Calculation of screened Coulomb potential matrices and its application to He bound and resonant states

We present two analytical methods, Taylor expansion and Gegenbauer expansion, to efficiently and accurately calculate the two-electron screened Coulomb potential matrix elements with Slater-type configuration-interaction basis functions. The former permits great advantages in fast computation of the potential matrices at small screening parameters and the latter allows accurate calculation of the matrices at all screening parameters. The bound and resonant states of a He atom embedded in the screening environment are calculated by employing the variational and complex-scaling methods, respectively, and the results are compared with other theoretical predictions. The expectation values of some physical quantities for He ground state are compared with the recent calculation of Ancarani and Rodriguez [Phys. Rev. A 89, 012507 (2014)] and extended to stronger screening environment. The energies and widths for the doubly excited resonant states are in good agreement with previous calculations, while the interelectronic angle $arccos\langle cos\left({\theta }_{12}\right)\rangle$ show significant discrepancies with the Feshbach projection calculation of Ordóñez-Lasso et al. [Phys. Rev. A 88, 012702 (2013)]. The expectation values of $\langle {\mathbf{p}}_{\mathbf{1}}·{\mathbf{p}}_{\mathbf{2}}\rangle$ are also calculated for the resonant states investigated here. We conclude that the present methods in the framework of complex scaling enable us to get reliable energy, width, and other physical quantities of the resonant states in a variety of screening conditions.

Figure 1

Relative error of Taylor expansion in calculating Eq. (18) with respect to the expansion number $n_{\max }$ at different Debye length $D$.

Figure 2

Variation of the He bound-state energies $(\Delta E=E_{\mathrm{He}}-E_{{\mathrm{He}}^{+}\left(1s\right)})$ against the screening parameter $\lambda$. (a) Ground state; (b) singly excited states, where $n=2$–5 represent lines from bottom to top. Present results are compared with the calculations of Kar and Ho [29], Lin et al. [32], Lam and Varshni [21], Saha et al. [25], Ancarani and Rodriguez [33], and Dai et al. [24].

Figure 3

Variation of the expectation values of some physical quantities for the He ground state against the screening parameter $\lambda$. (a) $\langle r_1\rangle$, $\langle r_{12}\rangle$, $\langle r_{<}\rangle$, $\langle r_{>}\rangle$, and $-\langle V\rangle /\langle T\rangle$; (b) $\langle {\mathbf{p}}_{\mathbf{1}}·{\mathbf{p}}_{\mathbf{2}}\rangle$, $\langle cos\left({\theta }_{12}\right)\rangle$, and $\langle {\mathbf{r}}_{\mathbf{1}}·{\mathbf{r}}_{\mathbf{2}}\rangle$. Present results are compared with the calculations of Ancarani and Rodriguez [33] when they are available.

Figure 4

Variation of the energy $(\Delta E=E_{\text{He}}-E_{{\text{He}}^{+}\left(Ns\right)})$ and width for the He ${}^1{S}^e$ resonant states associated with the He${}^{+}$$(N=2$–4) thresholds against the screening parameter $\lambda$. (a), (b) $N=2$; (c), (d) $N=3$; (e), (f), $N=4$. Present results are compared with the calculations of Kar and Ho [38], Ordóñez-Lasso et al. [44], and Chakraborty and Ho [41].

Figure 5

Variation of the energy $(\Delta E=E_{\text{He}}-E_{{\text{He}}^{+}\left(Ns\right)})$ and width for the He ${}^1{P}^o$ resonant states associated with the He${}^{+}$$(N=2$–4) thresholds against the screening parameter $\lambda$. (a), (b) $N=2$; (c), (d) $N=3$; (e), (f) $N=4$. Present results are compared with the calculations of Kar and Ho [39, 40] and Ordóñez-Lasso et al. [44].

Figure 6

Variation of $arccos\langle cos\left({\theta }_{12}\right)\rangle$ for the He ${}^1{P}^o$ resonant states associated with the He${}^{+}$$(N=2$–4) thresholds against the screening parameter $\lambda$. (a), (b) $N=2$; (c) $N=3$; (d) $N=4$. Present results are compared with the calculations of Lin et al. [58], Ordóñez-Lasso et al. [44], and Eiglsperger et al. [68].

Figure 7

Variation of $\langle {\mathbf{p}}_{\mathbf{1}}·{\mathbf{p}}_{\mathbf{2}}\rangle$ for the He ${}^1{P}^o$ resonant states associated with the He${}^{+}$$(N=2$–4) thresholds against the screening parameter $\lambda$. (a) $N=2$; (b) $N=3$; (c) $N=4$.