$\Delta$NO and the complexities of electron correlation in simple hydrogen clusters.

The $\Delta$NO two-electron density matrix (2-RDM) and energy expression are derived from a multideterminantal wave function. The approximate $\Delta$NO 2-RDM is combined with an on-top density functional and a double-counting correction to capture electron correlation. A trust-region Newton's method optimization algorithm for the simultaneous optimization of $\Delta$NO orbitals and occupancies is introduced and compared to the previous iterative diagonalization algorithm. The combination of $\Delta$NO and two different on-top density functionals, Colle-Salvetti (CS) and OF, is assessed on small hydrogen clusters and compared to density functional, single-reference coupled cluster, and multireference perturbation theory (MRMP2) methods. The $\Delta$NO-CS and $\Delta$NO-OF methods outperform the single-reference methods, and are comparable to MRMP2. However, there is a distinct qualitative error in the $\Delta$NO potential energy surface for H$_4$ compared to the exact. This discrepancy is explained through analysis of the $\Delta$NO orbitals, occupancies and the two-electron density.