The most accurate electronic structure methods based on N-particle wave functions are too expensive to be applied to large systems. It is clearer every day the need for treatments of electron correlation that scale favorably with the number of electrons. Among them, the Kohn-Sham formulation of the Density Functional Theory (DFT) has become very popular thanks to its relatively low computational cost. However, present-day functionals have several problems due mainly to the so-called “correlation kinetic energy”, but most importantly, currently available functionals are not N-representable.

A direction for improving DFT lies in the development of a functional theory based upon the one-particle reduced density matrix (1-RDM) rather than on the one-electron density. The 1-RDM is a much simpler object than the N-particle wave function, and the ensemble N-representability conditions are well-known. The existence and properties of the total energy functional of the 1-RDM are well-established. Its development may be greatly aided by imposition of multiple constraints that are more strict and abundant than their DFT counterparts. The major advantage of a 1-RDM formulation is that the kinetic energy is explicitly defined and does not require the construction of a functional. The unknown functional in a 1-RDM-based theory only needs to incorporate electron correlation. The 1-RDM functional incorporates fractional occupations in a natural way, which may provide a correct description of both dynamical and nondynamical correlation.

The 1-RDM functional is called Natural Orbital Functional (NOF) when it is based upon the spectral expansion of the 1-RDM. We have recently proposed an explicit antisymmetric form for the cumulant of the two-particle RDM in terms of two symmetric matrices, Δ and Π. The so-called D, Q and G positivity conditions have allowed us to propose the functional forms for the Δ and Π matrices. We refer to this approach as PNOF (Int. J. Quantum Chem. 106, 1093, 2006). This functional reduces to the exact expression for the total energy in two-electron systems. The PNOF depends only on the Coulomb (J), exchange (K) and exchange and time-inversion (L) integrals, thus can be referred to as JKL-only approximation. A spin-conserving NOF theory has been also formulated (J. Chem. Phys. 131, 021102, 2009). Validation tests of PNOF for predicting several atomic and molecular properties have been performed using the PNOFID code based on a novel iterative diagonalization method to obtain the natural orbitals (J. Comp. Chem. 30, 2078, 2009).

Please refer to the following publications for more information:

Review Article on NOFT:

  • M. Piris, Natural Orbital Functional Theory, in Reduced-Density-Matrix Mechanics: With Applications to Many-electron Atoms and Molecules, edited by D. Mazziotti, Adv. Chem. Phys. 134, 387 (2007).

Review Articles on PNOF:








  • M. Piris, J. M. Matxain, X. Lopez and J. M. Ugalde, The role of the positivity N-representability conditions in Natural Orbital Functional Theory, J. Chem. Phys. 133, 111101 (2010).
  • X. Lopez, F. Ruipérez, M. Piris, J. M. Matxain, J. M. Ugalde, Diradicals and diradicaloids in Natural Orbital Functional Theory, ChemPhysChem 12, 1061 (2011) .
  • X. Lopez, M. Piris, J. M. Matxain, F. Ruipérez, J. M. Ugalde, Natural orbital functional theory and reactivity studies of diradical rearrangements : ethylene torsion as a case study, ChemPhysChem 12, 1673 (2011).
  • J. M. Matxain, F. Ruipérez, M. Piris, “Computational Study of Be2 using Piris Natural Orbital Functionals“, J. Mol. Model. 19, 1967 (2013).
  • M. A. Rodríguez-Mayorga, E. Ramos-Cordoba, M. Via-Nadal, M. Piris, E. Matito, “Comprehensive benchmarking of density matrix functional approximations”, Phys. Chem. Chem. Phys. 19, 24029-24041 (2017).