The energetic representation of the molecular configuration in a dilute solution is introduced to express the solvent distribution around the solute over a one-dimensional coordinate specifying the solute-solvent interaction energy. In this representation, the correspondence is shown to be one-to-one between the set of solute-solvent interaction potentials and the set of solvent distribution functions around the solute. On the basis of the one-to-one correspondence, the Percus-Yevick and hypernetted-chain integral equations are formulated over the energetic coordinate through the method of functional expansion. It is then found that the Percus-Yevick, hypernetted-chain, and superposition approximations in the energetic representation determine the solvent distribution functions correctly to first-order with respect to the solute-solvent interaction potential and to the solvent density. The expressions for the chemical potential of the solute are also presented in closed form under these approximations and are shown to be exact to second-order in the solute-solvent interaction potential and in the solvent density.