The current work presents a new single-reference method for capturing at the same time the static and dynamic electron correlation. The starting point is a determinant wave function formed with natural orbitals obtained from a new interacting-pair model. The latter leads to a natural orbital functional (NOF) capable of recovering the complete intrapair, but only the static interpair correlation. Using the solution of the NOF, two new energy functionals are defined for both dynamic ($E^{\mathrm{dyn}}$) and static ($E^{\mathrm{sta}}$) correlation. $E^{\mathrm{dyn}}$ is derived from a modified second-order Møller-Plesset perturbation theory (MP2), while $E^{\mathrm{sta}}$ is obtained from the static component of the new NOF. Double counting is avoided by introducing the amount of static and dynamic correlation in each orbital as a function of its occupation. As a result, the total energy is represented by the sum ${\tilde{E}}_{\text{HF}}+E^{\mathrm{dyn}}+E^{\mathrm{sta}}$, where ${\tilde{E}}_{\text{HF}}$ is the Hartree-Fock energy obtained with natural orbitals. The new procedure called NOF-MP2 scales formally as $O(M^5)$ (where $M$ is the number of basis functions), and is applied successfully to the homolytic dissociation of a selected set of diatomic molecules, paradigmatic cases of near-degeneracy effects. The size consistency has been numerically demonstrated for singlets. The values obtained are in good agreement with the experimental data.

An explicit formulation of the Piris cumulant \documentclass[12pt]{minimal}\begin{document}$\mathbf{\lambda }\left(\bm {\Delta },\bm {\Pi }\right)$\end{document}λΔ,Π matrix is described herein, and used to reconstruct the two-particle reduced density matrix (2-RDM). Then, we have derived a natural orbital functional, the Piris Natural Orbital Functional 5, PNOF5, constrained to fulfill the D, Q, and G positivity necessary conditions of the N-representable 2-RDM. This functional yields a remarkable accurate description of systems bearing substantial (near)degeneracy of one-particle states. The theory is applied to the homolitic dissociation of selected diatomic molecules and to the rotation barrier of ethylene, both paradigmatic cases of near-degeneracy effects. It is found that the method describes correctly the dissociation limit yielding an integer number of electrons on the dissociated atoms. PNOF5 predicts a barrier of 65.6 kcal/mol for the ethylene torsion in an outstanding agreement with Complete Active Space Second-order Perturbation Theory (CASPT2). The obtained occupation numbers and pseudo one-particle energies at the ethylene transition state account for fully degenerate π orbitals. The calculated equilibrium distances, dipole moments, and binding energies of the considered molecules are presented. The values obtained are accurate comparing those obtained by the complete active space self-consistent field method and the experimental data.

The current work presents a natural orbital functional (NOF) for electronic systems with any spin value independent of the external potential being considered, that is, a global NOF (GNOF). It is based on a new two-index reconstruction of the two-particle reduced density matrix for spin multiplets. The emergent functional describes the complete intrapair electron correlation, and the correlation between orbitals that make up both the pairs and the individual electrons. The interorbital correlation is composed of static and dynamic terms. The concept of dynamic part of the occupation numbers is introduced. To evaluate the accuracy achieved with the GNOF, calculation of a variety of properties is presented. They include the total energies and energy differences between the ground state and the lowest-lying excited state with different spin of atoms from H to Ne, ionization potentials of the first-row transition-metal atoms (Sc-Zn), and the total energies of a selected set of 55 molecular systems in different spin states. The GNOF is also applied to the homolytic dissociation of selected diatomic molecules in different spin states and to the rotation barrier of ethylene, both paradigmatic cases of systems with significant multiconfigurational character. The values obtained agree with those reported at high level of theory and experimental data.

The natural orbital functional (NOF) theory is briefly reviewed. The meaning of the top‐down and bottom‐up approaches for the construction of a NOF is analyzed. A particular reconstruction of the two‐particle reduced density matrix (2‐RDM) based on the cumulant expansion is discussed. The cumulant is expressed by two auxiliary matrices, which are constrained to certain bounds due to the N‐representabilty conditions of the 2‐RDM. Appropriate forms of these matrices lead to different implementations known in the literature as PNOFi (i = 1–5). The strengths and weaknesses of PNOF5 are assessed. Its main strength is its ability to deal with the intrapair electron correlation at a reasonable computational cost. Its main limitation is the absence of the interpair electron correlation. The inclusion of the missing correlation via a multiconfigurational perturbation theory is shortly described. The growing interest in methods based on NOF theory points to a promising future in this field.