Operator Breit-Pauli

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... 

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. 

Breit-Pauli Operators Using General Cartesian Gaussian Functions. II. Two-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]. 

Since the factors ( , + mc2) l grow asymptotically (for p, -> oo, i.e. r, — 0) like I/ p, , all contributions of momentum operators in the numerator (leading to the 1/r3 divergence in the case of the Breit-Pauli operator) are cancelled asymptotically, and only a Coulomb singularity remains. Recently, Brummelhuis et al (2002) have formally proved that the operator is variationally stable. 

The Breit-Pauli operator may be recovered by expanding in powers of c-2... 

Fedorov and Gordon calculated the T/Sq spin-orbit coupling for X up to Pb and compared the values based on the full Breit-Pauli operator with their partial two-electron contribution (P2E) method. The approximation works for heavier elements (where the importance of the two-electron term ultimately vanishes), but for methylene it causes an error of 15%. Their best values for the spin-orbit coupling constant for XHg (X = C, Si, Ge) amount to 12.3, 57.1, and... 

In the case of a relativistic system, as a first (and useful) approximation, the zero-order spectrum can be taken as the nonrelativistic one, with Hq defined explicifly as fhe Coulomb Hamiltonian. Then, the perturbation V is also written explicitly as the relativistic Breit-Pauli operators, and it is this perturbation that turns the initially discrete state into a resonance. For example, this type of advanced calculation, with multichannel coupling included, has been shown to explain quantitatively the positions and lifetimes of the relativistic levels of mefastable states in negative ions [90]. However, if the more accurate four-component relativistic Dirac treatment for each electron is invoked for cases of high effective nuclear charge, then the stability against autoionization implies not only the exclusion of components representing decay to... 

Applying the Foldy-Wouthuysen transformation to the entire Hamiltonian including the Breit interaction (Eq. [75]), the Breit—Pauli operator is obtained ... 

We now turn to the Gaunt interaction, and use the terms from the modified Dirac representation in (15.54) to derive the Breit-Pauli operators. These terms need no renormalization, because they are all of order 1/c. The three classes of operators defined in (15.54) are considered in turn. 

It is easy to check that this operator goes over to the Breit-Pauli operator in the limit p 0, where Q -> 1/2 ... 

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