Beyond Atom Pairs: Multicenter Crystal Orbital Hamilton Populations for Molecules and Solids

The Crystal Orbital Hamilton Population (COHP) is a standard tool for energyresolved bonding analysis in solids and also applies to molecules, but it assigns the band energy only to atom pairs. A multicenter bond therefore has no energy of its own: across the five molecules and solids studied here, pairwise integrated COHP reports strong two-center bonding (−1.8 to −3.5 eV) and cannot distinguish an electrondeficient bridge from an electron-rich unit. We here formulate a multicenter COHP from the idempotency of the zero-temperature density matrix P in each spin channel, P 2 = P , which gives Tr(HP ) = Tr(HPn−1) for every n ≥ 2. Projecting the resulting cyclic matrix products onto ordered n-atom groups and replacing one density-matrix factor with the spectral function A(E) yields COHP (n) C1,...,Cn (E). Its energy integral, summed over all n-tuples, recovers the band energy independently for every n, and the n = 2 expression reduces to standard COHP without approximation. Across these systems, the three-center energy spans about 1320 meV: it is bonding in the B–H–B bridge of diborane (−820 meV), near zero in the Cr–O–Cr pathway of half-metallic