Gaunt interaction/term

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... 

From the four-component Dirac-Coulomb-Breit equation, the terms [99]—[102] can be deduced without assuming external fields. A Foldy-Wouthuysen transformation23 of the electron-nuclear Coulomb attraction and collecting terms to order v1 /c1 yields the one-electron part [99], Similarly, the two-electron part [100] of the spin-same-orbit operator stems from the transformation of the two-electron Coulomb interaction. The spin-other-orbit terms [101] and [102] have a different origin. They result, among other terms, from the reduction of the Gaunt interaction. 

The last term in is the gauge term. The addition to the Coulomb interaction in the Feynman gauge is called the Gaunt interaction, and in the Coulomb gauge it is the Breit interaction. 

Next, we consider the direct term from the Gaunt interaction ... 

The same kinds of reductions occur for the exchange terms from the Gaunt interaction. 

Including the Breit term for the electron-electron interaction in a scalar basis requires extensive additions to a Dirac-Hartree-Fock-Coulomb scheme. It is not possible to achieve the same reductions as for the Coulomb term, and the derivation of the Fock matrix contributions requires considerable bookkeeping. We will not do this in detail, but will provide the development for the Gaunt interaction as we did for the 2-spinor case. 

The gauge term, which is the difference between the Gaunt interaction and the Breit interaction, produces a spin-free operator that can be interpreted as an orbit-orbit interaction. Thus, both the Gaunt interaction and the gauge term of the Breit interaction give rise to spin-free contributions to the modified Dirac operator. We will use the developments of this section in chapter 17 to derive the Breit-Pauli Hamiltonian. 

We now turn to the Gaunt interaction, and use the terms from the modified Dirac representation in (15.54) to derive the Breit-Pauli operators. These terms need no renormalization, because they are all of order 1/c. The three classes of operators defined in (15.54) are considered in turn. 

These operators will be considered again later, when all the spin-free terms will be accumulated. We note here only that the Darwin term from the Gaunt interaction... 

The first two terms combine to give the orbit-orbit interaction, mentioned at the end of section 15.4, but they have been written in the present form so that the cancellation with the terms in the Gaunt interaction is more obvious. 

The spin-other-orbit interaction comes from the Gaunt interaction, as shown in chapter 15 and chapter 17. The derivation of these terms is somewhat more complicated because the kinematic factors Q can appear on either side of 1/r,- and they do not commute with it. The derivation gives us four terms,... 

Each of the operators in the gauge term consists of a scalar quadruple product, which may be rearranged as was the spin-spin term in the Gaunt interaction. 

This is called the Gaunt interaction. Obviously, if n=p and m=q in (19) and (20), w=0 and the retardation terms are idgn-ticially equal to zero. That is, for diagonal terms, h =h, and the Gaunt interaction is the correct two glectron interaction. Keeping the next term in the expansion of e, we find that the next contribution to h is... 

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. 

Most 4-component relativistic molecular calculations are based on the Dirac-Coulomb Hamiltonian corresponding to the choice g = Coulomb The Gaunt term of (173) has been written in a somewhat unusual manner. The speed of light has been inserted in the numerator which clearly displays that the Gaunt term has the form of a current-current interaction, contrary to the... 

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