Breit approximation

In this section, the spin-orbit interaction is treated in the Breit-Pauli [13,24—26] approximation and incoi porated into the Hamiltonian using quasidegenerate perturbation theory [27]. This approach, which is described in [8], is commonly used in nuclear dynamics and is adequate for molecules containing only atoms with atomic numbers no larger than that of Kr. 

In Table I, 3D stands for three dimensional. The symbol symbol in connection with the bending potentials means that the bending potentials are considered in the lowest order approximation as already realized by Renner [7], the splitting of the adiabatic potentials has a p dependence at small distortions of linearity. With exact fomi of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-elecbon counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

Modification of the potential operator due to the finite speed of light. In the lowest order approximation this corresponds to addition of the Breit operator to the Coulomb interaction. 

Perdew, J.P. and Cole, L.A. (1982) On the local density approximation for Breit interaction. Joumoi of Physics C, 15, L905-L908. 

Heavy atoms exhibit large relativistic effects, often too large to be treated perturba-tively. The Schrodinger equation must be supplanted by an appropriate relativistic wave equation such as Dirac-Coulomb or Dirac-Coulomb-Breit. Approximate one-electron solutions to these equations may be obtained by the self-consistent-field procedure. The resulting Dirac-Fock or Dirac-Fock-Breit functions are conceptually similar to the familiar Hartree-Fock functions the Hartree-Fock orbitals are replaced, however, by four-component spinors. Correlation is no less important in the relativistic regime than it is for the lighter elements, and may be included in a similar manner. 

The terms etc. represent the one-body mean-field potential, which approximates the two-electron interaction in the Hamiltonian, as is the practice in SCF schemes. In the DFB equations this interaction includes the Breit term (3) in addition to the electron... 

One of the purposes of this work is to make contact between relativistic corrections in quantum mechanics and the weakly relativistic limit of QED for this problem. In particular, we will check how performing plane-wave expectation values of the Breit hamiltonian in the Pauli approximation (only terms depending on c in atomic units) we obtain the proper semi-relativistic functional consistent in order ppl mc ), with the possibility of analyzing the separate contributions of terms with different physical meaning. Also the role of these terms compared to next order ones will be studied. 

Neglecting spin-orbit contributions (smaller than other relativistic corrections for the ground state of atoms, and zero for closed-shell ones), the Breit hamiltonian in the Pauli approximation [25] (weak relativistic systems) can be written for a many electron system as ... 

A fully relativistic extension of the scheme put forward in [12] has been introduced in [19], including the transverse electron-electron interaction (Breit +. .. ) and vacuum corrections. Restricting the discussion to the no-pair approximation [28] for simplicity, we here compare this perturbative approach to orbital-dependent Exc to the relativistic variant of the adiabatic connection formalism [29], demonstrating that the latter allows for a direct extraction of an RPA-like orbital-dependent functional for Exc- In addition, we provide some first numerical results for atomic Ec. 

Let us consider approximations in accounting for the Breit interaction, that we made when outer core and valence electrons are included in GRECP calculations with Coulomb two-electron interactions, but inner core electrons are absorbed into the GRECP. When both electrons belong to the inner core shells, the Breit effect is of the same order as the Coulomb interaction between them. Though Bff does not contribute to differential (valence) properties directly, it can lead to essential relaxation of both core and valence shells. This relaxation is taken into account when the Breit interaction is treated by self-consistent way in the framework of the HEDB method [33, 34]. 

Due to small relaxation of outer core shells in most processes of interest, these shells can be also considered as frozen when analyzing the Breit contributions and the Bed and Bev terms can be taken into account similarly to the Bfc and Bf ones. The error of this approximation will be additionally suppressed by relative weakness of the Breit interaction with the outer core electrons as compared to the inner core ones. We note... 

Zgp O.7, Zg 0.4, Zjg 0.3. Thus, Bcc>, Bcv, and B > contributions are negligible for the chemical accuracy of calculation. Therefore, the above made estimates provide us a good background for approximating the Breit interaction by a one-electron GRECP operator that should work well both for actinides and for superheavy elements. The numerical tests of the GRECPs accounting for the Breit effects are discussed in the next section. 

The most straightforward method for electronic structure calculation of heavy-atom molecules is solution of the eigenvalue problem using the Dirac-Coulomb (DC) or Dirac-Coulomb-Breit (DCB) Hamiltonians [4f, 42, 43] when some approximation for the four-component wave function is chosen. 

Calculation of the leading recoil corrections of order a Za) becomes now almost trivial. One has to take into account that in our approximation the analogue of the Breit Hamiltonian in (3.3) has the form [20]... 

In this review we shall first establish the theoretical foundations of the semi-classical theory that eventually lead to the formulation of the Breit-Pauli Hamiltonian. The latter is an approximation suited to make the connection to phenomenological model Hamiltonians like the Heisenberg Hamiltonian for the description of electronic spin-spin interactions. The complete derivations have been given in detail in Ref. (21), but turn out to be very involved and are thus scattered over many pages in Ref. (21). For this reason, we aim here at a summary that is as brief and concise as possible so that all relevant connections between different levels of approximation are evident. This allows us to connect present-day quantum chemical methods to phenomenological Hamiltonians and hence to establish and review the current status of these first-principles methods applied to transition-metal clusters. 

In the previous section we presented the semi-classical electron-electron interaction we treated the electrons quantum mechanically but assumed that they interact via classical electromagnetic fields. The Breit retardation is only an approximate treatment of retardation and we shall now consider a more consistent treatment of the electron-electron interaction operator that also provides a bridge to relativistic DFT, which is current-density functional theory. For the correct description we have to take the quantization of electromagnetic fields into account (however, we will discuss only old, i.e., pre-1940 quantum electrodynamics). This means the two moving electrons interact via exchanged virtual photons with a specific angular frequency u>... 

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