Fundamental statistical mechanical derivation of the relative binding free energy

Fundamental Statistical Mechanical Derivation of the Relative Binding Free Energy 

In this section an approximate form of the chemical potential of a species in solution is derived. This approximate chemical potential is then used to derive an equation for the relative binding free energy that is suitable for dielectric continuum calculations. Steps are then taken to refine the relative binding free energy such that the errors due to the approximations invoked are be minimized. The refined relative binding free energy is then cast at various levels of detail in order highlight its important contributions. 

In the context of ligand-protein interactions we start with the free energy change associated with the following process 

In this expression ct(X) is the symmetry number of the solute X (the number of indistinguishable orientations of the solute), C° is the standard state concentration of the solute, m is the mass of atom i of the solute, M(X) is the total number of atoms of the solute, P° is the standard pressure. 

Note that the term in the square brackets is the solvent averaged Boltzmann factor of the solute-solvent interaction energy. 

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