Bound and resonance states of the dipolar anion of hydrogen cyanide: Competition between threshold effects and rotation in an open quantum system

Bound and resonance states of the dipole-bound anion of hydrogen cyanide ${\mathrm{HCN}}^{-}$ are studied using a nonadiabatic pseudopotential method and the Berggren expansion technique involving bound states, decaying resonant states, and nonresonant scattering continuum. We devise an algorithm to identify the resonant states in the complex energy plane. To characterize spatial distributions of electronic wave functions, we introduce the body-fixed density and use it to assign families of resonant states into collective rotational bands. We find that the nonadiabatic coupling of electronic motion to molecular rotation results in a transition from the strong-coupling to weak-coupling regime. In the strong-coupling limit, the electron moving in a subthreshold, spatially extended halo state follows the rotational motion of the molecule. Above the ionization threshold, the electron's motion in a resonance state becomes largely decoupled from molecular rotation. The widths of resonance-band members depend primarily on the electron orbital angular momentum.

Figure 1

Illustration of the stability of the energies of the $J^{\pi }=2^{+}$ resonant states of ${\mathrm{HCN}}^{-}$ listed in Table 1 (large dots) when the nonresonant scattering contour is shifted. Here, the imaginary part of $k_{\mathrm{peak}}$ was varied from 0 to $-0.0001a_0^{-1}$. As a comparison, nonresonant eigenenergies are marked with tiny dots and exhibit significant shifts.

Figure 2

Similar as in Fig. 1 but zoomed in on the two threshold resonances (states 2 and 3 in Table 1). Here, the real part of $k_{\mathrm{peak}}$ is also varied from $0.14 a_0^{-1}$ to $0.16 a_0^{-1}$.

Figure 3

The convergence of $\mathrm{Im}\left(E\right)$ for $J^{\pi }=2^{+}$ resonances (a) 2, (b) 3, and (c) 1 of Table 1 as a function of ${\ell }_{\max }$. The quantum numbers $(\ell ,j)$ of the dominant channel are indicated.

Figure 4

Predicted energies of the ${\mathrm{HCN}}^{-}$ dipolar anion for $J^{\pi }=0^{+},1^{-},2^{+},3^{-},4^{+}$, and $5^{-}$ states in the complex-energy plane. Based on their complex energies, these states can be organized into five groups labeled $g_0$ to $g_4$. Bound states and near-threshold resonances belonging to $g_4$ and narrow resonances of $g_3$ are shown in the inset.

Figure 5

The intrinsic density of the valence electron in ${\mathrm{HCN}}^{-}$ in the ground-state rotational band $J^{\pi }=0_1^{+},1_1^{-},2_1^{+},3_1^{-},\cdots$ (All densities are in $a_0^{-1}$.)

Figure 6

Channel wave functions $(\ell ,j)$ of the (a) $J^{\pi }=0_1^{+}$ and (b) $4_1^{+}$ members of the ground-state rotational band in ${\mathrm{HCN}}^{-}$ in the adiabatic limit.

Figure 7

Density (19) of the valence electron in the bound states (a) $J^{\pi }=0_1^{+}$, (b) $1_1^{-}$, and (c) $2_1^{+}$ of ${\mathrm{HCN}}^{-}$. (All densities are in $a_0^{-1}$.)

Figure 8

Energy spectrum of the ${\mathrm{HCN}}^{-}$ anion for $J^{\pi }=0^{+},1^{-},2^{+},3^{-},4^{+}$, and $5^{-}$ shown as a function of $J(J+1)$. The dominant $K_J$ component (21) is indicated. If several components are present, the state is marked as “mix.”

Figure 9

Similar as in Fig. 7 but for ${\rho }_{J{K}_J=0}(r,\theta )$ (in ${10}^{-15}{a}_0$) in (a) $3_1^{-}$, (b) $4_1^{+}$, and (c) $5_1^{-}$.

Figure 10

Excitation energies of resonances of the $\mathrm{HCN}{}^{-}$ dipolar anion for various $J^{\pi }$ as a function of $j(j+1)$, where $j$ is the rotational angular momentum of the molecule in the dominant channel wave function for each considered state. Colors are related to groups of states in the complex-energy plane identified in Fig. 4. The symbols $\blacksquare ,•,▾$ and $+$ denote states with $J^{\pi }=2^{+},3^{-},4^{+}$, and $5^{-}$, respectively.

Figure 11

Intrinsic densities ${\rho }_{J{K}_J}(r,\theta )$ (in ${10}^{-10}{a}_0$) with $K_J=0,1,2$, for the two resonances $5_3^{-}$ and $5_{23}^{-}$ belonging to the group $g_2$, marked by arrows in Fig. 10. For both states, the dominant channel has $\ell =6$.

Figure 12

Similar as in Fig. 10 but for $(E_J-E_{\mathrm{bh}})\frac{2I}{j(j+1)}$, where $E_{\mathrm{bh}}$ is a band-head energy at $j=0$.

Figure 13

Similar as in Fig. 10 but for the resonance widths.