# Microscopic Pair Potentials and the Physical Properties of the Condensed Phases of Parahydrogen
## 4 Discussion and Conclusions  
The Silvera-Goldman pair potential has been the most common choice for studies of the condensed phases of \(p\)-H\({}_{2}\), and in most studies of small \(p\)-H\({}_{2}\) clusters. In general, spherically symmetric pair potentials are utilized in physical situations in which one may not be necessarily pursuing chemical accuracy, but rather seek a reasonable numerical account of the most experimentally important thermodynamic properties of the condensed phases of molecular hydrogen. In this work we have carried out first principle numerical simulations of liquid and solid \(p\)-H\({}_{2}\), based on the SG potential and on other pair potentials, including some _ab initio_ potentials developed over the past two decades. Our goal was that of carrying out an extensive comparison with experiment, in order to determine whether the SG remains \begin{table}
\begin{tabular}{l l l l} \hline
**Potential** & \(\epsilon_{K}\) **(K)** & **\(E\) (K)** & **Pressure (Bars)** \\ \hline [PERSON]–[PERSON] & \(70.15(9)\) & \(-87.94(4)\) & \(-6.9(1)\) \\ [PERSON] & \(71.37(5)\) & \(-94.03(5)\) & \(-26.6(2)\) \\ [PERSON] et al. & \(78.52(5)\) & \(-91.39(5)\) & \(50.8(2)\) \\ [PERSON]–[PERSON] & \(78.44(7)\) & \(-87.05(7)\) & \(66.5(3)\) \\ Experiment [34] & & \(-90.0\) & \\ Experiment [35] & & \(-93.5\) & \(\sim\)6 \\ \hline \end{tabular}
\end{table} 
Table 2: Simulation results for the kinetic (\(\epsilon_{K}\)) and total (\(E\)) energy per molecule (in K), and for the pressure (in bars) in the crystal (hcp) phase of \(p\)-H\({}_{2}\) at temperature \(T=4\) K and density \(n=0.0261\) Å\({}^{-3}\), computed using different intermolecular pair potentials. Statistical errors (in parentheses) are on the last digit. Also shown are experimentally extrapolated values for the ground state energy [34; 35], as well as the value of the pressure at \(T=4\) K, obtained by interpolating data in Table III of Ref. [35]. Figure 3: Static structure factor of hcp \(p\)-H\({}_{2}\) at density \(n=0.0261\) Å\({}^{-3}\) and temperature \(T=4\) K, computed in this work using the SG (circles) and P (boxes) intermolecular pair potentials. the best choice of spherical pair potential. The results of our first principle numerical simulations show that the _ab initio_ intermolecular pair potential by [PERSON] et al. provides a quantitatively more accurate description of the structure of the fluid, as shown by a comparison of computed and measured static structure factor. There are quantitative discrepancies between the estimates of the (kinetic) energy per molecule yielded by the SG and P potentials in the (liquid) solid phase. However, there are also significant, outstanding disagreements between different, independent experimental determinations of the kinetic energy in the liquid, as well as the energy in the crystal. It seems fair to state, however, that, in general, the _ab initio_ potential of [PERSON] et al. yields energy and pressure estimates closer to experiment than the Silvera-Goldman. The question arises of whether closer agreement with experiment can be achieved by including three-body terms. While the explicit inclusion of three-body contributions to the potential energy went beyond the scope of this work, some general considerations can be made, based on the experience accumulated with research on the condensed phases of helium. Three-body interaction potentials broadly include two distinct contributions, namely (_a_) the (mainly repulsive) triple-dipole Axilrod-Teller (AT) term [63], and (_b_) an attractive contribution, arising from electronic exchange taking place in triplets of neighboring molecules. The AT term has been shown to give a negligible contribution, at the thermodynamic conditions considered here [29], and therefore one can expect the attractive part, effectively included in the SG potential, to play the most important role.