Molecular Siegert states in an electric field

The Siegert states of atoms and molecules in a static electric field are the solutions of the stationary Schrödinger equation satisfying the regularity and outgoing-wave boundary conditions. Recently, an efficient method for calculating Siegert states in the single-active-electron approximation based on the adiabatic expansion in parabolic coordinates was proposed [P. A Batishchev et al., Phys. Rev. A 82, 023416 (2010); O. I. Tolstikhin et al., Phys. Rev. A 84, 053423 (2011)]. So far, this method has been implemented only for axially symmetric potentials, which corresponds to atoms and linear molecules aligned along the field. In the present work, we extend its implementation to a general class of soft-core molecular potentials. This makes it possible to calculate the Siegert eigenvalue $E=\mathcal{E}-i\Gamma /2$ defining the energy $\mathcal{E}$ and ionization rate $\Gamma$ of the corresponding state as functions of the electric field for arbitrarily oriented polyatomic molecules. The method is illustrated by calculations for the $1s\sigma$ and $2p\pi$ states of H${}_2^{+}$. Comparison of the results with the predictions of perturbation theory for $\mathcal{E}$ and weak-field asymptotic theory for $\Gamma$ is discussed.

Figure 1

Illustration of the unperturbed wave functions of H${}_2^{+}$ oriented under an angle $\beta$ with respect to the direction of the field in even $2p{\pi }^{+}$ and odd $2p{\pi }^{-}$ states. Different colors correspond to different signs of the wave function.

Figure 2

Energy $\mathcal{E}$ and ionization rate $\Gamma$ for the $1s\sigma$ state of the soft-core H${}_2^{+}$ with $R=2$ and $\epsilon =0.09$ as functions of the electric field $F$ for four representative orientation angles $\beta$. Solid (black) lines for $\mathcal{E}$ and $\Gamma$: exact results. Dashed (red) lines: results of perturbation theory and asymptotic theory, respectively. Bottom panels: ratio of the exact to the asymptotic results for $\Gamma$. Dash-dotted (blue) lines for $\beta =0^{\circ }$: exact results for a sharper potential with $\epsilon =0.0009$.

Figure 3

Energy $\mathcal{E}$ and normalized ionization rate $\Gamma /W_{00}\left(F\right)$ for the $1s\sigma$ state of H${}_2^{+}$ with $R=2$ and $\epsilon =0.09$ as functions of the orientation angle $\beta$ for two representative values of the field in tunneling ($F=0.06$) and overbarrier ($F=0.2$) regimes. Solid (black) lines: exact results. Note that for $F=0.2$ the results in the right panel are multiplied by 4. Dashed (red) lines for $\mathcal{E}$ and $\Gamma$: results of perturbation theory and asymptotic theory, respectively. For $F=0.06$, dashed and solid lines in the left panel cannot be distinguished at the scale of the figure.

Figure 4

Energy $\mathcal{E}$ and ionization rate $\Gamma$ for the $2p{\pi }^{\pm }$ states of H${}_2^{+}$ with $R=2$ and $\epsilon =0.09$ as functions of the electric field $F$ for $\beta =0^{\circ }$. Solid (black) lines for $\mathcal{E}$ and $\Gamma$: exact results. Dashed (red) lines: results of perturbation theory and asymptotic theory, respectively. Bottom panel: ratio of the exact to the asymptotic results for $\Gamma$.

Figure 5

Energy $\mathcal{E}$ and ionization rate $\Gamma$ for the even $2p{\pi }^{+}$ state of H${}_2^{+}$ with $R=2$ and $\epsilon =0.09$ as functions of the electric field $F$ for three representative orientation angles $\beta$. Solid (black) lines for $\mathcal{E}$ and $\Gamma$: exact results. Dashed (red) lines: results of perturbation theory and asymptotic theory, respectively. Bottom panels: ratio of the exact to the asymptotic results for $\Gamma$.

Figure 6

Energy $\mathcal{E}$ and normalized ionization rate $\Gamma /W_{00}\left(F\right)$ for the even $2p{\pi }^{+}$ state of H${}_2^{+}$ with $R=2$ and $\epsilon =0.09$ as functions of the orientation angle $\beta$ for four representative values of the field $F$. Solid lines: exact results. Dashed lines in left and right panels: results of perturbation theory and asymptotic theory, respectively.

Figure 7

Same as Fig. 5, but for the odd $2p{\pi }^{-}$ state.

Figure 8

Same as Fig. 6, but for the odd $2p{\pi }^{-}$ state.