The Many-Particle Hamiltonian and Degree of Freedom Reduction

2 The Many-Particle Hamiltonian and Degree of Freedom Reduction 

From a quantum mechanical perspective, the starting point for any analysis of the total energy is the relevant Hamiltonian for the system of interest. In the present setting, it is cohesion in solids that is our concern and hence it is the Hamiltonian characterizing the motions and interactions of all of the nuclei and electrons in the system that must be considered. On qualitative grounds, our intuition coaches us to expect not only the kinetic energy terms for both the electrons and nuclei, but also their mutual interactions via the Coulomb potential. In particular, the Hamiltonian may be written as 

The terms in this Hamiltonian are successively, the kinetic energy associated with the nuclei, the kinetic energy associated with the electrons, the Coulomb interaction between the electrons, the electron-nucleus Coulomb interaction and finally, the nuclear-nuclear Coulomb interaction. Note that since we have assumed only a single mass M, our attention is momentarily restricted to the case of a one-component system. 

As will be shown in coming sections, wide classes of total energy descriptions fall under this general heading including traditional pair potential descriptions, pair functionals, angular force schemes and cluster functionals. 

The second general class of schemes, which almost always imply a higher level of faithfulness to the full Hamiltonian, are those in which reference to the electrons is maintained explicitly. In this case, the mapping between the exact and approximate descriptions is of the form. 

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