Computational Methods

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 $^4$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 $1.0 \stackrel{\scriptstyle<}{\scriptscriptstyle\sim}\:r_s \stackrel{\scriptstyle<}{\scriptscriptstyle\sim}\:2.2$ have been studied using free particle nodes. In this work, we extend the investigation into the non-degenerate regime up to $r_s=14$. 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.