Nonrelativistic Coulombic interaction

Neglecting Vcpp in equation 6.1 for a moment, it is usually assumed that all relativistic effects are described by a suitable parametrization of the ECPs Vcv i-e- it sufficient to apply the nonrelativistic kinetic energy operator as well as the nonrelativistic Coulomb interaction between the valence electrons [19]. Besides the relativistic contributions, the ECP accounts for all interactions of the valence electrons with the nucleus and the (removed) core electrons, and it is given by [19]... 

The first (and still the foremost) quantum theory of stopping, attributed to Bethe [19,20], considers the observables energy and momentum transfers as fundamental in the interaction of fast charged particles with atomic electrons. Taking the simplest case of a heavy, fast, yet nonrelativistic incident projectile, the excitation cross-section is developed in the first Born approximation that is, the incident particle is represented as a plane wave and the scattered particle as a slightly perturbed wave. Representing the Coulombic interaction as a Fourier integral over momentum transfer, Bethe derives the differential Born cross-section for excitation to the nth quantum state of the atom as follows. 

The solution of the problem of the proper mass dependence of the relativistic corrections of order (Za) may be found in the effective Hamiltonian framework. In the center of mass system the nonrelativistic Hamiltonian for a system of two particles with Coulomb interaction has the form... 

In Eq. (3.47)

field operator of two-component structure, a are the Pauli matrices and the electron-electron interaction reduces to the Coulomb interaction, denoted by H e- As usual, the gauge term proportional to... 

Since the exact relativistic many-electron Hamiltonian is not known, the electron-electron interaction operators g(i, j) are taken to be of Coulomb type, i.e. 1/r,- . As a first relativistic correction to these nonrelativistic electron-electron interaction operators, the Breit correction, Equations (2.2) or (2.3), is used. For historical reasons, the first term in Equation (2.2) is called the Gaunt or magnetic part of the full Breit interaction. Since it is not more complicated than l/ri2, it is from an algorithmic point of view equivalent to the Coulomb interaction, therefore it has frequently been included in the calculations. The second term, the so-called retardation term, appears to be rather complicated and it has been considered less frequently. In the case of few-electron systems further quantum electrodynamical corrections, like self-energy and vacuum polarization, have also been considered and are reviewed in another part of this book (see Chapter 1). 

Recalling that the kinetic and interaction energies of a nonrelativistic Coulomb system are described by universal operators, we can also write Ev as... 

In the nonrelativistic limit, only the (LL LL) class of multi-centre integrals is needed this class is still the most important in relativistic calculations. In terms of the conventional expansion in powers of a, a DHF calculation at this level includes all the one-centre relativistic effects, screening by the Coulomb interaction, and some of the spin-other-orbit effects. The (LL 55) and (55 55)... 

Our decision in favor of combining nonrelativistic quantum mechanics with the nrl of electrodynamics becomes very important, when we consider the interaction between moving electrons (section 7). In the nrl there is only a nonretarded (instantaneous) Coulomb interaction, while both the magnetic interaction and the retardation of the Coulomb interaction are relativistic corrections and are therefore neglected. One needs to consider them only if one also includes relativistic corrections to the kinematics. 

One of the major fundamental difference between nonrelativistic and relativistic many-electron problems is that while in the former case the Hamiltonian is explicitly known from the very beginning, the many-electron relativistic Hamiltonian has only an implicit form given by electrodynamics [13,37]. The simplest relativistic model Hamiltonian is considered to be given by a sum of relativistic (Dirac) one-electron Hamiltonians ho and the usual Coulomb interaction term ... 

As for the all-electron case, nonrelativistic, quasirelativistic or relativistic formulations are also possible in pseudopotential calculations for the one- and two-electron valence-only operators and gy, respectively. The mutual Coulomb interaction between the nuclei has been replaced by the core-core interaction potential Fee. 

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. 

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