Pair-wise radial distribution functions

Fig. 2. Pair-wise radial distribution function between the oxygen atoms of methanol and water.

The thermodynamic variables of liquid systems, such as energy, pressure, etc., can be expressed in terms of the radial distribution function if one makes the assumption that the total potential energy is pair-wise additive, i.e.,... 

Using the pair-wise additivity of U(R), it is possible to integrate Eq. (18) over the equilibrium configurations of (N — 2) particles. If one then uses the definition of the radial distribution function, an expression for E in terms of g( r) and u(r) is obtained, and it is referred to as the energy equation... 

The choice of the adjustable parameters used in conjunction with classical potentials can result to either effective potentials that implicitly include the nuclear quantization and can therefore be used in conjunction with classical simulations (albeit only for the conditions they were parameterized for) or transferable ones that attempt to best approximate the Born-Oppenheimer PES and should be used in nuclear quantum statistical simulations. Representative examples of effective force fields for water consist of TIP4P (Jorgensen et al. 1983), SPC/E (Berendsen et al. 1987) (pair-wise additive), and Dang-Chang (DC) (Dang and Chang 1997) (polarizable, many-body). The polarizable potentials contain - in addition to the pairwise additive term - a classical induction (polarization) term that explicitly (albeit approximately) accounts for many-body effects to infinite order. These effective potentials are fitted to reproduce bulk-phase experimental data (i.e., the enthalpy at T = 298 K and the radial distribution functions at ambient conditions) in classical molecular dynamics simulations of liquid water. Despite their simplicity, these models describe some experimental properties of liquid... 

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