Breit correction

The sum of the main Fermi contribution and the Breit correction is given in the last line of Table 9.1. The uncertainty of the main Fermi contribution determines the uncertainty of the theoretical prediction of HFS in the ground state in muonium, and is in its turn determined by the experimental uncertainty of the electron-muon mass ratio. 

One not so obvious problem with the shape-consistent REP formalism (or any nodeless pseudoorbital approach) is that some molecular properties are determined primarily by the electron density in the core region (some molecular moments, Breit corrections, etc.) and cannot be computed directly from the valence-only wave function. For Phillips-Kleinman (21) types of wave functions, Daasch et al. (52) have shown that the core electron density can be approximated quite accurately by adding in the atomic core orbitals and then Schmidt orthogonalizing the valence orbitals to the core. This new set of orbitals (core plus orthogonalized valence) is a reasonable approximation to the all-electron set and can be used to compute the desired properties. This will not work for the shape-consistent case because / from Eq. (18) cannot be accurately described in terms of the core orbitals alone. On the other hand, it is clear from that equation that the corelike portion of the valence orbitals could be reintroduced by adding in fy (53),... 

The electron-electron interaction is usually supposed to be well described by the instantaneous Coulomb interaction operator l/rn. Also, all interactions with the nuclei whose internal structure is not resolved, like electron-nucleus attraction and nucleus-nucleus repulsion, are supposed to be of this type. Of course, corrections to these approximations become important in certain cases where a high accuracy is sought, especially in computing the term values and transition probabilities of atomic spectroscopy. For example, the Breit correction to the electron-electron Coulomb interaction should not be neglected in fine-structure calculations and in the case of highly charged ions. However, in general, and particularly for standard chemical purposes, these corrections become less important. 

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). 

For molecules the evaluation of the Breit correction to the Coulomb-type electron-electron interaction operator becomes computationally highly demanding and cannot be routinely evaluated, not even on the Dirac-Fock level. To test the significance of the Breit interaction, the Gaunt term is evaluated as a first-order perturbation. It turned out that it can be neglected in most cases as can be seen from the DF 4- Bmag calculations cited in Table 2.1. 

The fully self-consistent handling is compared with a perturbative evaluation of only the beyond-Breit terms and a perturbative treatment of the complete Ej. Even for the heaviest atoms the perturbative evaluation of the retardation corrections to the Breit term seems to be sufficient. On the other hand, use of first-order perturbation theory for the complete Ej leads to errors of the order of 1 eV for heavy atoms. An accurate description of inner shell transitions in these systems requires the inclusion of second-order Breit corrections. 

For open inner shells, the Breit correction is important and can be included perturbatively. QED effects can be allowed for by interpolation of tabulated data [14]. 

In the case of Coulomb-Breit and Breit-Breit corrections the reducible (reference state) contributions are not zero. The explicit expression for the reducible Coulomb-Breit correction originates from the graphs Fig.7c,d. In the case of the ground state it was first derived in [28] ... 

Quantum electrod)mamic (QED) effects are known to be very important for inner-shells, for example, in accurate calculations of X-ray spectra [61]. For highly charged few electron atoms they were found to be of similar size as the Breit correction to the electron-electron interaction [62]. Similar effects were also found for valence ns electrons of neutral alkali-metal and coinage metal atoms [63]. They are of the order of 1-2% of the kinetic relativistic effects there. The result for the valence ns electron is a destabilization, while for (n-l)d electron is an indirect stabilization. In the middle range (Z = 30-80) both the valence-shell Breit and the Lamb-shift terms behave similarly to the kinetic... 

The corresponding first-order Breit correction to the energy of a closed-shell atom is... 

Figure 6. The (Dirac) Hartree-Fock energy — hf and the first order Breit correction are given for helium-like ions. —grows approximately as and grows approximately as for large Z.

The first-order Breit correction, which grows approximately as Z , is compared with the Hartree-Fock energy for helium-like ions in Fig. 6. 

We present the three contributions to the Breit interaction, Brpa, which includes the first-order term together with higher-order RPA corrections, the residual second-order Breit correction, and the third-order Brueckner correction for 2s and 2p states of Uthium-hke ions, in Fig. 10. 

In Table 10, RCI [76] and MBPT [85] energies on the 2s—2pi/2 and Is — lpoji transitions in Li-like ions are compared with experiment. For these low- to imd-Z ions, higher-order Breit corrections are quite negligible and RCI and MBPT are in very good agreement... 

Other interesting calculations of Be ", Ne "", Be, and Ne atom have been carried out by Kenny et al. [85] in which they evaluated perturba-tionally the relativistic corrections to the total energies. In particular, they found that the Breit correction is systematically larger in the Dirac-Fock approximation and calculated the most accurate values of relativistic corrections for the Ne atom to date. These results demonstrate another useful capability of the correlated wavefunction produced by QMC to estimate relativistic effects. Similar study within the VMC method has been done on examples of Li and LiH [86]. We expect that an important future application will be to carry out similar calculations of transition... 

The Breit correction affects mostly the interaction of core electrons and interations between core and valence electrons. ... 

The first-order correction is known as the Breit term, and ai and velocity operators. Physically, the first term in the Breit correction corresponds to magnetic interaction between the two electrons, while the second term describes a retardation effect, since the interaction between distant particles is delayed relative to interactions between close particles, owing to the finite value of c (in atomic units, c -137). 

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