Electron self-energy, quantum electrodynamics

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 leading quantum electrodynamic effects to be accounted for in electronic structure calculations are the radiative corrections known as electron self-energy interaction and vacuum polarization. For the energy of electronic systems, the latter is usually small compared to the former, but only the latter can be expressed in terms of an effective additive potential to be included in the electronic structure calculations. The total vacuum polarization potential can be expanded into a double power series in the fine structure constant a and the external coupling constant Za. The lowest-order term, the Uehling potential, can be expressed as [110-112] ... 

This paper reviews progress in the application of atomic isotope shift measurements, together with high precision atomic theory, to the determination of nuclear radii from the nuclear volume effect. The theory involves obtaining essentially exact solutions to the nonrelativistic three- and four-body problems for helium and lithium by variational methods. The calculation of relativistic and quantum electrodynamic corrections by perturbation theory is discussed, and in particular, methods for the accurate calculation of the Bethe logarithm part of the electron self energy are presented. The results are applied to the calculation of isotope shifts for the short-... 

The Breit corrections are sometimes classified as nonradiative effects in contrary to the radiative affects which are true quantum-electrodynamical effects due to the electron self energy and vacuum polarization [30-32]. 

Emission and absorption of a virtual photon on the same electron is an effect that is not included in the Dirac theory and it is known as the electron self-energy (SE). This forms the major part of the Lamb shift, discovered experimentally by Lamb and Rutherford in 1947. This was the starting point for the development of modem quantum electrodynamic. The second most important part of the Lamb shift is the... 

Keywords Perturbation theory Quantum electrodynamics Electron correlation Electron self-energy Green s operator Covariant evolution operator... 

The procedures for many-body perturbation calculations (MBPT) for atomic and molecular systems are nowadays very well developed, and the dominating electrostatic as well as magnetic perturbations can be taken to essentially all orders of perturbation theory (see, for instance, [1]). Less pronounced, but in many cases still quite significant, are the quantum electrodynamical (QED) perturbations—retardation, virtual pairs, electron self-energy, vacuum polarization and vertex correction. Sophisticated procedures for their evaluation have also been developed, but for practical reasons such calculations are prohibitive beyond second order (two-photon exchange). Pure QED effects beyond that level can be expected to be very small, but the combination of QED and electrostatic perturbations (electron correlation) can be significant. However, none of the previously existing methods for MBPT or QED calculations is suited for this type of calculation. 

The anisotropy parameter is also strongly altered if Fig. 8 corrections are included. As to "Shadow" autoionization 3s- 4p state in Argon it may be presented rudely as double electron dipole excitations 3p 3d 3p. The interaction with "2 electron-2 hole" excitations is very essential in the continuous spectrum also. Near ns-threshold, this interaction enhances the influence of adjoining many-electron shells, nearly completely compensating the decrease of the ns cross-section because of the spectroscopic factor F. It means, that the calculation leads in fact to the same value of q, as it follows from (8). With increase of (a) the influence of connection with all "2 electron-2 hole" states but those taken into account by E rapidly decrease, so that becomes equal to F s that almost complete compensation of g and F influence at threshold is a consequence of a relation analogous to Word identity in Quantum electrodynamics ) which connects the photon-electron vertex and the electron self-energy part. Being valid for Oi->0, it links in our notations F and q in such a way that the equality between experimental and RPAE cross sections, which leads to (8), must hold. 

Quantum electrodynamic (QED) effects are not included in the present work. A recent calculation of these effects for s electrons [66] gave estimates of about 0.04 eV for the ionization potential of the Tl + 6s electron (self energy 0.05 eV, vacuum polarization —0.01 eV) and 0.06 eV for the El 11 Is electron (self energy 0.09 eV, vacuum polarization —0.03 eV). Since the p electrons responsible for the transitions discussed in this work exhibit much weaker penetration into the nucleus, QED effects here are expected to be considerably smaller, at most 0.01-0.02 eV, within the error limits of the method ( 0.05 eV) estimated by comparing calculated and experimental results for thorium (Tables 2.5 and 2.6). 

When innermost core shells must be treated explicitly, the four-component versions of the GREGP operator can be used, in principle, together with the all-electron relativistic Hamiltonians. The GRECP can describe here some quantum electrodynamics effects (self-energy, vacuum polarization etc.) thus avoiding their direct treatment. One more remark is that the... 

Such comparisons promise interesting tests of QED. Unfortunately, however, the theory of hydrogen is no longer simple, once we try to predict its energy levels with adequate precision [36]. The quantum electrodynamic corrections to the Dirac energy of the IS state, for instance, have an uncertainty of about 35 kHz, caused by numerical approximations in the calculation of the one-photon self-energy of a bound electron, and 50 kHz due to uncalculated higher order QED corrections. 

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. 

For further details the reader is referred to, e.g., a review article by Kutzel-nigg [67]. The Gaunt- and Breit-interaction is often not treated variationally but rather by first-order perturbation theory after a variational treatment of the Dirac-Coulomb-Hamiltonian. The contribution of higher-order corrections such as the vaccuum polarization or self-energy of the electron can be derived from quantum electrodynamics (QED), but are usually neglected due to their negligible impact on chemical properties. 

AE all-electron HF Hartree-Fock DHF DIrac-Hartree-Fock DC DIrac-Coulomb-Hamlltonlan +B Breit interaction in quasi-degenerate perturbation theory +QED quantum electrodynamic corrections (vacuum polarisation, self-energy) p.n. point nucleus f.n. finite (Fermi) nucleus exp. experimental data. 

---