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p-wave resonances in the exponential cosine screened Coulomb potential*

  • Yuan-Cheng Wang , 1, 2, 3, ** ,
  • Li Guang Jiao , 2, 3, 4, ** ,
  • Aihua Liu 5 ,
  • Yew Kam Ho 6 ,
  • Stephan Fritzsche 3, 4
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  • 1College of Physical Science and Technology, Shenyang Normal University, Shenyang 110034, China
  • 2College of Physics, Jilin University, Changchun 130012, China
  • 3 Helmholtz-Institut Jena, D-07743 Jena, Germany
  • 4GSI Helmholtzzentrum fur Schwerionenforschung GmbH, D-64291 Darmstadt, Germany
  • 5Institute of Atomic and Molecular Physics, Jilin University, Changchun 130012, China
  • 6Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan, China

**Author to whom any correspondence should be addressed.

Received date: 2023-11-02

  Revised date: 2024-03-25

  Accepted date: 2024-04-25

  Online published: 2024-06-12

Supported by

*National Natural Science Foundation of China(12174147)

Chinese Scholarship Council(202006175016)

Chinese Scholarship Council(202108210152)

Copyright

© 2024 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

We perform benchmark calculations of the p-wave resonances in the exponentially cosine screened Coulomb potential using the uniform complex-scaling generalized pseudo-spectral method. The present results show significant improvement in calculation accuracy compared to previous predictions and correct the misidentification of resonance electron configuration in previous works. It is found that the resonance states approximately follow an n2-scaling law which is similar to the bound counterparts. The birth of a new resonance would distort the trajectory of an adjacent higher-lying resonance.

Cite this article

Yuan-Cheng Wang , Li Guang Jiao , Aihua Liu , Yew Kam Ho , Stephan Fritzsche . p-wave resonances in the exponential cosine screened Coulomb potential*[J]. Communications in Theoretical Physics, 2024 , 76(7) : 075501 . DOI: 10.1088/1572-9494/ad432d

1. Introduction

The exponential cosine screened Coulomb potential (ECSCP) in the form $V(r)=-\tfrac{{{\rm{e}}}^{-\lambda r}}{r}\cos (\lambda r)$, where $\lambda =\sqrt{\tfrac{{m}_{e}{\omega }_{{pe}}}{{\hslash }}}$ is the screening parameter (${\omega }_{{pe}}=\sqrt{\tfrac{4\pi {n}_{0}{e}^{2}}{\epsilon {m}_{e}}}$ represents the electron plasma frequency, n0 is the electron number density, ε is the relative dielectric permeability, and me is the mass of electrons [1]), has been extensively employed in modeling the effective interaction between charged particles in strongly coupled dense quantum plasmas [113]. With gradually increasing screening strength, each bound state is eventually absorbed into the continuum at a particular screening parameter λc, which is generally referred to as the critical screening parameter for that particular state [14]. Several theoretical methods have been proposed for accurately calculating the bound state energy spectra of one-electron atoms under the ECSCP [1526]. For two-electron atoms embedded in strongly coupled dense quantum plasmas, investigation of the plasma screening effect on both bound states and auto-ionizing resonances has also attracted considerable interests in recent years [2736]. In our previous works [3739], we have successfully studied the asymptotic behavior of one-electron systems near the critical binding threshold and obtained highly accurate critical screening parameters for different bound states in a variety of short-range screened Coulomb potentials. For systems with non-zero orbital angular momenta, the bound state transforms into a shape-type resonance when the screening parameter exceeds the critical parameter, due to the existence of repulsive centrifugal potential barrier. Compared to the extensively investigated bound states where the Rayleigh–Ritz variational methods can be conveniently applied, accurate determination of the resonance parameters (position and width) is nevertheless a challenging task for different theoretical methods due to the asymptotic divergence of the resonance wave function [40].
Different approaches have been developed in the literature to investigate the resonance states of one-electron atoms under the ECSCP [4143]. Without loss of generality, in this work we focus on the p-wave (l = 1) states where several researchers have reported the positions and widths for some low-lying resonances. Singh and Varshni [41] first calculated the 2p resonance state for screening parameter λ from 0.15 to 0.18 by numerically integrating the Schrödinger equation and locating the resonance energy for which the scattering phase shift increases through π/2 (it is worth mentioning that the critical screening parameter for the 2p state is about 0.1482 [37]). Nasser et al [42] applied the J-matrix approach to predict the np resonance states with n = 2 − 6 for screening parameter λ from 0.025 to 0.7. Later, Bahlouli et al [43] reproduced these np resonances using the complex-scaling method based on the Laguerre basis set. A systematic investigation of the variation of resonances in the entire parameter space and the transformation from bound states to resonances in the ECSCP is not yet available in the literature.
In this work, we apply the uniform complex-scaling generalized pseudospectral (UCS-GPS) method [40, 4446] to calculate the p-wave resonances with high precision. This paper is structured as follows. In section 2, we give a brief introduction to the numerical method. In section 3, we present our results and compare them with previous calculations existing in the literature. A detailed discussion of the emergence of resonance states and the trajectories of resonance poles in the complex energy and momentum planes is also included in section 3. A summary of the present work is given in section 4. Atomic units (a.u.) are used throughout this paper.

2. Model and method

The non-relativistic radial Schrödinger equation for the one-electron systems under the ECSCP is given by
$\begin{eqnarray}\begin{array}{l}\left[-\displaystyle \frac{1}{2\mu }\displaystyle \frac{{{\rm{d}}}^{2}}{{\rm{d}}{r}^{2}}+\displaystyle \frac{l(l+1)}{2\mu {r}^{2}}+V(\lambda ,r)\right]{\psi }_{{nl}}(r)\\ \,=\,{E}_{{nl}}(\lambda ){\psi }_{{nl}}(r),\end{array}\end{eqnarray}$
where μ is the electron reduced mass and the potential V (λ, r) reads as
$\begin{eqnarray}V(\lambda ,r)=-\displaystyle \frac{Z}{r}{{\rm{e}}}^{-\lambda r}\cos (\lambda r),\end{eqnarray}$
in which λ is the screening parameter and Z is the nuclear charge. In this work, we focus on the screened one-electron system with an infinitely heavy nucleus, i.e., Z = μ = 1.
In the UCS-GPS method, the radial coordinates in the Hamiltonian are rotated by an angle θ, and then mapped onto [-1, 1] through the mapping function r = f(x)
$\begin{eqnarray}r\to r{{\rm{e}}}^{{\rm{i}}\theta }=f(x){{\rm{e}}}^{{\rm{i}}\theta }=L\displaystyle \frac{1+x}{1-x}{{\rm{e}}}^{{\rm{i}}\theta },\end{eqnarray}$
where the mapping parameter L determines the radius r = L at x = 0. The radial wave function is transformed by
$\begin{eqnarray}\phi (x,\theta )=\sqrt{f^{\prime} (x)}\psi (f(x){{\rm{e}}}^{{\rm{i}}\theta }).\end{eqnarray}$
The complex-scaled radial Schrödinger equation is then given by
$\begin{eqnarray}\begin{array}{l}\left[-\displaystyle \frac{1}{2f^{\prime} (x)}\displaystyle \frac{{{\rm{d}}}^{2}}{{\rm{d}}{x}^{2}}\displaystyle \frac{1}{f^{\prime} (x)}{{\rm{e}}}^{-2{\rm{i}}\theta }+\displaystyle \frac{l(l+1)}{2f{\left(x\right)}^{2}}{{\rm{e}}}^{-2{\rm{i}}\theta }+V(\lambda ,f(x){{\rm{e}}}^{{\rm{i}}\theta })\right]\phi (x,\theta )\\ =\,E(\lambda ,\theta )\phi (x,\theta ),\end{array}\end{eqnarray}$
where
$\begin{eqnarray}V(\lambda ,f(x){{\rm{e}}}^{{\rm{i}}\theta })=-\displaystyle \frac{{{\rm{e}}}^{-\lambda f(x){{\rm{e}}}^{{\rm{i}}\theta }}}{f(x){{\rm{e}}}^{{\rm{i}}\theta }}\cos (\lambda f(x){{\rm{e}}}^{{\rm{i}}\theta }).\end{eqnarray}$
In the GPS method, the second-order differential equation is solved in the discrete-variable representation by discretizing the variable x in terms of the abscissas in the Legendre–Gauss–Lobatto quadrature
$\begin{eqnarray}{x}_{0}=-1,{x}_{j,(j=1,...,N-1)}=\mathrm{zeros}\ \mathrm{of}\ {{L}}_{N}^{{\prime} },{{x}}_{N}=1,\end{eqnarray}$
where LN(x) is the Nth-order Legendre polynomial. The pseudospectral approximation
$\begin{eqnarray}\phi (x)\approx {\phi }_{N}(x)=\displaystyle \sum _{j=0}^{N}\phi ({x}_{j}){g}_{j}(x),\end{eqnarray}$
ensures that the expansion coefficients in terms of cardinal basis functions gj(x) are the wave functions themselves at the collocation points, i.e., φ(xj). In the present work, we simply use the cardinal function defined by
$\begin{eqnarray}{g}_{j}(x)=\displaystyle \frac{1}{N(N+1)}\displaystyle \frac{(1-{x}^{2})}{({x}_{j}-x)}\displaystyle \frac{P{{\prime} }_{N}(x)}{{P}_{N}({x}_{j})},\end{eqnarray}$
which can be easily verified that
$\begin{eqnarray}{g}_{j}({x}_{i})={\delta }_{{ij}}.\end{eqnarray}$
For a detailed implementation of the GPS method, readers are referred to [40, 4446].
After solving the complex symmetric non-Hermitian eigenvalue problem, the resonance poles are found on the right-hand side of the rotated continuum branch cuts in the complex energy plane [40], provided the rotational angle fulfills the condition
$\begin{eqnarray}\theta \gt \displaystyle \frac{1}{2}\arg ({E}_{\mathrm{res}}).\end{eqnarray}$
All system eigenenergies are rotated into the complex energy plane, where the positions of bound states do not change, the continua are rotated downward by an angle 2θ, and the resonance pole can be exposed from the continuum by acquiring a complex resonance energy ${E}_{\mathrm{res}}$, which reads
$\begin{eqnarray}{E}_{\mathrm{res}}={E}_{r}+{\rm{i}}{E}_{i}={E}_{r}-{\rm{i}}\displaystyle \frac{{\rm{\Gamma }}}{2},\end{eqnarray}$
in which Er and Γ are the resonance position and width, respectively.
By applying the variational procedure
$\begin{eqnarray}{\left.\displaystyle \frac{\partial {E}_{\mathrm{res}}}{\partial \theta }\right|}_{{\theta }_{\mathrm{opt}}}=\min ,\end{eqnarray}$
we can find the most stabilized resonance pole at the optimized angle of θopt. We set the maximum rotational angle to π/4 in the present calculations since π/4 is the maximum value for the complex-scaled ECSCP converging in the asymptotic region [46]. Although the optimized value of θ varies slightly by state and screening parameter, it is found that the use of a relatively large angle, e.g., a number close to π/4, is good enough for all situations investigated in the present work.
The convergence of our numerical calculations can be further improved by adjusting the mapping parameter L and increasing the number of collocation points N. This is symbolized by
$\begin{eqnarray}{\left.\displaystyle \frac{\partial {E}_{\mathrm{res}}}{\partial L}\right|}_{{L}_{\mathrm{opt}}}=\min ,\end{eqnarray}$
and
$\begin{eqnarray}{\left.\displaystyle \frac{\partial {E}_{\mathrm{res}}}{\partial N}\right|}_{{N}_{\max }}=\min .\end{eqnarray}$
In this work, the value of N is gradually increased to 5000, which is free of any numerical instabilities in quadruple precision. The UCS-GPS method is generally not sensitive to the mapping parameter L for a moderately large value of N. Throughout the calculations, we find very good convergence when using L = 1000.

3. Results and discussion

To illustrate the origin of resonances in the one-electron systems with the ECSCP, in figure 1 we show the effective potential energy curves defined by
$\begin{eqnarray}{V}_{\mathrm{eff}}(r)=\displaystyle \frac{l(l+1)}{2{r}^{2}}-\displaystyle \frac{Z}{r}{{\rm{e}}}^{-\lambda r}\cos (\lambda r),\end{eqnarray}$
where we restrict our discussion on the p-wave states, i.e., l = 1. The corresponding eigenstates associated with the potential energy curves are also included in figure 1. At λ = 0.13, the deep potential well traps a bound electron in the lowest 2p state with a negative energy at about ${E}_{\mathrm{bound}}^{2p}=-0.01068$. When λ is increased to the ‘critical' screening parameter, which is exactly located at ${\lambda }_{c}^{2p}\,=0.1482...$ [37], the system is in a zero-energy critical state where the bound-resonance quantum phase transition occurs. For λ = 0.17, the effective potential forms a barrier (due to the non-zero orbital angular momentum) that is high enough to support a shape-type resonance state at about ${E}_{\mathrm{res}}^{2p}\,=0.00971-{\rm{i}}0.002\,24$.
Figure 1. Effective potential energy curves Veff(r) defined in equation (12) for the p-wave states of the ECSCP at λ = 0.13, ${\lambda }_{c}^{2p}$, and 0.17. The potential well at λ = 0.13 can trap a 2p bound electron, while the high potential barrier at λ = 0.17 only supports a shape-type resonance state. At $\lambda ={\lambda }_{c}^{2p}$, the eigenenergy of the 2p state is exactly zero.
The critical screening parameters ${\lambda }_{c}^{{np}}$ for the lowest nine p-wave eigenstates (2p to 10p) are given in table 1. These results were obtained using the combination of the GPS method with Brent's minimization algorithm. The computational details can be found in [37]. For screening parameters being larger than ${\lambda }_{c}^{{np}}$, the np bound state transforms into a well-defined shape-type resonance, which can be accurately calculated using the present UCS-GPS method.
Table 1. Critical screening parameters for the lowest nine p-wave eigenstates of the ECSCP.
np ${\lambda }_{c}^{{np}}$
2p 0.148205032642758419285886459123
3p 0.687121436894544378282407881134(−1)
4p 0.392634011792194538641491183042(−1)
5p 0.253156253177013910985149600317(−1)
6p 0.176520702074135587212546717879(−1)
7p 0.130010639904740743652758680825(−1)
8p 0.997008724443218669747295568387(−2)
9p 0.788640558678603012644097745984(−2)
10p 0.639311481645614751381777756738(−2)
The UCS-GPS method explicitly depends upon the discretization dimension N. It is therefore necessary to estimate the convergence of numerical calculations regarding this parameter. In table 2, we show the UCS-GPS calculations of the 2p resonance state for the one-electron system with ECSCP at λ = 0.7, using the parameters of L = 1000, θ = 0.785, and N = 1000 to 5000. As seen from this table, the increase of the dimension of discretization N leads to a fast convergence of the resonance position and width. The present calculations are performed in quadruple precision (∼34 significant digits after the decimal point). Using N = 5000, an accuracy of 30 digits after the decimal point can be achieved, which is close to the limit of quadruple precision. In addition, table 2 includes the previous J-matrix calculations of Nasser et al [42] and the complex-scaling calculations of Bahlouli et al [43] using the Laguerre basis functions. It can be found that our results are in good agreement with the predictions of Nasser et al [42] for all the digits they reported. The last two digits reported by Bahlouli et al [43] are less accurate, which is probably due to the slow convergence in their complex-scaling calculations. The high accuracy of our results is well demonstrated by the full convergence in the present UCS-GPS calculations.
Table 2. Convergence of the 2p resonance state for the one-electron system with ECSCP at screening parameters of λ = 0.7, with L = 1000 and θ = 0.785. The results of Nasser et al [42] and Bahlouli et al [43] are included for comparison.
λ N Er Ei
0.7 1000 0.0254564308878703595566369476824 −0.405997619261475855140293094607
2000 0.0254564308878410068247375880331 −0.405997619261439084995429189949
3000 0.0254564308878410068247375870520 −0.405997619261439084995429170708
4000 0.0254564308878410068247375870476 −0.405997619261439084995429170721
5000 0.0254564308878410068247375870476 −0.405997619261439084995429170717

Nasser et al [45] 0.025 456 −0.40599
Bahlouli et al [45] 0.025 422 −0.405919
In figure 2, we show the variation of the real part of eigenenergies for the np states with n = 2 to 10 along with the increase of screening parameter. The numerical data is also given in tables 3 and 4 for further reference. All resonances can be classified by their counterparts in the bound region since the real parts of resonance energies connect smoothly with the negative bound state energies at the critical values of λc, i.e., each bound state transforms into a shape resonance located above the threshold at a corresponding critical screening parameter with continuously increasing λ. From figure 2 it is clear that a resonance is created at λc and its position increases, reaching a maximum, and finally becomes a close-to-zero-energy resonance. Surprisingly, it is found that there exist specific values of screening parameters at which two adjacent resonances cross each other, with the higher-lying resonances being slightly disturbed by the lower-lying ones. These crossing behaviors can be clearly seen in the inset of figure 2. Figure 3 displays the resonance width Γ for the same states as those shown in figure 2. The resonance width, which is two times that of the magnitude of the imaginary part of resonance energy, increases monotonically as the screening parameter increases. For the specific value of λ where two resonances share the same resonance positions, however, their resonance widths differ by orders of magnitude. In the complex energy plane as well as in the complex momentum plane, these two resonance poles are still far away from each other and therefore can be treated as non-degenerate states. In both figures 2 and 3, trajectory tracking allows us to precisely resolve the electron configuration of resonance states. As seen from table 3, both Nasser et al [42] and Bahlouli et al [43] incorrectly identify the 4p and 5p states at λ = 0.050. The same situation occurs at λ = 0.025 in the work of Nasser et al [42], who identified the resonance energy and width of the 7p resonance as those of the 6p resonance. In addition, Nasser et al [42] predicted an extremely low-lying 5p resonance at λ = 0.030, which is not reproduced in both Bahlouli et al [43] and the present calculations.
Figure 2. Bound state energies (lower) and resonance positions (upper) for the np eigenstates of ECSCP with n = 2 to 10. The inset shows an enlarged view of the high-lying resonances.
Figure 3. Resonance widths for the np states of ECSCP with n = 2 to 10. The inset shows an enlarged view of the widths for high-lying resonances.
Table 3. Resonance energies for the np states of ECSCP with n = 2 to 6. Numbers in parentheses represent powers of ten. The results of Nasser et al [42] and Bahlouli et al [43] are included for comparison when they are available.
np λ This work Nasser et al [41] Bahlouli et al [42]
2p 0.150 8.41463799507136(−4)-6.90016117390905(−5)i 8.4146380(−4)-6.90016(−5)i 8.414631(−4)-6.899(−5)i
0.155 3.19835469826691(−3)-4.17271267850758(−4)i 3.19835(−3)-4.17271(−4)i 3.19841(−3)-4.17602(−4)i
0.160 5.46594151270155(−3)-8.88396949705082(−4)i 5.4659415(−3)-8.88396(−4)i 5.4660(−3)-8.88864(−4)i
0.165 7.63002860494508(−3)-1.49387909741240(−3)i 7.63002860(−3)-1.4938791(−3)i 7.6302856(−3)-1.494354(−3)i
0.170 9.70695203299149(−3)-2.23603101512570(−3)i 9.706952033(−3)-2.23603101(−3)i 9.70709(−3)-2.23650(−3)i
0.175 1.17112300407746(−2)-3.10935956126523(−3)i 1.1711230040(−2)-3.109359561(−3)i 1.171123001(−2)-3.109359(−3)i
0.178 1.28834137420076(−2)-3.69323467301530(−3)i 1.288341374200(−2)-3.693234673(−3)i 1.28834137(−2)-3.6932346(−3)i
0.180 1.36533114432199(−2)-4.10642235903903(−3)i 1.36533114432(−2)-4.10642235903(−3)i 1.36533114(−2)-4.106422(−3)i
0.300 4.84423224621062(−2)-5.34073390579566(−2)i 4.844232246(−2)-5.3407339057(−2)i 4.8442322(−2)-5.34073391(−2)i
0.400 6.29067470676162(−2)-1.19343808546890(−1)i 6.29067470(−2)-1.193438085(−1)i 6.290674(−2)-1.1934380(−1)i
0.500 6.44225253344524(−2)-2.01523222601036(−1)i 6.442253(−2)-2.0152322(−1)i 6.442259(−2)-2.0152322(−1)i
0.600 5.24634779401062(−2)-2.97544963363380(−1)i 5.2463(−2)-2.97545(−1)i 5.246(−2)-2.97543(−1)i
0.700 2.54564308878410(−2)-4.05997619261439(−1)i 2.5456(−2)-4.0599(−1)i 2.5422(−2)-4.05919(−1)i

3p 0.075 2.34087579851323(−3)-3.56157943833318(−4)i 2.3408757985(−3)-3.5615794383(−4)i 2.34087603(−3)-3.56158(−4)i
0.080 3.85962887047523(−3)-9.79860400859225(−4)i
0.090 6.28423636914734(−3)-2.99789405697670(−3)i
0.100 7.99120736008921(−3)-5.82553278005339(−3)i
0.110 8.94858397361978(−3)-9.34650665922494(−3)i
0.120 9.07430035927160(−3)-1.36013301991694(−2)i
0.130 8.39654618124487(−3)-1.88258747777019(−2)i
0.140 7.32247886588654(−3)-2.51851313732973(−2)i
0.150 6.20578736593378(−3)-3.23345428562951(−2)i 6.2057(−3)-3.23345(−2)i 6.2057(−3)-3.23345(−2)i

4p 0.050 2.82404214024094(−3)-1.22240543251328(−3)i 1.221(−3)-8.035(−1)i 1.22(−3)-8.034(−3)i
0.055 3.46040354397717(−3)-2.46379020958530(−3)i
0.060 3.69009818410484(−3)-4.07346035370668(−3)i
0.065 3.54937316065712(−3)-6.15355044822490(−3)i
0.070 3.27423030205280(−3)-8.70856156108195(−3)i
0.075 2.94019754414923(−3)-1.14977196676793(−2)i 2.9401(−3)-1.14977(−2)i 2.940218(−3)-1.149772(−2)i
0.080 2.44050694521374(−3)-1.44134310547431(−2)i
0.085 1.70726275373763(−3)-1.74383483365434(−2)i

5p 0.027 5.81360458456676(−4)-3.46099941466531(−5)i
0.030 1.30178012581261(−3)-3.30094559435385(−4)i 1.0(−6)-8.0(−5)i
0.032 1.63307990843169(−3)-6.80826923566782(−4)i
0.035 1.92936190874223(−3)-1.40295006142746(−3)i
0.037 1.99613296099604(−3)-2.02791363535435(−3)i
0.040 1.98558854524646(−3)-3.21117430453898(−3)i
0.042 1.95597534162070(−3)-4.10351922070500(−3)i
0.045 1.82682094908000(−3)-5.50811533399168(−3)i
0.047 1.65086235056599(−3)-6.48618317102464(−3)i
0.050 1.22197782582441(−3)-8.03490388445078(−3)i 2.8240421402(−3)-1.2224054325(−3)i 2.824042(−3)-1.222404(−3)i

6p 0.020 6.92427903510190(−4)-9.70080858066130(−5)i
0.022 1.03624385178502(−3)-3.89514619100446(−4)i
0.025 1.25502387376253(−3)-1.16023747385265(−3)i 4.60(−4)-3.711(−3)i 1.25502(−3)-1.16023(−3)i
0.027 1.29563016381311(−3)-1.89729540898344(−3)i
0.030 1.23313336784715(−3)-3.14185200298618(−3)i 1.233(−3)-3.141(−3)i 1.2331(−3)-3.1418(−3)i
0.032 1.04441031869675(−3)-4.06019454769352(−3)i
0.035 5.51565165758984(−4)-5.62130286728890(−3)i
Table 4. Resonance energies for the np states of ECSCP with n = 7 to 10. Numbers in parentheses represent powers of ten.
np λ This work
7p 0.014 3.28947943545882(−4)-1.34833538327530(−5)i
0.015 5.53459397391246(−4)-9.10187780070191(−5)i
0.016 7.09883391798616(−4)-2.38283903876661(−4)i
0.017 8.08790073247991(−4)-4.49881154538681(−4)i
0.018 8.63918436100496(−4)-7.32402809325174(−4)i
0.019 9.03458649491975(−4)-1.07182076421291(−3)i
0.020 9.22726760736952(−4)-1.43814709485818(−3)i
0.021 9.05005741593023(−4)-1.82787946740459(−3)i
0.022 8.45075214955072(−4)-2.24771277922156(−3)i
0.023 7.46514777847301(−4)-2.70309695051456(−3)i
0.024 6.16566812718687(−4)-3.19311944834636(−3)i
0.025 4.59008576709142(−4)-3.71138911260040(−3)i

8p 0.010 1.15971795975842(−5)-1.16150675741718(−8)i
0.011 3.16061571394108(−4)-2.04410732941511(−5)i
0.012 4.95914048152250(−4)-1.31363610211629(−4)i
0.013 5.96362587422301(−4)-3.35695000604373(−4)i
0.014 6.55082116694703(−4)-6.31336533618891(−4)i
0.015 6.87013975671380(−4)-9.72394483919559(−4)i
0.016 6.67899516270094(−4)-1.35157127594063(−3)i
0.017 6.00387032236425(−4)-1.77788174976687(−3)i
0.018 4.93761733788663(−4)-2.24447283380143(−3)i
0.019 3.44472773060081(−4)-2.74140077060281(−3)i
0.020 1.45869837347042(−4)-3.26903789031866(−3)i

9p 0.008 4.30133228119992(−5)-2.17178914026093(−8)i
0.009 3.09195835539361(−4)-3.70956446968367(−5)i
0.010 4.38721115141968(−4)-1.97993722044232(−4)i
0.011 5.07340097737696(−4)-4.77025398483105(−4)i
0.012 5.30749538523244(−4)-8.14436126949938(−4)i
0.013 4.92705320168512(−4)-1.21079875111701(−3)i
0.014 4.02261644982922(−4)-1.65664265070585(−3)i
0.015 2.52428480947014(−4)-2.14499279279549(−3)i

10p 0.007 1.88888727147462(−4)-7.12117825187625(−6)i
0.008 3.40087820435550(−4)-1.36966956073477(−4)i
0.009 4.11135664655279(−4)-4.12011599015345(−4)i
0.010 4.16386333857880(−4)-7.62517291864373(−4)i
0.011 3.53605788699189(−4)-1.18030350367781(−3)i
0.012 2.18333249353177(−4)-1.65493956518500(−3)i
From both figures 2 and 3, one can see that it is quite difficult to display the variation of resonances with different principal quantum numbers in a single figure, because the low- and high-lying resonances vary in quite different energy scales. In figures 4 and 5, we alternatively show the n2-scaled resonance positions and widths, respectively, as a function of the n2-scaled screening parameter for the 2p, 3p, 5p, and 10p resonances. It is necessary to consider two facts in relation to this n2-scaling law. (I) In the unscreened situation, i.e., λ = 0, the one-electron system coincides with the free H atom, and therefore all bound state energies scale exactly with respect to n2. (II) The critical screening parameters λc scale approximately with respect to n2 in the high Rydberg limit [38], i.e., ${\lambda }_{c}^{{nl}}(n)\propto {n}^{-2}$. According to these two n2-scaling laws, we expect the resonance states with the same l angular symmetry to follow a similar trend. This can be seen from figures 4 and 5 for the resonance positions and widths, respectively, which roughly follow an n2-scaling law, especially for high-lying resonances.
Figure 4. n2-scaled bound state energies (lower) and resonance positions (upper) as a function of n2-scaled screening parameters for the 2p, 3p, 5p, and 10p states.
Figure 5. n2-scaled resonance widths as a function of n2-scaled screening parameters for the 2p, 3p, 5p, and 10p states. The inset shows an enlarged view of the n2Γ at small values of n2λ.
Although the n2-scaled resonances with different n generally show similar behavior in both shape and magnitude, there exist visible irregular structures in the resonance positions, which is attributed to the interference effect between adjacent resonances. Figure 6 shows the variation of complex momenta for the np resonances of the ECSCP with n = 2 to 10 as a function of λ. The relationship between the complex resonance energies and corresponding complex momenta is defined by ${k}_{\mathrm{res}}=\sqrt{2{E}_{\mathrm{res}}}$. When the 2p resonance appears at ${\lambda }_{c}^{2p}$, the trajectory line of the real momentum of the 3p resonance transforms from a convex curve to a concave curve, indicating the increasingly fast shift away of the resonance pole from the origin. Similar phenomena can be observed in the trajectories of higher-lying resonances as well. Let us recall the unsmooth characters of resonance energies shown in figures 2 and 4, the distortions actually start from the critical screening parameters where new resonances appear. Analogous to the Wigner threshold law ($\sigma (\omega )\propto {\left(\omega -{E}_{\mathrm{th}}\right)}^{l+\tfrac{1}{2}}$) in the quantum scattering theory [47], the resonance width in the critical region varies with an l-dependent power law ${{\rm{\Gamma }}}^{{nl}}(\lambda )\propto {\left(\lambda -{\lambda }_{c}^{{nl}}\right)}^{l+\tfrac{1}{2}}$ [37, 48]. The distortion effect due to the birth of a new resonance can probably be understood in a similar way as the channel-coupling effect when a new reaction channel is opened [48]. It is also observed in figure 6 that there exist situations where a resonance crosses two or more adjacent resonances in higher excitation levels, which means the new resonance may cause interference with two or more higher-lying resonances.
Figure 6. Variation of the complex momenta of np resonances with n = 2 to 10 as a function of screening parameter. The upper and lower sections show the real and imaginary parts of momentum, respectively. The vertical dashed lines indicate the critical screening parameters.

4. Conclusion

In this work, we carried out a systematic study of the p-wave resonances (2p to 10p) for the one-electron system with the ECSCP. The UCS-GPS method was employed to extract the resonance position and width with high precision. The variations of complex resonance energies and corresponding complex momenta with respect to the screening parameter were systematically investigated. The present results show significant improvement in computational convergence over previous predictions and correct the misidentification of resonance electron configuration in previous works. We found that the birth of a new resonance would interfere with adjacent higher-lying resonances and, furthermore, there exist situations where a resonance crosses two or more adjacent resonances in higher excitation levels. The nonlinear response of plasma-embedded atoms exposed in an external electric field or the microfield induced by local fluctuations in the densities of charged particles has attracted increasing interest in recent years [10, 49, 50]. The extension of the present numerical method to the resonance states of these systems is promising in future studies.
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