Rayleigh-Ritz Variational Method Research Articles

6Pe—6Po transitions in boron-like ions with Z = 8–13

The energies, fine structure splittings, transition rates, and lifetimes of inner-shell excited sextet states 1s2s2p2nl, (n = 2–7; l = s, p, d) and 1s2p33p of the boron isoelectronic sequence (Z = 8–13) are investigated using the multi-configuration Rayleigh–Ritz variation method. The mass polarization effect and relativistic corrections are included by first-order perturbation theory. Configuration structures of the high-n inner-shell excited sextet series 6Se,o(m) and 6Pe,o(m) (m = 1–5) of boron-like Na6+ ion are assigned. The wavelengths and transition rates of electric-dipole transitions between 6Pe(m) and 6Po(m) (m = 1–5) states are calculated. The quantum electrodynamics (QED) effects and higher order relativistic corrections are also considered to obtain more accurate transition wavelengths. The predicted transition wavelengths agree well with the available theoretical and experimental data. The lifetimes for the inner-shell excited sextet states 6Pe(m) (m = 1–5) are also reported and discussed with the increase of nuclear charge number, Z. These theoretical data are useful for the identification of spectral lines in experiments and the design of XUV and soft X-ray lasers.

Effect of Rashba spin–orbit interaction on the ground state energy of a D0 centre in a GaAs quantum dot with Gaussian confinement

The ground state energy of a neutral hydrogenic donor impurity (D0) trapped in a three-dimensional GaAs quantum dot with Gaussian confinement is calculated in the presence of Rashba spin–orbit interaction. The effect of the spin–orbit interaction is incorporated by performing a unitary transformation and retaining terms up to quadratic in the spin–orbit interaction coefficient. For the resulting Hamiltonian, the Rayleigh-Ritz variational method is employed with a simple wave function within the framework of effective-mass envelope function theory to determine the ground state energy and the binding energy of the donor complex. The results show that the Rashba spin–orbit interaction reduces the total GS energy of the donor impurity.

Energies, fine structures, and transitions of the core-excited sextet states 6Se,o(n) and 6Pe,o(n) (n=1–5) of B-like ions

A comprehensive theoretical study of atomic characteristics of energy levels and transitions for the core-excited 6Se,o(n) and 6Pe,o(n) (n=1–5) states of the boron isoelectronic sequence (Z=6–14) are investigated by the Rayleigh–Ritz variation method and multi-configuration interaction wavefunctions. The relativistic corrections and mass polarization effects are included by first-order perturbation theory. The configuration structures of the high-lying sextet series 6Se,o(n) and 6Pe,o(n) (n=1–5) of the B-like ions are assigned. The transition rates and wavelengths for the electric dipole transitions 6Se,o(n)—6Po,e(n) (n=1–5) of the B-like ions are calculated and compared with currently available theoretical and experimental data. Furthermore, the radiative transition rates and wavelengths for the important dipole transitions are discussed with the increase of nuclear charge number Z. The calculations will provide useful data for identification of spectral lines arising from the solar atmosphere and the experimental study in future work.

On the symmetry of three identical interacting particles in a one-dimensional box

We study a quantum-mechanical system of three particles in a one-dimensional box with two-particle harmonic interactions. The symmetry of the system is described by the point group D3d. Group theory greatly facilitates the application of perturbation theory and the Rayleigh–Ritz variational method. A great advantage is that every irreducible representation can be treated separately. Group theory enables us to predict the connection between the states for the small box length and large box length regimes of the system. We discuss the crossings and avoided crossings of the energy levels as well as other interesting features of the spectrum of the system.

Energies, expectation values, fine structures and hyperfine structures of the ground state and excited states for boron

The 1s 22s 22p 2P ground state and the 1s 22s 2 np (n = 3, 4) 2P, 1s 22s 23s 2S, 1s 22s2p 2 4P, 2L (L = S, P, D), 1s 22s2p3s 4Po, 1s 22s2p3p 4S, and 1s 22p 3 4S o, 2Lo (L = P, D) excited states of boron are calculated using the Rayleigh-Ritz variational method with multiconfiguration interaction wave functions. Energies, expectation values such as the electron density at the nucleus ρ(0) and radial moments 〈r n 〉, and fine-structure splittings are calculated. Hyperfine structure parameters and magnetic coupling constants of these states are also calculated. The present results are in good agreement with theoretical and experimental data available in the literature and should provide valuable reference data for future theoretical calculations and experimental measurements.

Energy, fine structure, hyperfine structure, and radiative transition rates of the high-lying multi-excited states for B-like neon

The Rayleigh-Ritz variational method with multiconfiguration interaction wave functions is used to obtain the energies of high-lying multi-excited quartet states 1s 22s2pnl and 1s 22p 2 nl 4Pe,o (n ≥ 2) in B-like neon, including the mass polarization and relativistic corrections. The fine structure and hyperfine structure of the excited quartet states for this system are investigated. Configuration structures of the high-lying multi-excited series are further identified by relativistic corrections and fine structure splittings. The transition rates and wavelengths are also calculated. Calculated wavelengths include the quantum electrodynamic effects. The results are compared with other theoretical and experimental data in the literature.

Group theoretical analysis of a quantum-mechanical three-dimensional quartic anharmonic oscillator

This paper illustrates the application of group theory to a quantum-mechanical three-dimensional quartic anharmonic oscillator with Oh symmetry. It is shown that group theory predicts the degeneracy of the energy levels and facilitates the application of perturbation theory and the Rayleigh–Ritz variational method as well as the interpretation of the results in terms of the symmetry of the solutions. We show how to obtain suitable symmetry-adapted basis sets.

ACOUSTOLECTRONIC SENSORS BASED ON LITHIUM NIOBATE RESONATORS

An acoustoelectronic sensor based on piezoelectric-semiconductor hybrid structure have been proposed in this study. The resonant frequencies of such a structure have been experimentally shown to decrease as the conductivity of a semiconductor layer increases. The effect observed is due to an interaction between charge carriers and piezoelectric fields at the piezoelectric-semiconductor interface. The surface distributions of both elastic and piezoelectric fields have been theoretically calculated by using the Rayleigh-Ritz variational method for some eigenfrequencies of a 3D LiNbO3 piezoelectric layer.

Rayleigh-Ritz variational method with suitable asymptotic behaviour

AbstractThis paper considers the Rayleigh-Ritz variational calculations with non-orthogonal basis sets that exhibit the correct asymptotic behaviour. This approach is illustrated by constructing suitable basis sets for one-dimensional models such as the two double-well oscillators recently considered by other authors. The rate of convergence of the variational method proves to be considerably greater than the one exhibited by the recently developed orthogonal polynomial projection quantization.

Energies, fine structure, and radiative transitions of the high-lying core-excited states [formula omitted] and [formula omitted] ([formula omitted]) for the boron isoelectronic sequence

The Rayleigh–Ritz variation method using a multiconfiguration interaction wavefunction is carried out on the high-lying core-excited states 6Se,o(n) and 6Pe,o(n) (n=1–5) for the boron isoelectronic sequence (Z=6–10), including mass polarization and relativistic corrections. Energy levels, fine structure, transition rates, and transition wavelengths for this system are calculated. Some energy levels of the high-n states are reported for the first time. Wavelengths of the electric-dipole transitions from these core-excited sextet states with quantum electro-dynamical effects and higher-order relativistic corrections are included. The lifetimes of the 1s2s2p23p6So,1s2p33s6So,1s2s2p23s6P, 1s2s2p23d6P, and 1s2p33p6P states for the boron isoelectronic sequence are also reported. Our calculated results agree well in most cases with available theoretical and experimental data in the literature.

Rates of convergence of the partial-wave expansion beyond the Kato cusp condition

The rates of convergence for the partial-wave expansion with odd-power ${r}_{12}$ terms for the ground-state energy of the helium atom are derived. For both the second-order $1/Z$ expansion and the Rayleigh-Ritz variational method, the energy increments of the partial-wave expansion converge as $O({L}^{\ensuremath{-}N\ensuremath{-}7})$, where $N$ is the highest odd-power ${r}_{12}$ function. The derivations require assumptions of the regularities for the ground-state helium wave function, which have not been established. Numerical results are presented for supporting the theoretical rates of convergence.

L’allergie au liquide séminal

The atomic parameters – energy levels, oscillator strengths, transition probabilities, and transition wavelengths have been presented for electric dipole transitions in C I and O III. The results include center-of-gravity energies of LS-terms of configurations 2s22p2, 2s22p3p for even parity and 2s2p3, 2s22p3s, 2s22p3d for odd parity, and probabilities of the corresponding important electric dipole transitions. The calculations are carried out using the multi-configuration Rayleigh-Ritz variation method with moderate-scale wavefunctions. The oscillator strength values in length and velocity gauges show fair agreement with each other, which indicates that the calculated wave functions are reasonably accurate. The calculated results are compared with existing experimental data. The present values show good agreement with the experimental data. The present transition probabilities should be useful for diagnostics as well as for precise spectral modeling in astrophysical and laboratory plasmas.

Accurate calculation of the eigenvalues of a new simple class of superpotentials in SUSY quantum mechanics

Abstract We obtain accurate eigenvalues for two recently derived SUSY partner Hamiltonians. We improve the Rayleigh-Ritz variational method proposed by the authors and show how to apply the Riccati-Padé method to those particular partner potentials.

Bound states for the quantum dipole moment in two dimensions

We calculate accurate eigenvalues and eigenfunctions of the Schrödinger equation for a two-dimensional quantum dipole. This model proved useful for the study of elastic effects of a single edge dislocation. We show that the Rayleigh–Ritz variational method with a basis set of Slater-type functions is considerably more efficient than the same approach with the basis set of point-spectrum eigenfunctions of the two-dimensional hydrogen atom used in earlier calculations.

Numerical solution of the time-independent Dirac equation for diatomic molecules:Bsplines without spurious states

Two numerical methods are used to evaluate the relativistic spectrum of the two-centre Coulomb problem (for the $H_{2}^{+}$ and $Th_{2}^{179+}$ diatomic molecules) in the fixed nuclei approximation by solving the single particle time-independent Dirac equation. The first one is based on a min-max principle and uses a two-spinor formulation as a starting point. The second one is the Rayleigh-Ritz variational method combined with kinematically balanced basis functions. Both methods use a B-spline basis function expansion. We show that accurate results can be obtained with both methods and that no spurious states appear in the discretization process.

Energies, fine structures, and radiative lifetimes for the multiexcited quartet states of B‐like oxygen

AbstractEnergies for the multiexcited states 1s22s2pnl and 1s22p2nl 4Pe,o (n ≥ 2) of B‐like oxygen are calculated using Rayleigh–Ritz variation method with configuration interaction. The mass polarization and relativistic corrections are obtained with first‐order perturbation theory. Configuration structures of the high‐lying multiexcited series are identified by energies and contribution to normalization of angular‐spin components. These structures are further checked by calculations of relativistic corrections and fine structure splittings. Hyperfine parameters and hyperfine coupling constants are calculated for the first time. Wavelengths including quantum electrodynamic effect and higher‐order relativistic corrections and lifetimes are also calculated. These results are compared with available results in the literature. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

Existence of Surface Waves and Band Gaps in Periodic Heterogeneous Half-spaces

We find a sufficient condition for the existence of surface (Rayleigh) waves based on the Rayleigh-Ritz variational method. When specialized to a homogeneous half-space, the sufficient condition recovers the known criterion for the existence of subsonic surface waves. A simple existence criterion in terms of material properties is obtained for periodic half-spaces of general anisotropic materials. Further, we numerically compute the dispersion relation of the surface waves for a half-space of periodic laminates of two materials and demonstrate the existence of surface wave band gaps.

Detailed calculations on low-energy positron-hydrogen-molecule and helium-antihydrogen scattering

In this paper, we consider two scattering processes: low-energy positron-hydrogen-molecule and helium-antihydrogen scattering. In the positron-hydrogen-molecule scattering calculations, we use the Kohn variational method to calculate Zeff, the number of target electrons available to the positron for annihilation. In the helium-antihydrogen scattering calculations, we use the Rayleigh-Ritz variational method to calculate a wave function for the leptons as a function of the distance between the helium and the antihydrogen. This is used, together with the associated nuclear wave function and the wave function for αp̄ + Ps−, to calculate the cross section for the rearrangement reaction He + H̄ → αp̄ + Ps−, using the T-matrix and a form of the distorted wave approximation. For both processes, positron-electron correlation is taken into account accurately using Hylleraas-type functions.

Rayleigh–Ritz variation method and connected-moments expansions

We compare the connected-moments expansion (CMX) with the Rayleigh–Ritz variational method in the Krylov space (RRK). As a benchmark model we choose the same two-dimensional anharmonic oscillator already treated earlier by means of the CMX. Our results show that the RRK converges more smoothly than the CMX. We also discuss the fact that the CMX is size consistent while the RRK is not.

Supersymmetric Quantum Mechanics, Excited State Energies and Wave Functions, and the Rayleigh−Ritz Variational Principle: A Proof of Principle Study

In addition to ground state wave functions and energies, excited states and their energies are also obtained in a standard Rayleigh-Ritz variational calculation. However, their accuracy is generally much lower. Using the super-symmetric (SUSY) form of quantum mechanics, we show that better accuracy and more rapid convergence can be obtained by taking advantage of calculations of the ground states of higher sector SUSY Hamiltonians, followed by application of the SUSY "charge operators". Our proof of principle study uses a general family of one-dimensional anharmonic oscillator models. We first obtain the exact, analytic ground states for a general family of anharmonic systems. We give the general, factorized form of the Hamiltonian for the hierarchy that arises in SUSY theory. The "charge" operators can then be used to convert states among the sectors. We illustrate the approach with two specific anharmonic oscillator models. Using the ground state of the second sector Hamiltonian, we show that the corresponding excited state energies and wave functions of the first sector are accurately obtained by applying the charge operators, using significantly smaller basis sets than are required in a standard variational approach applied to the original Schrodinger equation. This is a consequence of the higher accuracy of the Rayleigh-Ritz variational method when applied for ground states.