Density-functional perturbation theory for excited states from constrained DFT

Constrained DFT (cDFT) is a crude but computationally cheap approach for modeling excited states, in which non-equilibrium occupations are assigned to the Kohn-Sham states. Despite the availability of more sophisticated approaches, cDFT remains useful in contexts such as charge-transfer excitations, thermalized excited electron populations in solids, and TDDFT for excited-state absorption. Within ordinary DFT, vibrational and dielectric linear-response calculations have become routine with density-functional perturbation theory (DFPT), which needs only the occupied states. This technique works at zero temperature or with an effective temperature described through a smearing function, and can be used in both static and TDDFT calculations. However, the formalism has not been available for calculation of states with arbitrary occupations, as can arise in the cDFT description of excited states. I derive a simple modification to extend DFPT to arbitrary fractional occupations and show example applications for systems with non-equilibrium occupations.