Resonances for Coulombic potentials by complex scaling and free-reflection complex-absorbing potentials

Analytical expressions for the resonances of the long-range potential (LRP), $V\left(r\right)=a∕r-b∕r^2$, as a function of the Hamiltonian parameters were derived by Doolen a long time ago [Int. J. Quant. Chem. 14, 523 (1979)]. Here we show that converged numerical results are obtained by applying the shifted complex scaling and the smooth-exterior scaling (SES) methods rather than the usual complex coordinate method (i.e., complex scaling). The narrow and broad shape-type resonances are shown to be localized inside or over the potential barrier and not inside the potential well. Therefore, the resonances for Doolen LRP’s are not associated with the tunneling through the potential barrier as one might expect. The fact that the SES provides a universal reflection-free absorbing potential is, in particular, important in view of future applications. In particular, it is most convenient to calculate the molecular autoionizing resonances by adding one-electron complex absorbing potentials into the codes of the available quantum molecular electronic packages.

Figure 1

CS and SCS.

Figure 2

SFR-CAP using $r^{\prime }=15,r_0=13$. Note that the FR-CAP is simply the SFR-CAP, where $r^{\prime }=0$.

Figure 3

Eigenstates of the Doolen potential $V\left(r\right)=4\left(1∕r-4∕r^2\right)$ as a function of the box size. The number of basis functions is kept constant, $N=400$. In the inset the ground state is portrayed at a constant box size of $50\mathrm{a.u.}$ as a function of the number of basis functions $N$.

Figure 4

$\theta$ trajectory of the first resonance position and width relative error $\Delta E=E∕E_{\mathrm{exact}}-1$ where $E_{\text{exact}}$ is given in Table . $\Delta \Gamma =\Gamma ∕{\Gamma }_{\text{exact}}-1$, where ${\Gamma }_{\text{exact}}$ is given in Table .

Figure 5

First (dot dashed) and second (dashed) resonance wave functions shown on a base line at their respective real part of the energy.

Figure 6

Convergence of the SFR-CAP calculations by changing $r_0$. The results presented are for the first resonance position and width relative error $\Delta E=E∕E_{\mathrm{exact}}-1$, where $E_{\text{exact}}$ is given in Table . $\Delta \Gamma =\Gamma ∕{\Gamma }_{\mathrm{exact}}-1,$ ${\Gamma }_{\text{exact}}$ is given in Table .

Figure 7

First resonance wave function as $r_0$ changes. Showing the exponential divergent nature of the scaled broad resonance solution.

Figure 8

$r_0$ trajectory of the first bound state. FR-CAP refers to $\Delta V_f\ne 0$ and “FR-CAP” refers to $\Delta V_f=0$. $\Delta E=E∕E_{\mathrm{exact}}-1$ where $E_{\text{exact}}$ is given in Table . $\Gamma$ refers to ${\Gamma }_{\text{exact}}$ given in Table .

Figure 9

$r_0$ trajectory of the first bound state using the FR-CAP and increasing the number of basis functions used. $\Delta E=E∕E_{\mathrm{exact}}-1$ where $E_{\text{exact}}$ is given in Table . $\Gamma$ refers to ${\Gamma }_{\text{exact}}$ given in Table .