Nonrelativistic many-particle Hamiltonian

The nonrelativistic many-particle Hamiltonian of a collection of electrons and nuclei in the absence of additional external electromagnetic fields can be written (in Gaussian units) as. 

DHF calculations on molecules using finite basis sets require considerably more computational effort than the corresponding nonrelativistic calculations and cause several problems due to the presence of the Dirac one-particle operator. It is therefore desirable to find (approximate) relativistic Hamiltonians for many-electron systems which are not plagued by unboundedness from below and therefore do not cause problems like the variational collapse at the self-consistent field level or the Brown-Ravenhall disease at the configuration interaction level. It is also desirable to find forms in which the quality of a matrix representation of the kinetic energy is more stable than for the Dirac Hamiltonian, i.e., forms which are not affected by the finite basis set disease . 

A feature of the nonrelativistic external-field-free many-particle Hamiltonian is that its principles of construction are simple. First of all, kinetic energy operators for each particle in the system, i.e., for each electron and each atomic nucleus, are summed. Since the elementary particles can be well described as point-like particles, it is not necessary to consider electrostatic multipole... 

Having left the framework of field theory outlined in chapter 7 and thus having avoided any need for subsequent renormalization procedures, the mass and charge of the electron are now the physically observable quantities, and therefore do not bear a tilde on top. In contrast to quantum electrodynamics, the radiation field is no longer a dynamical degree of freedom in a many-electron theory which closely follows nonrelativistic quantum mechanics. Vector potentials may only be incorporated as external perturbations in the many-electron Hamiltonian of Eq. (8.62). From the QED Eqs. (7.13), (7.19), and (7.20), the Hamiltonian of a system of N electrons and M nuclei is thus described by the many-particle Hamiltonian of Eq. (8.66). In addition, we refer to a common absolute time frame, although this will not matter in the following as we consider only the stationary case. 

For the two-electron integrals, we want to divide the integrals into symmetry classes, as for the nonrelativistic integrals. We also want to divide the integrals into classes by time reversal symmetry, as we did for the one-electron integrals. Because of the structure of the Kramers-restricted Hamiltonian in terms of the one- and two-particle Kramers replacement operators, we hope to obtain a reduction in the expression for the Hamiltonian from time-reversal symmetry. The classification by time-reversal properties is also important for the construction of the many-electron Hamiltonian matrix, whose symmetry properties we consider in the next section. 

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