Breit Interaction in Polar Coordinates

By contrast to the transformation of standard Coulomb interaction operators to spherical coordinates, this task is much more involved in the case of the Breit interaction discussed in section 8.1. The Breit operator is symmetric with respect to an exchange of the full electron coordinate sets of electrons 1 and 2. This symmetry is partially lost if the radial terms are considered separately. 

Recall the frequency-dependent Breit operator in section 8.1.5 where co is the angular frequency of the exchanged (virtual) photon. 

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) 

It turned out [201] that a special s)mimetrized form of the Breit operator leads to more simplified matrix elements contributing to the electronic energy expression. The total Breit operator in its cu-dependent form and also in the long-wavelength limit is symmetric with respect to an interchange of the electron coordinates 1 and 2. For Bq (1, 2) this symmetry holds even for the radial and angular coordinates independently. However, this is not the case for the 

2) term, where we have B 1,2) = B 2,l) but not independently for the radial part and the angular and spin part. 

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