Jacek Karwowski

On the influence of the Debye screening on the spectra of two-electron atoms

Abstract The energies of the ground state and of the first three 1 S excited states of helium-like atoms with 2⩽ Z ⩽10 have been calculated under plasma-type confinement. The effect of the confinement has been described by the Debye screening potential. The energy levels show interesting behaviour under increased plasma strength, which is in conformity with recent experimental observations of the change in spectra of an impurity atom in liquid helium.

Spectra of confined two-electron atoms

Spectra of two electrons confined in a spherically symmetric potential of mixed Coulomb and harmonic form are studied using the Hartree-Fock and configuration interaction methods. The model studied corresponds to a two-electron atom confined in a harmonic oscillator potential. The spectral consequences of the interplay between the effects of the confinement due to the Hookes law and due to the nuclear attraction force are investigated in detail.

Application of the complex-coordinate rotation to the relativistic Hylleraas-CI method: a case study

It is shown that variational solutions corresponding to the bound states of the Dirac–Coulomb (DC) eigenvalue problem may be obtained using a complex-coordinate rotation method. The discrete eigenvalues of the DC Hamiltonian are treated as resonances coupled to the Brown–Ravenhall continuum (the superposition of the negative- and positive-energy one-electron continua). The method leads to a clean separation of the bound-state energies from the continuum. It is shown that the effects related to the resonance character of the bound states being an artefact of the DC approximation, in particular the instability of the ground state, are proportional to the third power of the fine structure constant α. Thus, the results correctly describe the physical reality up to the terms proportional to α2. The approach has been implemented for the case of the ground state of two-electron atoms described by the Dirac–Coulomb Hamiltonian in a space of the Hylleraas-type trial functions. The variational energies, calculated for Z = 2, 80, 90, correspond to the best available in the literature.

Quantum defect orbital study of the sodium isoelectronic sequence

The quantum defect orbital (QDO) method and its relativistic (RQDO) counterpart have been applied to the calculation of oscillator strengths for the 3s 2S → 3p 2Po, 3p 2Po → 3d 2D, 3d 2D → 4f 2Fo, and 3d 2D → 5f 2Fo transitions in sodium-like atoms. The consequences of including relativistic effects are demonstrated.

Matrix elements of one- and two-electron operators

A graphical representation of matrix elements of spin-free one- and two-electron operators is used for deriving a simple algorithm for the evaluation of their values. The method covers all the cases which may occur when wave functions are taken as mutually orthogonal antisymmetrized products of spinorbitals (which are assumed to form an orthonormal set) and are eigenfunctions of L2 and Lz operators. The resulting formulas are suitable as well for computer programming as for hand calculations.ZusammenfassungMit Hilfe einer graphischen Darstellung wird ein einfacher Algorithmus zur Bestimmung der Werte der Matrix-Elemente von spinfreien Ein- und Zweielektronenoperatoren abgeleitet. Durch diese Methode werden alle Fälle erfaßt, die auftreten, wenn die Wellenfunktionen wechselweise orthogonale, antisymmetrisierte Produkte von Spinorbitalen (von denen angenommen wird, daß sie einen orthogonalen Satz bilden) darstellen und Eigenfunktion der L2 und Lz Operatoren sind. Die erhaltenen Formeln eignen sich für Rechnungen mit und ohne Verwendung eines Computers.

A multireference direct CI program based on the symmetric group graphical approach

Specific features of an SGGA-based multireference direct CI program working in large internal spaces are discussed. In particular, advantages resulting from the explicit separation of the orbital and the spin spaces are explored. Concepts allowing for the efficient creation of a flexible and symmetry-adapted CI basis, for the high-speed generation of the coupling coefficients and for structuring a simple permutation driven algorithm to handle the orbital space are briefly discussed.

Complex coordinate rotation and relativistic Hylleraas-CI: helium isoelectronic series

A combination of the Hylleraas-CI (Hy-CI) and complex coordinate rotation (CCR) methods has been applied to study the nuclear charge dependence of the eigenvalues of the Dirac–Coulomb (DC) Hamiltonian corresponding to the ground states of helium isoelectronic series atoms. It has been shown that the CCR, due to the separation of the localized states from the unphysical Brown–Ravenhall continuum, removes the instabilities of the bound-state eigenvalues observed in large-basis set Hy-CI results. The Hy-CI–CCR results are in very good agreement with the most accurate ones available in the literature. Surprisingly, the difference between the DC Hy-CI–CCR eigenvalues and the eigenvalues of the positive-energy projected no-pair Hamiltonian is equal, up to the numerical accuracy of the results, to (Zα)3/6π, i.e. to (Zα)3 relativistic many-body perturbation theory contribution for electron–electron Coulomb interaction operator. An excellent agreement between the Hy-CI–CCR eigenvalues shifted by (Zα)3/6π and the no-pair ones confirms the very high accuracy achieved in both approaches. The numerical accuracy of the Hy-CI–CCR DC eigenvalues is estimated to eight significant figures.

Coupling constants in the direct configuration interaction method

A general algorithm of evaluation of the coefficients of molecular integrals (coupling constants) appearing in the direct configuration interaction method is derived. The configurations are assumed to be spin-adapted antisymmetrized products of orthonormal orbitals. No limitation is imposed either upon the reference state (the number of the singly occupied orbitals may be arbitrary) or upon the excitation multiplicity.

EIGENVALUES OF MODEL HAMILTONIAN MATRICES FROM SPECTRAL DENSITY DISTRIBUTION MOMENTS : THE HEISENBERG SPIN HAMILTONIAN

An approach aimed at approximating the extreme (the lowest and/or the highest) eigenvalues of matrices representing many-electron model Hamiltonians from a knowledge of several spectral density distribution moments is proposed. A detailed discussion of the Heisenberg spin Hamiltonian spectrum is presented as an example of an application.

AVERAGE ENERGY OF AN N-ELECTRON SYSTEM IN A FINITE-DIMENSIONAL AND SPIN-ADAPTED MODEL SPACE

An expression for the average energy of an N-electron system in a finite-dimensional, antisymmetric, and spin-adapted model space (as, e.g., a full-configuration interaction space) is derived using elementary properties of the Hamiltonian in the Fock space.