How Beneficial Is the Explicit Account of Doubly-Excited Configurations in Linear Response Theory?

In different branches of time-dependent density functional theory (TDDFT), the static and dynamic electron correlation enters in different ways. The standard spin-conserving linear response (LR-TDDFT) methodology includes explicitly the contributions of the singly-excited configurations; however, it relies on an implicit account of the electron correlation through an (approximate) exchange-correlation (XC) functional. In the mixed-reference spin-flip TDDFT (MRSF-TDDFT), a number of doubly-excited (DE) configurations are explicitly included in the description of their response states. Here, the importance of the explicit account of DE is investigated for the lowest four excited singlet states of all-trans-polyenes up to C24H26. For the optically bright 1B(u)(+) state, the DE contribution in MRSF-TDDFT approaches 10% with the increasing system size. For the optically dark 2A(g)(-) state, the DE contribution increases from ca. 13% (C4H6) to nearly 30% (C24H26). An even more considerable DE contribution (similar to 50%) is observed in the higher 1B(u)(-) states. As LRTDDFT misses these contributions entirely, its ability to accurately describe the excited states is limited by the XC functional. The hybrid XC functionals with a small fraction of the exact exchange, e.g., B3LYP, may mimic certain effects of DE through the self-interaction error (SIE). However, the description of the 1B(u)(+) state by LR-TDDFT remains poor. On the other hand, MRSF-TDDFT can flexibly take an implicit (through the XC functional) and an explicit (through DE) account of the electron correlation, which enables a more balanced description of various types of the excited states regardless of their character, thus reducing the chances of failure.