Coulomb Sturmian Research Articles

General analytical theory of three-photon decays for hydrogenic ions with Coulomb Sturmian basis functions

General analytical theory of three-photon decays for hydrogenic ions with Coulomb Sturmian basis functions

Spectral approaches in momentum space for dynamics of atoms in strong laser fields: application to long pulses and low-frequency regime

In this work, we present in detail a novel approach to solve the time-dependent Schrödinger equation in the momentum space (TDSE-). It is based on the expansion of the electron wavepacket and the nonlocal interacting Coulomb potential in a Coulomb Sturmian basis. It allows a spectral treatment of the TDSE- from which another formulation of our previous model potential is proposed. This latter enables us to go easily beyond the 1s + 2sp states, and accordingly, may help to have more insight on the role of the Coulomb potential or the atomic structure in the ionization process. In order to bring out the improvement of this approximation and as an illustration, the case of 3spd states supported by the Coulomb potential is derived analytically, leaving the other cases for later investigations. Physical observables such as oscillator strengths, bound states populations, ionization probabilities, energy spectra, momentum and energy-angle distributions, are described in detail and their formulas employed in our ab initio computations are given in compact matrix forms The calculations can be done either by using the ionized wave function or by projecting the wavepacket onto the incoming Coulomb wave function. For illustrative purpose, we apply these theories in the dipole approximation to two one-electron atomic targets initially prepared in the ground and excited electronic states, and driven by intense low-frequency electromagnetic fields. We consider several laser parameters for long pulses recently reported in the literature. Our results compare remarkably well with reference data of atomic structure, and their agreement with the atomic dynamics outcome of previous treatments is quite satisfactory.

Compton Profile of 1s2-State for 2 ≤ Z ≥ 10

Electron momentum distributions show a view of electronic structure that supplement the radial density function. The radial density function in momentum space, moments, and Compton profile of 1s2-state for 2 ≤ Z ≥ 10 are considered by configuration interaction (CI) wave function based on Coulomb Sturmian basis function. The realized results agree with the literature for Compton profiles of these atoms.

Two-dimensional hydrogen-like atom in magnetic field in the presence of Rashba spin–orbit coupling

Abstract We investigate the spin–orbit (SO) Rashba coupling effect on bound states of an electron confined to the two-dimensional (2D) hydrogen-like atom (TDHA) in the presence of an external magnetic field (B), applied perpendicularly to the atomic plane. Energy levels and wavefunctions for arbitrary B and SO Rashba coupling constant have been obtained by the diagonalization of the Schrodinger Hamiltonian with the SO Rashba coupling term, using the basis of discrete Coulomb Sturmian functions. The solutions are used to investigate the influence of the SO Rashba interaction on phase coherence of spin states in terms of the von Neumann (vN) entropy and the spin polarization of the system. We shown that there exists unique correspondence between the vN spin entropy and the magnitude of spin-polarization vector. We also study the influence of SO interaction on magnetic properties of the planar H-like exciton using parameters appropriate for InSb semiconductor nanostructure.

Application of the Economization of Power Series to Solving the Schrödinger Equation for the Gaussian Potential via the Asymptotic Iteration Method

This paper presents economized power series for the Gaussian function. The economization is accomplished by utilizing the “usual” and the “shifted” Chebyshev polynomials of the first kind. The resulting economized series are applied to the solution of the radial Schrodinger equation with the attractive Gaussian potential via the asymptotic iteration method (AIM). The obtained bound state energies are compared with those given by the same method when the Taylor expansion is used to approximate the Gaussian potential. We also compare them with those obtained from the exact Hamiltonian diagonalization on a finite basis of Coulomb Sturmian functions.

Quasirelativistic Coulomb Sturmian basis sets for atomic physics problems: illustration of their applicability to two-photon decay rates in hydrogen-like atoms

The continuum and discrete quasirelativistic one-electron Sturmian basis sets associated with the second-order Dirac–Coulomb equation are constructed by solving appropriate Sturm–Liouville problems. The potential-weighted orthogonality and closure relations obeyed by the members of such sets are discussed. The special case where both quasirelativistic Coulomb Sturmian basis sets are used to obtain solutions to the Gell-Mann–Feynman equation is presented. It is proved that for the continuum quasirelativistic Sturm–Liouville problem, the spectrum of potential strengths is continuous and covers the whole real axis. We have also introduced the complete outgoing continuum quasirelativistic Coulomb wavefunctions and presented the discretization of its radial components. The compact representation of the fully relativistic and quasirelativistic Coulomb Green’s functions is outlined. By making use of the Sturmian expansion of the Coulomb Green’s functions, closed-form expressions of two-photon spontaneous decay rates are derived for arbitrary multipole channels. For the sake of assessing the domain of applicability of quasirelativistic basis sets, an illustrative application is then carried out for two-photon decay rates for hydrogen-like ions. A comparison of present results with reference values previously obtained within a fully relativistic scheme (Bona et al 2014) is undertaken.

Quantum chemistry with Coulomb Sturmians: Construction and convergence of Coulomb Sturmian basis sets at the Hartree-Fock level

The first discussion of basis sets consisting of exponentially decaying Coulomb Sturmian functions for modelling electronic structures is presented. The proposed basis set construction selects Coulomb Sturmian functions using separate upper limits to their principle, angular momentum and magnetic quantum numbers. Their common Coulomb Sturmian exponent is taken as a fourth parameter. The convergence properties of such basis sets are investigated for second and third row atoms at the Hartree-Fock level. Thereby important relations between the values of the basis set parameters and the physical properties of the electronic structure are recognised. For example, an unusually large limit for the angular momentum quantum number in unrestricted Hartree-Fock calculations can be linked to the breaking of spherical symmetry in such cases. Furthermore, a connection between the optimal, i.e. minimum-energy, Coulomb Sturmian exponent and the average Slater exponents values obtained by Clementi and Raimondi (E. Clementi and D. L. Raimondi, J. Chem. Phys. 38, 2686 (1963)) is made. These features of Coulomb Sturmian basis sets emphasise their ability to correctly reproduce the physical features of Hartree-Fock wave functions.

Sturmian bases for two-electron systems in hyperspherical coordinates

We give a detailed account of an ab initio spectral approach for the calculation of energy spectra of two active electron atoms in a system of hyperspherical coordinates. In this system of coordinates, the Hamiltonian has the same structure as the one of atomic hydrogen with the Coulomb potential expressed in terms of a hyperradius and the nuclear charge replaced by an angle dependent effective charge. The simplest spectral approach consists in expanding the hyperangular wave function in a basis of hyperspherical harmonics. This expansion however, is known to be very slowly converging. Instead, we introduce new hyperangular Sturmian functions. These functions do not have an analytical expression but they treat the first term of the multipole expansion of the electron–electron interaction potential, namely the radial electron correlation, exactly. The properties of these new functions are discussed in detail. For the basis functions of the hyperradius, several choices are possible. In the present case, we use Coulomb–Sturmian functions of half integer angular momentum. We show that, in the case of H−, the accuracy of the energy and the width of the resonance states obtained through a single diagonalization of the Hamiltonian, is comparable to the values given by state-of-the-art methods while using a much smaller basis set. In addition, we show that precise values of the electric-dipole oscillator strengths for transitions in helium are obtained thereby confirming the accuracy of the bound state wave functions generated with the present method.

Three-body calculation of the1slevel shift in kaonic deuterium with realisticK¯Npotentials

The $1s$ level shift in kaonic deuterium was calculated using Coulomb Sturmian expansion of Faddeev equations. The convergence of the method yields an $\sim\ 1\ eV$ accuracy for the level shifts. We used three different, realistic, multichannel $\bar{K}N$ interactions reproducing all known experimental two-body $\bar{K}N$ data. The different results suggest, that the level shift should be in the range $\Delta E\sim(800\pm30)-(480\pm20){\rm i}\ \ eV$. The (almost) exact level shifts were compared with values, given by the commonly used approximations.

Dynamics of atoms in strong laser fields I: A quasi analytical model in momentum space based on a Sturmian expansion of the interacting nonlocal Coulomb potential

In this study we present a model that we have formulated in the momentum space to describe atoms interacting with intense laser fields. As a further step, it follows our recent theoretical approach in which the kernel of the reciprocal-space time-dependent Schrödinger equation (TDSE) is replaced by a finite sum of separable potentials, each of them supporting one bound state of atomic hydrogen (Tetchou Nganso et al. 2013). The key point of the model is that the nonlocal interacting Coulomb potential is expanded in a Coulomb Sturmian basis set derived itself from a Sturmian representation of Bessel functions of the first kind in the position space. As a result, this decomposition allows a simple spectral treatment of the TDSE in the momentum space. In order to illustrate the credibility of the model, we have considered the test case of atomic hydrogen driven by a linearly polarized laser pulse, and have evaluated analytically matrix elements of the atomic Hamiltonian and dipole coupling interaction. For various regimes of the laser parameters used in computations our results are in very good agreement with data obtained from other time-dependent calculations.

Impact of the valence band structure ofCu2Oon excitonic spectra

We present a method to calculate the exciton spectra of all direct semiconductors with a complex valence band structure. The Schr\"odinger equation is solved using a complete basis set with Coulomb Sturmian functions. This method also allows for the computation of oscillator strengths. Here we apply this method to investigate the impact of the valence band structure of cuprous oxide $\left(\mathrm{Cu_{2}O}\right)$ on the yellow exciton spectrum. Results differ from those in [Phys. Rev. Lett. 115, 027402 (2015)]; the differences are discussed and explained. The difference between the second and third Luttinger parameter can be determined by comparisons with experiments, however, the evaluation of all three Luttinger parameters is not uniquely possible. Our results are consistent with band structure calculations. Considering also a finite momentum $\hbar K$ of the center of mass, we show that the large $K$-dependent line splitting observed for the $1S$ exciton state [Phys. Rev. Lett. 91, 107401 (2003)] is not related to an exchange interaction but rather to the complex valence band structure of $\mathrm{Cu_{2}O}$.

On the evaluation of integrals with Coulomb Sturmian radial functions

It is presented the full evaluation of integrals involving Coulomb Sturmian functions already discussed in this Journal.

Matrix Continued Fraction Solution to the Relativistic Spin-0 Feshbach–Villars Equations

The Feshbach–Villars equations, like the Klein–Gordon equation, are relativistic quantum mechanical equations for spin-0 particles.We write the Feshbach–Villars equations into an integral equation form and solve them by applying the Coulomb–Sturmian potential separable expansion method. We consider boundstate problems in a Coulomb plus short range potential. The corresponding Feshbach–Villars CoulombGreen’s operator is represented by a matrix continued fraction.

Derivation, properties and application of Coulomb Sturmians defined in spheroidal coordinates

ABSTRACTCoulomb Sturmian amplitude functions are derived in prolate spheroidal coordinates and are presented in a closed algebraic form. Spheroidal Sturnian functions are revealed to be related to the polynomial solutions of Heun's confluent equation. A reduction of symmetry from spherical to axial leads to the coupling of spherical polar orbitals and the formation of hybrid orbitals. The contribution of each spherical orbital into a hybrid orbital depends strongly on distance R from a nucleus to the dummy centre, and substantially alters when R varies. At two limiting cases R = 0 and R → ∞ spheroidal Sturmians are purely atomic orbitals, whereas at intermediate R they contain many features intrinsic to diatomic molecular orbitals. Applications of spheroidal Sturmian basis are discussed; Coulomb spheroidal Sturmians are asserted to be the most appropriate basis functions for diatomic molecular calculations.

Coulomb Sturmians in spheroidal coordinates and their application for diatomic molecular calculations

Coulomb Sturmians are obtained in prolate spheroidal coordinates by separation of variables in the Schrödinger equation and direct solution of the appropriate one-dimensional equations. Molecular orbitals are expressed as linear combinations of the introduced Coulomb Sturmians and some low-lying energy terms and corresponding wave functions are calculated for one-electron diatomic molecules. It is shown that similarity of the one- and two-centre orbitals in spheroidal coordinates, combined with completeness and good convergence properties of Coulomb Sturmians, substantially speeds up convergence and makes the calculated results closer to the exact ones. Application of the elaborated calculating scheme for diatomic many-electron molecules is discussed.

Molecular integrals for slater type orbitals using coulomb sturmians

The use of Slater type orbitals in molecular calculations is hindered by the slowness of integral evaluation. In the present paper, we introduce a method for overcoming this problem by expanding STO’s in terms of Coulomb Sturmians, for which the problem of evaluating molecular integrals rapidly has been satisfactorily solved using methods based on the theory of hyperspherical harmonics.

An improvement on - exponential type orbitals for atoms in standard convention

The complete orthonormal sets of -exponential type orbitals (-ETOs) in LCAO approximation are investigated for the determination of the optimal values of integer α (− ∞ < α ⩽ 2) and non-integer α* (− ∞ < α* < 3) by minimising the total energies in atomic calculations. The Hartree–Fock–Roothaan calculations with the use of different values of indices α and α* are performed within the framework of the minimal basis sets approximation for the ground states of neutral atoms. It is found for non-integer values of α* that the efficiency of -ETOs in total energy calculations, electron density, and its derivative and cusp ratio at the nuclei is much better than the other integer values of α. It should be noted that the Coulomb–Sturmian and Lambda ETOs are special classes of ψ(α)-ETOs for α = 1 and α = 0, respectively. The performance of -ETOs in atomic energy calculations is also compared to those obtained by using other ETOs such as Slater and B functions. The optimal non-integer values of α* are also determined for each atom examined in this work. It is shown that the notably improvement in the efficiency of -ETOs can be obtained by the use of non-integer α* values.

Coulomb Sturmians as a basis for molecular calculations

Almost all modern quantum chemistry programs use Gaussian basis sets even though Gaussians cannot accurately represent the cusp at atomic nuclei, nor can they represent the slow decay of the wave function at large distances. The reason that Gaussians dominate quantum chemistry today is the great mathematical difficulty of evaluating interelectron repulsion integrals when exponential-type orbitals (ETOs) are used. In this paper we show that when many-centre Coulomb Sturmian ETOs are used as a basis, the most important integrals can be evaluated rapidly and accurately by means of the theory of hyperspherical harmonics. For the remaining many-centre integrals, Coulomb Sturmians are shown to have advantages over other ETOs. Pilot calculations are performed on N-electron molecules using the Generalized Sturmian Method.

Quantum Monte Carlo simulation of carbon monoxide reactivity when adsorbed at metal and oxide catalyst surfaces: Trial wave‐functions with exponential type basis and quasi‐exact three‐body correlation

AbstractReview publications show the advantages of using quantum Monte Carlo (QMC) methods when bonding interactions are modified and consequently, where the electron correlation energy varies and needs to be evaluated accurately. This article considers a model efficient metal catalyst for reactions of carbon monoxide. The ultimate aim is to help design industrially useful catalysts, since the reaction with water produces hydrogen gas selectively, that is, a clean fuel and a sustainable energy source. To narrow the gap between the industrial process and calculation, a model of platinum/oxide is studied. It is mimicked here by copper with and without adsorbed oxygen. Periodicity as well as a validated pseudo‐potential are used to limit the number of active electrons considered. A suitable code to carry out this task is CASINO. Furthermore, this code scales linearly to 100,000 processors and possesses a shared memory facility so that it is well‐suited to runs on the Blugene/P. It has been in production on such computers since early in 2011. A mixed plane‐wave/exponential type orbitals (ETO) basis is used for the systems studied. Plane waves are well‐suited to periodic solid substrates and a linear combination of ETOs for molecules. The atomic orbitals have direct physical interpretation, i.e., Coulomb Sturmians and hydrogen‐like orbitals. Their radial nodes are shown to be essential in obtaining the vitally important accurate QMC trial wave‐functions. Until 2008, ETO products on different atoms were difficult to manipulate for the evaluation of two‐electron integrals. Coulomb resolutions provide an excellent approximation that reduces these integrals to a sum of one‐electron overlap‐like integral products that each involve orbitals on at most two centers. They are thus readily evaluated. Only these integrals need to be re‐evaluated to change basis functions. In this article, QMC‐VMC variational optimisation is used with a quasi analytic two‐electron and nucleus correlation factor. The QMC diffusion Monte Carlo methodology is applied to adsorbed carbon monoxide and some of its reactions. © 2012 Wiley Periodicals, Inc.

Hydrogenoid orbitals revisited: From Slater orbitals to Coulomb Sturmians#

The simple connection between the Slater orbitals, venerable in quantum chemistry, and the Coulomb Sturmian orbitals, more recently employed in atomic and molecular physics, is pointed out explicitly in view of the renewed interest in both as basis sets in applied quantum mechanics. Research in Slater orbitals mainly concerns multicentre, many-body integrals, whereas that on Sturmians exploits their orthonormality and completeness with no need of continuum states. An account of recent progress is outlined, also with reference to relationships between the two basis sets, and with the momentum space and hyperspherical harmonics representations. The connection between the venerable Slater orbitals in quantum chemistry and the Coulomb Sturmain orbitals is pointed. An account of recent progress is outlined with reference to their relationships with the momentum space and hyperspherical harmonics representations.