Operator Pauli

Wouthuysen transformation that yields, up to second order in 1/c, the Pauli-operators... 

In contrast to the one-electron terms, the reduction of the 4x4 Dirac-Breit Hamiltonian to the 2x2 Breit-Pauli Hamiltonian is very tedious for the two-electron terms as each interaction term has to be transformed according to the Foldy-Wouthuysen protocol. As the derivation can be found for example in Refs. (56-58) and in detail in Ref. (21), we only present here the transformed terms and discuss their dimension. The two-electron Breit-Pauli operator gBP (i, j) reads... 

QR Method. The first relativistic method is the so-called quasi-relativistic (QR) method. It has been developed by Snijders, Ziegler and co-workers (13). In this approach, a Pauli Hamiltonian is included into the self-consistent solution of the Kohn-Sham equations of DFT. The Pauli operator is in a DFT framework given by... 

The Pauli operator of equations 2 to 5 has serious stability problems so that it should not, at least in principle, be used beyond first order perturbation theory (20). These problems are circumvented in the QR approach where the frozen core approximation (21) is used to exclude the highly relativistic core electrons from the variational treatment in molecular calculations. Thus, the core electronic density along with the respective potential are extracted from fully relativistic atomic Dirac-Slater calculations, and the core orbitals are kept frozen in subsequent molecular calculations. 

The comparison between the ZORA and QR methods is still rather limited (70), Table II. We are currently in the process of carrying out a more thorough comparison of the two approaches. The ZORA approach is clearly superior to the QR method on theoretical grounds since it avoids - rather than circumvents - the fundamental stability problems of the Pauli operator. 

O Pauli operator including spin-orbit coupling (variational procedure employing frozen cores), Reference 52... 

As may be seen by comparing Eqs. [103] and [105], the no-pair spin-orbit Hamiltonian has exactly the same structure as the Breit-Pauli spin-orbit Hamiltonian. It differs from the Breit-Pauli operator only by kinematical factors that damp the 1/rfj and l/r singularities. 

Pauli Operators Using General Cartesian Gaussian Functions. I. One-Electron Interactions. 

Surprisingly, the theoretical analysis of the extensive spectroscopic data for molecules and crystals in the 1940s and 1950s did not make use of the microscopic Breit-Pauli operator, but rather relied on various empirical effective SO operators. For impurity centers in crystals, for example, atom-like SO operators... 

The symmetry operations which commute with the non-relativistic (electrostatic) Hamiltonian Hes of a given system do not necessarily commute with the Breit-Pauli operator Hso. It is therefore appropriate to analyse the group of symmetry operators of Hso for each particular point-group symmetry of the electrostatic Hamiltonian. 

It is seen that the Breit-Pauli operator has the structure of (2) for each atomic center, but depends explicitly on the distances of the unpaired electron from the atomic centers, defined in (6). While the magnetic interaction energy is and thus of shorter range than the electrostatic interaction, it can nevertheless result in a non-negligible dependence of the SO operator on the nuclear coordinates. This effect is neglected when the empirical SO operators (2) or (3) are employed. 

It is useful for the symmetry analysis to write the Breit-Pauli operator (7) in determinantal form... 

When electronic states with more than one unpaired electron (triplet states, quartet states, etc.) are considered, the two-electron part of the Breit-Pauli operator becomes relevant. For a many-electron system with Dj,h symmetry, the complete Breit-Pauli operator reads... 

The analysis of the relativistic Renner coupling has been extended to 11 states, including the two-electron part of the Breit-Pauli operator, thus generalizing previous result of Hougen [45,46]. Other extensions of the theory are S — 11 coupling in the doublet manifold [47] and SO coupling in a half-filled n shell, as found, for example, in carbenes [48]. 

On the level of individual atoms the storage occurs within the three-state system consisting of 6), c+) and c ). We may safely neglect decoherence processes involving the excitation of other states. Then decoherence caused by individual and independent reservoir interactions can be described by the action of the two-level Pauli operators... 

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