Breit retardation operator

In the previous section we presented the semi-classical electron-electron interaction we treated the electrons quantum mechanically but assumed that they interact via classical electromagnetic fields. The Breit retardation is only an approximate treatment of retardation and we shall now consider a more consistent treatment of the electron-electron interaction operator that also provides a bridge to relativistic DFT, which is current-density functional theory. For the correct description we have to take the quantization of electromagnetic fields into account (however, we will discuss only old, i.e., pre-1940 quantum electrodynamics). This means the two moving electrons interact via exchanged virtual photons with a specific angular frequency u>... 

Note that the subscript on the a matrices refers to the particle, and a here includes all of the tlx, tty and components in eq. (8.4). The first correction term in the square brackets is called the Gaunt interaction, and the whole term in the square brackets is the Breit interaction. The Dirac matiices appear since they represent the velocity operators in a relativistic description. The Gaunt term is a magnetic interaction (spin) while the other term represents a retardation effect. Eq. (8.27) is more often written in the form... 

The terms H5+H6 are often written in the form of the sum of magnetic (Hm) and retarding (Hr) interactions, sometimes also called the relativistic Breit operator... 

SO coupling is a relativistic effect. The theory of the interaction of the magnetic moments of the electron spin and the orbital motion in one- and two-electron atoms has been formulated independently by Heisenberg and Pauli [12,13], shortly before the advent of the four-component Dirac theory of the electron [14]. Breit later has added the retardation correction [15]. The resulting Breit-Pauli SO operator, which can more elegantly be derived from the Dirac equation via a Foldy-Wouthuysen transformation [16], was thus well known for atoms since the early 1930s [17]. 

However, the interaction potential between two charged particles, nucleus-electron or electron-electron, is not just the Coulomb interaction, since in the relativistic description a retarded, velocity-dependent interaction must be considered. The full and general derivation of these interaction potentials is involved and approximate relativistic corrections to the Coulomb interaction are used in general. The frequency-dependent correction to the electron-electron Coulomb interaction, the Breit operator... 

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 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). 

The Breit two-electron corrections arises from the relativistic magnetic retardation between two electrons. The Breit operator, which describes this interaction, is... 

The current density only occurs implicitly as we have obtained a new interaction operator, the Coulomb-Breit operator, for the charge density in an essentially stationary picture. Hence, we describe the interaction energy by the stationary states only and have all retardation- and current-density-specific terms hidden in the operator. This picture will be generalized for many-electron systems later in section 8.8. 

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. 

Because of the importance of Darwin s expression for the classical electromagnetic interaction of two moving charges (section 3.5), we are particularly interested in the frequency-independent radial form of the Breit operator. This represents the consistent interaction term to approximately include the retarded electromagnetic interaction of the electrons in our semi-classic formalism that describes only the elementary particles (electrons) quantum mechanically. In this long-wavelength limit, m —> 0, the radial operator Vv l,2) in Eq. (9.16) becomes D (l, 2) — already known from the Coulomb case in Eq. (9.9)... 

Any electromagnetic perturbation of the set of N electrons in a molecule finally leads to sums of operators that may be classified according to the number of electronic coordinates involved as one- or two-electron operators. Of course, if an electron experiences a truly external field a one-electron operator will describe this interaction. If two electrons interact via retarded magnetic fields of the moving electrons as expressed in the frequency-independent Breit interaction and, hence, in the Breit-Pauli Hamiltonian, two-electron interaction operators arise. 

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