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

TL;DR

This paper derives a generalized tilted plane condition for the total energy surface of finite quantum systems, extending previous flat plane conditions and analyzing their implications for DFT and spectroscopic data.

Contribution

It introduces a generalized tilted plane condition for finite quantum systems' energy surfaces, expanding the flat plane concept to all electron counts and magnetizations.

Findings

Flat plane structure observed in oxygen atom

Derivative discontinuities can occur at non-integer electron counts

Tilted plane structures vary with d-orbital chemical coordination

Abstract

The piecewise linearity condition on the total energy with respect to the total magnetization of finite quantum systems is derived, using the infinite-separation-limit technique. This generalizes the well-known constancy condition, related to static correlation error, in approximate density functional theory (DFT). The magnetic analog of the DFT Koopmans' theorem is also derived. Moving to fractional electron count, the tilted plane condition is derived, lifting certain assumptions in previous works. This generalization of the flat plane condition characterizes the total energy surface of a finite system for all values of electron count N and magnetization M. This result is used in combination with tabulated spectroscopic data to show the flat plane-structure of the oxygen atom, among others. We find that derivative discontinuities with respect to electron count sometimes occur at non-integer values. A diverse set of tilted plane structures is shown to occur in d-orbital subspaces, depending on chemical coordination. General occupancy-based total-energy expressions are demonstrated thereby to be necessarily dependent on the symmetry-imposed degeneracies.