Breit-Pauli Hamiltonian with Electromagnetic Fields

Breit-Pauli Hamiltonian with External Electromagnetic Fields 

At this stage we should add the missing extemal-field-dependent operators to the Breit-Pauli Hamiltonian reviewed already by Bethe [72]. By contrast to what follows, these terms are also derived in the spirit of the ill-defined Foldy-Wouthuysen expansion in powers of 1 /c. However, since the molecular property calculation is carried out in a perturbation theory anyhow, we may utilize the complete field-dependent Breit-Pauli Hamiltonian in such calculations. 

Naturally, the field-dependent Breit-Pauli Hamiltonian automatically results as the low-order limit of the Foldy-Wouthuysen-transformed field-dependent Dirac-Breit Hamiltonian. Once this has been carried out along 

For the sake of simplicity the DKH property transformation has often been restricted to zeroth or first order, but higher-order property calculations are now routinely available [658,764,765]. The dilemma is that the DKH decoupling protocol of the one-particle states is involved and lengthy expressions emerge that have rather small numerical effects on the total observable. 

An important aspect of the DKH approach to molecular properties is to understand the necessity to start at the four-component Dirac framework with a Hamiltonian containing the property X under investigation. The evaluation of X within this four-component picture may then be accomplished either varia-tionally or by means of perturbation theory up to some well-defined order as discussed in section 15.1. The reduction to two-component formulations can be realized by suitably chosen DKH transformations for both the variational and the perturbative treatment of X. However, the unitary transformations to be applied are different for both schemes [764], which is to be shown in the following. Of course, this distinction holds irrespective of the specific features of X. The differences will only vanish for infinite-order perturbation theory. 

---

The Breit-Pauli Hamiltonian with an external field contains all standard one- and two-electron contributions as well as the magnetic interaction of the electrons and their interactions with an external electromagnetic field. We may group the various contributions in the Breit-Pauli Hamiltonian according to one-and two-electron terms,... 

In general, they can all be properly dealt with in the framework of perturbation (response) theory. According to the discussion in section 5.4, we may add external electromagnetic fields acting on individual electrons to the one-electron terms in the Hamiltonian of Eq. (8.66). Fields produced by other electrons, so that contributions to the one- and two-electron interaction operators in Eq. (8.66) arise, are not of this kind as they are considered to be internal and are properly accounted for in the Breit (section 8.1) or Breit-Pauli Hamiltonians (section 13.2). Although the extemal-field-free Breit-Pauli Hamiltonian comprises all internal interactions, such as spin-spin and spin-other-orbit terms, they may nevertheless also be considered as a perturbation in molecular property calculations. While our derivation of the Breit-Pauli Hamiltonian did not include additional external fields (such as the magnetic field applied in magnetic resonance spectroscopies), we now need to consider these fields as well. 

---