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##

Many-Particle Systems

The main purpose of the natural orbital analysis is to study many
particle systems. There, the situation is significantly more
complicated because one has several protons and it turns out to be
difficult to generalize the decomposition in Eq. 5.15 to the
many-proton case because one has no a priori criterion to which
proton a particular open path configuration belongs. The problem
exists for classical as well as for quantum mechanical
protons. However, there is a method that is conceptually correct but
not feasible for real applications. For a fixed configuration of
protons, one would store the electron two particle density matrix
as a
matrix where
is the number of grid points in a spatial direction. After a
sufficiently long MC simulation, one finds a converged results for
each matrix element and then can diagonalize the matrix. For low
temperature and low density the matrix is block diagonal and the
eigenvectors correspond to different localized electronic states, each
corresponding to a particular proton. This method shows how one would
in principle generate the eigenstates for a many particle
system. However, it is not practical because it requires the storage
of this enormous matrix and extremely good statistics to fill all
relevant matrix elements. The following sections discuss how to
approximate this result with computationally feasible methods.

**Subsections**

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Burkhard Militzer
2003-01-15