Endofullerenes represent specialized structures within fullerene molecules, typically characterized by a spherical arrangement formed by carbon atoms, harboring different atoms or molecules within their interior. These structures often consist of atoms or molecules placed inside the fullerene shell, typically with an inner filling. Endofullerenes acquire unique properties due to the materials encapsulated within them [1, 2]. The inner filling significantly alters or enhances the electronic, magnetic, and chemical properties of endofullerenes. These properties enable diverse applications of endofullerenes, such as in nanotechnology, drug delivery, catalysis, and materials science [3–6]. These structures, by manipulating properties at the molecular level, could potentially play a significant role in various industrial fields in the future. Endofullerenes are complex exotic molecular structures that can be synthesized in different geometries, symbolized in terminology as X@C_n [7]. The synthesis of endofullerenes is typically achieved using two main methods: endohedral metallofullerenes, or endohedral complementary amines. However, our focus in the present work leans more towards endofullerenes containing hydrogen atoms. The synthesis of hydrogen-containing endofullerenes generally involves the interaction of hydrogen plasma with carbon structures. This process typically occurs under high temperature and pressure conditions. One method involves the reaction of carbon structures (usually graphite or carbon nanotubes) with hydrogen plasma, occurring within an environment where hydrogen gas is used to create a high-energy plasma. Carbon structures allow hydrogen atoms to interact within this plasma, enabling the incorporation of hydrogen into the carbon structures [8]. In this process, mechanisms such as the diffusion or absorption of hydrogen into carbon structures can play a role. This process facilitates the synthesis of hydrogen-containing endofullerenes by incorporating hydrogen into the carbon structures. The resulting structures can possess various properties, depending on the position and distribution of hydrogen within the endofullerene. Such hydrogen-containing endofullerenes have the potential to be used in various applications as hydrogen storage technologies and energy storage systems [9]. The properties and potential applications of endofullerenes stimulate in-depth investigations in this area and inspire new experimental/theoretical studies. Endofullerenes come in various types with different numbers of carbon atoms, exhibiting diverse geometric properties, such as their sizes and superficial features. By considering these differences, the C_n model is proposed in the present work to analyze characteristics such as the width, depth, thickness, and smoothing of endofullerenes. Theoretical studies on endofullerenes have predominantly focused on research areas such as the ionization potential, delocalizations, quantum information, photoionization, orbital distortion, and magnetic, optical, and thermal properties for the X@C_n complex [10–16].

Plasma is a state typically composed of ionized gases found at high temperatures and energy levels. However, it can also be easily generated under atmospheric conditions. In this case, electrons stripped from the outer shells of atoms create ions and free electrons, allowing positively and negatively charged particles to move freely. Plasma is best characterized by ionized gases and is typically formed in conditions of high temperature, high pressure, or high electric fields. It can be produced not only in natural settings, such as the Sun and stars, but also in laboratories and industries. The high-energy level determines the many characteristics of plasmas. Due to these properties, plasmas find applications in various areas, such as lasers, semiconductor manufacturing, materials processing, space research, nuclear fusion, and a wide array of atomic and molecular interactions [17]. Plasmas play a significant role in applications within atomic and molecular physics. Processes such as electron collisions with atoms, and absorption and emission of photons form the basis of plasma physics. These processes can also vary under the influence of magnetic fields, highlighting the complex nature of plasmas that need to be studied and controlled [18, 19]. In this manner, the interactions of plasmas at the atomic and molecular levels, processes such as energy transfer and conversion, constitute a significant research area in modern physics and technology. Studies in this area have the potential to lead to groundbreaking innovations in various fields, such as the production of new materials, space exploration, and energy generation [20]. The plasma environment provides optimal conditions for exciting an atom or molecule. However, within this excitation process, processes such as oscillator strengths, phase shifts, localizations of energy levels, and photoionization can be managed through the plasma shielding function. In this context, the H@C_n molecule is considered embedded within the plasma to experience a strong shielding effect in a quantum plasma. When modeling quantum plasma interactions, our work considers a more general exponential cosine screened Coulomb (MGECSC) potential. For more details about this potential, please refer to [21–23]. Plasma-encased endofullerene systems are a significant research subject for two reasons. Firstly, the plasma environment is highly functional in synthesizing or modifying endofullerenes. The high temperature and energy levels within plasma allow carbon structures to react in various ways. Under these conditions, carbon structures can form different molecular configurations under the influence of plasma. Alternatively, free atoms or molecules within the plasma can penetrate the interior of carbon structures and reside within endofullerenes. The formation of such endofullerenes is being investigated in fields such as materials science, nanotechnology, and plasma chemistry, exploring their potential applications. The second reason is that plasmas' shielding properties can be utilized to control certain spectral, electronic, optical, and spectroscopic characteristics of endofullerenes. The energy levels, oscillator strengths, mean excitation energy, photoionization dynamics, and static dipole polarizability of a hydrogen atom enclosed in an endohedral cage under the influence of a weak plasma interaction have been examined using a finite difference approach [24]. In the present work, the C_n model for endohedral confinement has been considered within the framework of the Woods–Saxon potential, taking into account the well depth V₀, inner radius R₀, thickness D, and smoothing parameter γ. This research distinctly elucidates the interaction between plasma and an endohedral molecule, suggesting that plasma plays an effective role in the physical properties of the aforementioned endohedral molecule. According to the overall findings of the study, there is strong competition between the effects of plasma interaction and the static endohedral cavity, thereby altering the electronic characteristics of the system. Motivated by the functionality provided by the plasma shielding effect, a similar study has been conducted in nonideal classical plasma [25], explaining the effects of a nonideal classical plasma environment on the photoionization dynamics of hydrogen-containing endofullerene molecules. In this study as well, significant features of both components of the plasma–endofullerene system have been identified.

The change in the aforementioned physical properties of the plasma–endofullerene system has been sufficiently motivating in the study of quantities such as the persistent current (PC) and induced magnetic field (IMF). In atoms, molecular systems, and artificially low-dimensional structures such as quantum dots, electric currents arising from atomic orbits and the associated induced magnetic fields are fundamentally linked to the movement of electrons. Electrons revolving around the atom's nucleus carry momentum due to their speed within their orbits, constituting moving charges that generate a current. Electrons orbiting in a loop correspondingly produce a magnetic field associated with that current, following Ampere's law of circulating currents around a moving charge. These orbital currents create magnetic fields at the atomic level, influencing the magnetic properties of atoms and molecules. Particularly, these effects may involve directing or altering magnetic moments, forming the basis for many magnetic materials and devices. The magnetic effects of orbital currents find applications in various fields, ranging from medical imaging devices, such as magnetic resonance imaging, to magnetic memories and sensors. The PC at the atomic level and the resultant IMF play a significant role in various aspects of modern technology, emphasizing the crucial importance of precise control over them [26–29]. Various physical factors may be involved in the formation and control of the PC and IMF in atomic systems. In this context, twisted lights have recently garnered attention because they involve the interaction between the orbital angular momentum and the spatial modes of the electromagnetic field. What makes twisted lights intriguing is their ability to alter significant steady currents, even in situations where there are no externally applied magnetic fields [30–32]. Such beams provide a significant advantage in that the dynamics of the laser–matter interaction do not depend on traditional selection rules, and the PC and IMF can then be generated simply by changing the orbital angular momentum index of the beam. It should be pointed out that an optical vortex has a significant effect on the charge current and magnetism in fullerene [31]. The Rashba spin–orbit interaction is a significant factor in the generation and modification of the PC and IMF due to the spin-twisting itinerant motion of electrons. It stabilizes the alternating motion of electrons by fixing their spins [32]. Another significant factor affecting the PC and IMF is the use of laser pulses. In this manner, time-delayed light pulses on nanoscopic and mesoscopic ring structures [33] and picosecond-shaped laser pulses on valley currents and magnetization in graphene rings [34, 35] establish a functional mechanism. Taking into account electron spin, the effectiveness of short intense circularly polarized π laser pulses on the PC and IMF of hydrogen atoms and certain ions has been thoroughly explained [36]. Several factors can have a significant impact on the PC and IMF in atomic systems. Exploring these factors not only guides experimental researchers, but also motivates theoretical studies to test new models and mechanisms. In this context, some effects, such as noncentral interactions, laser pulses, external magnetic fields, external electric fields, and spherical confinement, offer significant outcomes [37–40]. The common outcome of most of these studies is to elucidate how the PC and IMF can be modified by external influences. These explanations benefit the development of new opto-magnetic devices and magneto-optic materials, allowing the manipulation of many physical quantities that are sensitive to magnetic fields at the subatomic level. Endofullerenes can exhibit different magnetic properties, depending on the filling material inside them. This can be controlled by selecting the interior fillings or through external influences (such as external magnetic fields). Moreover, it can lay the groundwork for magnetic storage systems and spintronic applications. Endofullerenes could serve as potential building blocks for nanoscale electronic devices. These systems enable the control of nano-sized currents and magnetic fields. Research in these fields can be seen as significant steps toward understanding and controlling magnetic specifications of endofullerenes, and translating them into applications. This could lead to significant discoveries and advancements in materials science, nanotechnology, and magnetism fields. Considering the advantages of plasma in the synthesis of endofullerenes, as mentioned earlier, and the spectral properties that are modifiable by plasma's screening effects, the analysis of PC and IMF in plasma-encased endofullerene molecules is highly motivating.

The spins of electrons result in magnetic moments, influencing the formation of the PC and IMF in atomic systems. Spin is associated with the PC, and spin-carrying currents can have an impact on magnetic moments. Such currents play a significant role in the design of spintronic devices and magnetic storage systems [41]. On the other hand, the PC creates spin-circulation effects. This is when electrons rotate around their spin, parallel or antiparallel to charge currents. Spin–orbit coupling affects the electronic properties of the material and the behavior of carriers. Furthermore, in certain topological materials, there are connections between the spins and orbital motions of electrons [42]. For these reasons, the spin effects of charge currents are regarded as a significant area of research and application in fields such as materials science, magnetism, and spintronics. Therefore, this study will provide a relativistic examination by considering the spin effects within the Dirac formalism. The expectation from this study is to elucidate the effects of relativistic impacts and spin orientations on the PC and IMF covering plasma effects, endofullerene encapsulation, and spherical confinement. In doing so, experienced models and theoretical procedures will be utilized to parametrize these effects.

The study is organized as follows: in section 2, the model and theoretical formalism are presented. Section 3 elucidates the obtained results. Section 4 is dedicated to a brief summary and conclusion of the work.

For atomic systems exhibiting spherical symmetric interactions, the time-independent Dirac Hamiltonian in atomic units reads aswhere V(r) is the spherical symmetric potential, α and β are four-component matrices in standard Dirac–Pauli notation, I is a 4 × 4 unity matrix, and c is the light speed. Finding the eigenvalues and eigenvectors of the Dirac Hamiltonian to investigate relativistic effects is quite challenging due to its mathematical complexity. Additionally, the Dirac equation can be analytically solved for only a few potentials. To determine the eigenvalues and eigenvectors of the Dirac Hamiltonian with central potential, one must perform separation of the variables in the relevant Dirac equation, and this is where the real difficulty begins. The series of operations, considering the angular momentum ($\hat{J}$) and parity operator ($\hat{P}$) that commute with the Hamiltonian with respect to the center of the coordinate system, results in coupled differential equations for radial wave functions [43]. However, considering the separability of the Dirac equation, highly functional and well-established formalisms have been developed for obtaining two decoupled second-order differential equations in the analysis of the Dirac equation, such as the SUSY approach [44], and coupling the relativistic equation into large and small components by squaring the Dirac Hamiltonian [45, 46]. Within the modification of spin and angular variables, for an atomic system with the potential interaction V(r), the Dirac second-order equation in atomic units reads as follows [47–51]with while E is the electron energy as the difference between the total and rest energy, and G_κ, F_κ are, respectively, the large and small component radial functions, ${I}^{{\prime} }$ is the 2 × 2 unit matrix. The relativistic momentum quantum number κ states are $\kappa =\mp (j+\tfrac{1}{2})$ and $j={\ell }\mp \tfrac{1}{2}$ in consideration of the orbital and total angular momentum quantum number ℓ and j, in which the mathematical formalism here works as κ = − ℓ for $j={\ell }-\tfrac{1}{2}$ and κ = ℓ + 1 for $j={\ell }+\tfrac{1}{2}$. The following nonunitary transformation matrix and r − independent when considering r²dV(r)/dr = η ≡ constant, is identified when considering Δ matrix elementswith s, which is a constant, andThe relevant equations can be decoupled, by employing M and Λ, as well as the relation M⁻¹ΔMwhich, in turn, provides the following coupling form of the second-order Dirac equationwhere F_±(r) are scalar amplitudes. Equation (8) is solved by employing the Runge–Kutta–Fehlberg method. For details, please refer to [52].

In a quantum plasma environment, an encapsulated H atom within a C_n fullerene is surrounded by a spherical encompassement with R₀ radius. When a relativistic hydrogen atom is compressed to a spherical region with R₀ radius under a pressure P, significant changes are observed in the atom's spectral energies. Therefore, studying the cases where the bound-state energy levels of the hydrogen atom change under such ultrarelativistic conditions is suitable for the ultrarelativistic condition and is important in understanding these situations. The pressure P referred to the case of when considering plasma-encased H@C_n under the spherical confinementwhere E₀ is the ground state energy, and ⟨V⟩ is the estimated value of the potential energy. In this case, the spherical confinement potential readsThe Woods–Saxon potential model, which is one of the suitable models for static trapping hydrogen atoms within a fullerene that is compatible with experimental data, is considered [53]. Then, the static endohedral entrapping potential model is given by [54]where V₀ is the confinement strength, D is the spherical shell thickness, R_c is the inner radius of the endohedral cage, and Γ is the smoothing parameter. The strong screening effect of quantum plasma, complicated correlations in the plasma environment, and various perturbations can be most comprehensively modeled using the MGECSC potential [21, 23, 55]. This model is presented as follows:where Z = 1 for the hydrogen atom, and a, b, and λ are the plasma shielding parameters. In such a scenario, by considering the combined effects of the spherical encompassement, quantum plasma, and endohedral trapping, the total interaction potential in equation (1) is written as follows:Under these conditions, within the context of spherical symmetry motivation, the eigenvalue equation that needs to be solved is H_D$\Psi$_nκm(r, θ, φ) = E$\Psi$_nκm(r, θ, φ), where $\Psi$_nκm(r, θ, φ) is written, by considering large and small component radial functions, aswhere χ_κm(θ, φ) is the spin angular function, expressed aswhere $\lt {\ell }m-\sigma \tfrac{1}{2}\sigma | {\ell }\tfrac{1}{2}{jm}\gt $ are the Clebsch–Gordan coefficients, ${Y}_{{\ell }}^{m-\sigma }(\theta ,\varphi )$ are the spherical harmonics, and φ^σ is the spin functions asand n, m are the principle quantum number and the projection of the angular quantum number j, on the z − axis, respectively.

When considering the $\Psi$_nκm(r, θ, φ) state, the probability current density is described asWith e being the electron charge, the orbital charge currents and induced magnetic field are depicted in the following form where μ₀ is the vacuum permeability. For details, please refer to [56–58].

In this study, we analyze the charge-current generations (PC: I(mA)) and IMF (B(T)) of spherical confined quantum plasma-encased H@C_n within the framework of relativistic formalism. This analysis involves the effects of eight parameters: plasma shielding parameters (λ, b, and a), endofullerene static encompassement parameters (V₀, D, R_c, and Γ), and the spherical confinement radius (R₀). Additionally, the impact of spin orientations is examined. The methodology employed in the study has been tested by comparing its results with those of other studies under certain limits. In plasma environments modeled by the MGECSC potential for cases in the absence of endohedral encapsulation (V₀ = 0) and without spherical confinement (R₀ → 0), the relativistic bound-state energies of the embedded hydrogen atom has been compared with the results produced by the direct perturbation method [59]. A perfect agreement between the relativistic bound-state energies in all relevant tables in [59] for Z = 1 and our results has been observed. Additionally, when considering the b = 0 and b = a = 0 limits in the MGECSC potential, a very good agreement has been found between our relativistic results for the ECSC and SC potentials and the nonrelativistic results produced by the Ritz method [60] and the 1/N method [61]. These observed numerical agreements are important and motivating for testing the employed methodology. Figure 1 presents the interaction potential profiles given in equation (13). Except for the profile in panel-h, all synchronized wave functions in the other profiles are oriented towards spin up. As depicted in figure 1(a), the variation of λ in the range of 0−150a₀ has a more pronounced effect in the weak regime of λ. An increase in λ in this weak regime makes the interaction potential more attractive. Table 1 shows the variation of the energy values of some quantum states synchronized with the parameter set of figure 1(a) as a function of λ. The attractiveness caused by λ in the interaction potential is also reflected in the localization of bound-state energies. As expected, an increase in λ in weak regimes significantly reduces the localization of bound states, followed by a subsequent stability in later increments of λ (See table 1). The dynamics of bound-state localizations are the primary factors influencing the PC (I) and IMF (B). Figure 2 depicts the variation of the PC and IMF as a function of λ. As seen in figure 2, an increase in λ within the range of 0−30a₀ initially leads to a noticeable decrease in the PC and IMF, followed by a subsequent stabilization. Figure 3 is synchronized with figure 2 and shows the radial density probabilities (RDP) of 2p, 3d, and 4f states for three different values of λ. Due to the intensified interaction potential as λ increases, the RDP of bound states decreases, leading to a decrease in I and B (see figure 3). The most pronounced response to the λ change is observed in the 2p state, resulting in the most dynamic effects in the I−B characteristics within the 2p state.

Figure 1(b) depicts the interaction potential profile as a function of the plasma shielding parameter b and radial variable r. As observed, an increase in b enhances the attractiveness of the interaction potential, stabilizing the system concerning bound states. Table 2 illustrates the energy values of certain quantum states for the variation of b within 0 − 2.5/a₀, when considering spin orientations. An increase in b leads to a more attractive interaction potential, resulting in a noticeable decrease in the localization of bound states. However, while the increase in the b parameter produces significant outcomes on energy levels, its effect on the PC and IMF is weak (See figure 4). Apart from the 2p level, the influence of b on the PC and IMF is quite uncertain in other states. Nevertheless, there is a remarkable effect of spin orientation on the values of I and B. Even though b has a strong impact on localizations, its weak effect on the I−B characteristics is somewhat unusual. However, as seen in figure 5(a), an increase in b mildly enhances the RDP of 2p, consequently leading to a slight increase in the PC and IMF for the 2p level. Similar elucidations hold true for the 3d and 4f states as well.

Figure 1(c) explains the effects of the plasma parameter a on the interaction potential. In this quantum system encompassed within a spherical region of radius R₀, there are no observable oscillations due to the a parameter, resulting in a weak repulsion. This weak repulsion leads to a decrease in the strength of the interaction potential, causing an inclination towards increased localization of bound states (See table 3). The weak impact of a on energy levels is reflected in the PC and IMF, as depicted in figure 6. An increase in a, most notably in the 2p state, causes a slight increase in the PC and IMF in other quantum states as well. This is because the RDP of 2p, 3d, and 4f states mildly increases as a increases (see figure 7). While the spin orientation does not alter the general character of I and B, there is a notable difference between up and down states, consistent with other findings. Figure 1(d) elucidates the variation in the interaction potential as a result that the depth of the endohedral cage increases. It is evident that as V₀ increases, the encompassing effect of the interaction potential also rises. Consequently, a decrease in the energy levels of all bound states is expected, which is distinctly confirmed in table 4. The outcomes derived from the 2p level in the I−B characteristics differ from those originating from 3d and 4f. As the strength of the endohedral capsulation increases, there is a tendency for I_2p and B_2p to decrease while the others tend to increase, except for a slight initial increase (see figure 8). A more detailed analysis is needed to understand the reason behind this pattern. In figure 9, the RDP for 2p, 3d, and 4f are shown for cases without endohedral capsulation and for V₀ = 20 eV and V₀ = 80 eV. With regard to the 2p state, the electron's RDP not only moves away from the orbital center (r = 0) but also increases in a more stable manner. While $\vec{J}$ is the primary factor for persistent currents, it alone is not sufficient. As the electron localizes farther from the orbital center, the PC decreases. In this regard, for the 2p state, it could be said that the influence of the electron moving away from the orbital center on the I−B characteristics is more dominant compared to the increase in the RDP. For I_3d and B_3d, as the strength of the endohedral cage increases, the electron moves closer to the orbital center and the RDP increases. This leads to a significant increase in both the PC and IMF. This increase results in the same outcome for 4f as well. However, for quantum levels such as 4f, which are farther from the orbital center, the change in the PC and IMF is not clearly visible due to the scale of the graph in figure 8.

Figure 1(e) illustrates the confinement effect of the interaction potential as a function of the endohedral enclosure width (D) and the radial variable (r). The increase in D yields significant outcomes on quantum levels. The impact of the endohedral capsulation width on the localizations of quantum levels is demonstrated in table 5. It is observed that as the endohedral cage width increases, energy levels decrease, leading the system towards greater stability. Figure 10 illustrates the PC and IMF responses originating from the 2p, 3d, and 4f levels for spin up and spin down orientations concerning the increase in the endohedral fullerene width D. As observed, there is a significant decrease in the I−B characteristics due to the increase in the endohedral fullerene width D. Figure 11 shows the RDP of 2p, 3d, and 4f as a function of r for increasing values of the endohedral fullerene width D. An increase in the endohedral fullerene width results in the electron localizing over a wider area, which is expected to cause a decrease in the RDP. Additionally, the electron moves radially away from the center, supporting the decrease in the PC and IMF due to the diminishing RDP, ultimately leading to the prominent I−B characteristics observed in figure 10.

Figure 1(f) depicts the variation in the interaction potential as function of the inner radius of the endohedral cage (R_c) and the radial variable (r). Table 6 presents the energy values of certain quantum levels, including spin up and spin down orientations. The change in R_c forms a repulsive potential characteristic for the 1s and 2p energy levels, an attractive characteristic for 4f and 5g, and an initially attractive (R_c ≃ 3a₀) and then repulsive characteristic for 3d. Figure 12 illustrates the behavior of the PC and IMF as a function of R_c. Although an increase in R_c creates an unstable response in the localization of certain energy levels, it causes a stable impact on the PC and IMF. The increase in the inner radius of the endohedral cage significantly decreases the PC and IMF (see figure 12). Figure 13 explains the behavior of the RDP of 2p, 3d, and 4f as a function of the radial variable (r) for different locations of the endohedral cage along the radial axis or, specifically, for an increasing inner radius. As observed, as the inner radius increases, both the electron's RDP decreases and it moves away from the radial center. These two outcomes, having a diminishing effect on the PC and IMF, complement each other, ultimately resulting in a notably decreasing I−B characteristic. Figure 1(g) explains the response due to the smoothing effect of the endohedral fullerene capsulation. As observed, as the smoothing increases, the interaction potential gains a repulsive characteristic, leading to a stable trend in terms of the localization of bound states. Table 7 illustrates the changes in the energy levels of quantum states due to smoothing. It is observed that as Γ increases, the endohedral fullerene capsulation exhibits a more unstable confining tendency, consequently resulting in a decrease in bound-state energies. However, this decrease is not very pronounced but rather follows a monotonic form. This monotonicity is also reflected in the PC and IMF characteristics (see figure 14). In a more repulsive potential, the probabilities of bound-state localizations are weaker. As depicted in figure 15, as Γ increases, the RDPs of 2p, 3d, and 4f show a slight decrease, which equally affects the PC and IMF characteristics (see figure 15). Table 8 presents the energies of certain quantum levels, depending on the increase in the spherical confinement effect (R₀). As observed, the effective range of the spherical confinement concerning the energy levels of bound states is approximately 0−9a₀. As R₀ increases, the quantum levels decrease. However, outside this effective range, there is almost no change in energy levels. Figure 16 demonstrates the changes in the PC and IMF as a function of the spherical confinement effect. It is evident that as the spherical confinement range decreases, the electron's RDP decreases significantly. This decrease results in very clear reductions in the PC and IMF within the effective range of R₀ (see figure 17).

Table 9 illustrates the effect of the increase in the magnetic quantum number (m) on the PC and IMF. As observed, this effect arising from spherical harmonics is weak because the interaction potential does not have angular dependence. An increase in the magnetic quantum number (m) leads to angular probability densities localizing more towards the xy-plane rather than the z-axis. Consequently, an increase in angular probability and, therefore, an increase in the PC and IMF could be expected. However, the current integration might not provide a stable response to the increase in m due to angular normalization coefficients and Clebsch–Gordan coefficients. Therefore, the results originating from some numerical cases might predominate in the increment of the angular probability when considering the PC and IMF. In summary, the study delves into the intricate interplay among different parameters affecting the PCs and IMFs of quantum plasma-encased endohedral fullerenes. The results show a nuanced dependence on these parameters, with certain regimes leading to enhanced or reduced PCs and magnetic fields. The interplay of plasma screening, endofullerene encapsulation, spherical confinement, and other factors adds a layer of complexity to the behavior of these quantum systems. The detailed analysis and presentation of results provide insights into the underlying physics and potential applications of such systems in quantum information processing and other fields.

As expected, while spin orientations create a very subtle difference in bound-state energies, there is a noticeable difference between the PC and IMF of spin up and down configurations when considering every parameter change. The values of I−B for spin up configurations are consistently larger. The difference between the spin up and down states is most significant for the PC and IMF of quantum levels closer to the hydrogen nucleus, diminishing as one moves away from the nucleus towards outer levels. The impact of the λ parameter is more pronounced on both the energies and I−B in weaker regimes, approximately around λ ≃ 0−40a₀. While the b screening parameter significantly affects bound-state energies, its effect on the PC and IMF is unexpectedly weak. Meanwhile, the a screening parameter has a weak effect on both the energies and the I−B characteristics. In terms of plasma functionality, the most effective and functional parameter for the PC and IMF is λ. This is advantageous from an experimental perspective as well because λ is a parameter that is dependent on plasma temperature T and plasma density n, allowing alterations in I−B characteristics with changes in n and T. An increase in the depth of endohedral fullerene encapsulation (V₀) affects the PC and IMF of 2p and 3d−4f levels in different ways. In this context, it has become evident in the analysis of V₀ that the I−B characteristic is not solely dependent on the current density ($\vec{J}$) but also on the electron's radial distance from the center. The effective range of V₀ for the PC and IMF has been determined to be 0−30 eV under the given conditions. As the endohedral fullerene expands (D), both the RDP decreases and the electron moves away from the radial center. Consequently, an increase in D effectively reduces I−B values. The inner radius of the endofullerene (R_c) stands out as the most significant and functional parameter for the PC and IMF. In this regard, it could be an alternative to D. The effect of smoothing (Γ) effect should be considered as a fine-tuning parameter for the PC and IMF of endofullerenes. This is because smoothing requires less experimental effort and has a very weak impact on I−B characteristics. The spherical confinement (R₀) also affects the PC and IMF. However, for the numerical values within the given parameter set, the functional range of R₀ is approximately 0−10a₀. Even without angular interactions, spherical harmonics influence the PC and IMF. However, the localization effect of ${Y}_{{\ell }}^{m}$ alone may not be the sole impactful factor in this influence.

Mustafa Kemal Bahar is the only author of this study.