On the calculation of Δ〈Sˆ2〉 for electronic excitations in time-dependent density-functional theory

Excited states are often treated within the context of time-dependent (TD) density-functional theory (DFT), making it important to be able to assign the excited spin-state symmetry. While there is universal agreement on how Δ〈Sˆ2〉, the difference between 〈Sˆ2〉 for ground and excited states, should be calculated in a wave-function-like formalism such as the Tamm-Dancoff approximation (TDA), confusion persists as to how to determine the spin-state symmetry of excited states in TD-DFT. We try to clarify the origins of this confusion by examining various possibilities for the parameters (σ1,σ2) in the formula Δ〈Sˆ2〉=[Δ〈SˆTDA2〉(X→)+Δ〈SˆTDA2〉(Y→∗)+σ1Δ〈Sˆmixed2〉(X→,Y→∗)]/(X→†X→+σ2Y→†Y→), where X→ is the particle-hole part and Y→ is the hole-particle part of the response theory vector. A first principles derivation leads directly to (σ1,σ2)=(+1,−1) which we argue is the best simple formula linking spin with energy, albeit approximately. On the other hand, if the desire is to recover wave-function-like values of Δ〈Sˆ2〉, then we argue that the choice (σ1,σ2)=(+1,+1) should be made. Additional examples are offered to justify that the choice of σ1=0 should also be made when seeking wave-function-like values of Δ〈Sˆ2〉.