Pair Correlation Functions

In this section, we compare the pair correlation functions for different densities and temperatures from PIMC simulations using free particle and VDM nodes. The pair correlation function is defined as (Allen and Tildesley, 1987),

$$g(r) = \frac{{V^{\!\!\!\!\!\!\:^\diamond}}}{N^2} \left<\righ... ...um_i \sum_{j\ne i} \delta({\bf r}-{\bf r}_{ij}) \left>\right..$$ (202)

Figure: Proton-proton pair correlation functions multiplied by the density $n$ from PIMC simulations of hydrogen using free particle nodes. The columns correspond to different $r_s$ values and the rows to different temperatures $T$.

   

Figure: Proton-proton pair correlation functions multiplied by the density $n$ as in Fig.  but for deuterium at higher densities.

   

It goes to 1 in the limit of large $r$ in an infinite system and to $(N-1)/N$ in a system of $N$ particles. The proton-proton pair correlation functions from PIMC simulations with free particle nodes are shown in Figs.  and . For $T \stackrel{\scriptstyle<}{\scriptscriptstyle\sim}\:20\,000\,\rm K$ a peak at the bond length of $1.4$ emerge, which clearly demonstrates the formation of molecules. In the low density region, we find it useful to multiply the pair correlation function by an extra density factor $N/{V^{\!\!\!\!\!\!\:^\diamond}}$ so that the area under the peak is proportional to the molecular fraction. For $r_s \stackrel{\scriptstyle>}{\scriptscriptstyle\sim}\:2$, the peak height gets smaller with decreasing density as a result of entropy dissociation. Thermal dissociation reduces the number of molecules with increasing temperature. For $r_s\stackrel{\scriptstyle<}{\scriptscriptstyle\sim}\:2$, pressure dissociation diminishes the peak with increasing density. For PIMC with free particle nodes, this process occurs within a small density interval, in which the system undergoes a transition from a molecular to a metallic regime (see Fig. ).

Figure: Proton-proton pair correlation function from PIMC simulations of deuterium using VDM nodes. The columns correspond to different $r_s$ values and the rows to different temperatures $T$.

Figure: Proton-proton pair correlation function as in Fig.  but for higher densities.

Fig.  and show the proton-proton pair correlation functions from PIMC simulations with VDM nodes using the standard normalization from Eq. . We also included simulations at higher densities corresponding to $r_s=1$. The molecular peak disappears gradually with increasing density, indicating that pressure dissociation leads to a smooth transition to a metallic regime.

Figure: Proton-electron radial correlation function multiplied by the density, $n\,r^2\,[g_{pe}(r)-1]$ from PIMC with free particle nodes. The columns correspond to different $r_s$ values and the rows to different temperatures $T$.

Figure: Proton-electron pair correlation function from PIMC simulations of deuterium using VDM nodes. The columns correspond to different $r_s$ values and the rows to various temperatures.

Figure: Proton-electron pair correlation function as in Fig.  but for higher densities.

Figure: Electron-electron pair correlation function from PIMC simulations of deuterium using VDM nodes. Solid lines correspond to pairs with parallel spins and dashed lines to anti-parallel spins. The columns belong to different $r_s$ values and the rows to various temperatures $T$.

Figure: Electron-electron pair correlation function as in Fig.  but for higher densities.

The proton-electron radial distribution function $r^2[g_{pe}(r)-1]$ from different simulations using free particle nodes is shown in Fig. . For non-interacting particles, this function would be identical to zero. The first peak shows an increased probability of finding a electron near a proton due to the attractive forces. At low temperature, the size of peak can also be interpreted as the occupation of bound electronic states. Also the unbound, scattering states lead to a smaller but non-zero contribution to this peak. However, one can still deduce that the degree of ionization, inversely related to the peak size, increases with temperature.

Alternatively, one can study the proton-electron pair correlation function as shown in Fig.  and from simulations using VDM nodes, where the same overall behavior is represented in a different form. Fig.  and show the electron-electron pair-correlation functions. The peak for pairs with anti-parallel electron spins indicates the formation of molecules, in which two electrons get very close, in cases where they realize the molecular binding. For same spin electrons, one always finds a strong repulsion due to the Pauli exclusion principle.