Grzegorz Pestka

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.

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.

Geminals in Dirac–Coulomb Hamiltonian eigenvalue problem

The applicability of the Dirac–Coulomb model in computational analysis of the properties of many-electron systems has been, since many years, a subject of dispute and controversy. The most common and numerically safe approach, based on the restriction of the variational space to the many-electron spinors spanning a subspace of the positive-energy part of the complete Hilbert space has been challenged by alternative models in which carefully selected both positive and negative energy functions are taken into account. However, these constructions are not possible when one goes beyond the one-electron model, e.g. when geminal-containing trial functions are used. Then the problem becomes particularly difficult and subtle. In this report several aspects specific for the geminal-based variational approach to the Dirac–Coulomb eigenvalue problem are discussed.

Hylleraas-CI Approach to Diraccoulomb Equation

The 1/r 12 singularity in the Hamiltonian describing a system of interacting electrons results in a cusp of the corresponding eigenfunctions. In the vicinity of r 12 = 0 the non-relativistic wavefunction is linear in r 12 [1].

Bound states of

Accurate non-relativistic energies of the only two bound states of have been computed variationally using basis sets of -correlated functions. The upper bounds to the non-relativistic energies of the and states are, respectively, -2.17807725 au and -0.72305872 au.

Dirac-Coulomb Equation: Playing with Artifacts

Recent developments based on the application of the Hylleraas-CI (Hy-CI) method to the variational solving the Dirac-Coulomb equation are reviewed. Three modes of the implementation of the Hy-CI method are discussed: a plain Hy-CI approach in which the effects related to the Brown-Ravenhall disease are treated by the appropriate selection of the model space, the Hy-CI approach combined with the complex coordinate rotation, in which these effects are controlled by the separation of the discrete and continuous spectra in the complex plane, and the positive-energy-space projected approach in which the influence of the effects of coupling with the continuum are removed by an appropriate restriction of the model space. Limits of applicability of these approaches as well as their advantages and disadvantages are discussed on the example of the ground states of helium isoelectronic series atoms.