# Microscopic Pair Potentials and the Physical Properties of the Condensed Phases of Parahydrogen
## 3 Results
### Liquid  
Table 1 shows our numerical results for the kinetic energy per molecule and the pressure of the equilibrium liquid phase of \(p\)-H\({}_{2}\), obtained in simulations with different intermolecular potentials, at four distinct thermodynamic conditions for which experimental results are available. We first note that the results for the kinetic energy are similar for the phenomenological Silvera-Goldman and Buck potentials, _and nearly identical_ for the first principle Patkowski and Diep-Johnson potentials, the values yielded by the first principle potentials being (rather consistently) \(\sim 10\%\) higher. Comparison with experiment for this particular physical quantity is complicated by the quantitative disagreement between the two most recent experimental determination of the kinetic energy per molecule at the same (or at sufficiently close) thermodynamic conditions, namely that by [PERSON] et al. [28] and that by [PERSON] et al. [36] (shown in Table 1); somewhat curious (and puzzling) is the fact that, while the result of [PERSON] et al. is relatively close to the estimate yielded by the SG potential, that of [PERSON] et al. agrees quantitatively with that based on the P potential. To our knowledge, this outstanding, significant inconsistency between different experimental results is yet to be resolved. In general, it appears as if the SG potential may yield results for \(e_{K}\) is closer agreement with experiment, although in some cases the size of experimental uncertainties prevents one from making a definitive conclusion. Thus, the situation at the present time is unclear; it is worth pointing out that the extraction of \(e_{K}\) from the measured momentum distribution is a rather delicate operation, and that similar disagreements between theory and experiments and/or different experimental estimates have been reported in other contexts, e.g., in helium mixtures [60; 61]. The pressure is consistently underestimated by the SG (and the Buck) potential, and overestimated by the P (and Diep-Johnson). The difference between the results yielded by the two potentials can be attributed in part to _a_) the softer repulsive core at short distance, and _b_) the presence in these potentials of an attractive term, proportional to the inverse ninth power of the intermolecular separation, which accounts in an effective way for three-body interactions. This term is not present in the P and Diep-Johnson potentials; this is because, in an _ab initio_ approach, non-additive terms describing the interaction of groups of three or more molecules are generally included as separate contributions to the total potential energy. We come back to this point in Section 4, when we discuss more extensively the possible role of three-body terms; for the moment, we note that, among the various pair-wise intermolecular potentials considered in this work, the Patkowski potential furnishes estimates of the pressure that are closest to the experimental results, as shown in Table 1.  
More cogent information is provided by the comparison of structural correlations, calculated with the two different potentials, for which a direct comparison with experiment is possible. Figure 2 shows the static structure factor \(S(q)\), obtained from the numerically computed pair correlation function \(g(r)\) through  
\[S(q)=n\ \int dr\ r\ g(r)\ sin(qr) \tag{2}\]