Photoionization of atoms and molecules studied by the Crank-Nicolson method

The Crank-Nicolson (C-N) method combined with a $B$-spline basis set can be used in both time-dependent and time-independent calculations of the photoionization cross sections of atoms and molecules. For time-independent systems, the imaginary-time propagation (ITP) method can usually only converge an arbitrary initial state to the ground state directly. Contrary to existing methods, it is found that the C-N method can converge an arbitrary initial state directly to not only the ground state but also excited and continuum states by controlling the time-step size. It is very useful if one is interested in only part of the spectral information since the computation is relatively cheap. The C-N method can also be directly applied in time-dependent calculations. During the time evolution, the spectral information, such as energy and momentum, can be retrieved by projecting the wave function on the eigenstates obtained using the ITP method. Both time-dependent and time-independent calculations agree very well with previous results. This method can also be extended to two-electron systems directly.

Figure 1

Prefactors of Eqs. (4) and (6) as a function of $x=E\Delta t/2$.

Figure 2

Comparison of hydrogen continuum radial state $r{\Psi }_E\left(r\right)$ obtained using the ITP method (with iterations $j=18$; solid line) with the analytic Coulomb wave function (dotted line) with $l=0$, $E=2$ a.u. Both are normalized on the energy scale.

Figure 3

One-photon ionization cross sections for ${\mathrm{H}}_2{}^{+}$ as a function of photoelectron energy. The initial state is the ground $1{\sigma }_g$ state. The laser polarization is parallel to the molecular axis. The internuclear distance $R=2$ a.u. The time-dependent results (solid line) are compared with those using the ITP method (circles).

Figure 4

Relative contribution of each partial cross section ${\sigma }_q$ to the total one-photon ionization cross section ${\sigma }_{\mathrm{total}}$. The initial state is the ground $1{\sigma }_g$ state.

Figure 5

Same as Fig. 3, but the initial state is the first excited state $1{\sigma }_u$.

Figure 6

Same as Fig. 4, but the initial state is the first excited state $1{\sigma }_u$.

Figure 7

One-photon ionization cross sections for ${\mathrm{HeH}}^{2+}$ as a function of photoelectron energy. The initial state is the ground $1s\sigma$ state. The laser polarization is parallel to the molecular axis. The internuclear distance $R=4$ a.u. The time-dependent results (solid line) are compared with those from the ITP method (circles).

Figure 8

Relative contribution of each partial cross section ${\sigma }_q$ to the total one-photon ionization cross section ${\sigma }_{\mathrm{total}}$. The initial state is the ground $1s\sigma$ state.

Figure 9

Same as Fig. 7, but the initial state is the first excited state $2p\sigma$.

Figure 10

Same as Fig. 8, but the initial state is the first excited state $2p\sigma$.