Coulomb interaction nonrelativistic Hamiltonian

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... 

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). 

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 ... 

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. 

The two-particle terms used in such calculations are either the nonrelativistic electrostatic Coulomb (C) interaction (yielding the Dirac-Coulomb DC) Hamiltonian correct to 0(a ))... 

The Hamiltonian of the two-electron atom already features all pair-interaction operators that are required to describe a system with an arbitrary number of electrons and nuclei. Hence, the step from one to two electrons is much larger than from two to an arbitrary number of electrons. For the latter we are well advised to benefit from the development of nonrelativistic quantum chemistry, where the many-electron Hamiltonian is exactly known, i.e., where it is simply the sum of all kinetic energy operators according to Eq. (4.48) plus all electrostatic Coulombic pair interaction operators. 

The traditional specification of a molecule in classical chemistry is in terms of atoms joined by bonds, and this accounts for the central fact of chemistry that the generic molecular formula is associated with the occurrence of isomers. Such an approach does not provide a useful basis for a physical theory since we do not know the general laws of interaction between atoms. Instead a more abstract description in terms of the particle constituents of a molecule, electrons and nuclei, is employed. We shall confine the discussion to the nonrelativistic level of theory with this proviso the interactions between electrons and nuclei are assumed to be fully specified by Coulomb s law, and this makes possible the explicit formulation of a molecular Hamiltonian. This so-called Coulomb Hamiltonian will be given explicitly (O Eq. 2.1) in the next section it forms the starting point of the chapter. 

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