Energy and information are two fundamental concepts in physics and chemistry. In density functional theory (DFT), all information pertaining to stability, reactivity, and other properties is encompassed in the ground state electron density. The basic theorems of DFT govern that energy is a universal functional of the density, and thus, it can be regarded as a special kind of information. In this work, we quantify the energetic information in terms of Shannon entropy and Fisher information for energetic distributions of atoms and molecules. Two identities are unveiled for an energetic density, its gradient, and Laplacian to rigorously satisfy. A new partition scheme to decompose atoms in molecules has been proposed using the energetic distribution. We also show that our approach can simultaneously quantify both two-body and many-body interactions. This new framework should provide new analytical tools for us to appreciate electronic properties of molecular systems, including stability and reactivity. More importantly, this work establishes the missing link in DFT between energy and information, the two most fundamental quantities in quantum theory.
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