From detailed considerations of two homogeneity postulates of the total molecular and electronic energies proposed by Parr and Gadre, a new homogeneity hypothesis of the total molecular energy W is presented: ∑iZi(∂W/∂Zi)N =k0W0+kRWR, where Zi is the atomic number, W0 and WR are the R-independent (R: internuclear distance) and R-dependent molecular energies, and k0 and kR are the local and nonlocal homogeneity parameters. Such a hypothesis is examined through derivation of potential constants for diatomic molecules and is shown to lead to exact formulas relating quadratic, cubic, quartic, and higher potential constants. Inhomogeneous linear first- and second-order differential equations, derived on the basis of the newly proposed homogeneity hypothesis, for diatomic molecules are solved to obtain some information on general potential forms of molecules. Approximate Hartree–Fock methods with the new homogeneity constraint are developed and discussed in some detail.