Breit interaction / operator frequency-independent

The spectrum of the single-electron Dirac operator Hd and its eigenspinors (/> for Coulombic potentials are known in analytical form since the early days of relativistic quantum mechanics. However, this is no longer true for a many-electron system like an atom or a molecule being described by a many-particle Hamiltonian H, which is the sum of one-electron Dirac Hamiltonians of the above kind and suitably chosen interaction terms. One of the simplest choices for the electron interaction yields the Dirac-Coulomb-Breit (DCB) Hamiltonian, where only the frequency-independent first-order correction to the instantaneous Coulomb interaction is included. 

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)... 

This expression can formally be evaluated without assuming any further simplification for the electron-electron interaction potential energy operator i.e., g i,j) is still defined by Eq. (8.69) as the sum of the Coulomb and frequency-independent Breit interactions. As in the case of the one-electron DKH transformation, g i,j) acts as an integral operator in momentum space and hence does not commute with any expression containing momentum operators such as Aj or R , for example. With the abbreviation gij = g i,j) the total two-electron interaction operator reads... 

The same discussion applies equally well for the Breit interaction. Only the four terms of the second line of Eq. (12.70) yield even operators for g i,j) = The corresponding two-component form of the free-particle Foldy-Wouthuysen-transformed frequency-independent Breit interaction Bq reads... 

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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