Upper and lower bounds for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg" display="inline" id="d1e152"><mml:mi>n</mml:mi></mml:math>th-order DFT energies

Density partitioning and the Inclusion-Exclusion principle are combined to expand a density-functional energy into its n th-order energies. For a class of functionals that we call monotonic, we construct rigorous lower and upper bounds for the n th-order energy and test their numerical performance for a number of Rung 1 functionals. • Partition-Inclusion-Exclusion (PIE) expansion of the DFT energy. • Proof that the PIE expansion of a monotonic functional is alternating. • Bounds for terms in the PIE expansion of a monotonic functional.