The exact density matrix is only known in a few cases. In
practice, applications have approximated the fermionic trial density
matrix
, most commonly by a
Slater determinant of single particle density matrices,

If one applies an operator that antisymmetrizes completely and then projects out states of magnetization one finds only configurations that can again be written as such a product of two determinants but with relabelled particles. From now on, we only consider spin unpolarized systems ( ). Enforcing the nodes means that each determinant stays positive all along the paths because configurations where both determinants flip signs simultaneously have zero measure.

Extensions of this type of nodes are necessary to describe a pairing
mechanism that permits the formation of Cooper pairs in super
conductors and electron-hole pairs in semi-conductors, which then can
Bose condense (Bouchard *et al.*, 1988). Simulations with pairing nodes have been
done at zero temperature by Gilgien (1997) and Zhu *et al.* (1996) and at finite
temperature by Shumway and Ceperley (1999).

The above trial density matrix has been extensively applied using the
free particle (FP) nodes (Ceperley, 1996) including applications to dense
hydrogen (Militzer *et al.*, 1999; Pierleoni *et al.*, 1994; Magro *et al.*, 1996). In this case, the density matrix of a
single FP in a periodically repeated box given by
Eq. 2.11 is used in Eq. 2.83.
It can be shown that for high temperatures, the interacting nodal
surface approaches the FP nodal surface. In addition, in
the limit of low density, exchange effects are negligible, the nodal
constraint has a small effect on the paths and therefore, its precise
shape is not important. The FP nodes also become exact in the limit of
high density when kinetic effects dominate over the interaction
potential. However, for high densities and high degeneracy,
interactions could have a significant effect on the fermionic density
matrix. To gain some quantitative estimate of the possible effect of
the nodal restriction on the thermodynamic properties, it is necessary
to try an alternative. In addition to FP nodes, a variational density
matrix (VDM) is derived in chapter
3 that already includes
interactions and atomic and molecular bound states. The effects on the
thermodynamic properties from using those as nodes will be discussed
in section .