Sturmian Basis Functions 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

Use of generalized hyperspherical Sturmian functions for a three-body break-up model problem

A hyperspherical Sturmian approach recently developed for three-body break-up processes is tested through an analytically solvable S-wave model. The scattering process is represented by a non-homogeneous Schrödinger equation in which the driven term is given by a Coulomb-like interaction multiplied by the product of a continuum wavefunction and a bound state in the particles' coordinates. The model contains most of the difficulties encountered in a real three-body scattering problem, e.g., non-separability in the electrons' spherical coordinates and Coulombic asymptotic behaviour, and thus provides an interesting benchmark for numerical methods. Since the Sturmian basis functions are constructed so as to include the correct asymptotic behaviour, a very fast convergence of the scattering wavefunction is observed. Excellent agreement is found with the analytical results for the associated transition amplitude. This holds true down to very low energies, a domain which is usually challenging as it involves huge spatial extensions. Within our method, such calculations can be performed without increasing significantly the computational requirements.

Use of uncorrelated Sturmian basis functions for three-body systems of finite masses

The Configuration Interaction method with uncorrelated Sturmian basis functions is applied to describe a variety of atomic and molecular three-body systems.

Generating optimal Sturmian basis functions for atomic problems

In this paper we discuss the optimization of Sturmian basis functions by studying bound atomic systems within the configuration interaction method. Our investigation clearly shows how the fulfillment of correct physical boundary conditions at short and large distances from the nucleus improves the convergence rate of the method. This is illustrated first through a one-electron atom, and then with the two-electron systems. For the ground state of the helium atom, and with 35 Sturmian functions per electron and angular momenta, we obtain an energy of $\ensuremath{-}2.903 712 820$ a.u., outperforming previous similar calculations [Bromley and Mitroy, Int. J. Quantum Chem. 107, 1150 (2007)].

Many‐center Coulomb Sturmians and Shibuya–Wulfman integrals

AbstractWhen momentum space is projected onto the surface of a unit 4‐D hypersphere by means of Fock's mapping, Coulomb Sturmian basis functions can be simply represented in terms of hyperspherical harmonics. The properties of these harmonics can be used to evaluate Shibuya–Wulfman integrals and other integrals that arise when the Sturmian basis functions are used in molecular calculations. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Sturmian basis functions for the harmonic oscillator

We define Sturmian basis functions for the harmonic oscillator and investigate whether recent insights into Sturmians for Coulomb-like potentials can be extended to this important potential. We also treat many body problems such as coupling to a bath of harmonic oscillators. Comments on coupled oscillators and time-dependent potentials are also made. It is argued that the Sturmian method amounts to a non-perturbative calculation of the energy levels, but the limitations of the method is also pointed out, and the cause of this limitation is found to be related to the divergence of the potential. Thus the divergent nature of the anharmonic potential leads to the Sturmian method being less acurate than in the Coulomb case. We discuss how modified anharmonic oscillator potentials, which are well behaved at infinity, leads to a rapidly converging Sturmian approximation.

Many-particle Sturmians

The weighted orthonormality relations for many-particle Sturmian basis functions are derived both in momentum space and in position space. It is shown that when these functions are used as a basis, the kinetic energy term disappears from the Schrodinger equation. A general method is developed for constructing many-electron Sturmian basis sets from one-electron Sturmians. This method is illustrated by applications to atoms and ions and to the H2 molecule. It is shown that the direct solution of the Schrodinger equation using many-particle Sturmianbasis functions offers a useful alternative to the Hartree–Fock approximation and configuration interaction; and it is shown that the Sturmian method leads to an automatic optimization of the orbital exponents.

Validity criteria for rotor-vibrator and independent-particle models of atoms: overlaps, oscillator strengths, and angular deviations

A comparison is made of Hartree-Fock, independent-particle (HF) and collective, molecule-like rotor-vibrator (RV) models for atoms with twovalenceelectrons. The two models are constructed from the same pseudopotential Hamiltonian. The criteria for the comparison are (a) overlap with well-converged configuration interaction (CI) wave functions, (b) comparison of oscillator strengths with values from well-converged CI functions (and, wherever possible, from experiment), and (c) comparison of the root-mean-square deviation of the angular distribution, Le., ((B12)2 - (812~))'/~. In the comparison of the overlaps, neither model is obviously superior to the other. The RV wave functions yield better oscillator strengths in more cases than the HF functions, despite the results of the overlap calculations. Qualitative comparisons of the spatial distributions of probability density suggest that the RV functions have angular distributions much more like the CI functions than do the HF functions. A simple model calculation of the effect of a small amount of configuration mixing elucidates this similarity. A brief comparison is made of energy optimization and overlap optimization. Many methods of modeling atoms use an independent-particle approach as a first approximation. In the independent-particle model, the electrons are assumed to maintain their individual angular momenta and energies, independent of all the other electrons in the atom. A traditional measure of how well the independent-particle model describes an atom is the correlation energy. The correlation energy in atoms, strictly the difference between the exact, nonrelativistic energy and the Hartree-Fack energy, can be fairly accurately determined by comparing the energy expectation values of well-converged CI (configuration interaction) and HF (Hartree-Fack) wave functions. Two new ways of classifying ground and low-lying excited states of pseudotwo-electron atoms have suggested some different zero-order approximations to the wave functions.Id One approach uses the hyperspherical coordinate system typically used in the adiabatic appro~imation.3.~ other uses the electron-nucleus-clectron angle and the electron-nucleus distances in rovibrational functi0ns5,~ as distinct from the functions developed by Hylleraas7q8 which include the interelectron distance explicitly and so correctly describe the cusp region. This paper shows the results of a comparison of the second of these two types of correlated wave functions, the RV functions with the independent-particle model. The HF method takes into account the mean field of the probability distribution of the charge of the electrons on each electron but does not include the individual interactions between electrons. Thus, the HF method, apart from the effects of particle symmetry, omits the correlation between the electrons present in any atom or molecule involving more than one electron. In this paper, we use newly calculated HFSCF (self-consistent field) wave functions as the independent-particle wave functions. Wave functions developed by Hunter and BerryS in the form of overlap optimized rotor-vibrator (RV) wave functions are used as the extreme of correlated wave functions. Although the CI wave functions are constructed from an independent-particle basis, the completeness of the expansion set allows a formally exact convergence, and tests performed previously6 justify the assump tion that the CI wave functions are well converged. The CI wave functions used are those calculated with Sturmian basis functions by Krause and Berry6 and use a p~eudopotential~ for the core electrons of atoms with more than two electrons. Overlaps between CI and RV wave functions have previously been calculated5 for several states, as have the oscillator strengthsl0

Hyperspherical Strumian basis functions

AbstractA Sturmian basis set is a set of solutions to the Schrödinger equation, with the potential scaled in such a way that all the members of the set correspond to the same value of the energy. We discuss, in particular, the set of Sturmian basis functions corresponding to solutions of the d‐dimensional hydrogenlike wave equation. These hydrogenlike Sturmian functions are expressed in terms of Laguerre polynomials and hyperspherical harmonics. When they are used as a basis for solving the many‐particle Schrödinger equation, the secular equations take on a simple form [Eq. (59)]. The necessary integrals are evaluated explicitly, and the possibility of combining the hyperspherical technique with dimensional scaling is discussed.

Numerical calculations of the ionization of one-dimensional hydrogen atoms using hydrogenic and Sturmian basis functions.

Exact integral expressions and simple analytical estimates for the bound-bound, bound-continuum, and continuum-continuum dipole matrix elements are derived by use of the momentum-space eigenfunctions for a one-dimensional model of a hydrogen atom. These results provide the essential ingredients for the numerical study of the quantum mechanisms responsible for the chaotic ionization of highly excited hydrogen atoms in intense microwave fields. A specific numerical algorithm for solving the Schr\odinger equation for a one-dimensional hydrogen atom in an oscillating electric field is described which uses these results for the dipole matrix elements along with a discrete representation of the continuum. In addition, the momentum-space representation of the Sturmian basis functions has been used to derive exact integral expressions and convenient analytical estimates for the projections of the Sturmian basis functions onto the hydrogenic bound and continuum states. These results are used to provide a direct comparison of numerical calculations for the ionization of one-dimensional hydrogen atoms using the hydrogenic and Sturmian bases.

The microscopic feshbach optical potential for a schematic coupled channel example

By means of a method of expanding the solutions of a system of coupled equations into a set of positive-energy sturmian basis functions, a separable approximation to the microscopic (Feshbach) elastic optical potential is obtained. The properties of this nonlocal potential are analysed in terms of the moments of the potential and of the wronskian of two independent solutions of the single-channel nonlocal Schroedinger equation. A numerical application is given for a schematic coupled channel example, representing a 60 MeV neutron incident on a schematic nucleus with up to 10 channels, coupled by phenomenological surface-peaked potentials. The nature of the nonlocality of this schematic potential is very different from the nonlocality due to exchange effects in that the wronskian is strongly angular momentum dependent and can be larger as well as smaller than unity. The range of the nonlocality is found to be larger than the wavelength of the elastic projectile.

Electron transfer inp−He+andHe2+-H collisions using a Sturmian basis

Total cross sections are reported for electron transfer into all bound states in collisions of protons with $^{4}\mathrm{He}^{+}$ ions and $^{4}\mathrm{He}^{2+}$ ions with H atoms at center-of-mass energies from 4 to 120 keV and 4 to 40 keV, respectively. The coupled-state calculations employ 19-24 Sturmian basis functions. Convergence of cross sections with respect to the size of this basis has been studied, and total cross sections are estimated to be converged to a few percent. A comparison is made with molecular-state, atomic-state, classical, and experimental results. In the energy region of overlap, Sturmian-state results and plane-wave-factor, molecular-state results agree to within 9% except for the $p\ensuremath{-}{\mathrm{He}}^{+}$ process at 14 keV, where the Sturmian results are 37% larger, probably reflecting the need to include the continuum which was omitted in the molecular calculations. There is also a similar discrepancy with the experimental results of Peart, Grey, and Dolder; these Sturmian results, however, do tie in with the experimental results of Angel, Sewell, Dunn, and Gilbody at higher energies.

Sturmian basis functions in the coupled state impact parameter method for H++H scattering

When Sturmian basis functions are used in the standard coupled state impact parameter method for proton-hydrogen atom scattering, the correct boundary conditions cannot, in general, be satisfied. This means that for most choices of the initial and final states the approximate transition amplitude, as usually defined, does not exist. This difficulty is resolved and an approximate transition amplitude is defined which is a variational estimate of the exact transition amplitude for arbitrary initial and final states. In the Sturmian representation the exchange matrix elements can be calculated more directly by solving a set of coupled differential equations which is somewhat different from the set of equations given by Cheshire (1967). A procedure that does not require storage and interpolation of the matrix elements in order to exploit their time-symmetry is also described.