Fractional occupations and density-functional energies and forces

The explicit inclusion of fractional occupation numbers in density-functional calculations is shown to require an additional term to make the energy functional variational. The contribution from this term to the density-functional force exactly cancels the correction term depending on changes in the occupation number. For occupation numbers obeying a Fermi distribution, the resulting functional is identical in form to the grand potential; other choices for the form of the occupation numbers will result in different functionals. These terms, although numerically small, should be included in practical calculations that allow for fractional occupation numbers