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Spin-flip methods provide access to certain electronic states having multireference character while retaining single-reference cost. However, conventional spin-flip time-dependent density functional theory (SF-TDDFT) often suffers from severe spin contamination that may cause inaccurate state ordering or engender ambiguous state character. For singlet excited states, this is largely rectified by a “mixed-reference” formulation (MRSF-TDDFT), while a spin-adapted formalism (SA-SF-TDDFT) addresses spin contamination in a general way for arbitrary multiplicities. Here, we revisit SA-SF-TDDFT and demonstrate that it significantly improves the agreement with reference data compared to other variants and also relative to conventional (spin-conserving) linear response TDDFT. Overall, SA-SF-TDDFT proves to be the most accurate among these methods, for excitation energies of both closed-shell molecules and doublet radicals as well as for singlet–triplet gaps. However, SF methods exhibit a notable limitation in the case of linear and quasi-linear doublet radicals, due to degeneracies in the high-spin quartet reference state. 

Abstract This experimental work explores the relationship between the properties and structure of mammalian fur from different habitats and the depth of water drop penetration when impacted in succession. For most mammals, water penetration depth reaches a saturation point, beyond which it no longer increases, creating a dry insulating air layer near the skin regardless of repeated water impacts. To understand this phenomenon, we define several dimensionless quantities representing fur macro-properties, such as guard hair and underfur densities, guard hair and underfur lengths, contact angles, and equivalent diameters. Additionally, we examine microscopic properties such as the aspect ratio and roughness of individual fiber scales. We establish connections between these macro- and microscopic characteristics, the thickness of the dry zone, the depth of water penetration, and the rate at which penetration depth decays exponentially. Our results show that the distal diameter influences the rate at which the penetration depth of water decays with additional impacts. Generally, a higher pelage density, larger guard hair diameter, and increased fur roughness contribute to a thicker dry zone. Using digital microscopy, we confirm that mammalian guard fur is hydrophilic, resisting dynamic water penetration, whereas the finer and denser underfur is hydrophobic, resisting static penetration. This dual-layer structure allows mammals to resist wetting during a heavy rainfall.