  Conical intersections constitute the conceptual bedrock of our working
understanding of ultrafast, nonadiabatic processes within photochemistry (and
photophysics). Accurate calculation of potential energy surfaces within the
vicinity of conical intersections, however, still poses a serious challenge to
many popular electronic structure methods. Multiple works have reported on the
deficiency of methods like linear-response time-dependent density functional
theory within the adiabatic approximation (AA LR-TDDFT) or algebraic
diagrammatic construction to second-order (ADC(2)) - approaches often used in
excited-state molecular dynamics simulations - to describe conical
intersections between the ground and excited electronic states. In the present
study, we focus our attention on conical intersections between excited
electronic states and probe the ability of AA LR-TDDFT and ADC(2) to describe
their topology and topography, using protonated formaldimine and pyrazine as
two exemplar molecules. We also take the opportunity to revisit the performance
of these methods in describing conical intersections involving the ground
electronic state in protonated formaldimine - highlighting in particular how
the intersection ring exhibited by AA LR-TDDFT can be perceived either as a
(near-to-linear) seam of intersection or two interpenetrating cones, depending
on the magnitude of molecular distortions within the branching space.
