In quantum chemistry, there is a debate about the existence of density functionals beyond the ground electronic state. Here, we apply the tool kit of density-functional theory (DFT) to analyze the characteristics of the eigenvalues of an arbitrary symmetric matrix. The latter can represent a model Hamiltonian, which is not necessarily obtained from a basis-set representation of the exact physical Hamiltonian. Based on the analogy with an exactly solvable physical model, we introduce a density functional for each eigenvalue of the matrix and show that, similar to an exactly solvable model, only the lowest and highest eigenvalues of an arbitrary symmetric matrix have well-defined single-valued functionals of density with unique convexity or concavity characteristics. All other eigenvalues lack such density functionals, as the respective quantities are multivalued density functions and lack unique convexity or concavity. The problem of the nonexistence of functionals for the excited states can be overcome by forming weighted sums of eigenvalues in analogy with the Gross-Oliveira-Kohn ensemble DFT, which results in well-defined density functionals with admissible characteristics.

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• Density functional theory
• Electronic structure
• Nonperturbative methods