Breit Pauli form

The last term on the last r.h.s. of (473) vanishes if (460) is satisfied exactly. In this case (473) can be rearranged to the Breit-Pauli form. 

The transformation of AE2 to Breit-Pauli form has been known for a long time [81, 82]. (There is a small difference in the operators published by Bethe-Salpeter [81] and Itoh [82], but this does not matter, since it has no effect on the expectation value with respect to a nonrelativistic reference function). A detailed presentation of this reformulation, which is unexpectedly tedious, has been given recently [83]. The result for the Gaunt interaction is ... 

One seemingly sensible approach to the relativistic electronic structure theory is to employ perturbation theory. This has the apparent advantage of representing supposedly small relativistic effects as corrections to a familiar non-relativistic problem. In Appendix 4 of Methods of molecular quantum mechanics, we find the terms which arise in the reduction of the Dirac-Coulomb-Breit operator to Breit-Pauli form by use of the Foldy-Wouthuysen transformation, broken into electronic, nuclear, and electron-nuclear effects. FVom a purely aesthetic point of view, this approach immediately looks rather unattractive because of the proliferation of terms at the first order of perturbation theory. To make matters worse, many of the terms listed are singular, and it is presumably the variational divergences introduced by these operators which are referred to in [2]. Worse still, higher-order terms in the Foldy-Wouthuysen transformation used in this way yield a mathematically invalid expansion. 

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. 

In Table I, 3D stands for three dimensional. The symbol p2A 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 p2A dependence at small distortions of linearity. With exact form of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-electron counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

The spin operators Sx, and Sg which occur in the Breit-Pauli Hamiltonian form a basis for the Lie algebra of SU(2). The concept of the electron as a spinning particle has arisen through the isomorphism between SU(2) and the angular momentum operators. This analogy is unnecessary and often undesirable. 

The Breit-Pauli spin-orbit Hamiltonian is found in many different forms in the literature. In expressions [101] and [102], we have chosen a form in which the connection to the Coulomb potential and the symmetry in the particle indices is apparent. Mostly s written in a short form where spin-same- and spin-other-orbit parts of the two-electron Hamiltonian have been contracted to a single term, either as... 

The dipolar spin-spin coupling operators are scalar operators of the form //1 . A 2 y.11. The tensorial structure of JCss becomes apparent if we write the Breit-Pauli spin-spin coupling operator as... 

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

The Breit-Pauli SOC Hamiltonian contains a one-electron and two-electron parts. The one-electron part describes an interaction of an electron spin with a potential produced by nuclei. The two-electron part has the SSO contribution and the SOO contribution. The SSO contribution describes an interaction of an electron spin with an orbital momentum of the same electron. The SOO contribution describes an interaction of an electron spin with the orbital momentum of other electrons. However, due to a complicated two-electron part, the evaluation of the Breit-Pauli SOC operator takes considerable time. A mean field approximation was suggested by Hess et al. [102] This approximation allows converting the complicated two-electron Breit-Pauli Hamiltonian to an effective one-electron spin-orbit mean-field form... 

In molecular property calculations the same mutual interplay of electron correlation, relativity and perturbation operators (e.g. external fields) occurs. For light until medium atoms relativistic contributions were often accounted for by perturbation theory facilitating quasirela-tivistic approximations to the Dirac-Hamiltonian [114-117]. It is well-known that operators like the Breit-Pauli Hamiltonian are plagued by essential singularities and therefore are not to be used in variational procedures. It can therefore be expected that for heavier elements per-turbational inclusion of relativity will eventually become inadequate and that one has to start from a scheme where relativitiy is included from the beginning. Nevertheless very efficient approximations to the Dirac equation in two-component form exist and will be discussed further below in combination with their relevance for EFG calculations. In order to calculate the different contributions to a first-order property as the EFG, Kello and Sadlej devised a multiple perturbation scheme [118] in which a first-order property is expanded as... 

Hay and Wadt (1985a, b) have published ECPs which are in form identical to the averaged RECPs of Christiansen, Ermler and co-workers. However, there are differences. First, the Hay-Wadt potentials are derived from the Cowan-GriflSn adaptation of the Breit-Pauli Hamiltonian into a variational computation of the atomic wave-function. From these solutions the ECPs are generated. It should be noted that the spin-orbit coupling is not included in the Hay-Wadt ECPs. Consequently, molecular calculations done using these ECPs would not include spin-orbit coupling. 

The explicit form of the property operators is derived in two steps. First, the scalar and vector fields are determined and, second, they are inserted into the Dirac-Breit [951,952] or any other quasi-relativistic operator derived from it such as the external-field-containing DKH operator, the external-field-containing ZORA Hamiltonian or the similar Breit-Pauli Hamiltonian. The theory of NMR parameters has been derived by Ramsey [953-955] from the nonrelativistic perspective and by Pyykko from the relativistic perspective [956,957]. 

An alternative formulation that one can use to obtain the Breit-Pauli Hamiltonian is to start with the Breit equation (Eq. [59]). The equation is first transformed by Fourier transformation to momentum space. The terms involving the positive and negative eigenstates into PauH functions v /+ and / are then written, and only the electronic part of the equations is kept. Then, expanding the energy (in momentum space) in powers of p/mc, and Fourier transforming back to the coordinate space, one finally arrives at a differential equation of the form containing the Breit-Pauli Hamiltonian. 

The two perturbations will have a different form for direct and Breit-Pauli perturbation theory. The wave function also will be different in form two components in Breit-Pauli theory and four components in direct perturbation theory. In addition, the metric must be expanded in the direet scheme,... 

The consequence is that we must treat the spin-orbit and the spin-other-orbit interactions separately we cannot combine them as in the Breit-Pauli Hamiltonian. The reason is that the functions on which the momentum operators operate are derived from the small component, and only in the nonrelativistic limit where the large and small components are related by kinetic balance can we rewrite the spatial part of the spin-other-orbit interaction in the same form as the spin-orbit interaction. The reader... 

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