ABSTRACT:The cumulant expansion gives rise to a useful decomposition of the two-matrix D in which the pair correlated matrix (cumulant) is disconnected from the antisymmetric product of the one-matrix ⌫. A new explicit antisymmetric approach for the two-particle cumulant matrix in terms of two symmetric matrices, ⌬ and ⌳, as functionals of the occupation numbers is proposed for singlet ground states of closedshell systems. It produces a natural orbital functional that reduces to the exact expression for the total energy in two-electron systems. The functional form of matrix ⌳ is readily generalized to any system with an even number of electrons. The diagonal elements of ⌬ equal the square of the occupation numbers, and the N-representability positivity necessary conditions of the two-matrix impose several bounds on the offdiagonal elements of matrix ⌬. The well-known mean value theorem and the partial sum rule obtained for the off-diagonal elements of ⌬ provide a prescription for deriving a practical functional. In particular, when the mean values { J * i } of the Coulomb interactions { J ij } for a given orbital i taking over all orbitals j i are assumed to be equal {K ii /2}, a functional close to self-interaction-corrected GU functional is obtained, but the two-matrix fermionic antisymmetric holds. An additional term for the matrix elements of ⌳ between HF occupied orbitals is proposed to ensure a correct description of the occupation numbers for the lowest occupied levels. The functional is tested in fully variational finite basis set calculations of 57 molecules. It gives reasonable molecular energies at the equilibrium geometries. The calculated values of dipole moments are in good agreement with the available experimental data.

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^{dyn}) and static (E^{sta}) correlation. E^{dyn} is derived from a modified second-order Møller-Plesset perturbation theory (MP2), while E^{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 E[over ˜]_{HF}+E^{dyn}+E^{sta}, where E[over ˜]_{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 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.