Operator Breit

Modification of the potential operator due to the finite speed of light. In the lowest order approximation this corresponds to addition of the Breit operator to the Coulomb interaction. 

If the relativistic effects are sufficiently large and therefore cannot be accounted for as corrections, then as a rule one has to utilize relativistic wave functions and the relativistic Hamiltonian, usually in the form of the so-called relativistic Breit operator. In the case of an N-electron atom the latter may be written as follows (in atomic units, in which the absolute value of electron charge e, its mass m and Planck constant h are equal to one, whereas the unit of length is equal to the radius of the first Bohr orbit of the hydrogen atom) ... 

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... 

Thus, we have expressed the non-relativistic Hamiltonian of a many-electron atom with relativistic corrections of order a2 in the framework of the Breit operator (formulas (1.15), (1.18)—(1.22)) in terms of the irreducible tensorial operators (second term in (1.15), formulas (19.5)—(19.8), (19.10)— (19.14), (19.20), respectively). 

Thus, introducing parameters a, / and T we can account for the essential part of the correlation effects. However, it turned out that in the framework of the semi-empirical approach, all relativistic corrections of the second order of the Breit operator improving the relative positions of the terms, are also taken into consideration (operators H2, and H s, described by formulas (1.19), (1.20) and (1.22), respectively). Indeed, as we have seen in Chapter 19, the effect of accounting for corrections Hj and H s in a general case may be taken into consideration by modifications of the integrals of electrostatic interaction, i.e. by representing them in form... 

In this respect, the single-configurational Hartree-Fock method looks more promising and universal when combined with accounting for the relativistic effects in the framework of the Breit operator and for correlation effects by the superposition-of-configurations or by some other method (e.g. by solving the multi-configurational Hartree-Fock-Jucys equations (29.8), (29.9)). 

In addition, V 2 should include corrections due to the finite speed of the electromagnetic interaction, as well as magnetic contributions present due to the electron spin. An approximate way to account for these effects, correct to order a2 a.u., is provided by the Breit operator [115]... 

Spin-orbit interaction Hamiltonians are most elegantly derived by reducing the relativistic four-component Dirac-Coulomb-Breit operator to two components and separating spin-independent and spin-dependent terms. This reduction can be achieved in many different ways for more details refer to the recent literature (e.g., Refs. 17-21). 

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... 

Judd, Crosswhite, and Crosswhite (10) added relativistic effects to the scheme by considering the Breit operator and thereby produced effective spin-spin and spin-other-orbit interaction Hamiltonians. The reduced matrix elements may be expressed as a linear combination of the Marvin integrals,... 

Note that (2.19) is a manifestly hermitian form of the orbit-orbit interaction familiar from atomic physics, usually described in texts as resulting from the reduction of the Breit operator (2.8) to n.r. form. This is unfortunate from a pedagogical point of view, since the Breit operator refers only to spin-1/2 particles. Spin has nothing to do with it I will return to the difference between (2.18) and (2.19) shortly. 

Having defined our starting point, the second quantized no-pair Hamiltonian, we may now take a closer look at the relations between the matrix elements. For future convenience we will also change the notation of these matrix elements slightly. Due to hermiticity of the Dirac Hamiltonian and the Coulomb-Breit operator we have... 

The latter two relations were derived using the fact that the operator K, Eq. (12), commutes with full Coulomb-Breit operator. It is also possible to apply time-reversal to individual particle coordinates if we take the different transformation character of the two parts of the two-electron interaction into account. This gives in addition... 

It is understood that the matrix elements of this operator should be evaluated with the relativistic 4-component wave function. The approximation (163) is called also the low-frequency approximation, since it arises when energy differences tend to zero AE — 0. An expression for the Breit operator suitable for the evaluations with the two-component (nonrelativistic) wave functions follows when we expand also the relativistic wave functions using Eqs(24)-(26) ... 

The expression (204) incorporates the one-electron corrections of order (aZ) eo that follow from the Dirac equation for the electron in the field of the nucleus. These corrections are included in the potentials f/jo. The Breit operator (164) is also included in the expression for Uik, the corresponding corrections are of order a Zeo- The radiative (order a aZ) eo) and nonradiative (order Q (aZ)eo) QED corrections are also incorporated in Eq(204). The radiative corrections, discussed in Sec.3.1 are included in the Bethe logarithm term and in the potential (7[Pg.452]

The Dirac Breit operator is only correct to 0(c ). It is, by no means, obvious how one should construct higher-order corrections in... 

The reduction of the Breit equation into a four-component form of interest is a complicated, tedious and not fully exact process since the Breit operator itself is precise only to the order of 1/c2 (the reader should consult more specialised literature [3-6]). For this purpose it is convenient to consider even operators of the form... 

The last term, however, brings a singularity when the rest mass of both the electrons is equal. Since we wish to apply first-order perturbation theory to the Breit operator, the (00 term will be neglected. 

The last two-electron Hamiltonian term involving the Breit operator... 

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