Sturmian Basis Sets Research Articles

Many–electron sturmians applied to atoms and ions

Goscinski's definition of Sturmian basis sets is generalized and applied to many–electron systems. It is shown that when such basis sets are used, the kinetic energy term disappears from the secular equation, the nuclear attraction potential is already in diagonal form in the secular equation, the orbital exponents are automatically optimized, and good accuracy, including correlation effects, can be obtained directly with very few basis functions and without the use of the self–consistent field approximation. As an illustration, the method is applied to the ground states and excited states of atoms and ions.

Positive energy Sturmian states for two-Coulomb-center problems

Properties of two-center Coulomb Sturmian basis sets are discussed. Analytic and numerical techniques to calculate these functions and coupling matrix elements are developed. A class of Sturmian functions is found that has no analytic continuation to negative energies and is not present in one-center potentials. Advantages of Sturmian sets over conventional eigenstate sets are emphasized. {copyright} {ital 1997} {ital The American Physical Society}

Generalized potential harmonics and contracted sturmians

A formalism is described for choosing computationally manageable Sturmian basis sets which are automatically adapted to the requirements of particular physical problems. The method is a generalization and improvement of the potential harmonic technique of Fabre de la Ripelle, which is extensively used in nuclear and atomic physics. The present generalization is not restricted to hyperspherical coordinates, and the basis functions are characterized by a physically motivated grand principal quantum number, rather than the grand angular momentum quantum number. As an illustration, the simple example of a two-electron atom is presented.

The Dirac - Coulomb Sturmians and the series expansion of the Dirac - Coulomb Green function: application to the relativistic polarizability of the hydrogen-like atom

Two relativistic Sturmian basis sets are constructed by solving first-order Sturm - Liouville problems for the Dirac - Coulomb equation. The Sturmian expansions of the Dirac - Coulomb Green function and the reduced Dirac - Coulomb Green function are derived. Their utility is illustrated by calculating static electric dipole polarizabilities for hydrogen-like atoms with .

The d-dimensional hydrogen atom: hyperspherical harmonics as momentum space orbitals and alternative Sturmian basis sets

Hydrogenoid orbitals, i.e. the solutions to the Schrödinger equation for a central Coulomb field, are considered in mathematical dimensions d = 2 and d > 3 different from the physical case, d = 3. Extending known results for d = 3, Sturmian basis sets in configuration (or direct) space — corresponding to variable separation in parabolic coordinates — are introduced as alternatives to the ordinary ones in spherical coordinates: extensions of Fock stereographic projections allow us to establish the relationships between the corresponding momentum (or reciprocal) space orbitals and the alternative forms of hyperspherical harmonics. Properties of the latter and multi-dimensional Fourier integral transforms are exploited to obtain the matrix elements connecting the alternative basis sets explicitly in terms of Wigner's rotation matrix elements for d = 2 and generalized vector coupling (or Hahn) coefficients for d > 3. The use of these orbitals as complete and orthonormal expansion basis sets for atomic and molecular problems is briefly commented.

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.

Single electron transfer and target excitation in (3s) collisions at 2 - in the coupled-Sturmian-pseudostate approach

A two-centre coupled-Sturmian-pseudostate approach is employed to calculate single electron transfer, target excitation and total ionization cross sections for the (3s) collision system at 2 - impact energies. The Sturmian basis set, together with an analytic Hartree - Fock potential for the Na core, are chosen carefully so that all three channels (transfer, excitation and ionization) are included in the coupled scheme. The present state-selective and total charge transfer cross sections compare fairly well with experimental data, as well as with previous theoretical results, particularly below . The target excitation cross sections for the sodium 3p and 4d states are in very good agreement with the available measurements, while significant discrepancy exists between theory and experiment for the excitation of the Na (4p) state.

A momentum-space picture of the chemical bond

The derivation of the reciprocal-space Schrödinger equation is reviewed, as well as Fock's method for solving it for hydrogenlike atoms. It is shown that Fock's solutions (which represent Fourier transformed hydrogenlike orbitals in terms of 4-dimensional hyperspherical harmonics) can be used as basis sets for solving other problems in quantum chemistry. Such basis sets are of the Sturmian type (i.e., all the members of the set correspond to the same energy), and they obey weighted orthonormality relations in both direct and reciprocal space. The kernel of the reciprocal-space Schrödinger equation is expanded in terms of Sturmian basis sets, and this expansion is used to solve the problem of a particle moving in a many-center potential. Both Coulomb and non-Coulomb potentials are treated, and a new method for evaluating the necessary integrals is discussed. © 1996 John Wiley & Sons, Inc.

Sturmian basis sets in momentum space

The properties of Sturmian basis sets in d-dimensional direct space and d-dimensional momentum space are reviewed, as well as the relationship between hydrogenlike Sturmians and hyperspherical harmonics. The kernel of the reciprocal-space Schrödinger equation is expanded in terms of Strumian basis sets. This expansion allows Shibuya and Wulfman's treatment of many-center Coulomb potentials to be extended to many-center potentials of a general form, and the method is also extended to the calculation of crystal orbitals and band structures. © 1996 John Wiley & Sons, Inc.

Electron transfer, target excitation, and ionization in H++Na(3s) and H++Na(3p) collisions in the coupled-Sturmian-pseudostate approach.

Total cross sections for electron transfer, target excitation, and ionization processes are reported for the ${\mathrm{H}}^{+}$+Na(3s) and ${\mathrm{H}}^{+}$+Na(3p) systems at 1--100 keV impact energies. We have employed a two-center coupled-Sturmian-pseudostate approach that has recently been developed and applied to several quasi-one-electron systems by Winter [Phys. Rev. A 48, 3706 (1993)]. The Sturmian basis set, together with an analytic Hartree-Fock potential, are chosen carefully so that all three channels (transfer, excitation, and ionization) are represented properly. We also discuss briefly the effects of the binding-energy correction on the calculated cross sections. We find that our various cross sections (${\mathrm{\ensuremath{\sigma}}}_{\mathrm{cap}}^{\mathrm{tot}}$, ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{cap}}^{\mathrm{H}(\mathrm{n}=2)}$, ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{cap}}^{\mathrm{H}(2\mathit{s})}$, ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{cap}}^{\mathrm{H}(2\mathit{p})}$, ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{exc}}^{\mathrm{N}\mathrm{a}(3\mathit{s}\ensuremath{\rightarrow}3\mathit{p})}$, ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{exc}}^{\mathrm{N}\mathrm{a}(3\mathit{s}\ensuremath{\rightarrow}3\mathit{d})}$, ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{ion}}^{\mathrm{tot}}$) and the ${\mathit{A}}_{20}$ alignment parameter for Na(3s\ensuremath{\rightarrow}3p) compare very well with most of the available experimental data, as well as with previous theoretical results.For all the above quantities, when a comparison is made between theory and measurement, our results are equally good or better than the earlier calculations employing different theoretical approaches. We still find a persistent discrepancy between theory (present as well as previous calculations) and measurements for the electron transfer cross sections into the metastable state of the H atom (${\mathrm{\ensuremath{\sigma}}}_{\mathrm{cap}}^{\mathrm{H}(2\mathit{s})}$). Our results for the orientation and alignment effects in the electron transfer process of ${\mathrm{H}}^{+}$+Na(3p) collisions are in good qualitative accord with recent observations and previous theoretical studies. In particular, we see a strong enhancement for the H(n\ensuremath{\ge}3) capture cross sections in the ${\mathrm{H}}^{+}$+Na(3p) collisions. The H(n=2) production cross sections in the ${\mathrm{H}}^{+}$+Na(3p) collisons follow a roughly similar trend, as observed experimentally when plotted against the impact energy.

Born-Floquet theory of laser-assisted electron-atom collisions

We present a non-Hermitian Born-Floquet theory of scattering of fast electrons by atoms in the presence of a strong monochromatic laser field. The interaction of the laser field with both the incident electron and the target atom is treated nonperturbatively, while the interaction of the incident electron with the target atom is treated in first Born approximation. Fluorescence is neglected. Detailed calculations are performed for the ''elastic'' scattering of 500 eV electrons by atomic hydrogen accompanied by the transfer of photons. The contribution of the entire spectrum of unperturbed atomic states to the dressing of the target is exactly taken into account by performing the calculations on a complex Sturmian basis set. In the nonresonant case, and for electric field strengths that are small with respect to the atomic unit, our Born-Floquet results are in agreement with those obtained using the semiperturbative approach of Byron and Joachain (in which target dressing is treated in first-order perturbation theory) even at intensities where multiphoton ionization is nonperturbative. The Born-Floquet approach is particularly useful to study resonant cases, where the laser frequency matches a transition frequency in the atom. Two such situations are analyzed.

Selected applications of hyperspherical harmonics in quantum theory

Hyperspherical harmonics are the d-dimensional generalization of ordinary d-dimensional spherical harmonics. In this paper, we review some properties of hyperspherical harmonics. We also discuss the generalization of Fock's reciprocal-space treatment of hydrogen-like atoms and show how it leads naturally to Sturmian basis sets. The hyperspherical Sturmian method yields a matrix whose roots are half the square root of the binding energy of the system. Diagonalization of the matrix determines not only the energy levels of the system, but also appropriate Slater exponents for the basis functions

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.

Resonances in helium photoionisation

The authors have developed an L2 analogue of R-matrix theory to calculate the bound and scattering states of helium using large Sturmian basis sets. The first application is to the lowest several doubly excited 1P resonances in the synchrotron photoionisation of the helium ground state. To facilitate comparison with experiment, the resonance shape parameters introduced by Fano were redefined in terms of Siegert boundary conditions to allow their ab initio calculation without fitting. Excellent agreement with recent highly precise experiments is obtained.

Nonperturbative calculation of partial differential rates for multiphoton ionization of a hydrogen atom in a strong laser field

We show how the multiphoton-ionization matrix element for an atom in a laser field at intensities beyond the regime of perturbation theory can be calculated from the standard Floquet expansion of the electron wave function. We expand the harmonic components in a discrete complex Sturmian basis set, and we evaluate the resulting (divergent) sum for the matrix element by using the Pad\'e method. We present results for partial rates and angular distributions for hydrogen.

The hydrogen atom in a magnetic field. Spectrum from the Coulomb dynamical group approach

The authors present some sample results for the problem of the hydrogen atom in a magnetic field. The energies have been computed for a typical Rydberg situation of atomic physics interest using limited computer facilities. The use of the Coulomb dynamical group allows a complete description of the symmetries and a rational choice of a Sturmian type basis set. Moreover, comparison with Rayleigh-Schrodinger perturbative expansions of the energies is performed.