# Microscopic Pair Potentials and the Physical Properties of the Condensed Phases of Parahydrogen
## 3 Results
### Solid  
At low temperature and at saturated vapor pressure, the equilibrium phase of \(p\)-H\({}_{2}\) is a hcp crystal, known experimentally [62] to undergo virtually no thermal expansion from \(T=0\) all the way to the melting temperature, namely \(T=13.8\) K, i.e., the density \(n\) remains very close to its ground state value, namely [34]\(n=0.0261\) A\({}^{-3}\). Moreover, at \(T=4\) K the system is very nearly in its ground state [32].  
In Table 2 we show the kinetic (\(e_{K}\)) and total (\(E\)) energy per molecule (in K), computed using various potentials; the results for the kinetic energy display the same trend already observed in the liquid phase, namely the _ab initio_ potentials yield a value of the kinetic energy \(\sim 10\%\) higher than the SG and Buck potentials. Now, the total energy per molecule for the SG and Buck potentials are in (nearly) perfect agreement with the \(T=0\) theoretical estimate of [PERSON] and [PERSON] [29], who computed them by Diffusion Monte Carlo simulations. The value yielded by the P potential is \(\sim 3.4\) K lower than that of the SG, whereas that afforded by the Diep-Johnson potential is actually closer to that yielded by the SG potential (only a fraction of a K higher); on the other hand, the Diep-Johnson and P potentials yield essentially identical estimates for \(e_{K}\), i.e., the difference in the estimates of \(E\) delivered by these two potentials is entirely due to the different attractive well depths. Comparison with experiment is again ambiguous, because in this case too, there exist two different estimates of the ground state energy, differing by roughly 3.5 K (see Table 2). In any case, the estimate yielded by the P potential falls roughly between the two conflicting experimental values.  
Figure 2: Static structure factor of liquid \(p\)-H\({}_{2}\) at density \(n=0.02295\) Å\({}^{-3}\) and temperature \(T=17.1\) K, computed in this work using the SG (doted line) and P (solid line) intermolecular pair potentials. Also shown (circles) are experimental results from Ref. [37]. Experimental uncertainties are not shown for clarity but they are typically of the order of 0.1 for \(q\geq 3\) Å\({}^{-1}\).  
Just like for the liquid phase, the SG potential yields a negative value of the pressure, while the P overestimates it by approximately 45 bars, based on an interpolation of results provided in Table III of Ref. [35]. The possible role of three-body terms in improving the agreement between theoretical and experimental results for the two potentials, both for the pressure and the energy, is discussed in Section 4.  
Figure 3 shows the static structure factor for the crystal, computed with the SG and P potentials, at the above-mentioned thermodynamic conditions. In this case, since the solid is not isotropic, \(S(q)\) cannot be obtained from the pair correlation function but rather must be computed directly, for a selected set of wave vectors. Unlike in the liquid phase, as expected \(S(q)\) displays in the crystal the characteristic, sharp (Bragg) peaks, whose strength greater for the P potential (by approximately 10%), as observed in the liquid.