Confined quantum systems exhibit significant changes to their structure, stability, reactivity, binding interactions, dynamics, and spectra as a consequence of modifications to the spatial boundary conditions in the presence of an extreme environment. Atoms or molecules within cavities, organic/inorganic host–guest complexes, quantum dots, fullerenes, and nanotubes are examples of real quantum confined systems. The main objective when studying confined quantum systems is to construct an accurate theoretical model that takes into account changes in the electronic wavefunction and energy levels due to the boundary conditions imposed by the surrounding environment. In this respect, theoretical calculations can be realized by adopting a suitable choice of boundary conditions. The pioneering work of Michels et al.¹ provided the first model of a hydrogen atom confined in an impenetrable spherical cavity to simulate the effect of pressure. In recent years, the topic of confined atoms has attracted much attention and has become a very active field of research.^2–5 Reviews with detailed discussion of the progress in this field can be found in Refs. 2–6 and references therein.

To describe the electronic structure and interactions of a multi-electron system, a variety of theoretical methods have been developed, including, among others, Hartree–Fock (HF) theory and density functional theory (DFT), and these have had great success in different problems in atomic physics.^6–11 Owing to the complexity of the N-body problem, some approximations have been implemented, in addition to having the system under confinement by spatial limitation of the electrons, which leads to complications in the one- and two-electron integrals. A widely used approach to the treatment of N-body systems is to consider a pseudopotential that describes the inner electronic structure of the atom. The basic idea of pseudopotentials is to take into account the multi-electron core interaction with a single valence electron by using a modified Coulomb potential.^7,8,11 Such pseudopotential approaches have been used to describe the electronic spectra of confined systems.^12,13 Lin and Ho¹² used a pseudopotential for the lithium atom to simulate the core interaction with the single valence electron with optimized parameters. They calculated the photoionization cross section of the 2s shell electron under confinement by a power exponential potential due to an endohedral cavity and found that multiple Cooper resonances emerged.¹⁴ Their results show the importance of cage thickness and a smooth shell boundary in the photoionization cross section. Another example is due to Sarsa et al.,¹³ who studied the effects of confinement on the outer valence electrons for the ground state configurations of carbon and iron atoms. In standard HF theory, the main complication that arises when dealing with quantum confined systems is the treatment of the one- and two-electron integrals, particularly the electron exchange integral, as shown by Ludeña.¹⁵ Fortunately, Slater¹⁶ proposed a simplified approach to treat the electron exchange operator in the HF method by replacing it by a term proportional to the charge density of the inner electrons. This approach has been fruitful in treating problems in atomic and molecular structure with satisfactory results,¹⁷ particularly in the development of DFT theory. However, to the authors’ knowledge, there have not been any studies of the effects of confinement on the excitation spectra of a multi-electron system in the context of HF theory by means of Slater’s X-α approach.

The goal of this work is to apply Slater’s X-α approach to the ground and excited states of Li-like atoms confined in an impenetrable spherical cavity. To show the strength of Slater’s X-α approach to the calculation of the excitation spectrum of atoms, we take as a benchmark example the ground states of lithium-like atoms in the initial electronic configuration 1s²2s, adopting a restricted HF approach. We focus on the dipole oscillator strength (DOS) and derived properties such as the static dipole polarizability and the mean excitation energy.

The remainder of this paper is organized as follows. In Sec. II, we present the theoretical approach used to study the influence of a spherical confinement cavity on lithium-like atoms. In Sec. III, we discuss our results and findings. In Sec. IV, we give our conclusions and perspectives. Note that we use atomic units (a.u.) throughout, unless physical units are explicitly stated.

In this section, we present the HF method to obtain the 1s²2s ground state energies of Li, Be⁺, and B²⁺ atoms, incorporating confinement conditions.

In an HF approach, the total wavefunction is defined as a Slater determinant Ψ(r) = N det{ψ_i}, where N is a normalization constant. The general restricted HF approach¹⁸ considers the two electrons in the 1s² core to be in the same orbital. To simplify the calculation of the excitation spectra, we assume that the inner electrons do not see the outer electrons. This approach is known as the frozen-core approximation, which is usually treated within a pseudopotential approach.^7,8,11 However, here we do not use any specific pseudopotential formula to describe the interaction of the inner electrons, since this is calculated explicitly for every confinement configuration in a self-consistent manner. Then, for the ground state of a lithium-like atom, the HF equations are(ĥ1+Ĵ1)ψ1r=ϵ1ψ1r,(1)(ĥ2+2Ĵ1−K^1)ψ2(r)=ϵ2ψ2(r),(2)where ϵ_j are the eigenvalues, ψ_j are the eigenfunctions, ĥ_j are the one-electron operators, and Ĵ_j and K^j are the two-electron Coulomb and exchange operators, respectively. These operators are defined asĥj(r)=−12∇j2−Zr+Vc(r),(3)Ĵj(r)ψi(r)=∫dr2ψj∗(r2)ψj(r2)r12ψi(r),(4)K^j(r)ψi(r)=∫dr2ψj∗(r2)ψi(r2)r12ψj(r),(5)where r₁₂ = |r − r₂|. The one-electron operator, Eq. (3), includes the confinement potential V_c(r) (see below). The total ground state energy of the confined system is given byEHF=2ĥ11+ĥ22+Ĵ11+2Ĵ12−K^12.(6)It can be seen that Eq. (1) is in eigenvalue form, so the excitation spectra of the core electron can be readily obtained. However, Eq. (2) is not, owing to the presence of the exchange operator, Eq. (5). To have an eigenvalue equation, we resort to Slater’s X-α approach,¹⁶ in which the exchange operator is replaced byK^1(r)ψ2(r)=αXρ11/3(r)ψ2(r),(7)where α_X is a parameter, and ρ(r) is the charge density due to the 1s² inner electrons¹⁸ and is given byρ1(r)=2|ψ1(r)|2,(8)with ψ₁(r) the eigenfunction of the core electrons. To solve the HF equations (1) and (2), we implement a finite-difference numerical approach within a self-consistent field (SCF) procedure to obtain the eigenvalues, eigenfunctions, and dipole-dependent properties. For each iteration, we calculate Eq. (5) explicitly and determine Slater’s X-α constant as^19,20αX=〈ψ2(r)|K^1(r)|ψ2(r)〉〈ψ2(r)|ρ11/3(r)|ψ2(r)〉,(9)finding the excitation spectrum of the 2s electron. The procedure is repeated until Eq. (6) has reached self-consistency within a 10⁻⁶ error difference for the total ground state energy.

Owing to the spherical symmetry of the system, we assume ψ(r)=Rl(r)Ylm(θ,φ) and R_l(r) = u_l(r)/r, where Ylm are spherical harmonics. This is done for both core and valence wavefunctions.

We consider the lithium-like atoms to be confined by an impenetrable spherical cavity, so the confinement potential is given by^1–6Vc(r)=0,r<R0,∞,r≥R0.(10)where R₀ is the confinement radius of the cavity and is commensurate with the confinement pressure (see below).

As the finite-difference approach has been reported previously by Cabrera-Trujillo and Cruz,²¹ here we just summarize our implementation to find the eigenvalues and eigenfunctions of Eqs. (1) and (2). The finite-difference approach consists in discretizing the function u(r) → u_k and r → r_k, known at the kth point on a numerical grid, with k = 0 corresponding to u₀ and k = N + 1 to u_N+1, which are the boundary conditions of the system.²² In our case, for the impenetrable cavity, u_N+1 = 0 and r_N+1 = R₀. We implement the finite-difference approach centered at the midpoint. With this, Eqs. (1) and (2) are rewritten asHϕ→=Eϕ→,(11)where H is a tridiagonal symmetric matrix with N eigenvalues and eigenfunctions, and ϕ→ is related to u→ by a linear transformation.²¹

We solve Eq. (11) in a grid box that extends from r = 0 to r = R₀, with a total of N = 2000 points spaced logarithmically in this range as a function of the confinement radius R₀. This logarithmic grid allows us to give a better description of the wavefunction cusp at the origin and a good number of continuum states. We have found that N = 2000 satisfies the Thomas–Reiche–Kuhn (TRK) sum rule up to five decimal digits. The accuracy of our finite-difference approach can be controlled by the number of points in the grid and their spacing; for example, for the free lithium atom, we obtain eigenvalues with precision up to the fifth decimal place. This approach gives a total of N excited states to describe the DOS electronic properties for each spherical cavity with radius R₀, per electron. The values of R₀ are chosen between 0.5 a.u. and 100 a.u. Our approach is implemented in a Fortran 95 code that calculates the eigenvalues, eigenfunctions, and physical properties of the system.

The DOS accounts for the transition probability from an initial state to a final excited state and is defined asfn0=2(EHFn−EHF0)|〈Ψn(r)|∑i=1Nri⋅ϵ^|Ψ0(r)〉|2,(12)where ϵ^ is the direction of momentum transfer or the polarization vector of the electromagnetic radiation. Here, EHF0 and EHFn are respectively the initial and final total energies, with Ψ₀ and Ψ_n respectively the initial and final total wavefunctions for a given transition. Owing to the presence of just a single determinant and for an electron operator, as in the case of the dipole operator Ô = Σ_i r_i, the DOS are reduced to single-electron transitions from either the core or the valence electron.¹⁸ Under the spherical symmetry of the system, Eq. (12) becomes^23,24fn0i=23(ϵni−ϵ0i)〈Rni(r)|r|R0i(r)〉2,(13)where the i stands for electron i = 1 (core electron transitions) or i = 2 (valence electron transitions). To confirm that our numerical approach has rendered a complete set of states, the TRK sum rule,²⁵ Σ_nf_n0 = N_e, must be satisfied, i.e., N_e = 3. Note that for absorption, Eq. (13) is positive, but for emission, i.e., when ϵni<ϵ0i, the DOS becomes negative.

The static dipole polarizability is defined through the DOS [Eq. (12)], and is given byαs=∑nfn0(EHFn−EHF0)2=∑i,nfn0i(ϵni−ϵ0i)2,(14)where it exhibits an explicit dependence on the single-electron dipole oscillator strengths. Consequently, it can be rewritten asαs=2αs1s+αs2s,(15)where αsi is the contribution from the core (i = 1) or the valence (i = 2) electron. For photon absorption, the polarizability is positive, but for photon emission, it becomes negative.

A parameter that characterizes the amount of energy loss when a swift heavy ion penetrates a target is provided by the mean excitation energy I₀, as defined by Bethe:²⁶lnI0=∑nfn0⁡ln(EHFn−EHF0)∑nfn0=∑i,nfn0i⁡ln(ϵni−ϵ0i)∑i,nfn0i.(16)Using the TRK sum rule,²⁵ Eq. (16) can be rewritten as3⁡lnI0=2∑nfn01s⁡ln(ϵn1s−ϵ01s)+∑nfn02s⁡ln(ϵn2s−ϵ02s)=2⁡ln⁡I01s+ln⁡I02s,(17)which is the orbital decomposition or Bragg rule for the mean excitation energy, as used by Oddershede and Sabin²⁷ in energy loss deposition studies.

To show the reliability of our approach when applied to a multi-electron system, we present in Table I the results for the unconfined Li, Be⁺, and B²⁺ atoms. We show the core electron ground and excited orbital energies ϵ01s and ϵ02p, the total HF energy E_HF, the DOS fis2pi for the first dipole transition, the polarizability αsi, and the mean excitation energy I0i. The same quantities are also reported for the valence (2s) electron. In the case of the ϵ01s and E_HF energy values for the Li, Be⁺, and B²⁺ atoms, we observe good agreement up to four-decimal precision when compared with the results of Froese-Fischer.²⁸ For the Li atom mean excitation energy I01s, we observe a difference of less than 5% with respect to the value reported by Oddershede and Sabin.²⁷ For the valence (2s) electron, we obtain orbital energy values of ϵ02s,Li=−0.20116 a.u., ϵ02s,Be+=−0.67282 a.u., and ϵ02s,B2+=−1.39722 a.u., for the Li, Be⁺, and B²⁺ atoms, respectively, with a difference of less than 3% in comparison with the results of Froese-Fischer.²⁸ However, for the total HF energy, we obtain values with a difference of less than 1% with respect to the results of Froese-Fischer.²⁸ For the dipole polarizability, we obtain αs2s,Li=171.188 a.u. and αs2s,Be+=27.3836 a.u., in comparison with values of 164.05 a.u. and 24.4966 a.u. reported by Schwerdtfeger and Nagle³¹ and Tang et al.,³² respectively.

Thus, our approach using Slater’s X-α allows us to account for the ground state properties ϵ0is and E_HF of the Li, Be⁺, and B²⁺ atoms, with good agreement with available theoretical results.

In Fig. 1, we show the wavefunctions for the 1s and 2s orbitals as functions of position r for unconfined Li, Be⁺, and B²⁺ atoms. For comparison, we also show the results of Froese-Fischer.²⁸ We observe excellent agreement for the 1s orbital for all ranges. For the 2s electron, we also obtain very good agreement for electron distances r > 1 a.u. However, we observe that small deviations appear for the inner part of the wavefunction, r < 1 a.u., compared with the Froese-Fischer results.

In Fig. 2, we show the 1s, 2s, 2p, 3s, and 3p energy levels for Li, Be⁺, and B²⁺ atoms confined by an impenetrable spherical cavity as a function of the confinement radius R₀. For comparison, we also show the results of Weiss³³ for the free atom energy levels at R₀ = 30 a.u. We can see that the energy levels increase, reaching the continuum, as the confinement radius decreases. Furthermore, we observe the appearance of crossing points for the 2s and 2p energy levels, as well as for the 3s and 3p levels, which are highlighted by circles for better visualization. Our results confirm the energy level behavior and crossing already reported from other approaches.^34,35Figure 2(a) shows the 3s and 3p states of the Li atom. We find that the energy levels reach the continuum for cavities with R₀ < 10 a.u. and R₀ < 12 a.u., respectively. Here, the 3s energy level is deeper than the 3p state. As the confinement radius decreases, the 3s and 3p energy levels increase until a crossing point around R₀ ∼ 6 a.u. For values of R₀ < 6, the 3p energy is lower than the 3s energy. For the 3s and 3p states of the Be⁺ and B²⁺ atoms, we find a similar trend as for the Li atom. We can see that the crossing points between the 3s and 3p states of the Li, Be⁺, and B²⁺ atoms are in the positive spectrum. For the Be⁺ atom, we find that the crossing point occurs at R₀ ∼ 4.6 a.u.

In Fig. 2(b), we show the 2s and 2p energy levels, and we can again see an energy increase and the emergence of crossing points as R₀ decreases. We find that the 2p energy level of the lithium atom becomes positive for R₀ < 4.6 a.u. and the 2s energy level becomes positive for R₀ < 4.4 a.u. In this case, the crossing point is present for R₀ ∼ 3.4 a.u. This cavity size corresponds to a pressure of 85 GPa (see below), which is lower than the 210 GPa reported by Rahm et al.³⁶ The discrepancy is attributed to the different confinement models. The same occurs for the Be⁺ atom at a confinement radius of R₀ ∼ 2.5 a.u. and for B²⁺ at R₀ ∼ 2.1 a.u. We obtain positive energy values for the 2s and 2p levels of the Be⁺ atom for R₀ < 2.4 a.u. and R₀ < 2.3 a.u., respectively. For the 2s and 2p states of the B²⁺ atom, we find positive energies for R₀ < 1.71 a.u. and R₀ < 1.55 a.u., respectively. Note that since our initial ground state electronic configuration occupies the 2s orbital level, for cavities with R₀ lower than the crossing-point radius, one would have photon emission instead of photon absorption for the initial electronic configuration, 1s²2s. We should stress that for R₀ smaller than the critical cavity radius, the lowest-energy state of the Li-like systems becomes p-type, and hence the photon emission from the initial s to the final p states brings the excited electron to its ground state. Once this transition has taken place, as the cavity radius is reduced, the p-state evolution lies energetically below the corresponding s state, and hence the DOS become positive for excitations.

In Fig. 2(c), we show the 1s energy levels as functions of R₀. We find that the effect of the cavity on the 1s ground state energy is minimal for R₀ > 2 a.u., R₀ > 1.5 a.u., and R₀ > 1 a.u. for Li, Be⁺, and B²⁺ atoms, respectively. For the Li atom, we observe a sudden change in the energy for R₀ < 2 a.u. and it reaches a positive value for the 1s core state for R₀ < 0.77 a.u. A similar situation occurs for the Be⁺ atom for R₀ < 0.555 a.u. and for B²⁺ for R₀ < 0.427 a.u.

The increase in energy is explained as follows. For an impenetrable cavity, the electrons remain localized within the cavity. As the pressure increases, so does the electron kinetic energy (as a consequence of the Heisenberg uncertainty principle), and the total energy can become positive for a critical pressure,³⁶ as we have just shown, but the system is still bounded.

In Fig. 3, we show the results for the total HF energy for the Li, Be⁺, and B²⁺ atoms in the initial 1s²2s configuration, confined by an impenetrable spherical cavity, as functions of the confinement radius. For comparison, we also show, in the case of Li, the theoretical results of Sañu-Ginarte et al.,²⁹ Le Sech and Banerjee,³⁷ Sarsa and Le Sech,³⁸ and Sarsa et al.,¹³ and we can see that there is excellent agreement. In addition, the HF results for the unconfined atoms obtained by Weiss³³ are shown at R₀ = 5 a.u. For the free case, when R₀ → ∞ a.u., we obtain total energies of EHFLi=−7.43749 a.u., EHFBe+=−14.2838 a.u., and EHFB2+=−23.3826 a.u., in good agreement with the values reported by Froese-Fischer²⁸ and Sañu-Ginarte et al.²⁹ As the confinement radius decreases, the HF energy increases, for all atoms, as previously reported by Connerade et al.³⁹ For the Li, Be⁺, and B²⁺ atoms, we observe in Fig. 3 that the effect of the cavity is minimal for R₀ > 7.3 a.u., R₀ > 5.5 a.u., and R₀ > 2.5 a.u., respectively. For the Li atom, for R₀ < 1.3 a.u., the HF energy becomes positive. A similar situation occurs for the Be⁺ and B²⁺ atoms for R₀ < 0.94 a.u. and R₀ < 0.75 a.u., respectively.

From Figs. 2 and 3, we conclude that the effect of the confinement cavity is stronger on the valence electrons that on the core electrons.

The order of magnitude of the pressure that the cavity exerts on the atomic system as R₀ is shrunk is given by the static pressureP=−∂EHF∂v=−14πR02∂EHF∂R0,(18)where v is the volume of the spherical cavity. In Fig. 4, we show the results for the pressure as a function of the spherical cavity radius R₀ for Li, Be⁺, and B²⁺ atoms. For comparison, we show some characteristic pressures found in nature. We first note that for the same cavity radius, the pressure is lowest for B²⁺, increases for Be⁺, and it is highest for Li. This is a consequence of the ionic character of the system. The lithium atom is more diffuse in its 2s orbital, so the same cavity radius induces a higher pressure, while, owing to the high nuclear charge, the boron ion has already compacted its 2s electron, so the same cavity radius induces a smaller pressure on the ionic system. The figure also shows the cavity size and pressure for which the 2s → 2p transition occurs in our approach. For Li, we find it at 85 GPa (R₀ = 3.4 a.u.), for Be⁺ at 350 GPa (R₀ = 2.5 a.u.), and for B²⁺ at 690 GPa (R₀ = 2.1 a.u.). These results are within an order of magnitude of those reported by Rahm et al.,³⁶ which were obtained using a different confinement model, thus confirming the suitability of our approach. Rahm et al. reported a higher pressure, probably because their model considers penetrable confinement conditions.

In Fig. 5(a), we show the DOS for the electronic transitions 1s → 2p (core excitations) and 2s → 2p (valence excitations) for Li, Be⁺, and B²⁺ atoms confined by an impenetrable spherical cavity as a function of the confinement radius R₀. In the case of the Li atom, for R₀ < 15 a.u., f2s2p2s begins to decrease, showing a change near R₀ ∼ 9 a.u., where we find a value of f2s2p2s=0.58895 a.u. At R₀ = 6.5 a.u., we obtain a DOS of 0.460 46 a.u., and at R₀ = 3.5 a.u. a value of f2s2p2s=0.02320 a.u., near the radius for which the crossing point occurs. For the Be⁺ atom, we observe similar behavior. The DOS for the transition decreases rapidly near R₀ ∼ 5 a.u., with f2s2p2s=0.35781 a.u., and then reaches a value of f2s2p2s=0.01523 a.u. at R₀ = 2.6 a.u., which is near the crossing point. For the B²⁺ atom, we observe the same DOS reduction as R₀ is decreased until the crossing point at R₀ = 2.1 a.u. When R₀ is decreased, the DOS for the valence electron excitation, f2s2p2s, is reduced until the 2s electron reaches the crossing point between the 2s and 2p energy levels, so that at that pressure the DOS become zero (ϵ_2s = ϵ_2p). Crossing occurs for confinement radii R₀ ∼ 3.4 a.u., 2.4 a.u., and 2.1 a.u. for Li, Be⁺ and B²⁺ atoms, respectively. For confinement radii less than the crossing point, the 2p energy levels have lower values than those found for the 2s energy level, and there is photon emission induced by the pressure cavity. We should note here that in a sudden approximation perturbation, for a shrinking of the cavity from radius R₀ to R₀ + ΔR₀, the probability of finding the system in the 2p state is zero owing to symmetry arguments (orthogonal states). Thus, there is a higher probability for the system to remain in the same s symmetry state and then proceed to the 2p state by photon emission. Consequently, for a cavity radius lower than the critical crossing point, f2s2p2s becomes negative owing to photon emission, and some other transitions must increase its DOS value to satisfy the TRK sum rule. In Fig. 5(a), we also show the core results for the f1s2p1s transition, and we can see that the DOS increases as R₀ is reduced. f1s2p1s shows an abrupt change near R₀ ∼ 5 a.u., 3.5 a.u., and 3 a.u. for the Li, Be⁺, and B²⁺ atoms, respectively. For lower values of R₀, the DOS transition increases, reaching values near 1 as consequence of confinement, thus becoming a dominant intensity line.

In Fig. 5(b), we show the 1s → 3p and 2s → 3p DOS for Li, Be⁺, and B²⁺ atoms as functions of the cavity radius. As can be seen, for cavities with radius lower than the critical crossing point, f2s3p2s becomes larger than unity, although the TRK sum rule is satisfied for all cavity radii. Thus, 2s → 3p becomes the strongest transition, so there is a change in luminosity in the atom as the pressure increases, but in this case due to photon emission induced by the change in pressure, similar to piezoluminescence.⁴⁰

In Fig. 6, we show the static dipole polarizabilities αs2s and αs1s for the valence and core states for Li, Be⁺, and B²⁺ atoms confined by an impenetrable spherical cavity, as a function of the confinement radius R₀. The crossing points are highlighted by vertical lines. Note that owing to the small contribution of the core electrons, the total atomic polarizability is dominated by the valence contribution for all pressures. For comparison, Fig. 6(a) also shows the unconfined Li and Be⁺ results as reported by Schwerdtfeger and Nagle¹¹ and Tang et al.³² at R₀ = 30 a.u., and it can be seen that there is good agreement with our results. For R₀ → ∞, we obtain the dipole polarizabilities for unconfined lithium-like atoms as αs2s,Li=171.188 a.u., αs2s,Be+=27.3836 a.u., and αs2s,B2+=8.99440 a.u., which exhibit a difference of ∼4% with respect to HF results.¹¹ From Fig. 6, we observe that as the confinement radius decreases, so does the polarizability, until the s–p crossing point is reached. In Fig. 6(a) for the Li atom, for a cavity with radius R₀ = 4.4 a.u., the polarizability decreases to 54.2926 a.u., which is about 30% of the free value. As R₀ is reduced, the 2s and 2p energy levels become positive, and the αs2s polarizability increases, diverging at R ∼ 3.4 a.u., which is at the critical crossing point of the 2s–2p energy levels. For lower values of R₀, αs2s becomes negative owing to the transition to photon emission. In the case of the Be⁺ atom, at R₀ = 6.5 a.u., we observe a value of αs2s=26.2083 a.u., and then the polarizability decreases for lower values of the confinement radius until R₀ ∼ 3.4 a.u., where a minimum value of αs2s=16.3509 a.u. is found. Then, for values of R₀ < 3.4 a.u., the polarizability increases rapidly, diverging at R₀ ∼ 2.4 a.u., and it then becomes negative for lower values of R₀. A similar situation occurs for the Be²⁺ atom, but with a minimum value of 6.772 95 a.u. at R₀ ∼ 2.8 a.u. and a divergence at the crossing point R₀ ∼ 2.1 a.u. In Fig. 6(b), we show the results for the core contribution αs1s, where the effect of the impenetrable cavity starts for R₀ < 3 a.u. A decrease in αs1s is observed for lower values of R₀, where the energy levels become positive. As noted already, the total static dipole polarizability αs=2α1s+αs2s is dominated by the contribution of the valence electron, αs2s.

In Fig. 7, we show the mean excitation energies I01s, I02s, and I₀, for Li, Be⁺, and B²⁺ atoms confined by an impenetrable spherical cavity as functions of the confinement radius R₀. We can see that at R₀ = 30 a.u., the results for the free mean excitation energies are in good agreement with previous HF results from Oddershede and Sabin,²⁷ Kamakura,⁴¹ and Dehmer et al.⁴² From Fig. 7(a), we can see that as R₀ decreases, I02s increases, showing an abrupt change near R₀ ∼ 10 a.u., 5 a.u., and 4 a.u. for the Li, Be⁺, and B²⁺ atoms, respectively, with I02s,Li=3.87967 eV, I02s,Be+=14.73933 eV, and I02s,Be2+=28.5561 eV. For the Li valence electron, we observe an increase of ∼40% with respect to the free mean excitation energy at R₀ = 4.4 a.u. For Be⁺, we find an increase of ∼11%, and for B²⁺ an increase of ∼9% for the same confinement. Figure 7(a) also shows I01s for Li, Be⁺, and B²⁺ atoms, and we observe an increase in the mean excitation energy as R₀ is reduced. In Fig. 7(b), we show results for the total mean excitation energy I₀. We find I₀ = 33.708 90 a.u., in good agreement with the value of 34.004 13 a.u. obtained by Oddershede and Sabin,²⁷ Eq. (17), and the value of 34 a.u. reported by Dehmer et al.⁴² for Li atoms. Note that as a consequence of 2s–2p energy level crossing, the photon emission produces a negative energy transfer, so the logarithmic contribution is undetermined, as defined by Eqs. (12) and (13). This is observed in the I02s contribution and in the total I₀ mean excitation energy for R₀ less than the critical cavity radius at the s–p crossing energy levels. Thus, a different approach may be required to determine it, such as that proposed by Smith et al.⁴³

One advantage of Slater’s X-α approach is that we can estimate the electron exchange contribution to the energy for a confined quantum system through a single parameter. In Fig. 8, we show the behavior of Slater’s X-α parameter α_X as a function of the cavity confinement radius R₀. We find that the largest contribution occurs for the Li atom, followed by the Be⁺ ion and then the B²⁺ ion for low-pressure cavities. The contribution increases as R₀ decreases, reaching a maximum, and it then decreases as the cavity becomes small. For the ions, the 2s electrons are tighter and the electron exchange parameter is lower for large spherical cavities. However, this behavior is inverted as the cavity increases the pressure. For cavities whose radius is smaller than the critical radius, the α_X parameter is largest for B²⁺, followed by Be⁺, and then Li. So, electron exchange is important as long as the valence electrons remain bounded.

In Tables II and III, for reference purposes, we show the 1s, 2s, and 2p energy levels, the total HF energy, the first allowed DOS transition 2s → 2p, the dipole polarizability, the mean excitation energy, and Slater’s α_X parameter [Eq. (9)] for selected values of the confinement radius R₀ for Li, Be⁺, and B²⁺ atoms,

We have studied lithium-like atoms confined by an impenetrable spherical cavity of radius R₀. We find good to excellent agreement when comparing orbital and total energies, as well as when determining dipole transitions, static polarizability, and mean excitation energies for the unconfined systems. For the lithium atom, we find excellent agreement for confined ground state energies in comparison with available theoretical results.

We confirm that, as a consequence of the confinement, the system orbital and total energies increase as the pressure increases owing to a reduction in cavity size. However, the first allowed dipole transition, 2s → 2p, decreases, while 2s → 3p increases. Consequently, as the pressure increases, the intensity of light emitted by the atom in the cavity is shifted. However, there is a crossing point (critical pressure) at which the 2s and 2p energy levels are inverted; consequently, the DOS for that transition becomes zero at that critical pressure. For higher pressures, the DOS become negative owing to photon emission. In addition, the 2s → 3p DOS reach values larger than unity for high pressures, and the 2s → 2p DOS becomes negative. Thus, we can confirm that the static dipole polarizability is reduced as the pressure increases, as the electrons become highly localized within the cavity and less prone to be polarized, and diverges at the point of transition from photon absorption to photon emission. We also find that the mean excitation energy, which measures the ability of the atom to absorb energy due to excitations, increases as the pressure is increased, with implications for material damage under extreme conditions. As a result of the existence of the crossing point, the valence and total mean excitation energy become undetermined owing to a logarithmic indeterminacy, and thus a different approach may be required.

Our work shows the reliability of Slater’s X-α approach in the context of HF theory to study confined N-electron quantum systems. This approach has the advantage that it can be extended to larger systems to provide excitation spectra in different confinement environments, thus shedding light on the behavior of N-body quantum systems under extreme conditions.