### On the calculation of $Δ$〈Ŝ2〉 for electronic excitations in time-dependent density-functional theory.

Excited states are often treated within the context of time-dependent (TD) d.-functional theory (DFT), making it important to be able to assign the excited spin-state symmetry. While there is universal agreement on how $Δ$〈Ŝ2〉, the difference between 〈Ŝ2〉 for ground and excited states, should be calcd. in a wave-function-like formalism such as the Tamm-Dancoff approxn. (TDA), confusion persists as to how to det. the spin-state symmetry of excited states in TD-DFT. We try to clarify the origins of this confusion by examg. various possibilities for the parameters ($\sigma$1,$\sigma$2) in the formula $Δ$〈S ̂2〉=[$Δ$〈S ̂TDA2〉(X→)+$Δ$〈S ̂TDA2〉(Y→*)+$\sigma$1$Δ$〈S ̂mixed2〉(X→,Y→*)]/(X→†X→+$\sigma$2Y→†Y→), where X⃑ is the particle-hole part and Y⃑ is the hole-particle part of the response theory vector. A first principles derivation leads directly to ($\sigma$1,$\sigma$2) = ( + 1, - 1) which we argue is the best simple formula linking spin with energy, albeit approx. On the other hand, if the desire is to recover wave-function-like values of $Δ$〈Ŝ2〉, then we argue that the choice ($\sigma$1,$\sigma$2) = ( + 1, + 1) should be made. Addnl. examples are offered to justify that the choice of $\sigma$1 = 0 should also be made when seeking wave-function-like values of $Δ$〈Ŝ2〉. [on SciFinder(R)]

### Références

Titre
On the calculation of $Δ$〈Ŝ2〉 for electronic excitations in time-dependent density-functional theory.
Type de publication
Article de revue
Année de publication
2017
Revue
Comput. Phys. Commun.
Volume
213
Pagination
72–91
ISSN
0010-4655
Soumis le 12 avril 2018