In this section, we will give a brief overview of different
computational methods used to study hot, dense hydrogen.
The first simulations used quasi-classical methods based on effective
potential (Hansen, 1973). In recent years a variety of simulation
techniques have been developed from first principles, which include
density-functional-theory molecular dynamics
(Kohanoff and Hansen, 1995; Galli *et al.*, 2000; Hohl *et al.*, 1993; Lenosky *et al.*, 2000) and tight-binding molecular dynamics
(Collins *et al.*, 1995; Lenosky *et al.*, 1997a; Kwon *et al.*, 1994; Lenosky *et al.*, 1997b). The density function theory and
the tight-binding approach are used to describe the electron-electron and
the electron-proton interactions, while classical dynamics is used for
the protons. The PIMC equation of state will be compared with results
from these methods. Also wave packet molecular dynamics has been
applied to dense hydrogen (Klakow *et al.*, 1994b; Ebeling and Militzer, 1997; Klakow *et al.*, 1994a; Nagel *et al.*, 1998). It employs a
Hartree-Fock type ansatz for the wave function and time-dependent
variational principle to describe its evolution.

Furthermore there is PIMC, which represent an exact quantum-statistical method to determine equilibrium properties, which relies on an approximation for the nodal surfaces in order to deal with the fermion sign problem. It treats protons and electrons quantum-mechanically. A general description of the path-integral formalism can be found in (Kleinert, 1990; Schulman, 1981; Feynman, 1972).

The path integral technique as a numerical method in the form used in
this work has been developed in series of works including
(Pollock and Ceperley, 1987,1984). It has been applied to the variety of different
bosonic systems including the study of the lambda phase transition in
He (Ceperley, 1995), hard-sphere bosons
(Grüter *et al.*, 1997), the melting transition of molecular hydrogen surfaces
(Wagner and Ceperley, 1996), and conditions of superfluidity of molecular hydrogen
(Gordillo and Ceperley, 1997). Also a number of fermionic systems have been studied: the
crystallization of the one-component plasma Jones and Ceperley (1996), electronic
forces on molecules (Zong and Ceperley, 1998) and the electron hole plasma (Shumway and Ceperley, 1999).

First simulations of dense hydrogen have been by Pierleoni *et al.* (1994) and
Magro *et al.* (1996). Densities corresponding to
have been studied using free particle nodes. In this work, we extend
the investigation into the non-degenerate regime up to . Then we
perform a time step analysis and study finite size
effects. Furthermore, a variational density matrix method is
developed, which is used to replace the free particle nodes. Using
this improved nodal surface we reexamine the PPT predicted by
Magro *et al.* (1996). Furthermore, we extended the restricted PIMC method
to open paths and calculate the momentum distribution as well as
natural orbitals.