Pauli operators factorization

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. 

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. 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

When the two systems A and B are brought from infinity to their equilibrium postions, the wavefunctions FA and TB of the subsystems will be overlapping. The Pauli principle is obeyed by explicitly antisymmetrizing (operator A) and renormalizing (factor N) the product wavefunction ... 

In the Breit-Pauli Hamiltonian, the one-electron Pauli Hamiltonians and the two-dimensional Breit operator can be clearly identified. The factor eh/ 2m.eC) in these operators represents the Bohr magneton ji-q, which we have already encountered in section 5.4.3, and the gradient of the nuclear potential is the electric field strength... 

To compare the operator with the terms from the Pauli Hamiltonian, we need to expand the various kinematic factors in powers of 1/c up to second order, as follows ... 

The variational problems with the Pauli Hamiltonian stem not only from the mass-velocity operator, which is negative, but also from the spin-orbit operator, which behaves as 1 /r and can be negative. In the Hamiltonian as p becomes larger the kinematic factors A and V become progressively smaller, with the result that the potential energy terms are reduced as the momenrnm increases. In the large momenrnm limit, when the electron is close to the nucleus, Ep cp, /2, and IZ oe -p/p. 

This is the two-electron spin-orbit interaction operator, and reduces to the Breit-Pauli form when the limit p 0 is applied in the kinematic factors. 

The reader can easily verify that in the limit p -> 0 in the kinematic factors this yields the expression obtained in the Breit-Pauli Hamiltonian. Thus, for the Douglas-Kroll Hamiltonian, calculating the primitive integrals over basis funetions for these operators will involve the same work as for the Breit-Pauli Hamiltonian, but at the same time the kinematic factors will have to be accounted for. 

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