Breit Interaction from Quantum Electrodynamics

After we have presented the semi-classical derivation of the Breit interaction, in which the electrons are described quantum mechanically while their interaction is considered via classical electromagnetic fields, we should discuss how this interaction can eventually be derived from quantum electrodynamics [212]. Consequently, we revisit the basics already introduced in section 7.2.2. 

there exists a well-defined QED basis to the semi-classical theory. The frequency-dependent interaction energy in Eq. (8.32) is the most general form, and consequently a frequency-dependent form of the Breit interaction which cannot be derived within the semi-classical theory, has been in use and may be considered to be somewhat more fundamental. For this reason, we present it here. The general expression in use is based on a formulation by Bethe and Salpeter [111,135], 

The connection to such a momentum-space representation as given above can also be made by starting from the cosine expression of the retarded interaction derived in the last section — through Eq. (8.35) — which finally produces [211] 

Depending on gauge choices, Quiney et al. write two expressions became popular in atomic physics and quantum chemistry [213] the interaction operator in Lorenz (Feynman) gauge in configuration-space representation as 

Note the imprinted form of the Breit interaction plus an independent exponential retardation factor. Grant and co-workers [213] refer in this context to earlier work by Brown [214] who essentially embeds the Rosenfeld arguments into the QED language with the final result being the Coulomb plus Gaunt operators times the retardation factor as derived in the previous section. 

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This statement implies that not only the Coulomb interaction is included in Er and Exc but also the (retarded) Breit interaction. It thus points at the fact that a consistent and complete discussion of many-electron systems and consequently of RDFT must start from quantum electrodynamics (QED). RDFT necessruily has to reflect the various features of QED, both on the formal level and in the derivation of explicit functionals. The most important differences to the noiu-elativistic situation arise from the presence of infinite zero point energies and ultraviolet divergencies. In addition, finite vacuum corrections (vacuum polarization, Casimir energy) show up in both fundamental quantities of RDFT, the four current and the total energy. These issues have to be dealt with by a suitable renormalization procedure which ultimately relies on the renormalization of the vacuum Greens functions of QED. The first attempt to take... 

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. 

It now remains to expand the operators in (3.235) using the definitions given in (3.230) but before we do so we must draw attention to a difficulty with (3.235). The final term, containing the operator (00)2 is not obtained if a more sophisticated treatment starting from the Bethe Salpeter equation is used. The reader will recall our earlier comment that the interaction term in the Breit Hamiltonian is acceptable provided it is treated by first-order perturbation theory. Rather than launch into quantum electrodynamics at this stage, we shall proceed to develop (3.235) but will omit the (00)2 term without further comment. 

Other electron nuclear interaction terms involving 7ra rather than Ia arise from this treatment. However, these terms have all been dealt with in the previous chapter and we do not repeat them here.) The terms in (4.23) are the same as those obtained previously starting from the Dirac equation. Equation (3.244) will yield both the electron and nuclear Zeeman terms and a Breit equation for two nuclei, reduced to non-relativistic form, would yield the nuclear-nuclear interaction terms. Although many nuclei have spins other than 1/2, and even the proton with spin 1 /2 has an anomalous magnetic moment which does not fit the simple Dirac theory, the approach outlined here is fully endorsed by quantum electrodynamics provided that only terms involving M l are retained (see equation (4.23)). The interested reader is referred to Bethe and Salpeter [11] for further details. In our present application we see that the expressions for both... 

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). 

Therefore, we may here start from the Breit interaction to derive the pseudo-relativistic Hamiltonians instead of following the somewhat meandering historical path from 1926 to about 1932, which was mainly based on classical considerations. As usual, a quantum-electrodynamical derivation is also possible and has been presented by Itoh [678], but the sound basis of our semi-classical theory, which we pursue throughout this book, is necessarily the Breit equation. Needless to say, the rigorous transformation approach to the Dirac-Coulomb-Breit Hamiltonian yields results identical to those from the QED-based derivation. 

The operators o(i) and /3(/) are just like those used in (11.2.7), but they are taken to act on the spin variables of the ith particle only. The interaction term arising from / is called the Darwin-Breit term after its discoverers, and its form is confirmed by more advanced quantum-electrodynamic considerations. 

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