Results from Many-Particle Simulations

In this section, we report results from VDM Monte Carlo simulation with 32 pairs of protons and electrons in the temperature and density range of $5\,000\,$K $\,\leq T \leq 250\,000\,$K and $1.75 \leq r_s \leq 4.0$. Particle configurations are generated by sequencing over all particles, giving the particle a uniform displacement, computing the new density matrix from Eqs. 3.19 and 3.13, and accepting or rejecting the new configuration by the Metropolis algorithm. This is completely analogous to the usual Monte Carlo ground state variational calculations except for the additional work of determining the variational parameters based on the proposed configuration.

Although the Gaussian ansatz VDM will be seen to provide a reasonable model for hydrogen over the full density and temperature regime, the main purpose in presenting these results is to serve as a base for documenting future improvements from better VDMs and the application to PIMC.

Figure 3.6: Proton-proton pair correlation function from VDM (solid line) and PIMC (dashed lines at $r_s$=1.75, 2.0, and 4.0 for $T \leq 125\,000\,$K).

The proton-proton pair correlation functions are shown in Fig. 3.6. For temperatures below $20\,000\,$K, a peak emerges near $1.4$ that demonstrates clearly the formation of molecules. The comparison with PIMC simulations (Militzer and Ceperley, 2000; Magro et al., 1996) at low density shows that the peak positions agree well but PIMC predicts a significantly bigger height indicating a larger number of molecules. This could be explained by the missing correlations in the VDM ansatz.

At a density of $r_s=2.0$, proton-proton pair correlation functions from PIMC and VDM are almost identical. If the peak is sufficiently separated from the remaining curve, the area under the peak multiplied by the density gives an estimate for the molecular fraction. By comparing the estimate for different densities at $5\,000$K, one finds that the molecular fraction is diminished when the density is lowered below that corresponding to $r_s=2.0$. This effect is well-known and is a result of the increased entropy of dissociated molecules, which leads to complete dissociation and ionization in the low density limit at non-zero temperatures.

Considerable differences between the proton-proton pair correlation functions are found at $r_s=1.75$ below $T=20\,000\,K$ where VDM shows a fair number of molecules while PIMC predicts a metallic fluid where all bonds are broken as a result of pressure dissociation (Militzer et al., 1999; Magro et al., 1996). This effect has to be verified by PIMC simulations with VDM nodes because free particle nodes could enhance the transition to a metallic state. Proton-proton pair correlation functions from additional VDM simulations for $r_s=1.5, 1.25, {\rm and} 1.0$ are shown in figure 3.7. The VDM method exhibits a smooth transition from a molecular to a atomic structure, in which the molecular binding is gradually reduced with increasing density.

The position of the peak of the proton-proton pair correlation functions shifts from $1.45$ at the lowest density, corresponding to $r_s=4.0$, to $1.3$ at $r_s=1.75$. The same trend has been found in the PIMC simulations (Magro et al., 1996) but the opposite was reported in (Galli et al., 2000; Rescigno, 1999).

Figure 3.7: Proton-proton pair correlation function from VDM (solid line) and PIMC (dashed lines at $r_s$=1.75 for $T \leq 125\,000\,$K).

Figure 3.8: Proton-electron pair correlation functions from VDM (solid line) and PIMC (dashed lines at $r_s$=1.75, 2.0, and 4.0 for $T \leq 125\,000\,$K).

In the proton-electron pair correlation functions shown in Fig. 3.8, one finds a strong attraction present even at high temperatures such as $250\,000\,$K. At low temperatures, the electrons are bound in atoms and molecules. This pair correlation function does not show a clear distinction between the two cases. From studying the height of the peak at the origin multiplied by the density, one can make comparisons of the number of bound electrons at low temperature. Similar to the molecular fraction, one finds a reduction of bound electrons with decreasing density below that corresponding to $r_s=2.0$. The comparison with PIMC shows that VDM underestimates the height of the peak. This is probably a result of the Gaussian ansatz, which does not satisfy the cusp condition at the proton.

Figure 3.9: Electron-electron pair correlation function for electrons with parallel spin from VDM (solid line) and PIMC (dashed lines at $r_s$=1.75, 2.0, and 4.0 for $T \leq 125\,000\,$K).

Figure 3.10: Electron-electron pair correlation function for electrons with anti-parallel spin from VDM (solid line) and PIMC (dashed lines at $r_s$=1.75, 2.0, and 4.0 for $T \leq 125\,000\,$K). Note the change in scale in the last row.

Fig. 3.9 shows the effect of the Pauli exclusion principle leading to a strong repulsion for electrons in the same spin state. This effect is not present in the interaction of electrons with anti-parallel spin displayed in Fig. 3.10. There one observes the effect of the Coulomb repulsion at high temperature. At low temperature, one finds a peak at the origin as a result of the formation of molecules, in which two electrons of opposite spin are localized along the bond. The differences from the PIMC graphs can be interpreted as a consequence of the different molecular fractions observed in Fig. 3.6.

Figure 3.11: Average squared width of the Gaussian single particle density matrices as a function of temperature for different densities

The average squared width $w$ of the Gaussian is shown in Fig. 3.11 as a function temperature and density. At high temperature and low density, one finds only small deviations from the free particle limit. These become more significant with increasing density and decreasing temperature. At low temperature, the attraction to the protons dominates, which leads to a decreasing average width. Finally bound states form and the width approaches a finite limit. At low densities, this is close to the ground state squared width of the isolated molecule, $3.138$. It should be noted that in the limit of very high density, one expects the Gaussians orbitals to be almost as delocalized as the free particle solution because the Coulomb interaction is then a correction to the dominating kinetic terms. This limit does not seem to be represented correctly in this VDM ansatz. The current VDM orbital are too localized in the limit of high density. We interpret this as an effect of the insufficiently accurate treatment of the exchange terms described in section 3.7. In particular, one would need to include corrections to the norm matrix, which where left out because of the drastic increase in the numerical requirements.

Figure 3.12: Internal energy per atom versus temperature from the VDM using the thermodynamic (TE, Eq. 3.50) and direct estimator (DE, Eq. 3.49) compared with PIMC results.

In Fig. 3.12, we compare the internal energy from the thermodynamic estimator in Eq. 3.50 and the direct estimator in Eq. 3.49. Both agree fairly well at low density. Differences build up with increasing density and decreasing temperature. Comparing with PIMC simulations, one finds that the VDM energies are generally too high. The magnitude of this discrepancy shows the same density and temperature dependence as the difference between the two VDM estimators. The difference from the PIMC results could be explained by the missing correlation effects in the VDM method.

At high temperature, the thermodynamic estimator always gives lower energies than the direct estimator. Below $T=25\,000\,$K, the ordering is reversed. This is consistent with the results from the isolated atom and molecule. The consequence is that the direct estimator is actually closer to the value expected from PIMC simulations. However, it should be noted that this estimator is not thermodynamically consistent (see section 3.5.2).

Figure 3.13: Pressure versus temperature in high and low temperature range. VDM pressure is calculated from virial relation using both the direct (DE, Eq. 3.49) and thermodynamic (TE, Eq. 3.51 and 3.52) estimators for kinetic and potential energy.

In Fig. 3.13, we compare pressure as a function of temperature and density from the two VDM estimators with PIMC results. At low density, the agreement is remarkably good. With increasing density and decreasing temperature, the difference grows. For densities over $r_s=2.0$ below $10\,000\,$K, one finds a significant drop in the direct estimator for the pressure. We interpret this effect as a result of the thermodynamic inconsistency.

Figure 3.14: Comparison of experimental Hugoniot functions with VDM and PIMC results.

Fig. 3.14, compares the Hugoniot from laser shock wave experiments (Da Silva, 1997; Collins et al., 1998) with VDM and PIMC results. VDM direct estimator (DE, full diamonds, Eq. 3.49) and VDM thermodynamic estimator (TE, full circles, Eq. 3.50-3.52)). The long dashed line indicates the theoretical high pressure limit $\rho=4 \rho_0$ of the fully dissociated non-interacting plasma. In the experiments, a shock wave propagates through a sample of precompressed liquid deuterium characterized by its initial state, ($E_0$,  ${V^{\!\!\!\!\!\!\:^\diamond}}_0$, $p_0$). Assuming an ideal shock front, the variables of the shocked material ($E$,  ${V^{\!\!\!\!\!\!\:^\diamond}}$, $p$) satisfy the Hugoniot relation (Zeldovich and Raizer, 1966) (see section for details),

$$H = E-E_0+\frac{1}{2}({V^{\!\!\!\!\!\!\:^\diamond}}-{V^{\!\!\!\!\!\!\:^\diamond}}_0)(p+p_0)=0 \quad.$$ (191)

The initial conditions in the experiment were $T=19.6\,\rm {K}$ and $\rho=0.171\,\rm {g/cm^3}$. We set $p_0 = 0$ because $p_0 \ll p$. We show two VDM curves based on the thermodynamic and direct estimators. For $E_0$, we use the corresponding value of the ground state of the isolated hydrogen molecule, $E_0^{TE}=-0.955$ and $E_0^{DE}=-1.124$.

We expect the difference of the two estimators to give a rough estimate of the accuracy of the VDM approach. At high temperature, the difference is relatively small and agreement with PIMC simulations is reasonable. Both VDM estimators indicate that there is maximal compressibility around 1.5 Mbar. Furthermore, significant deviations are found from the experiments except for may be the lowest pressure point of $0.25\rm {Mbar}$. However, in this regime of high density and relatively low temperature a more careful study seems unavoidable. In section , we give a more detailed discussion on the Hugoniot that include predictions from other method and results from PIMC simulations using the VDM nodal surface to restrict the paths.