The theorem of Hohenberg and Kohn, that the electron density is a unique functional of the external potential, applies to a closed system with a fixed number of electrons. Transferability of the electron density of an atom between two systems, necessary to account for the fundamental role of a functional group with characteristic properties in chemistry, however, is a problem in the physics of an open system. Transferability in chemistry requires a new theorem stated in terms of the density: that the electron density of an atom in a molecule or crystal determines its additive contribution to all properties of the total system, its transferability being determined by a paralleling degree of transferability in the atom's virial field, the virial of the Ehrenfest force exerted on its electron density. Transferability of the virial field is found in spite of unavoidable changes in the external potential that occur on transfer. The properties of an atom in a molecule are determined by the Heisenberg equation of motion for a "proper open system" derived from the principle of stationary action. The theorem is grounded in the common sense observation that two atoms that "look the same", i.e., have the same charge distribution, must possess identical properties. Transferability is discussed in terms of the properties of the electron density, the one-density and the two-density matrix, the latter demonstrating that both the Coulombic and exchange contributions to the energy of a group are separately transferable. It is demonstrated that the exchange indices determined by the two-density matrix provide a readily determinable measure of the effective range of the exchange energy.