Effect of potential screening on the ${\mathrm{H}}_2$ autoionizing states

We study the behavior of autoionizing states of the hydrogen molecule subject to screened Coulomb interactions using an ab initio Feshbach configuration interaction method. Special attention is given to the algorithms developed for the evaluation of (i) screened molecular orbitals expressed in terms of one-center expansions using B-spline polynomial basis functions and (ii) screened two-electron integrals between configurations expressed in terms of such molecular orbitals, by solving the screened Poisson equation. As an illustration of the method we focus on the lowest Feshbach resonance of the ${\mathcal{Q}}_1{}^1{\mathrm{\Sigma }}_g^{+}$ series of doubly excited states of ${\mathrm{H}}_2$, which lies between the first ${\mathrm{H}}_2{}^{+}$ ($1s{\sigma }_g$) and the second ${\mathrm{H}}_2{}^{+}$ ($2p{\sigma }_u$) ionization thresholds. We show that Coulomb screening in the electron-proton interaction and between electrons may significantly alter the resonance position and autoionizing decay as a function of internuclear distance. In general, screening increases the resonance lifetime. However, when electron-proton screening dominates over electron-electron screening, we find that the ${\mathcal{Q}}_1$ resonance acquires a pronounced shape-resonance character at internuclear distances where the resonance approaches the lower ionization threshold, thus leading to a pronounced decrease of its lifetime.

Figure 1

Electronic energy of the lowest $Q_1{}^1{\mathrm{\Sigma }}_g^{+}$ doubly excited resonance state (solid lines) along with the first ${\mathrm{H}}_2{}^{+}$ ($1s{\sigma }_g$) and the second ${\mathrm{H}}_2{}^{+}$ ($2p{\sigma }_u$) ionization thresholds (dashed lines) as a function of internuclear distance for different values of ${\lambda }_p$ and ${\lambda }_e$. Thick red lines correspond to the unscreened case, ${\lambda }_p={\lambda }_e=0$; green lines to electron-proton screening, ${\lambda }_p=0.11$ a.u. and ${\lambda }_e=0$ a.u.; and thin blue lines to the case when both electron-proton and electron-electron interactions are screened, with ${\lambda }_p={\lambda }_e=0.25\mathrm{a}.\mathrm{u}.$

Figure 2

Total width $\mathrm{\Gamma }$ (panel a, solid lines) and partial widths ${\mathrm{\Gamma }}_{\ell =0}$ (solid lines) and ${\mathrm{\Gamma }}_{\ell =2}$ (dashed lines) (panel b) of the lowest $Q_1{}^1{\mathrm{\Sigma }}_g^{+}$ resonance as a function of the internuclear distance for different values of ${\lambda }_p$ and ${\lambda }_e$. Thick red lines correspond to the unscreened case, ${\lambda }_p={\lambda }_e=0$, green lines to the electron-proton screened case, with ${\lambda }_p=0.11$ a.u. with ${\lambda }_e=0$ a.u., and thin blue lines to electron-proton plus electron-electron screening, with ${\lambda }_p={\lambda }_e=0.25$ a.u. The partial widths ${\mathrm{\Gamma }}_{\ell =4}$ and ${\mathrm{\Gamma }}_{\ell =6}$ are approximately zero and are not shown here for the sake of clarity. Widths are plotted only up to the internuclear distance $R$ at which the resonance energies plotted in Fig. 1 cross the lower threshold ${\mathrm{H}}_2{}^{+}$ ($1s{\sigma }_g$).

Figure 3

Weights of the four most important CI configurations of the lowest resonance ${\mathcal{Q}}_1{}^1{\mathrm{\Sigma }}_g^{+}$ as a function of the internuclear distance. At short internuclear distances ${\left(2s{\sigma }_g\right)}^2$ dominates over $\left(2p{\pi }_u^{+}\right)\left(3p{\pi }_u^{-}\right)$ (both with solid lines). Above $R\approx 1$ a.u. the configuration ${\left(2p{\sigma }_u\right)}^2$ (dashed lines) dominates over $\left(2p{\sigma }_u\right)\left(3p{\sigma }_u\right)$ (dotted lines). Thick red lines correspond to the unscreened case, ${\lambda }_p={\lambda }_e=0$, green lines to the electron-proton screening, with ${\lambda }_p=0.11\mathrm{a}.\mathrm{u}.$ and ${\lambda }_e=0$, and thin blue lines to electron-proton plus electron-electron screening, with ${\lambda }_p={\lambda }_e=0.25\mathrm{a}.\mathrm{u}.$

Figure 4

Approximate partial widths ${\mathrm{\Gamma }}_{\ell }^{\text{model}}$ after Eq. (51), plotted as a function of the internuclear distance for partial waves $\ell =0$ (solid lines) and $\ell =2$ (dashed lines). Thick red lines correspond to the unscreened case, ${\lambda }_p={\lambda }_e=0$, green lines to the electron-proton screening, with ${\lambda }_p=0.11\mathrm{a}.\mathrm{u}.$ and ${\lambda }_e=0$, and thin blue lines to electron-proton plus electron-electron screening, with ${\lambda }_p={\lambda }_e=0.25\mathrm{a}.\mathrm{u}.$

Figure 5

(Panel a) Direct potential $V_d$ and its components $V_Z$ and $V_{1s{\sigma }_g}$ seen by the ionizing electron within the static exchange approximation at the internuclear distance $R=3.7\mathrm{a}.\mathrm{u}.$ and as a function of $z$ (along the internuclear axis). The black solid line corresponds to the $1s{\sigma }_g$ wave function. Solid lines: direct potential $V_d\left(\mathbf{r}\right)=V_Z\left(\mathbf{r}\right)+V_{1s{\sigma }_g}\left(\mathbf{r}\right)$. Dashed lines: potential due to the ${\mathrm{H}}_2{}^{+}$ ground state $V_{1s{\sigma }_g}\left(\mathbf{r}\right)$ and potential generated by the nuclei $V_Z\left(\mathbf{r}\right)$. Thick red lines correspond to the unscreened case, ${\lambda }_p={\lambda }_e=0$, green lines to the electron-proton screening, with ${\lambda }_p=0.11\mathrm{a}.\mathrm{u}.$ and ${\lambda }_e=0$, and thin blue lines to electron-proton plus electron-electron screening, with ${\lambda }_p={\lambda }_e=0.25\mathrm{a}.\mathrm{u}.$ (Inset, panel b) Detail of the direct potential $V_d$ where a potential barrier appears only when the electron-proton interaction is screened. Note that $V_Z$ only depends on ${\lambda }_p$ and $V_{1s{\sigma }_g}$ only on ${\lambda }_e$, so that only two curves are drawn for both potentials.