# Microscopic Pair Potentials and the Physical Properties of the Condensed Phases of Parahydrogen
## 1 Introduction  
Hydrogen, the simplest and most abundant element in the universe, continues to motivate a lot of experimental and theoretical research, because of its fundamental interest and technological relevance [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11]. In particular, predicting quantitatively the equation of state (EOS) of the condensed phases of hydrogen, in a broad range of thermodynamic conditions, remains a worthwhile theoretical goal, of significant potential relevance to a wide variety of pure and applied fields of science. The _ab initio_ theoretical study of the EOS of hydrogen in which all the elementary constituents of the system (namely, electrons and protons), interacting through electrostatic Coulomb forces, are individually taken into account, constitutes a challenging quantum-mechanical many-body problem. Most calculations have been typically carried out within the framework of Density Functional Theory (DFT) [12; 13; 14] or quantum Monte Carlo (QMC) [15; 16; 17; 18]. Valuable as they are, _ab initio_ approaches become impractical or unviable in specific physical regimes, due to the simultaneous presence of widely different energy scales; in those cases, one typically resorts to approximations. For example, when the system is in the plasma phase, at relatively high pressure and temperature, it is acceptable to treat the heavier protons as classical particles, and carry out semi-classical molecular dynamics or Monte carlo simulations, in which forces are evaluated either by DFT or QMC [19; 20; 21; 22; 23; 24; 25]. On the other hand, in molecular phases it is legitimate to adopt the Born-Oppenheimer approximation and regard individual hydrogen molecules as elementary particles, describing their interaction by means of a static potential, the simplest choice being that of a pair-wise, central potential. Besides being obviously unable to capture processes involving electronic transfer, such a crude model treats hydrogen molecules as spherical objects; this is known to be an oversimplification [26], even for the most nearly spherical (and abundant) isotope, namely parahydrogen (\(p\)-H\({}_{2}\)). Equally absent are, of course, energy contributions of interactions involving, e.g., triplets of molecules. However, the use of static pair potentials renders the computation much faster andconceptually simpler than _ab initio approaches_, allowing for the computation of thermodynamic properties of molecular hydrogen in the condensed phase virtually without any uncontrolled approximation, e.g., by means of QMC simulations. Thus, the development and testing of increasingly reliable pair potentials, affording a quantitatively accurate account for energetic and structural properties of the condensed phase of molecular hydrogen in a wide range of thermodynamic conditions, is a worthwhile goal of quantum chemistry, one with likely impact in various areas of condensed matter physics and materials science. The (mainly phenomenological) Silvera-Goldman (SG) [27] is the most commonly adopted pair potential; it has been shown to afford a reasonable, semi-quantitative description of low temperature structure and energetics of the equilibrium liquid and solid phases of \(p\)-H\({}_{2}\) (the only species considered in this paper). Specifically, the computed (kinetic) energy per particle [28; 29; 30; 31; 32] is in agreement with _some_ of the reported experimental measurements [28; 33; 34; 35; 36], while the pressure is generally significantly underestimated. There are also quantitative differences between theory and experiment regarding structural correlations [37]. Another popular pair potential is the Buck [38], featuring a slightly deeper attractive well. These potentials are similar, in that the same general functional form is parametrized to reproduce some experimental data.