Few-electron systems

J. Rychlewski, Electron correlation in few-electron systems, in Handbook of Molecular Physics and Quantum Chemistry (ed. S. Wilson), Wiley, New York, 2003, Vol. 2, pp.199-218. 

Parr, R.G. and Bartolotti, L.J. (1983). Some remarks on the density functional theory of few-electron systems, J. Phys. Chem. 87, 2810-2815. 

In the implementation of density functional methods for atoms and molecules, accurate wave functions can be utilized to test or construct the energy density functional[l]. Except for one-electron systems, however, these wave functions are necessarily approximate. Beyond hydrogen, the most precise wave functions available are obtained by the use of variational methods. For the simplest, few-electron systems, such calculations are capable of producing energies and wave functions of very high accuracy, more than sufficient for the present requirements of density functional theory. In this article we review the development and present state of accurate, variational calculations on simple atomic and molecular systems. In order to facilitate comparison of various alternative... 

First-principles simulations of one- or few-electron systems involve quantum systems of the lowest dimensionality we will consider in this section. These might entail the smallest error intrinsic in the calculation of the wave function. However, the blessing of small dimensionality is compensated by the tendency of systems in this class usually involve electrons in highly irregular potentials. Furthermore, the potential is usually a pseudopotential which describes the effect of atomic cores and/or solvent molecules on the quantum system of interest. Unfortunately, pseudopotentials introduce errors which are difficult to calibrate. 

The extended geminal models have two main advantages. First, the conceptual structure which facilitates interpretation. This property is utilized in several studies on intermolecular interactions where energy decomposition schemes illuminate the character of the bonding. Second, the models are highly accurate. This feature is related to the FCI corrections on which the models are based. The reported calculations on few-electron systems illustrate this point. However, as demonstrated by the calculation on the neon dimer reported in this work, a high accuracy of a calculation on larger systems, require that at least triple pair corrections are included. 

The electronic many-body Hamiltonian in equation (1) is treated in the framework of the independent-electron frozen-core model. This means that there is only one active electron, whereas the other electrons are passive (no dynamic conelation is accounted for) and no relaxation occurs. In this model the electron-electron interaction is replaced by an initial-state Hartree-Fock-Slater potential [37]. This treatment is expected to be highly accurate for heavy collision systems at intermediate to high incident energies. The largest uncertainties of the independent-electron model will show up for low-Z few-electron systems, such as H -F H and H + He° or for high multiple-ionization probabilities. 

Bound-state QED provides a proper and practicable description of few-electron systems. Both QED-radiative corrections and electron-electron interactions may be treated perturbatively with respect to the coupling a — e2, counting the number of virtual photons involved, while the interaction with the external nuclear fields is included to all orders in Za. 

Since the exact relativistic many-electron Hamiltonian is not known, the electron-electron interaction operators g(i, j) are taken to be of Coulomb type, i.e. 1/r,- . As a first relativistic correction to these nonrelativistic electron-electron interaction operators, the Breit correction, Equations (2.2) or (2.3), is used. For historical reasons, the first term in Equation (2.2) is called the Gaunt or magnetic part of the full Breit interaction. Since it is not more complicated than l/ri2, it is from an algorithmic point of view equivalent to the Coulomb interaction, therefore it has frequently been included in the calculations. The second term, the so-called retardation term, appears to be rather complicated and it has been considered less frequently. In the case of few-electron systems further quantum electrodynamical corrections, like self-energy and vacuum polarization, have also been considered and are reviewed in another part of this book (see Chapter 1). 

The bound states of high-Z few-electron systems are not only particularly well suited for the study of QED in strong Coulomb fields. They are also most sensitive... 

Highly-ionized atoms DHF calculations on isoelectronic sequences of few-electron ions serve as the starting point of fundamental studies of physical phenomena, though many-body corrections are now applied routinely using relativistic many-body theory. Relativistic self-consistent field studies are used as the basis of investigations of systematic trends in ionization energies [137-144], radiative transition probabilities [145-148], and quantum electrodynamic corrections [149-151] in few-electron systems. Increased experimental precision in these areas has driven the development of many-body methods to model the electron correlation effects, and the inclusion of Breit interaction in the evaluation of both one-body and many-body corrections. 

Chapter 3 Accurate Relativistic Calculations Including QED Contributions for Few-Electron Systems... 

Later, in Sec. 4, we will give a detailed discussion of the need for the no-pair Hamiltonian in relativistic calculations, its limitations, and its relation to QED. To establish a foundation for our studies of few-electron systems, we start in Sec. 2 with a discussion of the one-electron central-field Dirac equation and radiative corrections to one-electron atoms. In Sec. 3 we describe many-body perturbation theory (MBPT) calculations of few-electron atoms, and finally, in Sec. 4 we turn to relativistic configuration-interaction (RCI) calculations. 

Experiments of rather fundamental importance are those performed in the few-electron systems for which theory can provide accurate predictions of spectroscopic data. The first collinear-beam experiment on the simplest molecule and the possibility of a redetermination of the... 

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