-electron atoms, and particularly apply it to the treatment of photoionization processes in atoms. The method is presented in a very general formulation and incorporates a wide range of commonly used approximation schemes, such as the single-active electron approximation, time-dependent configuration interaction with single-excitations, or the time-dependent $R$-matrix method. We prove the applicability of the method by calculating the photoionization cross sections of helium and beryllium, as well as the x-ray–IR pump-probe ionization of beryllium.

Figure 1

Illustration of the restricted-active-space scheme for a number of $P=4$ partitions of the single-particle basis $\mathcal{B}$. In each partition, one imposes certain restrictions on the allowed particle numbers. The total $N$-particle wave function $|\Psi \rangle$ is constructed as the tensor product of the $N_i$-particle wave functions $|{\Psi }_i\rangle$, with ${\sum }_i{N}_i=N$.

Figure 2

Special cases of the TD-RASCI scheme for the example of beryllium. The numbers $N_i$ label the allowed particle numbers in the partitions ${\mathcal{B}}_i$ (black boxes). Active electrons are shown as open (red) circles. Left: Single-active-electron approximation with an active $2s$ orbital. The gray-shaded orbitals are fixed, and no more than a single electron is allowed in the continuum. Right: Time-dependent configuration-interaction singles. All electrons are active, but only single excitations from the ground state are included.

Figure 3

Spatial partitioning of the active space used in the RASCI treatment of photoionization processes. ${\mathcal{B}}_1$ marks the region in which the ground state $|{\Psi }_0\rangle$ is localized, ${\mathcal{B}}_2$ the continuum. The example shows a four-electron atom (e.g., beryllium).

Figure 4

Radial part of the single-particle basis (18) used in this work, which consists of a combination of Hartree-Fock orbitals in the vicinity of the atom (left element), and FEDVR basis functions in the continuum (two elements on the right). In each element, the basis functions are constructed on a Gauss-Lobatto grid with 20 points. In the first element, the $1s$ and $2s$ orbitals are emphasized.

Figure 5

Single-ionization cross sections of helium, calculated for a laser field with intensity $I={10}^{12}{\mathrm{W}/\mathrm{cm}}^2$ and a squared-sine envelope of duration $T=400$a.u. The black dots show experimental results of Samson [60], red squares the TD FCI reference results and blue triangles the TD-RASCI results, which all agree almost perfectly. The results of TD-CIS calculations are depicted by green triangles.

Figure 6

Single-ionization cross sections of helium for the same field parameters as in Fig. 5. Black squares are the experimental results of Samson et al. [60], and red arrows mark the resonance positions of ${}^1{P}^o$ symmetry according to Scrinzi and Piraux [61].

Figure 7

Single-ionization cross section for beryllium as a function of photon energy. Black squares denote the experimental results of Wehlitz et al. [63, 64], the dashed curve the time-dependent calculations of Laulan and Bachau [5]. Both are compared to TD-CIS (green) and TD-RASCI (blue) calculations.

Figure 8

Single-photoionization cross section for beryllium between the ${\mathrm{Be}}^{+}\left(2p\right)$ and ${\mathrm{Be}}^{+}\left(3s\right)$ thresholds. TD-CIS and TD-RASCI calculations are compared to the results of Kim et al. [66], which were obtained with the $R$-matrix method.

Figure 9

Photoionization cross section of beryllium around the $1s$ threshold. The red dashed line shows the results of Laulan and Bachau [5] obtained through a time-dependent method and fixation of the $2s$ orbitals, the green and blue curves our TD-RASCI and TD-CIS results. The black lines mark the resonance and ionization energy due to the NIST database.

Figure 10

Total ionization yield of beryllium subjected to the three different x-ray–IR pump-probe pulses shown on top. Shown are the results from a single-active-electron calculation ($2s$) and the more accurate ($5s,3p$) TD-RASCI approximation (see text). For better comparison, the horizontal lines at the end of the pulses mark the total ionization yield.

Figure 11

Angular distribution of the photoionization yield of beryllium subjected to the x-ray–IR pulse on top. Shown are the results of different RAS approximations (see text).