Kimika Teorikoa

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 wavefunction, 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 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 DoNOF 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:

• M. Piris, I. Mitxelena, DoNOF: an open-source implementation of natural-orbital-functional-based methods for quantum chemistry, Comp. Phys. Comm. 259, 107651 (2021), Code Ocean Capsule, arXiv:2004.06142 [physics.comp-ph].
• J. F. Huan Lew-Yee, M. Piris, J. M. del Campo, Resolution of the identity approximation applied to PNOF correlation calculations, J. Chem. Phys. 154, 064102 (2021); arXiv:2012.15662 [physics.chem-ph].
• L. Franco, I. A. Bonfil-Rivera, J. F. H. Lew-Yee, M. Piris, J. M. del Campo, R. A. Vargas-Hernández, “Softmax parameterization of the occupation numbers for natural orbital functionals based on electron pairing approaches“, J. Chem. Phys. 160, 244107 (2024); arXiv:2403.09463 [physics.chem-ph].

Review Articles 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).
• M. Piris, The electron pairing approach in Natural Orbital Functional Theory, in Quantum Chemistry at the Dawn of the 21st Century, edited by Ramon Carbó-Dorca and Tanmoy Chakraborty. Series: Innovations in Computational Chemistry. Chapter 22. pp. 593-620. Apple Academic Press (2018).
• M. Piris, The role of the N-representability in one-particle functional theories, in Many-body approaches at different scales: a tribute to N. H. March on the occasion of his 90th birthday, edited by G. G. N. Angilella and C. Amovilli. Chapter 22, pp. 261-278, New York: Springer (2018).
• I. Mitxelena, M. Piris, Jesus M. Ugalde, Advances in Approximate Natural Orbital Functional Theory, in State of The Art of Molecular Electronic Structure Computations: Correlation Methods, Basis Sets and More, edited by Philip Hoggan and Ugo Ancarani, Adv. Quantum Chem. 79, 155-177 (2019)
• M. Piris, Advances in Approximate Natural Orbital Functionals: From Historical Perspectives to Contemporary Developments, in Novel treatments of strong correlations, edited by Ramon A. Miranda-Quintana & John F. Stanton, Adv. Quantum Chem. 90, Chapter 2, pp.15-66 (2024). arXiv:2312.07163.
• M. Piris, Exploring the Potential of Natural Orbital Functionals, Chem. Sci. 15, 17284-17291 (2024); chemrxiv-2024-qhg51

Review Articles on PNOF:

• M. Piris, A natural orbital functional based on an explicit approach of the two-electron cumulant, Int. J. Quantum Chem. 113, 620 (2013).
• M. Piris, J. M. Ugalde, Perspective on Natural Orbital Functional Theory, Int. J. Quantum Chem. 114, 1169 (2014).

GNOF:

• M. Piris, “Global Natural Orbital Functional: Towards the Complete Description of the Electron Correlation“, Phys. Rev. Lett. 127, 233001 (2021); arXiv:2112.02119[physics.chem-ph].
• I. Mitxelena, M. Piris, “Benchmarking GNOF against FCI in challenging systems in one, two and three dimensions“, J. Chem. Phys. 156, 214102 (2022); arXiv:2203.12447 [physics.chem-ph].
• J. F. Huan Lew-Yee, J. M. del Campo, M. Piris, “Electron correlation in the Iron(II) Porphyrin by NOF approximations“, J. Chem. Theory Comput. 19, 211–220 (2023); arXiv:2212.01640 [physics.chem-ph].
• J. F. Huan Lew-Yee, M. Piris, J. M. del Campo, “Outstanding Improvement in Removing the Delocalization Error by Global Natural Orbital Functional“, J. Chem. Phys. 158, 084110 (2023); arXiv:2212.01597 [physics.chem-ph].
• L. Franco, J. F. Huan Lew-Yee, J. M. del Campo, “Correlation balance for describing carbenes: An NOF study“, AIP Advances 13, 065213 (2023).
• J. M. Mercero, R. Grande-Aztatzi, J. M. Ugalde, M. Piris, “Natural Orbital Functional Theory Studies of All-Metal Aromaticity. The Al3 anion“, Adv. Quantum Chem. 88, 229-248 (2023).
• J. F. H. Lew-Yee, I. A. Bonfil-Rivera, M. Piris, J. M. del Campo, “Excited states by coupling Piris natural orbital functionals with extended random phase approximation“, J. Chem. Theory Comp. 20, 2140–2151 (2024); arXiv:2311.05504 [physics.chem-ph].
• A. Rivero-Santamaría, M. Piris, “Time evolution of natural orbitals in ab initio molecular dynamics“, J. Chem. Phys. 160, 071102 (2024); arXiv:2311.04802 [physics.chem-ph].
• I. Mitxelena, M. Piris, “Assessing the Global Natural Orbital Functional Approximation on Model Systems with Strong Correlation“, J. Chem. Phys. 160, 204106 (2024); arXiv:2403.03554 [physics.chem-ph].
• M. Piris, X. Lopez, J. M. Ugalde “Time-Resolved Chemical Bonding Structure Evolution by Direct-Dynamics Chemical Simulations”, J. Phys. Chem. Lett. 15, 12138−12143 (2024).
• J. F. H. Lew-Yee,1, J. M. del Campo, M. Piris, “Advancing Natural Orbital Functional Calculations Through Deep Learning-Inspired Techniques for Large-Scale Strongly Correlated Electron Systems“, … …, … (2024); ; arXiv:2411.18493 [physics.chem-ph].

NOF-MP2, NOF-MBPT:

• M. Piris, “Global Method for Electron Correlation”, Phys. Rev. Lett. 119, 063002 (2017); arXiv:1708.03719 [physics.chem-ph].
• M. Piris, “Dynamic electron-correlation energy in the NOF-MP2 method from the orbital-invariant perturbation theory”, Phys. Rev. A 98, 022504 (2018); arXiv:1808.06070 [physics.chem-ph].
• X. Lopez, M. Piris, “Performance of the NOF-MP2 method in hydrogen abstraction reactions“, Theor. Chem. Acc. 138, 89 (2019); arXiv:1906.04432 [physics.chem-ph].
• M. Piris, “Natural Orbital Functional for Multiplets“, Phys. Rev. A 100, 032508 (2019); arXiv:1908.05501 [physics.chem-ph].
• R. Quintero-Monsebaiz, L. I. Perea-Ramírez, M. Piris, A. Vela, “Spectroscopic properties of open shell diatomic molecules using Piris Natural Orbital Functionals“, Phys. Chem. Chem. Phys. 23, 2953-2963 (2021).
• J. M. Mercero, J. M. Ugalde, M. Piris, “Chemical Reactivity Studies by the Natural-Orbital-Functional 2nd-Order-Møller-Plesset (NOF-MP2) method. Water Dehydrogenation by the Scandium Cation“, Theor. Chem. Acc. 140, 74 (2021); arXiv:2012.13202 [physics.chem-ph].
• M. Rodríguez-Mayorga, I. Mitxelena, F. Bruneval, M. Piris, “Coupling Natural Orbital Functional Theory and Many-Body Perturbation Theory by Using Nondynamically Correlated Canonical Orbitals“, J. Chem. Theory Comput. 17, 7562–7574 (2021).

PNOF7:

• M. Piris, “Global Method for Electron Correlation”, Phys. Rev. Lett. 119, 063002 (2017); arXiv:1708.03719 [physics.chem-ph].
• I. Mitxelena, M. Piris, M. A. Rodríguez-Mayorga, “On the performance of Natural Orbital Functional Approximations in Hubbard model”, J. Phys. Condens. Matter 29, 425602 (2017); “Corrigendum”, J. Phys. Condens. Matter 30, 089501 (2018).
• M. 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).
• I. Mitxelena, M. Piris, “Analytic second-order energy derivatives in natural orbital functional theory”, J. Math. Chem. 56, 1445-1455 (2018); arXiv:1802.05887[physics.chem-ph].
• I. Mitxelena, M. Rodríguez-Mayorga, M. Piris, “Phase Dilemma in Natural Orbital Functional Theory from the N-representability Perspective”, Eur. Phys. J. B 91, 109 (2018); arXiv:1804.06282[physics.chem-ph].
• M. Piris, “Dynamic electron-correlation energy in the NOF-MP2 method from the orbital-invariant perturbation theory”, Phys. Rev. A 98, 022504 (2018); arXiv:1808.06070 [physics.chem-ph].
• R. Quintero-Monsebaiz, I. Mitxelena, M. Rodríguez-Mayorga, A. Vela, M. Piris, “Natural orbital functional for spin-polarized periodic systems”, J. Phys.: Condens. Matter 31, 165501 (2019); arXiv:1901.06942 [physics.chem-ph].
• M. Piris, “Natural Orbital Functional for Multiplets“, Phys. Rev. A 100, 032508 (2019); arXiv:1908.05501 [physics.chem-ph].
• I. Mitxelena, M. Piris, “An efficient method for strongly correlated electrons in one dimension“, J. Phys.: Condens. Matter 32, 17LT01 (2020); arXiv:1912.09312[cond-mat.str-el].
• I. Mitxelena, M. Piris, “An efficient method for strongly correlated electrons in two-dimensions“, J. Chem. Phys. 152, 064108 (2020); arXiv:1911.10157[cond-mat.str-el].
• I. Mitxelena, M. Piris, “Analytic gradients for spin multiplets in natural orbital functional theory“, J. Chem. Phys. 153, 044101 (2020) arXiv:2005.02333 [physics.chem-ph].
• R. Quintero-Monsebaiz, L. I. Perea-Ramírez, M. Piris, A. Vela, “Spectroscopic properties of open shell diatomic molecules using Piris Natural Orbital Functionals“, Phys. Chem. Chem. Phys. 23, 2953-2963 (2021).
• J. F. Huan Lew-Yee, M. Piris, J. M. del Campo, Resolution of the identity approximation applied to PNOF correlation calculations, J. Chem. Phys. 154, 064102 (2021); arXiv:2012.15662 [physics.chem-ph].
• J. M. Mercero, J. M. Ugalde, M. Piris, “Chemical Reactivity Studies by the Natural-Orbital-Functional 2nd-Order-Møller-Plesset (NOF-MP2) method. Water Dehydrogenation by the Scandium Cation“, Theor. Chem. Acc. 140, 74 (2021); arXiv:2012.13202 [physics.chem-ph].
• L. Franco, I. A. Bonfil-Rivera, J. F. H. Lew-Yee, M. Piris, J. M. del Campo, R. A. Vargas-Hernández, “Softmax parameterization of the occupation numbers for natural orbital functionals based on electron pairing approaches“, J. Chem. Phys. 160, 244107 (2024); arXiv:2403.09463 [physics.chem-ph].
• J. F. H. Lew-Yee,1, J. M. del Campo, M. Piris, “Advancing Natural Orbital Functional Calculations Through Deep Learning-Inspired Techniques for Large-Scale Strongly Correlated Electron Systems“, … …, … (2024); ; arXiv:2411.18493 [physics.chem-ph].

PNOF6:

• M. Piris, “Interacting pairs in natural orbital functional theory”, J. Chem. Phys. 141, 044107 (2014).
• X. Lopez, M. Piris, F. Ruiperez, J. M. Ugalde “Performance of PNOF6 for Hydrogen Abstraction Reactions”, J. Phys. Chem. A 119, 6981 (2015).
• M. Piris, N. H. March, “Low-lying Isomers of Free-space Halogen Clusters with Tetrahedral and Octahedral Symmetry in Relation to Stable Molecules Such as SF₆”, J. Phys. Chem. A 119, 10190 (2015) .
• E. Ramos-Cordoba, X. Lopez, M. Piris, E. Matito, “H4: A Challenging System For Natural Orbital Functional Approximations”, J. Chem. Phys. 143, 164112 (2015); arXiv:1507.08244 [physics.chem-ph].
• J. Cioslowski, M. Piris, E. Matito, “Robust validation of approximate 1-matrix functionals with few-electron Harmonium atoms”, J. Chem. Phys. 143, 214101 (2015); arXiv:1511.06564 [physics.chem-ph].
• M. Piris, X. Lopez, J. M. Ugalde, “The Bond Order of C2 from an Strictly N-Representable Natural Orbital Energy Functional Perspective“, Chemistry -­ A European Journal 22, 4109 (2016).
• A. Deveson, D. Cremer, G. Frenking, M. Piris, S. Shaik, “Why Does C2 Cause so Many Problems?”,  ChemistryViews (2016).
• I. Mitxelena, M. Piris, “Molecular Electric Moments calculated by using Natural Orbital Functional Theory”,  J. Chem. Phys. 144, 204108 (2016).
• I. Mitxelena, M. Piris, “Analytic gradients for natural orbital functional theory”, J. Chem. Phys. 146, 014102 (2017); arXiv:1612.04673 [physics.chem-ph].
• M. 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).

PNOF5-PT2:

• M. Piris, “Interpair electron correlation by second-order perturbative corrections to PNOF5”, J. Chem. Phys. 139, 064111 (2013).
• M. Piris, F. Ruipérez, J. M. Matxain, “Assessment of the second-order perturbative corrections to PNOF5”, Mol. Phys. 112, 711 (2014).
• M. Piris, X. Lopez, J. M. Ugalde, “The Bond Order of C2 from an Strictly N-Representable Natural Orbital Energy Functional Perspective“, Chemistry -­ A European Journal 22, 4109 (2016).
• A. Deveson, D. Cremer, G. Frenking, M. Piris, S. Shaik, “Why Does C2 Cause so Many Problems?”,  ChemistryViews (2016).

PNOF5e:

• M. Piris, J. M. Matxain, X. Lopez, “The intrapair electron correlation in natural orbital functional theory“,  J. Chem. Phys. 139, 234109 (2013).
• M. Piris, N. H. March, “Is the Hartree-Fock prediction that the chemical potential μ of non-relativistic neutral atoms is equal to minus the ionization potential I sensitive to electron correlation?”, Physics and Chemistry of Liquids 53, 696 (2015).
• M. Piris, N. H. March, “Chemical and Ionization Potentials: Relation via the Pauli Potential and NOF Theory”, Int. J. Quantum Chem. 116, 805-818 (2016).
• I. Mitxelena, M. Piris, “Analytic gradients for natural orbital functional theory”, J. Chem. Phys. 146, 014102 (2017); arXiv:1612.04673 [physics.chem-ph].
• 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).

PNOF5:

• M. Piris, X. Lopez, F. Ruipérez, J. M. Matxain, J.M. Ugalde, A natural orbital functional for multiconfigurational states, J. Chem. Phys. 134, 164102 (2011).
• J. M. Matxain, M. Piris, F. Ruipérez, X. Lopez, J. M. Ugalde, Homolytic molecular dissociation in natural orbital functional theory, Phys. Chem. Chem. Phys. 13, 20129 (2011).
• J. M. Matxain, M. Piris, J. M. Mercero, X. Lopez, J. M. Ugalde, “sp3 hybrid orbitals and ionization energies of methane from PNOF5″, Chem. Phys. Lett. 531, 272 (2012).
• J. M. Matxain, M. Piris, J. Uranga, X. Lopez, G. Merino, J. M. Ugalde, “The Nature of the Chemical Bonds from PNOF5 calculations″, ChemPhysChem. 13, 2297 (2012).
• M. Piris, J. M. Matxain, X. Lopez, J. M. Ugalde, “The extended Koopmans’ theorem: vertical ionization potentials from Natural Orbital Functional Theory”, J. Chem. Phys. 136, 174116 (2012).
• X. Lopez, F. Ruipérez, M. Piris, J. M. Matxain, E. Matito, J. M. Ugalde, “Performance of PNOF5 for radical formation reactions: Hydrogen atom abstraction, C-C and O-O homolytic bond cleavage in selected molecules”, J. Chem. Theory Comput. 8, 2646 (2012).
• M. Piris, “Bounds on the PNOF5 natural geminal occupation numbers“, Comput. Theor. Chem. 1003, 123 (2013).
• J. M. Matxain, F. Ruipérez, M. Piris, “Computational Study of Be2 using Piris Natural Orbital Functionals“, J. Mol. Model. 19, 1967 (2013).
• M. Piris, J. M. Matxain, X. Lopez, J. M. Ugalde, “The one-electron picture in the Piris Natural Orbital Functional 5 (PNOF5)“, Theor. Chem. Acc. 132, 1298 (2013).
• F. Ruipérez, M. Piris, J. M. Ugalde, J. M. Matxain, “The natural orbital functional theory of the bonding in Cr2, Mo2 and W2”, Phys. Chem. Chem. Phys. 15, 2055 (2013).
• J. M. Matxain, F. Ruipérez, I. Infante, X. Lopez, J. M. Ugalde, G. Merino, M. Piris, “Communications: Chemical bonding in carbon dimer isovalent series from the natural orbital functional theory perspective”, J. Chem. Phys. 138, 151102 (2013).
• X. Lopez, M. Piris, M. Nakano, B. Champagne, “Natural Orbital Functional Calculations of Molecular Polarizabilities and Second Hyperpolarizabilities. Hydrogen Molecule as a Test Case“, J. Phys. B:  At. Mol. Opt. Phys. 47, 015101 (2014).
• E. Ramos-Cordoba, P. Salvador, M. Piris, E. Matito, “Two new constraints for the cumulant matrix”, J. Chem. Phys. 141, 234101 (2014).
• M. Piris, N. H. March, “Weizsäcker inhomogeneity kinetic energy term for the inhomogeneous electron liquid characterizing some thirty homonuclear diatomic molecules at equilibrium and insight into Teller’s theorem in Thomas-Fermi statistical theory”, Physics and Chemistry of Liquids 52, 804 (2014).
• X. Lopez, M. Piris, “PNOF5 Calculations Based on the “Thermodynamic Fragment Energy Method”: CnH2n+2 (n=1,10) and (FH)n (n=1,8) as test cases“, Theor. Chem. Acc. 134, 151 (2015).
• M. Piris, X. Lopez, J. M. Ugalde, “The Bond Order of C2 from an Strictly N-Representable Natural Orbital Energy Functional Perspective“, Chemistry -­ A European Journal 22, 4109 (2016).
• A. Deveson, D. Cremer, G. Frenking, M. Piris, S. Shaik, “Why Does C2 Cause so Many Problems?”,  ChemistryViews (2016).
• M. Piris, N. H. March, “Potential energy curves for P2 and P2+ constructed from a strictly N-representable natural orbital functional”, Physics and Chemistry of Liquids 54, 797 (2016) .
• I. Mitxelena, M. Piris, “Analytic gradients for natural orbital functional theory”, J. Chem. Phys. 146, 014102 (2017); arXiv:1612.04673 [physics.chem-ph].

PNOF4:

• 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).

PNOF3:

• M. Piris, J. M. Matxain, X. Lopez, and J. M. Ugalde, Accurate description of atoms and moecules by NOFT, J. Chem. Phys. 132, 031103, (2010).
• X. Lopez, M. Piris, J. M. Matxain, and J. M. Ugalde, Performance of PNOF3 for reactivity studies: X[BO] and X[CN] isomerization reactions (X=H,Li) as a case study, Phys. Chem. Chem. Phys. 12, 12931, (2010).
• J. M. Matxain, M. Piris, X. Lopez, J. M. Ugalde, Complete Basis Set Calculations by PNOF3, Chem. Phys. Lett. 499, 164 (2010).
• 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).

PNOF2:

• M. Piris, X. Lopez and J. M. Ugalde, Dispersion interactions within the PNOF theory: the helium dimer, J. Chem. Phys. 126, 214103 (2007).
• M. Piris, X. Lopez, J.M. Ugalde, Natural orbital functional description of van der Waals interactions: A case study of the effect of the basis set for the helium dimer, Int. J.Quantum Chem. 108, 1660 (2008).
• M. Piris, X. Lopez, J.M. Ugalde, Correlation holes for the helium dimer, J. Chem. Phys. 128, 134102 (2008).
• M. Piris, J.M. Matxain, J.M. Ugalde, Piris natural orbital functional study of the dissociation of the radical helium dimer, J. Chem. Phys. 129, 014108 (2008).

PNOF1:

• M. Piris, A new approach for the Two-Electron Cumulant in Natural Orbital Functional theory, Int. J. Quantum Chem. 106, 1093 (2006).
• P. Leiva and M. Piris, Natural Orbital Functional study for the electric response properties of molecules, P. Leiva and M. Piris, J. Theo. Comp. Chem. 4, 1165 (2005).
• P. Leiva and M. Piris, Assessment of a new approach for the two-electron cumulant in natural-orbital-functional, J.Chem. Phys. 123, 214102 (2005).
• P. Leiva and M. Piris, Calculation of vertical ionization potentials with the Piris Natural Orbital Functional, P. Leiva and M. Piris, J.  Mol.Struct.: THEOCHEM 770, 45, (2006).
• P. Leiva and M. Piris, Description of high-spin restricted open-shells with the Piris Natural Orbital Functional, Int. J. Quantum Chem. 107, 1 (2007).
• M. Piris, J.M Matxain, X. Lopez, J.M. Ugalde, Spin conserving natural orbital functional theory, J. Chem. Phys. 131, 021102 (2009).