Interaction Gaunt

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 transformation of AE2 to Breit-Pauli form has been known for a long time [81, 82]. (There is a small difference in the operators published by Bethe-Salpeter [81] and Itoh [82], but this does not matter, since it has no effect on the expectation value with respect to a nonrelativistic reference function). A detailed presentation of this reformulation, which is unexpectedly tedious, has been given recently [83]. The result for the Gaunt interaction is ... 

We can consider both the case that (1,2) is just the Coulomb repulsion or that it contains the Breit (or Gaunt) interaction. 

The partial wave contributions for the Gaunt and the Breit interaction are similar, but have different prefactors, for which, so far, no analytic expressions have been derived. The counterpart of B for the Gaunt interactions is roughly —2.7 times that for the Coulomb interaction (note the difference in sign). 

The DCB Hamiltonian is covariant to first order, and the presence of the Breit (or approximate Gaunt) interaction serves to increase the accuracy of calculated spectroscopic splittings and core binding energies. 

In the context of Breit operator (4.19) one should also mention its approximate form known as the Gaunt interaction V ... 

The effect of the Breit and Gaunt interactions has been investigated in many atomic systems and as will be demonstrated later in this chapter, it is known that they are very important in high-resolution atomic X-ray spectroscopy. 

Some examples of the molecular calculations are known as well. For instance, the effect of the magnetic electron-electron Gaunt interaction on bond length as been demonstrated in reference is CH [0.0pm] 517/4 [0.0pm] GeH [0.1pm, 326 ppm] ... 

As in the case of the unretarded Gaunt interaction, we choose the velocity operator expression r ca (instead of the momentum operator) to obtain consistent results as before and obtain the interaction energy of Eq. (3.252) promoted to the operator calculus of quantum theory. 

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. 

If the Coulomb and Gaunt interactions are combined into a single interaction, there is of course a loss of one of these relations, and then the only relation between the integrals involves permutation of both pair indices. However, the factor of 2 lost by this procedure is regained by the use of a single interaction instead of two separate interactions. 

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. 

We can also show that the direct contribution from the Gaunt interaction vanishes, as we saw in the 2-spinor formalism, and we are left with... 

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