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Excited-state descriptors based on the one-particle transition density matrix referring to the exciton picture have been implemented for time-dependent density functional theory. State characters such as local, extended pp*, Rydberg, or charge transfer can be intuitively classified by simple comparison of these descriptors. Strong effects of the choice of the exchange-correlation kernel on the physical nature of excited states can be found and decomposed in detail leading to a new perspective on functional performance and the design of new functionals. [ABSTRACT FROM AUTHOR]