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The piecewise linearity condition on the total energy with respect to totalmagnetization of open quantum systems is derived, using theinfinite-separation-limit technique. This generalizes the well known constancycondition, related to static correlation error, in approximate densityfunctional theory (DFT). The magnetic analog of the DFT Koopmans theorem isalso derived. Moving to fractional electron count, the tilted plane conditionis derived, lifting certain assumptions in previous works. This generalizationof the flat plane condition characterizes the total energy curve of a finitesystem for all values of electron count N and magnetization M. This result isused in combination with tabulated spectroscopic data to show the flatplane-structure of the oxygen atom, as an example. A diverse set of tiltedplane structures is shown to occur in d-orbital subspaces, depending on theirchemical coordination. General occupancy-based expressions for total energiesare demonstrated thereby to be necessarily dependent on the symmetry-imposeddegeneracies.

The tilted-plane structure of the energy of open quantum systems