Doubly excited states (DESs) of two-electron atom is a topic of active interest in recent times, both from theoretical and experimental aspects.^[1-9] The abundance of such DESs is noted in various astrophysical observations as well as in high temperature laboratory plasma.^[10-19] The DESs of two-electron atom having unnatural parity ($\pi = (-1)^{L+1}$, $L$ is the total angular momentum quantum number) and lying below the second ionization threshold, are metastable bound. These states favorably can decay to a lower state via radiative process rather than decaying through non-radiative autoionization channel. Examples of such DESs of unnatural parity are $^{1,3}$P$^e$, $^{1,3}$D$^o$, $^{1,3}$F$^e$ states arising out of dominant $pp$, $pd$, $pf$ configurations respectively.

Atomic systems under external environments have been studied by various researchers during the past several decades, as they provide useful information about the environment. A large number of investigations^[20-30] are there in the literature on the modified properties of plasma embedded atomic systems. Extensive review articles^[31-32] are available on this topic. Plasma coupling strength ($\Gamma$) is defined as the ratio between average inter-particle electrostatic energy to the average thermal kinetic energy. The high temperature and low density classical plasma are categorized as weakly coupled ($\Gamma <1$). According to the Debye-Hückel theory^[33] a short range Yukawa-type or screened Coulomb model potential is considered to mimic the modified inter-particle interaction under WCP environment. Due to its simplicity and effectiveness, such screened Coulomb potential has been used widely by researchers for the investigation of spectral and structural properties of atomic systems under WCP environment. In this model, plasma electron density ($n_{e}$) and temperature ($T$) are combinedly expressed through the plasma screening length ($D$).^[33] As the screened Coulomb potential is more positive in nature than the "pure" Coulomb potential, in general, with the decrease of plasma screening length $(D)$, the energy levels are pushed up and the gap between two successive energy level decreases.^[20] This causes the transition energy to decrease and a red shift^[21] may be observed. However, it is remarkable that for some specific transitions between two doubly excited energy levels, the wavelengths get blue shifted or show a pattern with both red and blue shift w.r.t. the plasma screening length $(D)$.^[21]

In the present work, we have estimated the non-rel-ativistic energy eigenvalues of doubly excited metastable bound $2pnf\,(n=4$--6) ($^{1,3}$F$^e$) states of two-electron ions ($Z=2$--4) as well as the $2s$ and $2p$ states of the respective one-electron ions under WCP environment. According to the Debye-Hückel theory,^[33] in a two-electron Hamiltonian, the effect of plasma screening should be reflected in both the one-particle electron-nucleus attraction terms and the electron-electron repulsion term of the total potential. Although the effect of screening on the electron-nucleus attraction term predominates overthe electron-electron repulsion term in determining the properties of plasma embedded two electron atom, we have considered the effect of screening on both attractive electron-nucleus part and repulsive electron-electron part in the potential. Computationally, it is difficult to include the effect of screening in repulsive electron-electron part even for a partially correlated CI type basis constructed with Slater-type orbitals as the analytic solution of the corresponding basis integrals becomes extremely cumbersome.^[34-35] However, we have been able to develop the methodology to estimate the basis integral for the trial wavefunction is expanded in multi-exponent Hylleraas type basis set in a way that the effect of screening in repulsive electron-electron part has been considered fully without any perturbative approximation. Ritz variational method is used to determine the energy eigenroots. The wavelengths for the dipole allowed transitions between doubly excited metastable bound states $2pnf$ [$n=4$--6] $(^{1,3}$F$^e)$ and $2pn'd$ ($n'=3$--6) $(^{1,3}$D$^o)$ are determined for different values of plasma screening length ($D$). The non-relativistic energy-eigenvalues of $2pnd$ ($n=3$--6) ($^{1,3}$D$^o$) states are taken from an earlier work of Saha et al.^[21] The details of the methodology are given in Sec. 2 followed by the discussion on the results in Sec. 3 and finally concluded in Sec. 4.

The non-relativistic Hamiltonian (in a.u.) of a two-electron atom immersed in WCP environment may be written as

where, in case of screening by both ions and electrons, the Debye screening length ($D$) reads as^[33]

For a fully ionized plasma comprising of a single nuclear species, the effective nuclear charge is $Z_{e}=Z$ whereas in case of screening by electrons only, $Z_{e}=0$. After seperation of the centre of mass coordinates, the wave function of $^{1,3}$F$^{e}$ states due to dominant $pf$ configuration of a two-electron atom can be written in terms of six co-ordinates $(r_1, r_2, \theta_{12}; \theta ,\varphi ,\psi )$ as,^[36-37]

where, $D_{L}^{\kappa \pm}$ are the rotational harmonics and functions of three Eulerian angles ($\theta$, $\varphi$, $\psi$) that define the orientation of the triangle formed by the two electrons and the nucleus in space; $\kappa$ is the angular momentum quantum number about the body fixed axis of rotation.^[36] The radial parts of the wavefunction are given by $f_{3}^{0}=-F_{1}\sin\theta_{12}$, $f_{3}^{2+}=({\sqrt{15}}/{6})F_{1}\sin 2\theta_{12}$ and $f_{3}^{2-}=({\sqrt{15}}/{6})F_{2}(1-\cos 2\theta_{12})$; where, $F_{1}=(f\mp\tilde{f})$, $F_{2}=(f\pm\tilde{f})$ with the condition $\tilde{f}=f(r_{2},r_{1})$ and $\theta_{12}$ is the angle between $\vec{r}_1$ and $\vec{r}_2$. The upper sign corresponds to the singlet state and the lower sign to the triplet state. The trial radial wave function corresponding to $pf$ configurations is expanded in Hylleraas basis set as

with the features: (a) The powers of $r_{1}$, $r_{2}$ and $r_{12}$ satisfies $(l_{i},m_{i},n_{i})\geq(0,0,0)$; (b) $A$ is the total number of ($l_{i},m_{i},n_{i}$) set considered in the calculation; (c) $\eta_i (j) = e^{-\rho_i r_j}$ are the Slater-type orbitals where $\rho$'s are the non-linear parameters; (d) $p$ denotes the total number of non-linear parameters; (e) In the double sum of Eq. (4), $k_{1} < k_{2}$; (f) $C_{ik_{1}k_{2}}$ are the linear variational parameters. The effect of the radial correlation is incorporated through different $\rho$'s in the wave function whereas, the angular correlation effect is taken care of through different powers of $r_{12}$. The number of terms in the basis set expansions for the trial radial wave function $f$ is therefore $N=[{p(p+1)}/{2}]\times A$. In the present case, we have considered a nine-exponent ($p=9$) basis set where the non-linear parameters are taken in a geometrical sequence following $\rho_{i}=\rho_{i-1}\gamma$, $\gamma$ is the geometrical ratio. After choosing the proper trial radial wave function, the energy eigenvalues are obtained by solving the generalized eigenvalue equation.^[38] The details regarding the analytic evaluation of the correlated basis integrals are discussed in Dutta et al.^[38]

The variational equation for the $nl$-state of the respective one-electron atoms under WCP environment can be written as

The radial function $f(r)$ is expanded in terms of a pure exponential basis set as

We have used 101 number of terms in the basis set and the exponents are taken in a geometrical sequence $\sigma_{i}=\sigma_{i-1}\beta$, $\beta$ is the geometrical ratio. The energy eigenvalues $E$'s and linear variational coefficients $C_{i}$'s are determined by matrix diagonalization procedure. All calculations are carried out in quadruple precision. Such procedure is repeated for different plasma screening length ($D$) considered in the present case.

Table 1 shows the convergence behavior of the energy eigenvalues of $2pnf$ ($n=4$--6) $(^{1,3}$F$^e)$ states of He with respect to the total number of terms $N=540$ and $N=675$ in the 9-exponent basis set for three different Debye screening lengths $D=100$, 50, and 20 (in a.u.). It can be seen from Table 1 that, all the energy eigenvalues converge at least up to sixth decimal place for $D=100$ a.u. and $D=50$ a.u. whereas, energies of $2p5f$ and $2p6f$ converge up to fourth and third decimal places respectively for $D=20$ a.u. The energy values of $2pnf$ ($n=4$--6) $(^{1,3}$F$^e)$ states of two-electron ions $(Z=2$--4) in the presence of WCP environment are given in Tables 2--4 respectively. Only the values obtained from the wave function of maximum basis size ($N=675; A=15$) are reported in Tables 2--4. It is observed that as the plasma screening length ($D$) decreases, the two-electron energy levels are pushed towards the continuum. Such behaviour is quite consistent with the fact that the screened Coulomb potential becomes more and more positive with respect to the decrease in plasma screening length ($D$). Moreover, Tables 2--4 show that for all the ions the singlet states are more bound than the triplet states from low to moderate plasma screening. At high screening (i.e. at low values of screening length $D$), we see that the singlet and triplet states become exactly or nearly degenerate. As the plasma screening increases, the two-electron energy levels become largely affected by the continuum embedded states through configuration interactions. At very high screening region, the energy values of two-electron states come very close to the one-electron continuum and tend to merge into the $2p$ threshold of the respective one-electron system.

The $2s$ and $2p$ threshold energies of respective one-electron atoms are also included in Tables 2--4 for a comprehensive analysis of the position of two-electron energy levels. The departure from Coulomb potential facilitates the removal of $l$-degeneracy in the one-electron atoms and it is evident that the $2s$ level remains more bound compared to the $2p$ level as $D$ decreases. Figure 1 illustrates the comparative behavior of different doubly excited triplet $2p4p$ (P$^e)$, $2p4d$ (D$^o)$ and $2p4f$ (F$^e)$ states below He$^+(2p)$ threshold. The energy values of $2p4p$ ($^3$P$^e)$ and $2p4d$ $(^3$D$^o)$ states of helium, immersed in WCP environment have been taken from Refs. ^[20] and ^[21] respectively. In Fig. 1, we have shown the position of triplet $2p4p$, $2p4d$, $2p4f$ energy levels of helium along with the $2s$ and $2p$ thresholds of He$^+$ at different plasma screening strength. We note that at low screening regions when the system is almost equivalent to a free system, the one-electron $2s$ and $2p$ levels are merged on each other due to their $l$-degeneracy. These levels are split when $l$-degeneracy is sufficiently lifted at a higher screening in presence of plasma environment which is evident from the diagram. It is seen from Fig. 1 that the $2p4p$ $(^3$P$^e)$ and $2p4d$ $(^3$D$^o)$ states always lie below both the $2s$ and $2p$ thresholds of He$^+$, but the $2p4f$ $(^3$F$^e)$ level crosses the $2s$ threshold of He$^+$ when the plasma screening length ($D$) is sufficiently small. Hence, at a low value of $D$, the $2p4f$ $(^3$F$^e)$ level of helium merges to the one-electron continuum.

We have also estimated the energy (in meV) corresponding to the

$$ 2pnf \;(^{1,3}{\rm F}^e) \rightarrow 2pn'd (^{1,3}{\rm D}^o) $$

transitions $(n=4$--6; $n'=3$--6) for different two-electron atoms ($Z=2$--4) embedded in WCP environment. The $2pn'd (^{1,3}$D$^o)$ energy values are taken from Saha et al.^[21] and the results are exhibited in Tables 5--7 for $Z=2$--4 respectively. We mention that the absolute values of the difference between the position of the energy levels are given. The sequence for transition we maintain in the table is $2pnf \rightarrow 2pnd$ whereas in all the cases the $2pnf$ states are not high lying. For instance, in the case of triplet states of Li$^+$, $2p4f$ state lies energetically higher than $2p3d$ and $2p4d$ states but lower than the $2p5d$ and $2p6d$ states. We have used the conversion relation 1 a.u. of energy $= 27.21138$ eV.^[39] It is worthwhile to mention that for $2pn'd (^3$D$^o) \rightarrow 2p3p (^3$P$^e)$ transitions in WCP environment, an initial blue shift followed by a red shift with respect to decreasing plasma screening length was reported in Ref. ^[21] whereas in the present case no such behavior is seen for $$ 2pnf (^3{\rm F}^e) \rightarrow 2pn'd (^3{\rm D}^o)$$ transitions. The transition energies, in a systematic manner, follow either a blue shift or a red shift for a particular transition scheme. For example, the $2p4f (^3$F$^e) \rightarrow 2p3d (^3$D$^o)$ line for $Z=4$ gets a gradual red shift with respect to decreasing plasma screening length ($D$) and a blue shift is observed for the $2p4f (^3$F$^e) \rightarrow 2p4d (^3$D$^o)$ of the same ion under similar conditions. Such features are evident from Fig. 2 where the $2p4f(^{1,3}$F$^e)\rightarrow 2pnd(^{1,3}$D$^o)$ transition energies $(n=3$--6) of $Z=4$ are plotted as a function of Debye screening length ($D$).

We report the behaviour of doubly excited energy levels of helium-like ions in WCP environment considering screened Coulomb potential. The two-electron energy levels as well as the respective one-electron thresholds become more positive as the plasma screening length decreases. The position of different doubly excited states has been compared extensively. The transition wavelengths between doubly excited states are found to undergo a gradual blue shift or a red shift with respect to the variation in plasma screening length. Such features have implications in interpreting complex atomic spectra like those of laboratory plasma experiments or astrophysical observations.