Hydrogen-like Wave Functions Research Articles

In English

Using a variational method within the framework of the effective mass approximation, using a triangular coordinate system of an electron, hole, and exciton moving in a titanium dioxide quantum dot, the exciton energy spectrum was obtained as a function of the radius a of the quantum dot. The variational wave function of the exciton contained factors that took into account the motion of an electron and a hole in a potential well of infinite depth of a quantum dot, as well as the form of a hydrogen-like wave function. It is shown that the occurrence of an exciton in a quantum dot has a threshold character. An exciton, as a bound state of an electron and a hole, is formed starting from a certain critical radius ac, the value of which exceeds the Bohr radius of the exciton in titanium dioxide. The exciton energy levels are located in the band gap of the titanium dioxide quantum dot. In this case, with an increase in the radius a of the quantum dot (so that a≥ac), a band of exciton states appears in the band gap of the titanium dioxide quantum dot. The mechanism for the formation of optical absorption spectra in nanosystems containing titanium dioxide nanocrystals are presented. It is found that the optical absorption of anatase NC, which was observed under the experimental conditions, was due to the appearance of an exciton in the NC. Using the variational calculation of the energy spectrum of an exciton in NC, the position of the absorption peak of NC anatase was determined. This absorption peak differs slightly from the absorption peak, which was obtained in the experimental work.

The spin Hall effect in polycrystalline samples of nonmagnetic fifth- and sixth-period metals

In this paper we propose a method to estimate the coefficient of spin Hall effect for polycrystalline samples of pure nonmagnetic metals. The transverse resistivities characterizing the spin Hall effect were calculated for different metals of the 5th and 6th periods. It is shown that the result within a statistical error agrees with the experimental data. Keywords: spin Hall effect, spin-orbit interaction, hydrogen-like wave functions.

Correction of a relativistic impulse approximation expression used to obtain Compton profiles from photon scattering doubly differential cross sections

A Compton profile (CP) can provide useful information about the electron populations and distributions in atoms, molecules, and ions to assess many physical properties of matter. However, a CP cannot be measured directly, but must be obtained from scattering data. The CPs discussed in this study are derived from photon-atom doubly differential cross sections (DDCS) via the following expression which is derived from an impulse approximation (IA) theory given by DDCS = KJ, where K represents a kinematic factor and J represents the CP. A relativistic version of this expression (i.e., RKJ)—an approximation of the full relativistic IA expression—is used for relativistic regimes; however, it does not yield accurate results for the inner and middle shells of moderate to heavy atoms. In this study, expressions from nonrelativistic (NR) hydrogen-like wavefunctions with a relativistic QED kinematic factor K rel and relativistic electron energy were derived to correct the RKJ expression for the K, L, M, and N atomic subshells. This derivation made it possible for relativistic contributions and screening effects to largely cancel, for any regime of energy angle and Z. Thus, the RKJ error which can be greater than 30% is reduced to within few percent over 99% of the electron momentum distribution range of any subshell CP when compared to published tabulated theoretical values. Two simple versions of the relativistic QED kinematic component of the corrected RKJ expressions were obtained and tested: one valid at high photon energies, and the other at small scattering angles. RKJ corrections were applied to the extraction of CP from the full spectrum K-N shell DDCS, which resulted in much improved accuracy for the K-shell. Good agreement was observed with tabulated beyond K-shell CPs around the tail regions, but systematic differences were found to occur at the maxima. The details of this phenomenon are illustrated and discussed in this article.

Proposal for measuring line profiles and angular distributions of the radiative-electron-capture transition in low-energy ion-atom collisions: Beyond the impulse approximation

We propose to measure line profiles and angular distributions of the radiative-electron-capture transition in collisions of sub-MeV/$u {\mathrm{Ar}}^{18+}$ with helium. We estimate the line profile and the angular distribution using hydrogenlike and Hartree-Fock wave functions and show that the asymmetry of the line profile and the deviation of the angular distribution from the ${sin}^{2}{\ensuremath{\theta}}_{\mathrm{t}}$ law are magnified in the sub-MeV/$u$ region. Finite detector resolution and ion-beam monochromaticity are considered, and the results show that by using a usual bent-crystal spectrometer and a group of commercial silicon drift x-ray detectors one will be able to obtain sufficiently precise line profile and angular distribution to distinguish different theoretical descriptions of target atoms. The enhanced sensitivity to different theoretical approaches at low energies suggests that the proposed experiment will provide insights into the electromagnetic transition of an electron between a pair of relatively moving nuclei.

Application of the Screened Hydrogenic Model to Light Atoms

The purpose of this work was to develop a Screened Hydrogenic Model (SHM) to accurately calculate the electron energies for light atoms and ions with up to ten electrons for atomic numbers up to 18. The total energy of an atom or ion was calculated with effective nuclear charges and screening parameters for each electron type (1s, 2s and 2p) within a specific electron configuration. Multiple energy states, centered at the total energy, were calculated for electron configurations that have Russell-Saunders coupling. The energy of each electron included its relativistic energy, EREL, but close overall agreement between the calculated and experimental energies of multi-electron configurations required that the one-electron expression for EREL be modified in a simple manner. In the present work, 98% of the 587 calculated energies for light atoms/ions have a relative error within ±0.1% of the corresponding experimental energies. The effective nuclear charges described in this work allow hydrogen-like wave functions to be defined for the electrons within a multi-electron configuration. The SHM, described in this work, is available for future calculations involving light atoms and ions.

Relativistic configuration interaction calculations of transitions for low-lying states in the helium isoelectronic sequence

The excitation energy and the oscillator strength values have been computed for the 1s2–1s2p, 1s2s–1s2p, 1s2s–1s3p transitions in the helium isoelectronic sequence (Z=2 through Z=116). The importance of the Breit interaction and radiative corrections to the oscillator strength as well as transition energies have been studied. The method of calculations was based on the relativistic hydrogen-like (with non-integer nuclear number) wavefunctions.

Morse quantum well modulated by nonresonant intense laser field: Binding energy and optical absorption related to shallow donor impurities

Binding energies of the lowest two s-symmetric impurity states and linear, third order nonlinear, and total optical absorption coefficients corresponding to the 1s → 2s transitions for electrons in a Morse quantum well are investigated. Influence of a non-resonant intense laser field is considered for different values of the structure parameter, impurity position, and laser field intensity. In order to find the eigenvalues and eigenfunctions of the unbound electron confined within the Morse quantum well under the intense laser field, the time independent Schrödinger equation is solved by using the effective mass and parabolic band approximations. The impurity binding energy is determined by means of a variational procedure, using hydrogen-like trial wave functions, and the optical response is treated with the use of a two-level approach in the density matrix expansion. Our results show that it is possible to design suitable devices described by a Morse-like confinement pattern in the desired wavelength by changing the mentioned parameters.

Excitation of high orbital angular momentum Rydberg states with Laguerre–Gauss beams

We consider the excitation of Rydberg states through photons carrying an intrinsic orbital angular momentum degree of freedom. Laguerre–Gauss modes, with a helical wave-front structure, correspond to such a set of laser beams, which carry units of orbital angular momentum in their propagation direction, with ℓ0 the winding number. We demonstrate that, in a proper geometry setting, this orbital angular momentum can be transferred to the internal degrees of freedom of the atoms, thus violating the standard dipole selection rules. Higher orbital angular momentum states become accessible through a single photon excitation process. We investigate how the spacial structure of the Laguerre–Gauss beam affects the radial coupling strength, assuming the simplest case of hydrogen-like wavefunctions. Finally we discuss a generalization of the angular momentum coupling, in order to include the effects of the fine and hyperfine splitting, in the context of the Wigner–Eckart theorem.

One-electron model of the high-frequency dielectric function of dense plasmas

The approach is developed to calculate reflectivity and high-frequency dielectric permeability of dense plasmas. The Kubo-Greenwood formula is applied. One-electron wave functions are used. Plane waves are substituted for the wave functions of the free states. Hydrogen-like wave functions are substituted for the wave functions of the bound states. The restriction of the excited pair bound states is included in the approach. Calculations are performed for the shock-compressed xenon. The results are in the fair agreement with the experimental data for densities ρ > 2 g/cm3 whereas the remarkable discrepancy appears at low densities. The discrepancy can be attributed to a finite width of the shock front. The width increases with the decrease of density and can contribute to the reflectivity at low densities.

Trends in Atomic Parameters for Crystals and Free Ions across the Lanthanide Series: The Case of LaCl3:Ln(3+).

Analyses of the crystal field energy levels of the series LaCl3:Ln(3+) using a semiempirical Hamiltonian shows that only five ions (Pr, Nd, Pm, Dy, Ho) meet the criteria to avoid overfitting of the atomic part. A new parameter (SNES) has been introduced to represent the strength of the normalized electrostatic repulsion for these ions. This parameter varies linearly (R(2)adj = 0.9994, N = 5) with the reciprocal of the radius of the tripositive lanthanide ion, as expected from the form of repulsive Coulomb interaction. The Slater parameters from the crystal field analyses, F(k)(corr) (i.e., corrected for the effects of the two-particle component of the three-body operator associated with the T(2) parameter), exhibit an exponential variation with the number of electrons, n, in 4f(n). This is explained by reference to the radial part of a hydrogen-like wave function. The ratio of F(k)(corr) with the ab initio free ion Slater parameter F(k)(ab initio) varies linearly with n. Fitted parameters F(k)(corr: free ion) from the free ion data for Pr(3+) and Nd(3+) show that the corresponding ab initio values are between 14 and 27% too high. The spin-orbit coupling constant from crystal field analyses (ζ4f) exhibits a quartic variation with atomic number, and the ratio ζ4f/ζ4f(ab initio) follows an exponential growth model with n. The results serve to confirm the hypothesis that smooth trends can be observed across the Ln(3+) series for the fitted parameters despite the fact that the majority of experimental data is lacking.

Helium atom in an external electric field: Exact diagonalization

An exact diagonalization method is applied to solve the quantum-mechanical problem of spinless helium atom in an external electric field of arbitrary magnitude. The basis set for two-electron problem is built from different pair combinations ψnalama(αra)ψnblbmb(αrb) of orthonormalized single-particle hydrogen-like wave functions ψnml(r) belonging to any possibly bound states of the individual a- and b-electrons in the Coulomb central field renormalized by the scale parameter α > 0. Within the selected basis the matrix elements of the total Hamiltonian allows an exact analytical representation in the form of finite numerical sums. The diagonalization procedure is performed by Jacobi algorithm for N × N square Hermitian matrix built on the basis of dimension N = 25. The systematics and the numerical values of the low-lying energy levels at zero field are in good agreement with known experimental data. The field dependences of low-lying levels (Stark effect) and polarizability in the ground state of helium a...

On the role of correlation energy produced by the spin-spin interaction in the formation of binding energy for two-electron atomic systems

An algorithm is proposed for computing the correlation energy produced by the spin-spin interaction VSS of electrons in He-like atomic systems. The algorithm takes into account the singular nature of VSS and the formation of a compact finite motion of electrons in the range of distances Ze2/mc2 < r12 < ħ/mc between the electrons under the action of the magnetic fields of spins for a singlet ground state. Good agreement with experimental values of the ionization potential is attained for a wide set of two-electron atomic systems without resorting to variational procedures, but only using hydrogen-like wavefunctions and correctly taking into account the singular nature of the spin-spin interaction of electrons.

Characteristic plateau in theL1-subshell ionization cross section of Ag induced by proton collisions

A characteristic dependence of the proton-induced 2s subshell ionization cross section generated by the nodal structure of the 2s wave function was probed via high-resolution measurements of the Lbeta{sub 3} (L{sub 1}-M{sub 3}) and Lbeta{sub 6} (L{sub 3}-N{sub 1}) x-ray lines of Ag. The intensity ratio of the measured lines depends strongly on the proton energy exhibiting a pronounced maximum around 0.4 MeV. The experimental intensities are critically compared to the theoretically predicted values. The latter were obtained using the ionization cross sections calculated within the semiclassical approximation employing hydrogenlike and Dirac-Fock electron wave functions.

A comparison of the hydrogenlike dipole radial matrix elements with overlap integrals and a step toward explicit expressions of the multipole matrix elements

For nonrelativistic radial hydrogenlike wavefunctions Rnl(r), the ratio ⟨Rνλ|r|Rnl⟩/⟨Rνλ|Rnl⟩ of the dipole to the overlap integral is proportional to (l ⋅ ΔE)−1 for l = λ + 1. ΔE is the energy difference between the shells ν and n. This property holds for discrete or continuous states, and can be derived as a particular case of the Feneuille recurrence formula (Feneuille 1970 C. R. Acad. Sci., Paris B 271 992). This relation and Feinberg's comparable one for ⟨Rνλ|r−2|Rnl⟩ suggest the existence of explicit expressions for the matrix elements ⟨Rνλ|rt|Rnl⟩. For ν ≠ n we are not yet able to write down a general formula valid for any l and λ, but we show with examples (λ = 0, l = 1 and λ = 1, l = 0) how explicit formulae can be obtained for fixed values of l and λ.

A variational density-functional calculation of the total atomic binding energy with recently proposed kinetic-energy and exchange-energy functionals

Recently, DePristo and Kress have presented a Padé-approximation formula for the kinetic-energy functional of an atom. Also recently, using the DePristo–Kress results, Cedillo, Robles, and Gázquez obtained a Padé-approximation formula for the exchange-energy functional of such a system. These functionals have been adopted in the present work in a variational density-functional calculation of the total atomic binding energy, using the Ne atom as an example. The total-energy functional used in the present work contains the kinetic-energy functional, the functional describing the interaction of the electrons with the atomic nucleus, the functional describing the classical or direct part of the interaction among the electrons, and the exchange-energy functional. The electron (number) density of the Ne atom is modeled by using hydrogenlike one-electron wave functions (with the 2s function orthogonalized to the 1s function) containing three variational parameters. It is found that the DePristo–Kress Padé-approximation formula for the kinetic-energy functional together with the Cedillo–Robles–Gázquez Padé-approximation formula for the exchange-energy functional leads to a total-energy value that is in good agreement with the single-ζ Hartree–Fock total-energy value for the Ne atom.

Total atomic binding energy via the density functional theory

Following a suggestion by Lieb, the (original) Weizsäcker inhomogeneity term is introduced into the Thomas-Fermi-Dirac energy-density functional by a correction factor of 1/5.376. Choosing the Ne atom as an example, and constructing the electron (number) density of this atom from hydrogen-like one-electron wavefunctions containing three variational parameters, the energy-density functional has been minimized. The resulting total binding energy (−128.463 a.u.) is in very good agreement with the near Hartree-Fock value (−128.574 a.u.). For comparison, calculations have also been carried out with the Thomas-Fermi-Dirac energy-density functional augmented by the (original) Weizsäcker term multiplied by a correction factor of 1/9 that has been argued for by Kirzhnits and Kompaneets and Pavlovskil. The resulting total binding energy (−137.375 a.u.) is not in good agreement with the near HF value (−128.547 a.u.) though it still represents an agreement over the value (−153.225 a.u.) obtained without the inhomogeneity correction. It has been found that the radial electron (number) density computed with the factor advanced by Lieb (1/5.376) is in better agreement with the Hartree-Fock (double zeta) radial electron (number) density than the one computed with the factor advanced by Kirzhnits and Kompaneets and Pavlovskil (1/9).

A new approach to the quantum mechanics of atoms and small molecules

When the Schrodinger equation for a system of N particles interacting through Coulomb forces is expressed in terms of hyperspherical coordinates, it closely resembles the d-dimensional generalization of the hydrogen atom wave equation (where d = 3N). The d-dimensional hydrogenlike wave functions can be found exactly, and a set of these functions, all corresponding to the same energy (but to different charges), fulfills a potential weighted orthonormality relation. When such a set of “generalized Coulomb Sturmians” is used as a basis set for bound states, solution of the resulting secular equations yields a spectrum of parameters k0, which are proportional to the square roots of the binding energies. These parameters also govern the asymptotic behavior of the basis set appropriate for the expansion of the bound states to which they correspond.

Excitons in conjugated polymers: Wavefunctions, symmetries, and quantum numbers

We introduce a mapping from configuration interaction singles wavefunctions, expressed as linear combinations of particle-hole excitations between Hartree-Fock molecular orbitals, to real-space exciton wavefunctions, expressed as linear combinations of particle-hole excitations between localized Wannier functions. The exciton wavefunction is a two-dimensional amplitude for the exciton center-of-mass coordinate, R, and the electron-hole separation (or relative coordinate), r, having an exact analogy to one-dimensional hydrogenlike wavefunctions. We describe the excitons by their appropriate quantum numbers, namely, the principle quantum number, n, associated with r and the center-of-mass pseudomomentum quantum number, j, associated with R. In addition, for models with particle-hole symmetry, such as the Pariser-Parr-Pople model, we emphasize the connection between particle-hole symmetry and particle-hole parity. The method is applied to the study of excitons in trans-polyacetylene and poly(para-phenylene).

Ion structure for x-ray Thomson scattering in dense fusion plasmas

The properties of the ion feature of the Thomson scattering signal are investigated. Firstly, the description of the atomic form factor by hydrogen-like wave functions is reviewed and better screening charges are obtained. Then the ionic structure in systems with several ion species is calculated from the HNC integral equation.

Direct and indirect exciton states in GaAs-(Ga, Al)As double quantum wells under crossed electric and magnetic fields

A theoretical study of the direct and indirect exciton states in GaAs/Ga1-xAlxAs coupled double quantum wells under crossed electric and magnetic fields is presented. The setup of the system under consideration consists of an electric field perpendicular to the layers and an in-plane applied magnetic field. For calculations we use a variational procedure (by using a simple hydrogen-like envelope wave-function), in the effective-mass and parabolic-band approximations. Present theoretical results are found in fair agreement with available experimental measurements in double quantum well heterostructures under applied electric and magnetic fields.