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The exact conditions for density functionals and density matrix functionalsin terms of fractional charges and fractional spins are known, and theirviolation in commonly used functionals has been shown to be the root of manymajor failures in practical applications. However, approximate functionals arenot normally expressed in terms of the fractional variables. Here we develop ageneral framework for extending approximate density functionals andmany-electron theory to fractional-charge and fractional-spin systems. Ourdevelopment allows for the fractional extension of any approximate theory thatis a functional of $G^{0}$, the one-electron Green's function of thenon-interacting reference system. The extension to fractional charge andfractional spin systems is based on the ensemble average of the basic variable,$G^{0}$. We demonstrate the fractional extension for the following theories:(1) any explicit functional of the one-electron density, such as the LDA andGGA; (2) any explicit functional of the one-electron density matrix of thenon-interacting reference system, such as the exact exchange functional andhybrid functionals; (3) many-body perturbation theory; and (4) random-phaseapproximations. A general rule for such an extension has also been derivedthrough scaling the orbitals and should be useful for functionals where thelink to the Green's function is not obvious. The development thus enables theexamination of approximate theories against known exact conditions on thefractional variables and the analysis of their failures in chemical andphysical applications in terms of violations of exact conditions of the energyfunctionals. The present work should facilitate the calculation of chemicalpotentials and fundamental band gaps with approximate functionals andmany-electron theories through the energy derivatives with respect to thefractional charge.

Extension of many-electron theory and approximate density functionals to fractional charges and fractional spins