# Microscopic Pair Potentials and the Physical Properties of the Condensed Phases of Parahydrogen
## 2 Model and Methodology  
The system is described as an ensemble of \(N\) point-like, identical particles with mass \(m\) equal to that of a \(p\)-H\({}_{2}\) molecule, and with spin zero, thus obeying Bose statistics. The system is enclosed in a cell of volume \(\Omega\), so that \(n=N/\Omega\) is the nominal density. The cell is shaped like a cube (cuboid) for simulations of the system in the liquid (solid) phase; periodic boundary conditions are used in all three directions. The quantum-mechanical many-body Hamiltonian reads as follows:  
\[\hat{H}=-\lambda\sum_{i}\
abla_{i}^{2}+\sum_{i<j}v(r_{ij}) \tag{1}\]  
where the first (second) sum runs over all particles (pairs of particles), \(\lambda\equiv\hbar^{2}/2m=12.031\) KA\({}^{2}\), \(r_{ij}\equiv|\mathbf{r}_{i}-\mathbf{r}_{j}|\) and \(v(r)\) is the pair potential which describes the interaction between two molecules, for which the various versions mentioned above have been utilized, for the purpose of comparing predictions for various thermodynamic quantities. We performed first principles QMC simulations of the system described by Equation (1), based on the canonical version [48; 49] of the continuous-space Worm Algorithm [50; 51] based on [PERSON]'s space-time approach to quantum statistical mechanics [52] Since this methodology is extensively described in the literature, it will not be reviewed here. We carried out the bulk of our simulations by treating \(p\)-H\({}_{2}\) molecules as _distinguishable_; this is because, although quantum exchanges have detectable effects on the momentum distribution of the liquid at equilibrium, near freezing [31], they do not affect to a significant degree the quantities of interest here. Indeed, quantum exchanges are strongly suppressed in the bulk phases of molecular hydrogen, due to the relatively large radius of the repulsive core of the pair-wise interaction at short intermolecular separations [53]. This is why all credible scenarios of possible \(p\)-H\({}_{2}\) superfluidity involve small cluster (comprising of the order of 30 molecules), not bulk phases [43; 55; 54]. Details of the simulation are standard; the short-time approximation to the imaginary-time propagator used here is accurate to fourth order in the time step \(\tau\) (see, for instance, Ref. [56]). We have carried out numerical extrapolation of the estimates to the \(\tau\to 0\) limit, and observed convergence of the thermal averages for a value of \(\tau=2.5\times 10^{-3}\) K\({}^{-1}\) for all quantities of interest here, with the exception of the pressure, which we computed using the well-known virial estimator [57]; for this particular quantity, four times shorter a time step was required in order to arrive at a converged estimate. Physical quantities of interest calculated in this study include, besides pressure and energetics, structural correlations such as the static structure factor. The kinetic energy per molecule \(e_{K}\) and the static structure factor \(S(q)\) can afford the most direct comparison of theory with experiment, as they are accessible through neutron scattering measurements [36; 37; 58]. We have carried out simulations for systems of varying size, the smallest (largest) comprising \(N=256\) (\(N=4096\)) molecules. In general, we found that the numerical estimates of \(e_{K}\) obtained on a simulated system comprising \(N=256\) molecules are indistinguishable, within statistical uncertainties, from those obtained on systems of larger sizes, in both the liquid and the solid phase. On the other hand, some noticeable size dependence was observed for the other quantities; at any rate, the range of system sizes considered in this work gives us reasonable confidence in our ability to gauge finite-size effects, i.e., that the numerical values quoted herein are representative of the thermodynamic limit, within their statistical uncertainties.