Mario Piris

Ikerbasque - Basque Foundation for Science · Donostia International Physics Center

The natural orbital functional theory (NOFT) has emerged as an alternative formalism to both density functional (DF) and wavefunction methods. In NOFT, the electronic structure is described in terms of the natural orbitals (NOs) and their occupation numbers (ONs). The approximate NOFs have proven to be more accurate than those of the density for sy...

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 int...

Natural orbital functional (NOF) theory offers a promising approach for studying strongly correlated systems at an affordable computational cost, with an accuracy comparable to highly demanding wavefunction-based methods. However, its widespread adoption in cases involving a large number of correlated electrons has been limited by the extensive ite...

Direct-dynamics simulations monitor atomic nuclei trajectories during chemical reactions, where chemical bonds are broken and new ones are formed. While they provide valuable information about the ongoing nuclear dynamics, the evolution of the chemical bonds is customarily overlooked, thus, hindering key information about the reaction mechanism. He...

In recent years, Natural Orbital Functional (NOF) theory has gained increasing significance in quantum chemistry, successfully addressing one of the field's most challenging problems: providing an accurate and balanced description of systems with strong electronic correlation. The quest for NOFs that strike the delicate balance between computationa...

Natural Orbital Functional (NOF) theory has become a cornerstone of quantum chemistry in recent years, successfully addressing one of the field's most challenging problems: providing an accurate and balanced description of systems with strong electronic correlation. The quest for NOFs that strike the delicate balance between computational tractabil...

Within the framework of natural orbital functional theory, having a convenient representation of the occupation numbers and orbitals becomes critical for the computational performance of the calculations. Recognizing this, we propose an innovative parametrization of the occupation numbers that takes advantage of the electron-pairing approach used i...

CNDOL is an a priori, approximate Fockian for molecular wave functions. In this study, we employ several modes of singly excited configuration interaction (CIS) to model molecular excitation properties by using four combinations of the one electron operator terms. Those options are compared to the experimental and theoretical data for a carefully s...

This chapter provides a comprehensive review of fundamental concepts related to approximate natural orbital functionals (NOFs), emphasizing their significance in quantum chemistry and physics. Focusing on fermions, the discussion excludes considerations of finite temperature and systems with a variable number of particles. The theoretical foundatio...

In the past decade, natural orbital functional (NOF) approximations have emerged as prominent tools for characterizing electron correlation. Despite their effectiveness, these approaches, which rely on natural orbitals and their associated occupation numbers, often require hybridization with other methods to fully account for all correlation effect...

In this work, we extend the Piris natural orbital functionals (PNOFs) to excited states by coupling their reconstructed second-order reduced density matrices with the extended random-phase approximation (ERPA). We have named the general implementation as PNOF-ERPA, and we refer to specific cases as PNOF-ERPA0, PNOF-ERPA1 and PNOF-ERPA2, according t...

Within the framework of natural orbital functional theory, having a convenient representation of the optimization function becomes critical for the computational performance of the calculation. Recognizing this, we propose an innovative parametrization of the occupation numbers that takes advantage of the electron-pairing approach used in Piris nat...

In the past decade, natural orbital functional (NOF) approximations have emerged as prominent tools for characterizing electron correlation. Despite their effectiveness, these approaches, which rely on natural orbitals and their associated occupation numbers, often require hybridization with other methods to fully account for all correlation effect...

This work combines for the first time ab initio molecular dynamics (AIMD) within the Born–Oppenheimer approximation with a global natural orbital functional (GNOF), an approximate functional of the one-particle reduced density matrix. The most prominent feature of GNOF-AIMD is its ability to display the real-time evolution of natural orbitals, prov...

In this work, we explore the use of Piris natural orbital functionals (PNOFs) to calculate excited-state energies by coupling their reconstructed second-order reduced density matrix with the extended random-phase approximation (ERPA). We have named the general method PNOF-ERPA, and specific approaches are referred to as PNOF-ERPA0, PNOF-ERPA1, and...

This chapter provides a comprehensive review of fundamental concepts related to approximate natural orbital functionals (NOFs), emphasizing their significance in quantum chemistry and physics. Focusing on fermions, the discussion excludes considerations of finite temperature and systems with a variable number of particles. The theoretical foundatio...

In this work, we extend the Piris natural orbital functionals (PNOFs) to excited states by coupling their reconstructed second-order reduced density matrices with the extended random-phase approximation (ERPA). We have named the general implementation as PNOF-ERPA, and we refer to specific cases as PNOF-ERPA0, PNOF-ERPA1 and PNOF-ERPA2, according t...

This work combines for the first time ab initio molecular dynamics (AIMD) within the Born-Oppenheimer approximation , with a global natural orbital functional (GNOF), an approximate functional of the one-particle reduced density matrix. The most prominent feature of GNOF-AIMD is the ability to display the real-time evolution of natural orbitals, pr...

The Al3− triangle singlet, triplet, and quintet states have been characterized. This study is developed from the point of view of an approximate natural orbital functional (NOF), for which the energy of the ground state is given as a functional of the natural orbitals and their occupation numbers. Specifically, the recently proposed global NOF (Pir...

This work assesses the performance of the recently proposed global natural orbital functional (GNOF) against the charge delocalization error. GNOF provides a good balance between static and dynamic electronic correlation leading to accurate total energies while preserving spin, even for systems with a highly multi-configurational character. Several...

The relative stability of the singlet, triplet, and quintet spin states of iron(II) porphyrin (FeP) represents a challenging problem for electronic structure methods. While it is currently accepted that the ground state is a triplet, multiconfigurational wave function-based methods predict a quintet, and density functional approximations vary betwe...

The relative stability of the singlet, triplet, and quintet spin states of Iron(II) porphyrin (FeP) represents a challenging problem for electronic structure methods. While it is currently accepted that the ground state is a triplet, multiconfigurational wavefunction-based methods predict a quintet, and density functional approximations vary betwee...

This work assesses the performance of the recently proposed global natural orbital functional (GNOF) against the charge delocalization error. GNOF provides a good balance between static and dynamic electronic correlation leading to accurate total energies while preserving spin, even for systems with a highly multi-configurational character. Several...

This work assess the reliability of the recently proposed [Phys. Rev. Lett. 127, 233001, 2021] global natural orbital functional (GNOF) in the treatment of the strong electron correlation regime. We first use an H10 benchmark set of four hydrogen model systems of different dimensionalities and distinctive electronic structures: a 1D chain, a 2D rin...

This work assess the reliability of the recently proposed [Phys. Rev. Lett. 127, 233001, 2021] global natural orbital functional (GNOF) in the treatment of the strong electron correlation regime. We use a benchmark set of four hydrogen model systems of different dimensionalities and distinctive electronic structures: a 1D chain, a 2D ring, a 2D she...

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 int...

The reliability of the recently proposed natural orbital functional supplemented with second-order Møller–Plesset calculations (NOF-MP2) has been assessed for the mechanistic studies of elementary reactions of transition metal compounds by investigating the dehydrogenation of water by the scandium cation. Both high- and low-spin state potential ene...

In this work, the required algebra to employ the resolution of the identity approximation within the Piris Natural Orbital Functional (PNOF) is developed, leading to an implementation named DoNOF-RI. The arithmetic scaling is reduced from fifth-order to fourth-order, and the memory scaling is reduced from fourth-order to third-order, allowing signi...

Spectroscopic properties such as equilibrium distances, vibrational constants, rotational constants, dissociation energies, and excitation energies are calculated for nine heteronuclear diatomic molecules (PH, NF, NH, NO, CS, AlF, ClF, BeO and CF) using an interactive pair model (PNOF7s), that has been generalized for spin multiplet states, and its...

In this work, the required algebra to employ the resolution of the identity approximation within Piris Natural Orbital Functional (PNOF) is developed, leading to an implementation named DoNOF-RI. The arithmetic scaling is reduced from fifth-order to fourth-order, and the memory scaling is reduced from fourth-order to third-order, allowing significa...

The reliability of the recently proposed natural orbital functional supplemented with second-order M{\o}ller-Plesset calculations, (NOF-MP2), has been assessed for the mechanistic studies of elementary reactions of transition metal compounds by investigating the dehydrogenation of water by the scandium cation. Both high- and low-spin state potentia...

Analytic energy gradients with respect to nuclear motion are derived for non-singlet compounds in the natural orbital functional theory. We exploit the formulation for multiplets in order to obtain a simple formula valid for any many-electron system in its ground mixed state with a total spin S and all possible spin projection Sz values. We demonst...

Analytic energy gradients with respect to nuclear motion are derived for non-singlet compounds in the natural orbital functional theory. We exploit the formulation for multiplets in order to obtain a simple formula valid for any many-electron system in its ground mixed state with a total spin S and all possible spin projection Sz values. We demonst...

The natural orbital functional theory (NOFT) has emerged as an alternative formalism to both density functional (DF) and wavefunction methods. In NOFT, the electronic structure is described in terms of the natural orbitals (NOs) and their occupation numbers (ONs). The approximate NOFs have proven to be more accurate than those of the density for sy...

This work deals with the problem of strongly correlated electrons in two-dimensions. We give a reduced density matrix (RDM) based tool through which the ground-state energy is given as a functional of the natural orbitals and their occupation numbers. Specifically, the Piris Natural Orbital Functional 7 (PNOF7) is used for studying the 2D Hubbard m...

The one-particle reduced density matrix functional theory in its natural orbital functional (NOF) version is used to study strongly correlated electrons. We show the ability of the Piris NOF 7 (PNOF7) to describe non-dynamic correlation effects in one-dimensional (1D) systems. An extensive study of 1D systems that includes Hydrogen (H) chains and t...

The one-particle reduced density matrix functional theory in its natural orbital functional (NOF) version is used to study strongly correlated electrons. We show the ability of the Piris NOF 7 (PNOF7) to describe non-dynamic correlation effects in one-dimensional (1D) systems. An extensive study of 1D systems that includes Hydrogen (H) chains and t...

This work deals with the problem of strongly correlated electrons in two-dimensions (2D). We give a reduced density matrix (RDM) based tool through which the ground-state energy is given as a functional of the natural orbitals and their occupation numbers. Specifically, the Piris Natural Orbital Functional 7 (PNOF7) is used for studying the 2D Hubb...

A natural orbital functional for electronic systems with any value of the spin is proposed. This energy functional is based on a new reconstruction for the two-particle reduced density matrix (2RDM) of the multiplet, that is, of the mixed quantum state that allows all possible spin projections. The mixed states of maximum spin multiplicity are cons...

A natural orbital functional for electronic systems with any value of the spin is proposed. This energy functional is based on a new reconstruction for the two-particle reduced density matrix (2RDM) of the multiplet, that is, of the mixed quantum state that allows all possible spin projections. The mixed states of maximum spin multiplicity are cons...

The recently proposed natural orbital functional second-order M{\o}ller-Plesset (NOF-MP2) method is capableof achieving both dynamic and static correlation even for those systems with significant multiconfigurational character. We test its reliability to describe the electron correlation in radical formation reactions, namely, in the homolytic X-H...

The recently proposed natural orbital functional second-order Møller–Plesset (NOF-MP2) method is capable of achieving both dynamic and static correlations even for those systems with a significant multiconfigurational character. We test its reliability to describe the electron correlation in radical formation reactions, namely in the homolytic X–H...

Natural orbital functional theory is considered for systems with one or more unpaired electrons. An extension of the Piris natural orbital functional (PNOF)based on electron pairing approachis presented, specifically, we extend the independent pair model, PNOF5, and the interactive pairmodel PNOF7 to describe spin-uncompensated systems. An explicit...

Natural orbital functional theory is considered for systems with one or more unpaired electrons. An extension of the Piris natural orbital functional (PNOF) based on electron pairing approach is presented, specifically, we extend the independent pair model, PNOF5, and the interactive pair model PNOF7 to describe spin-uncompensated systems. An expli...

The basic concepts relevant to the approximate natural orbital functional (NOF)theory are presented. We discuss in detail the reconstruction that leads to Piris NOF (PNOF)approximations focusing on the electron pairing, namely, the independent pair model PNOF5 and the inter-pair electron correlation model PNOF7. It is shown that PNOF7 is an ideal c...

The original formulation (Phys. Rev. Lett. 119, 063002, 2017) of the natural orbital functional - second-order M{\o}ller-Plesset (NOF-MP2) method is based on the MP2 that uses the canonical Hartree-Fock molecular orbitals. The current work presents a reformulation of the dynamic energy correction based on the orbital-invariant MP2, which allows to...

The original formulation [M. Piris, Phys. Rev. Lett. 119, 063002 (2017)] of the natural-orbital-functional second-order-Møller-Plesset (NOF-MP2) method is based on the MP2 that uses the canonical Hartree-Fock molecular orbitals. The current work presents a reformulation of the dynamic energy correction based on the orbital-invariant MP2, which allo...

Any rigorous approach to first-order reduced density (1RDM) matrix functional theory faces the phase dilemma, that is, having to deal with a large number of possible combinations of signs in terms of the electron-electron interaction energy. This problem was discovered by reducing a ground-state energy generated from an approximate N-particle wavef...

The analytic energy gradients in the atomic orbital representation have recently been published (Mitxelena and Piris in J Chem Phys 146:014102, 2017) within the framework of the natural orbital functional theory (NOFT). We provide here an alternative expression for them in terms of natural orbitals, and use it to derive the analytic second-order en...

Any rigorous approach to first-order reduced density (1RDM) matrix functional theory faces the phase dilemma, that is, having to deal with a large number of possible combinations of signs in terms of the electron-electron interaction energy. This problem was discovered by reducing a ground-state energy generated from an approximate N-particle wavef...

The analytic energy gradients in the atomic orbital representation have recently been published (J. Chem. Phys. 146, 014102, 2017) within the framework of the natural orbital functional theory (NOFT). We provide here an alternative expression for them in terms of natural orbitals, and use it to derive the analytic second-order energy derivatives wi...

(Figure Presented). There was an error in the results corresponding to PNOF7 for the homogeneous four site square and six site hexagon Hubbard models reported in the article. Accordingly, the left side of figures 4-6 have been corrected in this corrigendum. Note that PNOF7 energies always remain above FCI (figure 4). For the four site square, the P...

The purpose of this chapter is to analyze the role of the N-representability in one-particle functional theories, that is, in theories where the ground-state energy is represented in terms of the first-order reduced density matrix (1RDM) \(\Gamma \) or simply its diagonal part: the density \(\rho \). I have chosen to write on this topic to honor No...

Hubertus J. J. van Dam {[}Phys. Rev. A 93, 052512, 2016{]} claims that the one-particle reduced density matrix (1RDM) of an interacting system can be represented by means of a single-determinant wavefunction of fictitious non-interacting particles. van Dam introduced orbitals within a mean-field framework that produce energy levels similar to Hartr...

Hubertus J. J. van Dam [Phys. Rev. A 93, 052512, 2016] claims that the one-particle reduced density matrix (1RDM) of an interacting system can be represented by means of a single-determinant wavefunction of fictitious non-interacting particles. van Dam introduced orbitals within a mean-field framework that produce energy levels similar to Hartree-F...

Strongly correlated materials are now under intense development, and Natural Orbital Functional (NOF) methods seem to be able to capture the physics of these systems. We present a benchmark based on the Hubbard model for a class of commonly used NOF approximations (also known as reduced density matrix functional approximations). Our findings highli...

The energy usually serves as a yardstick in assessing the performance of approximate methods in computational chemistry. After all, these methods are mostly used for the calculation of the electronic energy of chemical systems. However, computational methods should be also aimed at reproducing other properties, such strategy leading to more robust...

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 wavefunction 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 intra-...

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 wavefunction 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 intra-...

The analytic energy gradients with respect to nuclear motion are derived for natural orbital functional (NOF) theory. The resulting equations do not require to resort to linear-response theory, so the computation of NOF energy gradients is analogous to gradient calculations at the Hartree-Fock level of theory. The structures of 15 spin-compensated...

The analytic energy gradients with respect to nuclear motion are derived for natural orbital functional (NOF) theory. The resulting equations do not require to resort to linear-response theory, so the computation of NOF energy gradients is analogous to gradient calculations at the Hartree-Fock level of theory. The structures of 15 spin-compensated...

A theoretical framework for modelling the electronic structure of absorbing molecules that may contribute to explain and even predict phenomena where a singlet exciton is converted to triplet is reported. The pathway is suggested to occur when the excited singlet and triplet electron density maps are similarly shaped at Frank Condon states. Calcula...

The performance of the “thermodynamic fragment energy method” (FEM) in the context of natural orbital functional theory (NOFT) in its PNOF5 implementation is assessed. Two test cases are considered: the linear chains \(\mathrm{C}_{n}\mathrm{H}_{2n+2}(n=1,10)\) and the hydrogen-bonded \(\mathrm{(FH)}_{n}(n=1, 8)\) clusters. Calculations show a fast...

The molecular electric dipole, quadrupole and octupole moments of a selected set of 21 spin-compensated molecules are determined employing the extended version of the Piris natural orbital functional 6 (PNOF6), using the triple-$\zeta$ Gaussian basis set with polarization functions developed by Sadlej, at the experimental geometries. The performanc...

The potential energy curves of P2 and P2+ have been calculated using an approximate, albeit strictly N-representable, energy functional of the one-particle reduced density matrix: PNOF5. Quite satisfactory accord is found for the equilibrium bond lengths and dissociation energies for both species. The predicted vertical ionization energy for P2 by...

The molecular electric dipole, quadrupole, and octupole moments of a selected set of 21 spin-compensated molecules are determined employing the extended version of the Piris natural orbital functional 6 (PNOF6), using the triple-ζ Gaussian basis set with polarization functions developed by Sadlej, at the experimental geometries. The performance of...

This small modest molecule has been the subject of a lot of attention over the past years as to what the bonding situation is between these two atoms, as can be seen from the selection of references given here [1–8]. To put it in very simple terms, there appear two main theories—a quadruple bond and not a quadruple bond (in fact somewhere in betwee...

The potential energy curves of P2 and P2+ have been calculated using an approximate, albeit strictly N-representable, energy functional of the one-particle reduced density matrix: PNOF5. Quite satisfactory accord is found for the equilibrium bond lengths and dissociation energies for both species. The predicted vertical ionization energy for P2 by...

The bond order of the ground electronic state of the carbon dimer has been analyzed in the light of natural orbital functional theory calculations carried out with an approximate, albeit strictly N-representable, energy functional. Three distinct solutions have been found from the Euler equations of the minimization of the energy functional with re...

In recent years, the natural orbital functional (NOF) theory has emerged as an alternative approach to both density functional theory (DFT) and wave-function (WFN) methods for electronic structure investigations. Several NOFs have been proposed, for which validation is necessary. A well-known tool for calibration, testing, and benchmarking of an ap...

The performance of the “thermodynamic fragment energy method” (FEM) in the context of natural orbital functional theory (NOFT) in its PNOF5 implementation is assessed. Two test cases are considered: the linear chains Cn H2n+2 (n = 1, 10) and the hydrogen-bonded (FH)n (n = 1, 8) clusters. Calculations show a fast convergence of the PNOF5-FEM method,...

A simple comparison between the exact and approximate correlation components U of the electron-electron repulsion energy of several states of few-electron harmonium atoms with varying confinement strengths provides a superior validation tool for 1-matrix functionals. The robustness of this tool is clearly demonstrated in a survey of 14 known functi...

Hartree-Fock (HF) theory makes the prediction that for neutral atoms the chemical potential (μ) is equal to minus the ionization potential (I). This has led us to inquire whether this intimate relation is sensitive to electron correlation. We present here therefore some discussion of the predictions for neutral atoms and atomic ions, and some homon...

Motivated by earlier work involving one of us (NHM) on some 20 stable tetrahedral (t) and octahedral (o) molecules, including XF4 (X = C, Si, Ge), the natural orbital functional PNOF6 (J. Chem. Phys. 141, 044107, 2014) is here used to study the free-space halogen cluster t-F4. We consider an extended functional PNOF6(Nc) by coupling Nc orbitals (Nc...

The correct description of nondynamic correlation by electronic structure methods not belonging to the multireference family is a challenging issue. The transition of $D_{2h}$ to $D_{4h}$ symmetry in H$_4$ molecule is among the most simple archetypal examples to illustrate the consequences of missing nondynamic correlation effects. The resurge of i...

Radical formation through homolytic X-H bond cleavage in LiH, BH, CH4, NH3, H2O and HF is investigated using natural orbital functional theory in its recent PNOF6 implementation, which includes inter electron-pair correlation, and the results compared to PNOF5 level of theory, CASSCF wavefunction methods and experimental data. It is observed that P...

Numerical estimates of the chemical (mu) and ionization (I) potentials from natural orbital functional (NOF) theory of one of us (MP) are presented for neutral atoms ranging from H to Kr, and compared with the corresponding experimental ionization energies. The semi-empirical treatment by Vela et al. relates mu approximately to the measured I. By a...

We suggest new strict constraints that the two-particle cumulant matrix should fulfill. The constraints are obtained from the decomposition of ⟨Ŝ(2)⟩, previously developed in our laboratory, and the vanishing number of electrons shared by two non-interacting fragments. The conditions impose stringent constraints into the cumulant structure without...

A natural orbital functional theory for non-relativistic quantum chemistry due to one of us [MP] is here utilised to calculate the von Weizsäcker inhomogeneity kinetic energy [Inline formula] at the equilibrium bond length (Re) for some 30 homonuclear diatomic molecules. Tw/N 2, where N is the total number of electrons in the molecule considered, b...

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 c...

An interacting-pair model is considered to attain a new natural orbital functional (NOF). The new approach, which will be termed PNOF6, belongs to the [Formula: see text]-only family of NOFs known in the literature as PNOFi. Accordingly, the approximated two-particle reduced density matrix (RDM) is reconstructed from the one-particle RDM, consideri...

In a recent paper (J. Chem. Phys. 139, 064111, 2013), an antisymmetrised product of strongly orthogonal geminals with the expansion coefficients explicitly expressed by means of the occupation numbers was used to generate the Piris natural orbital functional 5 (PNOF5). This functional describes most of the non-dynamical effects, but also an importa...

The longitudinal polarizability (α) and second hyperpolarizability (γ) of the H2 molecule as a function of the internuclear distance are calculated using the PNOF5 level of theory. It is shown that PNOF5 gives accurate results for the longitudinal α and γ over the whole curve range, even for those structures exhibiting a high degree of diradical ch...

A previously proposed [M. Piris, X. Lopez, F. Ruipérez, J. M. Matxain, and J. M. Ugalde, J. Chem. Phys. 134, 164102 (2011)] formulation of the two-particle cumulant, based on an orbital-pairing scheme, is extended here for including more than two natural orbitals. This new approximation is used to reconstruct the two-particle reduced density matrix...

TM@ZniSi nanoclusters have been characterized by means of the Density Functional Theory, in which Transition Metal (TM) stands from Y to Cd, and i = 12 and 16. These two nanoclusters have been chosen owing to their highly spheroidal shape which allow for favored endohedral structures as compared to other nanoclusters. Doping with TM is chosen due t...

An antisymmetrized product of strongly orthogonal geminals with the expansion coefficients explicitly expressed by the occupation numbers is used to generate the Piris natural orbital functional 5 (PNOF5). Second-order corrections to PNOF5 are derived and implemented using the multiconfigurational perturbation theory size consistent at second-order...