Relativistic Wave Functions Research Articles

Unified Implementation of Relativistic Wave Function Methods: 4C-iCIPT2 as a Showcase.

In parallel to the unified construction of relativistic Hamiltonians based solely on physical arguments (J. Chem. Phys. 2024, 160, 084111), a unified implementation of relativistic wave function methods is achieved here via programming techniques (e.g., template metaprogramming and polymorphism in C++). That is, once the code for constructing the Hamiltonian matrix is made ready, all the rest can be generated automatically from existing templates used for the nonrelativistic counterparts. This is facilitated by decomposing a second-quantized relativistic Hamiltonian into diagrams that are topologically the same as those required for computing the basic coupling coefficients between spin-free configuration state functions (CSF). Moreover, both time reversal and binary double point group symmetries can readily be incorporated into molecular integrals and Hamiltonian matrix elements. The latter can first be evaluated in the space of (randomly selected) spin-dependent determinants and then transformed to that of spin-dependent CSFs, thanks to simple relations in between. As a showcase, we consider here the no-pair four-component relativistic iterative configuration interaction with selection and perturbation correction (4C-iCIPT2), which is a natural extension of the spin-free iCIPT2 (J. Chem. Theory Comput. 2021, 17, 949), and can provide near-exact numerical results within the manifold of positive energy states (PES), as demonstrated by numerical examples.

Finite Nuclear Size Effect on the Relativistic Hyperfine Splittings of 2s and 2p Excited States of Hydrogen-like Atoms

Hyperfine splittings play an important role in quantum information and spintronics applications. They allow for the readout of the spin qubits, while at the same time providing the dominant mechanism for the detrimental spin decoherence. Their exact knowledge is thus of prior relevance. In this work, we analytically investigate the relativistic effects on the hyperfine splittings of hydrogen-like atoms, including finite-size effects of the nucleis’ structure. We start from exact solutions of Dirac’s equation using different nuclear models, where the nucleus is approximated by (i) a point charge (Coulomb potential), (ii) a homogeneously charged full sphere, and (iii) a homogeneously charged spherical shell. Equivalent modelling has been done for the distribution of the nuclear magnetic moment. For the 1s ground state and 2s excited state of the one-electron systems H1, H2, H3, and He+3, the calculated finite-size related hyperfine shifts are quite similar for the different structure models and in excellent agreement with those estimated by comparing QED and experiment. This holds also in a simplified approach where relativistic wave functions from a Coulomb potential combined with spherical-shell distributed nuclear magnetic moments promises an improved treatment without the need for an explicit solution of Dirac’s equation within the nuclear core. Larger differences between different nuclear structure models are found in the case of the anisotropic 2p3/2 orbitals of hydrogen, rendering these excited states as promising reference systems for exploring the proton structure.

Matrix methods for opacity calculations

Atomic structure calculations are the first step in an opacity calculation. They provide the self-consistent electronic potential and electron wavefunctions that are needed to evaluate the Slater integrals and dipole matrix elements. In this paper, we show that simple matrix methods can be used to perform Hartree–Fock–Slater or LDA calculations on standard atomic structure grids, including relativistic corrections, yielding bound and continuum wavefunctions that can be used, in conjunction with statistical broadening techniques, to obtain realistic average-atom opacities.

Study on regeneration mechanism of composite adsorbent by Mg-MOF-74-based modified biochar

In this paper, composite adsorbent was prepared from biochar and Mg-MOF-74 by in-situ growth method to investigate regeneration mechanism. The effects of O2 and temperature on regeneration characteristics were investigated by CO2 adsorption properties and characterization techniques, and the optimal regeneration conditions were determined. Regeneration mechanism of adsorbent was revealed by adsorption kinetics and elemental valence analysis. The related wave function parameters were calculated based on DFT to reveal the repair mechanism of the failed oxidation sites from the microscopic level. The mechanism of CO2 adsorption by the repaired oxidation sites was explored based on the regenerated adsorption configuration. It was found that the regeneration performance of the adsorbent exhibited a trend of increasing and then decreasing with the increase of O2 concentration and temperature, and the optimal regeneration conditions were determined to be 5 % O2 concentration and 200 °C. At optimal regeneration conditions, a synergistic interaction between O2 and poly-metals was generated to enhance the adsorbent polarity. O2 also reacted with the adsorbent in a redox reaction to produce new oxygen-containing functional groups and cause pore expansion, the mass transfer and diffusion was enhanced. The oxidation site adsorbed O2 to undergo electron rearrangement and release the adsorbed CO2. Due to the nature of common orbital hybridization between metals, the metals underwent conjugation and synergistic effects with O2 to form tetrahedral co-coordination structures with lower energies. The electron density and electric field effects of the system were enhanced. The former enhanced interaction with CO2 to form carbonate. The latter increased the activity of the neighboring N atom, which in turn generated a stable ring structure with carbonate, and CO2 adsorption was enhanced.

Accurate ab initio calculation and neural network prediction of the atomic properties of Os LXXIII, Ir LXXIV, Pt LXXV, and Au LXXVI

In this paper, we present an ab initio theoretical calculation of the atomic parameters, including energy levels, wavelengths, lifetimes, and transition parameters, associated with the 1s22snl and 1s22pnl (n= 2–4, l≤n−1) configurations of Be-like sequence: Os LXXIII, Ir LXXIV, Pt LXXV, and Au LXXVI. Our analysis is based on the relativistic wavefunctions derived from the multi-configuration Dirac–Fock (MCDF) method, which are implemented in the relativistic atomic structure package GRASP2018. The correlations within the (principal quantum number) n= 10 complex are accounted for. The Breit interaction and quantum electrodynamical effects are added in the relativistic configuration interaction (RCI) calculation. Our results are compared with the existing data. The uncertainties of the dipole transition line strengths are assessed. In addition, we propose a feed forward neural network to predict the energy levels of Au LXXVI. This is a powerful machine learning tool capable of learning from existing data and predicting unknown data. The network is trained using theoretical energy levels of Os LXXIII, Ir LXXIV, and Pt LXXV obtained from the MCDF method. Our neural network exhibits average relative deviations of approximately 0.01% and 0.02% for trained and predicted energy levels, respectively, when compared to the MCDF results. This level of deviation suggests that our results are reasonably accurate. The data set presented in this study is useful for modeling fusion plasmas.

Computation of the near-infrared electro-absorption in GeSn/SiGeSn step quantum wells

In this study, we propose a theoretical simulation of the type-I step quantum well obtained from GeSn/SiGeSn to scan a wide range of telecommunication wavelengths and obtain near-infrared optical modulators. At T = 300 K, the band discontinuities and energy gap between stretched Ge1−xSnx and relaxed Si0.1Ge0.9−ySny due to the acquisition of the heterostructure were calculated.Then, optimization of this heterostructure based on (Si) GeSn was performed using the solid theory model to balance out the composition y of Si0.1Ge0.9-ySny relaxed and thickness of Ge0.91Sn0.09 QWs. The eigenenergies and their related wavefunctions are computed by solving the Schrödinger equation using the finite difference method under the framework of the effective mass approximation. Depending on the y concentration, the energy levels of the electron and the heavy hole, the change of transition energies and oscillator strength were examined for different well widths. Additionally, the absorption coefficient with y concentration and structure parameters were examined.From the findings obtained, it was determined that this material group is very important to obtain high efficiency from electro-absorption modulators covering the 1.55 μm wavelength range.

Relativistic Quantum Chemical Investigation of Actinide Covalency Measured by Electron Paramagnetic Resonance Spectroscopy.

We investigate actinide covalency effects in two [AnCptt3] (An = Th, U) complexes recently studied with pulsed electron paramagnetic resonance spectroscopy, using the Hyperion package to obtain relativistic hyperfine coupling constants from relativistic multiconfigurational wave functions. 1H and 13C HYSCORE simulations using the computed parameters show excellent agreement with the experimental data, highlighting the accuracy of modern relativistic ab initio methods. The extent of covalency indicated from the calculations on [ThCptt3] is in agreement with the original report based on traditional spectral fitting methods, while the covalency in [UCptt3] is found to be previously overestimated. The latter is due to the paramagnetic spin-orbit effect that arises naturally in a relativistic theory of hyperfine coupling and yet was not accounted for in the original study, thus highlighting the necessity of relativistic approaches for the interpretation of magnetic resonance data pertaining to actinides.

Production of Ds0(2590)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_{s0}(2590)^+$$\\end{document} in B nonleptonic decays

In 2021, a new charm-strange meson, Ds0(2590)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_{s0}(2590)^+$$\\end{document}, was discovered, which is believed to be the Ds+(21S0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_s^+(2^1S_0)$$\\end{document}. However, its low mass and wide width are challenged by theoretical results. Given the small branching ratio of the current production channel, resulting in a small number of events and large errors, we suggest searching for the Ds0(2590)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_{s0}(2590)^+$$\\end{document} in the B meson nonleptonic decays, Bq→Dq(∗)Ds0(2590)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_q\\rightarrow D^{(*)}_qD_{s0}(2590)^+$$\\end{document} (q=u,d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=u,d$$\\end{document}), followed by Ds0(2590)+→D∗K\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_{s0}(2590)^+\\rightarrow D^*K$$\\end{document}. We find that Br(Bq→Dq(∗)Ds0(2590)+)×Br(Ds0(2590)+→D∗K)=(2.16∼2.82)×10-3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Br(B_q\\rightarrow D^{(*)}_qD_{s0}(2590)^+)\ imes Br(D_{s0}(2590)^+\\rightarrow D^{*}K)=(2.16\\sim 2.82)\ imes 10^{-3}$$\\end{document} is very large, and the result is not sensitive to the mass of Ds0(2590)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_{s0}(2590)^+$$\\end{document}. Due to the large branching ratio, numerous Ds0(2590)+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D_{s0}(2590)^+$$\\end{document} events are expected. This study is based on the framework of the instantaneous Bethe–Salpeter equation, and the relativistic wave functions used for mesons contain different partial waves. The contributions of the different partial waves are also studied.

A cosmological bootstrap for resonant non-Gaussianity

Recent progress has revealed a number of constraints that cosmological correlators and the closely related field-theoretic wavefunction must obey as a consequence of unitarity, locality, causality and the choice of initial state. When combined with symmetries, namely homogeneity, isotropy and scale invariance, these constraints enable one to compute large classes of simple observables, an approach known as (boostless) cosmological bootstrap. Here we show that it is possible to relax the restriction of scale invariance, if one retains a discrete scaling subgroup. We find an infinite class of solutions to the weaker bootstrap constraints and show that they reproduce and extend resonant non-Gaussianity, which arises in well-motivated models such as axion monodromy inflation. We find no evidence of the new non-Gaussian shapes in the Planck data. Intriguingly, our results can be re-interpreted as a deformation of the scale-invariant case to include a complex order of the total energy pole, or more evocatively interactions with a complex number of derivatives. We also discuss for the first time IR-divergent resonant contributions and highlight an inconsequential inconsistency in the previous literature.

Radiative dynamics of laser-driven Li@C n embedded in quantum plasma

This work considers a guest Lithium atom (Li@C n ) in an endohedral fullerene, embedded in a quantum plasma modeled by the more general exponential cosine screened Coulomb (MGECSC) potential, under the influence of a spherical confinement and laser radiation field. The system is examined in nonrelativistic form and the related wave equation is solved using the tridiagonal matrix method (TMM), thus obtaining the discrete-continuum spectrum and related wave functions. The numerical values of the relevant parameters in this process are physically accessible values. The effects of the plasma, laser field and endohedral cavity on the photoionization cross section (PCS) are analysed in detail. The shielding effect of the plasma medium and the pulsating effect of the laser field modify the effective potential energy of the system, affecting the localizations of the 2s and continuum states, causing various overlapping cases. Considering different values of the endohedral encompassement parameters, which means that different types of fullerenes are taken into account, overlapping cases occur for different spectra and wave functions. Scrutinising these overlappings, the confinement and Cooper resonances of the PCSs are analysed. This analysis provides many details for the radiative dynamics of an artificial system Li@C n . The relevant ranges and critical values of plasma, laser field, and endohedral encapsulation parameters in the formation process of these resonances and PCSs are explained, as well as the cross-section curves, resonance positions, effective photoelectron energy range, and general PCS behavior, which can be important for potential experiments in addition to other theoretical investigations.

General-relativistic wave–particle duality with torsion

We propose that the four-velocity of a Dirac particle is related to its relativistic wave function by ui=ψˉγiψ/ψˉψ . This relativistic wave–particle duality relation is demonstrated for a free particle related to a plane wave in a flat spacetime. For a curved spacetime with torsion, the momentum four-vector of a spinor is related to a generator of translation, given by a covariant derivative. The spin angular momentum four-tensor of a spinor is related to a generator of rotation in the Lorentz group. We use the covariant conservation laws for the spin and energy–momentum tensors for a spinor field in the presence of the Einstein–Cartan torsion to show that if the wave satisfies the curved Dirac equation, then the four-velocity, four-momentum, and spin satisfy the classical Mathisson–Papapetrou equations of motion. We show that these equations reduce to the geodesic equation. Consequently, the motion of a particle guided by the four-velocity in the pilot-wave quantum mechanics coincides with the geodesic motion determined by spacetime. We also show how the duality and the operator form of the Mathisson–Papapetrou equations arise from the covariant Heisenberg equation of motion in the presence of torsion.

Decay property of the X(3842) as the ψ3(1D33) state

In this paper, the new particle X(3842) discovered by the LHCb Collaboration is identified to be the ψ3(13D3) state. We study its strong decays with the combination of the Bethe-Salpeter method and the P30 model. Its electromagnetic (EM) decay is also calculated by the Bethe-Salpeter method within the Mandelstam formalism. The strong decay widths are Γ[X(3842)→D0D¯0]=1.27 MeV and Γ[X(3823)→D+D−]=1.08 MeV, and the ratio is B[X(3842)→D+D−]/B[X(3823)→D0D¯0]=0.84. The EM decay width is Γ[X(3842)→χc2γ]=0.29 MeV. We also estimate the total width to be 2.87 MeV, which is in good agreement with the experimental data 2.79−0.86+0.86 MeV. Since the used relativistic wave functions include different partial waves, we also study the contributions of different partial waves in electromagnetic decay. Published by the American Physical Society 2024

Effect on incomplete fusion of breakup fragments of the weakly bound projectile 8B in reactions with targets 27Al, 28Si, 208Pb and 209Bi

Calculations of total, complete and incomplete fusion cross sections are presented as function of the incident energy of the weakly bound projectile 8B in reactions with light 27Al, 28Si and heavy 208Pb and 209Bi mass targets. In particular, the relative effect of the breakup fragments; proton and 7Be of 8B on incomplete fusion is determined. Also, the effect of an additional proton to the nuclear potential of the targets 28Si, 209Bi respect to 27Al and 208Pb is determined as a function of the incident energy. It is found that the proton absorption becomes more important than for 7Be for light targets for energies around and above the barrier, however, the inverse occurs for heavy targets. The calculations are performed with two different methods. The CDCC to determine the relative projectile-target radial wave functions which subsequently are used in the angular momentum-dependent model of fusion probabilities to calculate complete, incomplete and total fusion.

Importance of elastic scattering in nucleon-exotic nucleus experiments

In any experiment involving the interaction of two hadronic, many-body systems, the underlying interaction is conventionally described by an optical potential. Usually, this is of phenomenological, local, form, with real and imaginary parts each of which has parameters which are determined by fits to elastic scattering data. In order to predict observables for a wide range of targets, those data must cover a wider range of nuclei so that experiments with individual targets may be described appropriately. Further, for other experiments, obtaining the relative wave functions between the interacting systems requires the optical potential. Unfortunately, for nuclei far from the valley of stability such a global approach is inappropriate given the large differences in density at the surfaces due to halos and skins. In that case, phenomenology requires the measurements of elastic scattering data for the specific interacting systems. An alternative approach is to use microscopic formulations of the optical potential, such as the Melbourne g-folding potential. Aspects of both of these approaches for intermediate energies, given the interest in establishing facilities for exotic nuclear experiments at intermediate energies, will be discussed.

First-Principles Calculations of the Optical Spectra of Uranium Ions in Oxide Crystals

Recently U6+-doped phosphors are attracting attentions as potential green phosphors for the white LED application. However, the origins of the peaks in the optical spectra in some U-doped phosphors have not been clarified yet. For the development of the novel uranium phosphors, detailed analysis of the absorption and emission mechanisms of these materials is indispensable.In this study, we have performed the first-principles electronic structure calculations of U in oxide crystals using the relativistic discrete-variational multi-electron (DVME) code which is a configuration-interaction calculation program based on the four-component fully relativistic wave functions obtained by the relativistic discrete-variational Xα (DV-Xα) molecular orbital method. The optical spectra due to the charge transfer transitions of U6+ in oxides were calculated using the explicitly obtained many-electron wave functions. For comparison, the 5f-6d transitions of U3+ in oxide crystals were also calculated and the influence of the valency was investigated. The theoretical spectra were compared with the experimental spectra and the origins of the peaks were clarified based on the configuration analysis of the corresponding relativistic many-electron wave functions.

Study of characteristics elementary atomic processes in the neon-like multicharged ions plasma within an energy approach

We present the theoretical foundations of an advanced relativistic approach to computing the main energy, spectral characteristics of radiative-collisional processes in the plasma (in particular, the Debye plasma) of atomic as the example, neon-like) ions with simultaneous, quantitatively consistent consideration of the complex relativistic, interelectron exchange-correlation and plasma environment effects. The approach is based on the combination of a relativistic energy approach (S-matrix Gell-Mann and Low formalism), the relativistic gauge-invariant many-body perturbation theory with optimized Dirac-Fock-Sturm and Debye-Hückel approximations with accounting for the plasma environment effects with possible generalization on the presence of an additional external electromagnetic field. The fundamental point of our approach is the selection of the optimized Dirac-Fock-Sturm zeroth approximation and application of the consistent procedure for constructing a one-quasiparticle representation (basis’s of relativistic wave functions) in compliance with the principle of gauge invariance, in particular, by minimizing a gauge-noninvariant contributions to the radiative widths of the atomic (ionic) levels due to the complex exchange-correlation effects. The electron-ion collision strength and the dioelect5ron capture rate etc are determined within the presented theory.

Study of Electron Impact Excitation of Na-like Kr Ion for Impurity Seeding Experiment in Large Helical Device

In this work, a krypton gas impurity seeding experiment was conducted in a Large Helical Device. Emission lines from the Na-like Kr ion in the extreme ultraviolet wavelength region, such as 22.00 nm, 17.89 nm, 16.51 nm, 15.99 nm, and 14.08 nm, respective to 2p63p(2P1/2o)−2p63s(2S1/2), 2p63p(2P3/2o)−2p63s(2S1/2), 2p63d(2D3/2)−2p63p(2P3/2o), 2p63d(2D5/2)−2p63p(2P3/2o), and 2p63d(2D3/2)−2p63p(2P1/2o) transitions, are observed. In order to generate a theoretical synthetic spectrum, an extensive calculation concerning the excitation of the Kr25+ ion through electron impact was performed for the development of a suitable plasma model. For this, the relativistic multiconfiguration Dirac–Hartree–Fock method was employed along with its extension to the relativistic configuration interaction method to compute the relativistic bound-state wave functions and excitation energies of the fine structure levels using the General Relativistic Atomic Structure Package-2018. In addition, another set of calculations was carried out utilizing the relativistic many-body perturbation theory and relativistic configuration interaction methods integrated within the Flexible Atomic Code. To investigate the reliability of our findings, the results of excitation energies, transition probabilities, and weighted oscillator strengths of different dipole-allowed transitions obtained from these different methods are presented and compared with the available data. Further, the detailed electron impact excitation cross-sections and their respective rate coefficients are obtained for various fine structure resolved transitions using the fully relativistic distorted wave method. Rate coefficients, calculated using the Flexible Atomic Code for population and de-population kinetic processes, are integrated into the collisional-radiative plasma model to generate a theoretical spectrum. Further, the emission lines observed from the Kr25+ ion in the impurity seeding experiment were compared with the present plasma model spectrum, demonstrating a noteworthy overall agreement between the measurement and the theoretical synthetic spectrum.

Source function from two-particle correlation through deblurring

In heavy-ion collisions, low relative-velocity two-particle correlations have been a tool for assessing space-time characteristics of particle emission. Those characteristics may be cast in the form of a relative emission source related to the correlation function through the Koonin-Pratt (KP) convolution formula that involves the relative wave-function for the particles in its kernel. In the literature, the source has been most commonly sought by parametrizing it in a Gaussian form and fitting to the correlation function. At times the source was more broadly imaged from the function, still employing a fitting. Here, we propose the use of the Richardson-Lucy (RL) optical deblurring algorithm for deducing the source from a correlation function. The RL algorithm originally follows from probabilistic Bayesian considerations and relies on the intensity distributions for the optical object and its image, as well as the convolution kernel, being positive definite, which is the case for the corresponding quantities of interest within the KP formula.

The graviton four-point function in de Sitter space

We compute the tree-level late-time graviton four-point correlation function, and the related quartic wavefunction coefficient, for Einstein gravity in de Sitter spacetime. We derive this result in several ways: by direct calculation, using the in-in formalism and the wavefunction of the universe; by a heuristic derivation leveraging the flat space wave-function coefficient; and by using the boostless cosmological bootstrap, in particular the combination of the cosmological optical theorem, the amplitude limit, and the manifestly local test. We find agreement among the different methods.

Few-body model approach to the lowest bound S-state of non-symmetric exotic atoms and ions

Lowest bound S-state energy of Coulombic three-body systems (N Z+ μ − e −) consisting of a positively charged nucleus of charge number Z (N Z+), a negatively charged muon (μ −) and an electron (e −), is investigated in the framework of few-body (i.e., two- and three-body) cluster model approach. For the three-body cluster model, we adopted the hyperspherical harmonics expansion (HHE) method. An approximated two-body model calculation is also performed for all the three-body systems considered here. A Yukawa-type screened Coulomb potential with an arbitrary screening parameter (λ) is chosen for the two-body subsystems of the three-body system. In the resulting Schrödinger equation (SE), the three-body relative wave function is expanded in the complete set of hyperspherical harmonics (HH). The use of the orthonormality of HH in the SE leads to a set of coupled differential equations (CDEs) which are solved numerically for a manageable basis size to get the energy (E). The pattern of convergence in energy relative to increasing basis size is also investigated. Results are compared with some of those found in the literature.