Analogy to Zero Temperature Methods

Considerable effort has been devoted to systems where finite temperature ions (treated either classically or quantum mechanically by path integral methods) are coupled to degenerate electrons on the Born-Oppenheimer surface. In contrast, the theory for similar systems with non-degenerate electrons ($T$ a significant fraction of $T_{F}$) is relatively underdeveloped except at the extreme high $T$ limit where Thomas-Fermi and similar theories apply. In this chapter, we present a variational approach for systems with non-degenerate electrons analogous to the methods used for ground state many body computations.

Although an oversimplification, we may usefully view the ground state computations as consisting of three levels of increasing accuracy (Hammond et al., 1994).

1. At the first level, the ground state wave function consists of determinants, for both spin species, of single particle orbitals often taken from local density functional theory
   

   $$\psi_{GS}({\bf R})=\left\vert \begin{array}{ccc} \phi_{1}({\... ...&\ldots&\phi_{N}({\bf r}_{N})\\ \end{array}\right\vert \quad.$$ (119)

   
   
   The majority of ground state condensed matter calculations stop at this level.

2. If desired, additional correlations may be included by multiplying the above wave function by a Jastrow factor, $\prod_{i,j}f(r_{ij})$, where $f$ will also depend on the type of pair (electron-electron, electron-ion). Computing expectations exactly (within statistical uncertainty), with this type of wave function now requires Monte Carlo methods.

3. Finally diffusion Monte Carlo (Foulkes et al., 1999; Ceperley and Mitas, 1996) methods using the nodes of this wave function to avoid the Fermion problem may be used to calculate the exact correlations consistent with the nodal structure.

The finite temperature theory proceeds similarly. Rather than the ground state wave function a thermal density matrix Eq. 2.5 is needed to compute the thermal averages of operators as shown in Eq. 2.7.

1. At the first level, this many body density matrix may be approximated by determinants of one-body density matrices, for both spin types, as well as the ions
   

   $$\rho({\bf R},{\bf R'};\beta)=\left\vert \begin{array}{ccc} \... ...({\bf r}_{N},{\bf r}'_{N};\beta) \end{array}\right\vert \quad.$$ (120)

   
   

2. The Jastrow factor can be extended to finite temperatures and the above density matrix multiplied by $\prod_{i,j}f(r_{ij},r'_{ij};\beta)$. In particular, the high temperature density matrix used in path integral computations has this form.

3. Finally, the nodal structure from this variational density matrix (VDM) will be used in restricted path integral Monte Carlo simulations as described in chapter . This method has been extensively applied using the free particle nodes (Magro et al., 1996; Pierleoni et al., 1994). One aim of the approach is to provide more realistic nodal structures as input to PIMC.

This chapter considers the first level in this approach. The next section is devoted to a general variational principle which will be used to determine the many body density matrix. The principle is then applied to the problem of a single particle in an external potential and compared to exact results for the hydrogen atom density matrix. After a discussion of some general properties, many body applications are considered starting with a hydrogen molecule and then proceeding to warm, dense hydrogen. It is shown that the method and the ansatz considered can describe dense hydrogen in the molecular, the dissociated and the plasma regime. Structural and thermodynamic properties for this system over a range of temperatures (T$=5\,000$ to $250\,000\,K$) and densities (electron sphere radius $r_s=1.75$ to $4.0$) are presented.