Breit-Pauli Hamiltonian Correction Term

The two parts of this formula are derived from the same QED Feynman diagram for interaction of two electrons in the Coulomb gauge. The first term is the Coulomb potential and the second part, the Breit interaction, represents the mutual energy of the electron currents on the assumption that the virtual photon responsible for the interaction has a wavelength long compared with system dimensions. The DCB hamiltonian reduces to the complete standard Breit-Pauli Hamiltonian [9, 21.1], including all the relativistic and spin-dependent correction terms, when the electrons move nonrelativistically. 

Table 3.1. Contributions of various physical effects (non-relativistic, Bieit, QED, and beyond QED, distinct physical contributions shown in bold) to the ionization energy and the dipole polarizability a of the helium atom, as well as comparison with the experimental values (all quantities are expressed in atomic units i.e.. e = 1. fi = 1, mo = 1- where iiiq denotes the rest mass of the electron). The first column gives the symbol of the term in the Breit-Pauli Hamiltonian [Eq. (3.72)] as well as of the QED corrections given order by order (first corresponding to the electron-positron vacuum polarization (QED), then, beyond quantum electrodynamics, to other particle-antiparticle pairs (non-QED) li,7T,. ..) split into several separate effects. The second column contains a short description of the effect. The estimated error (third and fourth columns) is given in parentheses in the units of the last figure reported. 

Only recently has the work of Bauschlicher, Walch and Siegbahn showed the need to include d correlation, whereas the work of Werner and Martin and of Scharf, Brode and Ahlrichs stressed the importance of cluster corrections and relativistic corrections. The results of Werner et al. were obtained using CEPA-1 to account for cluster contributions, while Scharf et al. used the CPF approach. Both groups accounted for relativistic corrections by employing first-order perturbation theory, i.e. by evaluating the Cowan-Griffin operator which consists of the mass-velocity and the one-electron Darwin term of the Breit-Pauli Hamiltonian. 

This chapter is devoted to the development of perturbation expansions in powers of 1 /c from the Dirac equation. In the previous chapter, the Pauli Hamiltonian was developed using the Foldy-Wouthuysen transformation. While this is an elegant method, it is probably simpler to make the derivation from the elimination of the small component with expansion of the denominator, and it is this approach that we use here. Another convenient approach is to make use of the modified Dirac equation in the limit of equality of the large and pseudo-large components. This approach enables us to draw on results from the modified Dirac approach in developing the two-electron terms of the Breit-Pauli Hamiltonian. We then demonstrate how the use of perturbation theory for relativistic corrections requires that multiple perturbation theory be employed for correlation effects and for properties. The last sections of this chapter are... 

The reduction of the remaining contributions from the gauge term is extremely tedious, and in fact gives a zero contribution to the operator to 0(c ). Thus, there is no spin-dependent contribution from the gauge term for the relativistic correction to the electron-electron interaction to the Breit-Pauli Hamiltonian. 

A number of static perturbations arise from internal interactions or fields, which are neglected in the nonrelativistic Born-Oppenheimer electronic Hamiltonian. The relativistic correction terms of the Breit-Pauli Hamiltonian are considered as perturbations in nonrelativistic quantum chemistry, including Darwin corrections, the mass-velocity correction, and spin-orbit and spin-spin interactions. Some properties, such as nuclear magnetic resonance shielding tensors and shielding polarizabilities, are computed from perturbation operators that involve both internal and external fields. 

For the shielding constants of the heavy elements themselves, relativistic effects arising from the other terms in the Breit-Pauli Hamiltonian (see O section The Molecular Breit— Pauli Hamiltonian ) are in general found to be more important than the spin-orbit corrections (Manninen et al. 2003). In general, relativistic effects on the heavy-atom shielding are due to a large number of contributions (Fukui et al. 1996 Ruiz de Aziia et al. 2003) in contrast to... 

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. 

In principle, we should have an infinite series in because there are relativistic corrections to the Hamiltonian at all orders in n, but since we are developing a method appropriate for the Breit—Pauli approxiination, we need only consider the lowest-order terms. Technically, we ought to write Ho as Hoo to indicate that it is of zeroth order in both perturbations, but we will omit the second zero. 

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