Information theory thermodynamics of molecules and their Hirshfeld fragments

The Hirshfeld stockholder partitioning of a molecular density into subsystem densities, e.g., of atoms-in-molecules, is derived from the Fisher entropy for locality (intrinsic accuracy) referenced to the isolated atom densities. This complements the previous derivation using the entropy deficiency of Kullback and Leibler, thereby strongly suggesting that the Hirshfeld result is independent of the information measure applied to assimilate the reference atomic information. Several properties of such subsystems are examined. It is shown that they represent equilibrium, stable pieces of the molecular density, which minimize and reduce to zero the nonadditivity of the missing information relative to the isolated atom reference. The equilibrium criteria for the optimum partitioning in both the entropy and energy representations are derived and interpreted in terms of local entropies of subsystems and their related "temperatures" of information. Generalized variational principles in the entropy and energy representations are proposed, which link the entropy of the information theory with energetical parameters of molecular systems. The associated information temperature becomes infinite, when a single ground-state density is the density of interest.