Translating Spin-Adapted RPA to Spin-Adapted TDDFT.

Linear-response TDDFT based on open-shell references may produce spin-mixed excited states, particularly when the response space is not spin complete. Spin-adapted RPA was developed using tensor equation-of-motion and applying the Wigner-Eckart theorem with tensor decoupling. Casting the RPA Fock matrix and kernels as energy derivatives gives a TDDFT extension. To restore spin-component degeneracy and eliminate multicounting of spin correlation in pure exchange-correlation functionals, we use hybrids combining HF exchange with spin-unpolarized pure XC parts. Given that the mapping from spin-adapted RPA to spin-adapted TDDFT is ad hoc, we introduce a first-principles framework for evaluating the expectation value ⟨Ŝ2⟩ of excited states in order to rigorously prove their spin purification. In addition, we derive explicit working equations for the computation of the single-particle density matrix of excited states, thereby enabling a natural orbital analysis of their electronic structure. Benchmarks, diradicals/triradicals, Cr2 dissociation, and phenol O-H conical intersections demonstrate the method.