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Some theorems on derivatives of the Coulomb density functional with respectto the coupling constant $\lambda$ are given. Consider an electron density$n_{GS}({\bf r})$ given by a ground state. A model Fermion system with thereduced coupling constant, $\lambda<1$, is defined to reproduce $n_{GS}({\bfr})$ and the ground state energy. Fixing the charge density, possible phasetransitions as level crossings detected in a value of the reduced densityfunctional happen only at discrete points along the $\lambda$ axis. If thedensity is $v$-representable also for $\lambda<1$, accumulation of phasetransition points is forbidden when $\lambda\rightarrow 1$. Relevance of thetheorems for the multi-reference density functional theory is discussed.

Theorems on ground-state phase transitions in Kohn-Sham models given by the Coulomb density functional