A new approach for the two-electron cumulant in natural orbital functional theory

Variational minimization scheme for the one-particle reduced density matrix functional theory in the ensemble N-representability domain

The one-particle reduced density-matrix (1-RDM) functional theory is a promising alternative to density-functional theory (DFT) that uses the 1-RDM rather than the electronic density as a basic variable. However, long-standing challenges such as the lack of the Kohn-Sham scheme and the complexity of the pure N-representability conditions are still impeding its wild utilization. Fortunately, ensemble N-representability conditions derived in the natural orbital basis are known and trivial such that almost every functional of the 1-RDM is actually natural orbital functional, which does not perform well for all the correlation regimes. In this work, we propose a variational minimization scheme in the ensemble N-representable domain that is not restricted to the natural orbital representation of the 1-RDM. We show that splitting the minimization into the diagonal and off-diagonal parts of the 1-RDM can open the way toward the development of functionals of the orbital occupations, which remains a challenge for the generalization of site-occupation functional theory in chemistry. Our approach is tested on the uniform Hubbard model using the Müller and the Töws-Pastor functionals, as well as on the dihydrogen molecule using the Müller functional.

Softmax parameterization of the occupation numbers for natural orbital functionals based on electron pairing approaches

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 in Piris natural orbital functionals through the adoption of the softmax function, a pivotal component in modern deep-learning models. Our approach not only ensures adherence to the N-representability of the first-order reduced density matrix (1RDM) but also significantly enhances the computational efficiency of 1RDM functional theory calculations. The effectiveness of this alternative parameterization approach was assessed using the W4-17-MR molecular set, which demonstrated faster and more robust convergence compared to previous implementations.

Exploiting the Hessian for a Better Convergence of the SCF-RDMFT Procedure

One-body reduced density matrix functional theory provides an alternative to density functional theory, which is able to treat static correlation while keeping a relatively low computation scaling. Its disadvantageous cost comes mainly from a slow convergence of the self-consistent energy optimization. To improve on that problem, we propose in this work the use of the Hessian of the energy, including the coupling term. We show that using the exact Hessian is very effective at reducing the number of iterations. However, since the exact Hessian is too expensive to use in practice, we propose an approximation based on an inexpensive exact part and BFGS updates.

del Campo J. Excited States by Coupling Piris Natural Orbital Functionals with the Extended Random-Phase Approximation

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 PNOF-ERPA2, according to the way the excitation operator is built. The implementation has been tested in the first excited states of H2, HeH+, LiH, Li2, and N2 showing good results compared to the configuration interaction (CI) method. As expected, an increase in accuracy is observed on going from ERPA0 to ERPA1 and ERPA2. We also studied the effect of electron correlation included by PNOF5, PNOF7, and the recently proposed global NOF (GNOF) on the predicted excited states. PNOF5 appears to be good and may even provide better results in very small systems, but including more electron correlation becomes important as the system size increases, where GNOF achieves better results. Overall, the extension of PNOF to excited states has been successful, making it a promising method for further applications.

Self-consistent field method for open-shell systems within the density-matrix functional theory

The unrestricted Hartree-Fock method is extended to correlation calculation within the density-matrix functional theory. The method is derived from an entropic cumulant functional for the correlation energy. The eigenvalue equations for the spin-orbitals are modified by the orbital occupation numbers. The Euler equation for the occupation numbers results in the Fermi-Dirac distribution, which is very efficient to update as soon as the orbital eigenvalue equations are solved. The method is demonstrated on the ground state of O2 .

Effective local potentials for density and density-matrix functional approximations with non-negative screening density

A way to improve the accuracy of the spectral properties in density functional theory (DFT) is to impose constraints on the effective, Kohn-Sham (KS), local potential [J. Chem. Phys. 136, 224109 (2012)]. As illustrated, a convenient variational quantity in that approach is the "screening" or "electron repulsion" density, ρrep, corresponding to the local, KS Hartree, exchange and correlation potential through Poisson's equation. Two constraints, applied to this minimization, largely remove self-interaction errors from the effective potential: (i) ρrep integrates to N - 1, where N is the number of electrons, and (ii) ρrep ≥ 0 everywhere. In this work, we introduce an effective "screening" amplitude, f, as the variational quantity, with the screening density being ρrep = f2. In this way, the positivity condition for ρrep is automatically satisfied, and the minimization problem becomes more efficient and robust. We apply this technique to molecular calculations, employing several approximations in DFT and in reduced density matrix functional theory. We find that the proposed development is an accurate, yet robust, variant of the constrained effective potential method.

del Campo J. Charge delocalization error in Piris Natural Orbital Functionals

Piris Natural Orbital Functionals (PNOF) have been recognized as a low-scaling alternative to study strong correlated systems. In this work, we address the performance of the fifth functional (PNOF5) and the seventh functional (PNOF7) to deal with another common problem, the charge delocalization error. The effects of this problem can be observed in charged systems of repeated well-separated fragments, where the energy should be the sum of the charged and neutral fragments, regardless of how the charge is distributed. In practice, an energetic overstabilization of fractional charged fragments leads to a preference for having the charge delocalized throughout the system. To establish the performance of PNOF functionals regarding charge delocalization error, charged chains of helium atoms and the W4-17-MR set molecules were used as base fragments and their energy, charge distribution and correlation regime were studied. It was found that PNOF5 prefers localized charge distributions, while PNOF7 improves the treatment of interpair static correlation and tends to the correct energetic limit for several cases, although a preference for delocalized charge distributions may arise in highly strong correlation regimes. Overall, it is concluded that PNOF functionals can simultaneously deal with static correlation and charge delocalization errors, resulting in a promising choice to study charge-related problems.

Benchmarking GNOF against FCI in challenging systems in one, two, and three dimensions

This work assesses the reliability of the recently proposed [M. Piris, Phys. Rev. Lett. 127, 233001 (2021)] global natural orbital functional (GNOF) in the treatment of the strong electron correlation regime. First, we use an H10 benchmark set of four hydrogen model systems of different dimensionalities and distinctive electronic structures: a 1D chain, a 2D ring, a 2D sheet, and a 3D close-packed pyramid. Second, we study two paradigmatic models for strongly correlated Mott insulators, namely, a 1D H50 chain and a 4 × 4 × 4 3D H cube. We show that GNOF, without hybridization to other electronic structure methods and free of tuned parameters, succeeds in treating weak and strong correlation in a more balanced way than the functionals that have preceded it.

Density Functional Theory Transformed into a One-Electron Reduced-Density-Matrix Functional Theory for the Capture of Static Correlation

Density Functional Theory (DFT), the most widely adopted method in modern computational chemistry, fails to describe accurately the electronic structure of strongly correlated systems. Here we show that DFT can be formally and practically transformed into a one-electron reduced-density-matrix (1-RDM) functional theory, which can address the limitations of DFT while retaining favorable computational scaling compared to wave function based approaches. In addition to relaxing the idempotency restriction on the 1-RDM in the kinetic energy term, we add a quadratic 1-RDM-based term to DFT's density-based exchange-correlation functional. Our approach, which we implement by quadratic semidefinite programming at DFT's computational scaling of O(r³), yields substantial improvements over traditional DFT in the description of static correlation in chemical structures and processes such as singlet biradicals and bond dissociations.

Self-Consistent-Field Method for Correlated Many-Electron Systems with an Entropic Cumulant Energy

A self-consistent field method is presented within density matrix functional theory. The computational cost for a correlated many-electron calculation is reduced to that of the self-consistent-field Hartree-Fock method, while the accuracy still reaches that of sophisticated configuration interaction based methods. In this method, the two-electron cumulant energy is measured with an information entropy associated with the Fermi-Dirac distribution of the occupation numbers. An eigenvalue equation for the orbitals is obtained, with the eigenvalues (orbital energies) connected to the occupation numbers through the Fermi-Dirac distribution. The occupation numbers for the strongly occupied orbitals are very close to the natural orbital occupation numbers from wave function methods. It covers in a single scheme the nondynamical correlation in weak or breaking bonds as well as the dynamical correlation at all distances. The method is well suited to large-scale potential energy surface calculation and molecular dynamics simulation.

Density inversion method for local basis sets without potential auxiliary functions: inverting densities from RDMFT

A density inversion method is presented, to obtain the constrained, optimal, local potential that has a prescribed asymptotic behaviour and reproduces optimally any given ground-state electronic density. This work builds...

Resolution of the identity approximation applied to PNOF correlation calculations

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 significant computational time savings. After the DoNOF-RI calculation has fully converged, a restart with four-center electron repulsion integrals can be performed to remove the effect of the auxiliary basis set incompleteness, quickly converging to the exact result. The proposed approach has been tested on cycloalkanes and other molecules of general interest to study the numerical results, as well as the speed-ups achieved by PNOF7-RI when compared with PNOF7.

Quantum Information and Algorithms for Correlated Quantum Matter

Discoveries in quantum materials, which are characterized by the strongly quantum-mechanical nature of electrons and atoms, have revealed exotic properties that arise from correlations. It is the promise of quantum materials for quantum information science superimposed with the potential of new computational quantum algorithms to discover new quantum materials that inspires this Review. We anticipate that quantum materials to be discovered and developed in the next years will transform the areas of quantum information processing including communication, storage, and computing. Simultaneously, efforts toward developing new quantum algorithmic approaches for quantum simulation and advanced calculation methods for many-body quantum systems enable major advances toward functional quantum materials and their deployment. The advent of quantum computing brings new possibilities for eliminating the exponential complexity that has stymied simulation of correlated quantum systems on high-performance classical computers. Here, we review new algorithms and computational approaches to predict and understand the behavior of correlated quantum matter. The strongly interdisciplinary nature of the topics covered necessitates a common language to integrate ideas from these fields. We aim to provide this common language while weaving together fields across electronic structure theory, quantum electrodynamics, algorithm design, and open quantum systems. Our Review is timely in presenting the state-of-the-art in the field toward algorithms with nonexponential complexity for correlated quantum matter with applications in grand-challenge problems. Looking to the future, at the intersection of quantum information science and algorithms for correlated quantum matter, we envision seminal advances in predicting many-body quantum states and describing excitonic quantum matter and large-scale entangled states, a better understanding of high-temperature superconductivity, and quantifying open quantum system dynamics.

Interacting Quantum Atoms-A Review

The aim of this review is threefold. On the one hand, we intend it to serve as a gentle introduction to the Interacting Quantum Atoms (IQA) methodology for those unfamiliar with it. Second, we expect it to act as an up-to-date reference of recent developments related to IQA. Finally, we want it to highlight a non-exhaustive, yet representative set of showcase examples about how to use IQA to shed light in different chemical problems. To accomplish this, we start by providing a brief context to justify the development of IQA as a real space alternative to other existent energy partition schemes of the non-relativistic energy of molecules. We then introduce a self-contained algebraic derivation of the methodological IQA ecosystem as well as an overview of how these formulations vary with the level of theory employed to obtain the molecular wavefunction upon which the IQA procedure relies. Finally, we review the several applications of IQA as examined by different research groups worldwide to investigate a wide variety of chemical problems.

An efficient method for strongly correlated electrons in one dimension

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 the 1D Hubbard model with periodic boundary conditions is provided. Different filling situations and large sizes with up to 122 electrons are considered. Compared to quasi-exact results, PNOF7 is accurate in different correlation regimes for the 1D Hubbard model even away from the half-filling, and maintains its accuracy when the system size increases. The symmetric and asymmetric dissociations of the linear H chain composed of 50 atoms are described to remark the importance of long-range interactions in presence of strong correlation effects. Our results compare remarkably well with those obtained at the density-matrix renormalization group level of theory.

Recent Progress in Lattice Density Functional Theory

Recent developments in the density-functional theory of electron correlations in many-body lattice models are reviewed. The theoretical framework of lattice density-functional theory (LDFT) is briefly recalled, giving emphasis to its universality and to the central role played by the single-particle density-matrix γ . The Hubbard model and the Anderson single-impurity model are considered as relevant explicit problems for the applications. Real-space and reciprocal-space approximations to the fundamental interaction-energy functional W [ γ ] are introduced, in the framework of which the most important ground-state properties are derived. The predictions of LDFT are contrasted with available exact analytical results and state-of-the-art numerical calculations. Thus, the goals and limitations of the method are discussed.

Natural orbital functional for spin-polarized periodic systems

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 explicit form for the two-electron cumulant of high-spin cases is only taken into account, so that singly occupied orbitals with the same spin are solely considered. The rest of the electron pairs with opposite spins remain paired. The reconstructed two-particle reduced density matrix fulfills certain N-representability necessary conditions, as well as guarantees the conservation of the total spin. The theory is applied to model systems with strong non-dynamic (static) electron correlation, namely, the one-dimensional Hubbard model with periodic boundary conditions and hydrogen rings. For the latter, PNOF7 compares well with exact diagonalization results so the model presented here is able to provide a correct description of the strong-correlation effects.

Diverging Exchange Force and Form of the Exact Density Matrix Functional

For translationally invariant one-band lattice models, we exploit the ab initio knowledge of the natural orbitals to simplify reduced density matrix functional theory (RDMFT). Striking underlying features are discovered. First, within each symmetry sector, the interaction functional F depends only on the natural occupation numbers n. The respective sets P_{N}^{1} and E_{N}^{1} of pure and ensemble N-representable one-matrices coincide. Second, and most importantly, the exact functional is strongly shaped by the geometry of the polytope E_{N}^{1}≡P_{N}^{1}, described by linear constraints D^{(j)}(n)≥0. For smaller systems, it follows as F[n]=[under ∑]i,i^{'}V[over ¯]_{i,i^{'}}sqrt[D^{(i)}(n)D^{(i^{'})}(n)]. This generalizes to systems of arbitrary size by replacing each D^{(i)} by a linear combination of {D^{(j)}(n)} and adding a nonanalytical term involving the interaction V[over ^]. Third, the gradient dF/dn is shown to diverge on the boundary ∂E_{N}^{1}, suggesting that the fermionic exchange symmetry manifests itself within RDMFT in the form of an "exchange force." All findings hold for systems with a nonfixed particle number as well and V[over ^] can be any p-particle interaction. As an illustration, we derive the exact functional for the Hubbard square.

Systematic High-Accuracy Prediction of Electron Affinities for Biological Quinones

Quinones play vital roles as electron carriers in fundamental biological processes; therefore, the ability to accurately predict their electron affinities is crucial for understanding their properties and function. The increasing availability of cost-effective implementations of correlated wave function methods for both closed-shell and open-shell systems offers an alternative to density functional theory approaches that have traditionally dominated the field despite their shortcomings. Here, we define a benchmark set of quinones with experimentally available electron affinities and evaluate a range of electronic structure methods, setting a target accuracy of 0.1 eV. Among wave function methods, we test various implementations of coupled cluster (CC) theory, including local pair natural orbital (LPNO) approaches to canonical and parameterized CCSD, the domain-based DLPNO approximation, and the equations-of-motion approach for electron affinities, EA-EOM-CCSD. In addition, several variants of canonical, spin-component-scaled, orbital-optimized, and explicitly correlated (F12) Møller-Plesset perturbation theory are benchmarked. Achieving systematically the target level of accuracy is challenging and a composite scheme that combines canonical CCSD(T) with large basis set LPNO-based extrapolation of correlation energy proves to be the most accurate approach. Methods that offer comparable performance are the parameterized LPNO-pCCSD, the DLPNO-CCSD(T₀ ), and the orbital optimized OO-SCS-MP2. Among DFT methods, viable practical alternatives are only the M06 and the double hybrids, but the latter should be employed with caution because of significant basis set sensitivity. A highly accurate yet cost-effective DLPNO-based coupled cluster approach is used to investigate the methoxy conformation effect on the electron affinities of ubiquinones found in photosynthetic bacterial reaction centers. © 2018 Wiley Periodicals, Inc.

Density functional approximations for orbital energies and total energies of molecules and solids

The relation of Kohn-Sham (KS) orbital energies to ionization energies and electron affinities is different in molecules and solids. In molecules, the local density approximation (LDA) and generalized gradient approximations (GGA) approximate the exact ionization energy (I) and affinity (A) rather well with self-consistently calculated (total energy based) I^LDFA and A^LDFA, respectively. The highest occupied molecular orbital (HOMO) energy and lowest unoccupied molecular orbital (LUMO) energy, however, differ significantly (by typically 4-6 eV) from these quantities, ϵ_H^LDFA(mol)>-I(mol)≈-I^LDFA(mol), ϵ_L^LDFA(mol)<-A(mol)≈-A^LDFA(mol). In solids, these relations are very different, due to two effects. The (almost) infinite extent of a solid makes the difference of orbital energies and (L)DFA calculated ionization energy and affinity disappear: in the solid state limit, ϵ_H^(L)DFA(solid)=-I^(L)DFA(solid) and ϵ_L^(L)DFA(solid)=-A^(L)DFA(solid). Slater's relation ∂E/∂n_i = ϵ_i for local density functional approximations (LDFAs) [and Hartree-Fock (HF) and hybrids] is useful to prove these relations. The equality of LDFA orbital energies and LDFA calculated -I^LDFA and -A^LDFA in solids does not mean that they are good approximations to the exact quantities. The LDFA total energies of the ions with a delocalized charge are too low, hence I^LDFA(solid) < I and A^LDFA(solid) > A, due to the local-approximation error, also denoted delocalization error, of LDFAs in extended systems. These errors combine to make the LDFA orbital energy band gap considerably smaller than the exact fundamental gap, ϵ_L^LDFA(solid)-ϵ_H^LDFA(solid)=I^LDFA(solid)-A^LDFA(solid)<I-A (the LDFA band gap problem). These results for density functional approximations are compared to exact KS and to HF and hybrids. For the exact KS HOMO energy, one has ϵ_H^KS=-I. The exact KS LUMO energy does not approximate the experimental -A (neither in molecules nor in solids), but is considerably below, which is the main reason for the exact KS HOMO-LUMO energy gap being considerably below the fundamental gap I - A (the exact KS band gap problem).

The correct and unusual coordinate transformation rules for electromagnetic quadrupoles

Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat space-time due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordinate transformation and an integral. Transformations involving integrals have not been seen before. This is significantly different from the familiar transformation rules for a dipole, where the components transform as tensors. It enables quadrupoles to be correctly defined in general relativity and to prescribe the equations of motion for a quadrupole in a coordinate system adapted to its motion and then transform them to the laboratory coordinates. An example is given of another unusual feature: a quadrupole which is free of dipole terms in polar coordinates has dipole terms in Cartesian coordinates. It is shown that dipoles, electric dipoles, quadrupoles and electric quadrupoles can be defined without reference to a metric and in a coordinates-free manner. This is particularly useful given their complicated coordinate transformation.

Comprehensive benchmarking of density matrix functional approximations

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 approximations with a wider range of applicability. In this study, we suggest a battery of ten tests with the aim to analyze density matrix functional approximations (DMFAs), including several properties that the exact functional should satisfy. The tests are performed on a model system with varying electron correlation, carrying a very small computational effort. Our results not only put forward a complete and exhaustive benchmark test for DMFAs, currently lacking, but also reveal serious deficiencies of existing approximations that lead to important clues in the construction of more robust DMFAs.

On the performance of natural orbital functional approximations in the Hubbard model

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 highlight the importance of imposing ensemble N-representability conditions in order to obtain consistent results in systems with either weak or strong electronic correlation, such as the Hubbard system with a varying two-particle interaction parameter. Based on the accuracy of the results obtained using PNOF7, which retrieves a large amount of the total strong nondynamic correlation, the Hubbard model points out that N-representability gives solid foundations for NOF development.

Global Method for Electron Correlation

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.

Analytic gradients for natural orbital functional theory

The analytic energy gradients with respect to nuclear motion are derived for the natural orbital functional (NOF) theory. The resulting equations do not require resorting 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 systems, composed of first- and second-row atoms, are optimized employing the conjugate gradient algorithm. As functionals, two orbital-pairing approaches were used, namely, the fifth and sixth Piris NOFs (PNOF5 and PNOF6). For the latter, the obtained equilibrium geometries are compared with coupled cluster singles and doubles calculations and accurate empirical data.

Reliability of interacting quantum atoms (IQA) data computed from post-HF densities: impact of the approximation used

The BBC1 approximation is recommended for IQA calculations; MP2/BBC1 and CCSD/BBC1 produced highly comparable FAMSEC-based interpretations of intramolecular interactions.

Molecular electric moments calculated by using natural orbital functional theory

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 the PNOF6 is established by carrying out a statistical analysis of the mean absolute errors with respect to the experiment. The calculated PNOF6 electric moments agree satisfactorily with the corresponding experimental data and are in good agreement with the values obtained by accurate ab initio methods, namely, the coupled-cluster single and doubles and multi-reference single and double excitation configuration interaction methods.

H4: A challenging system for natural orbital functional approximations

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 H4 molecule is among the most simple archetypal examples to illustrate the consequences of missing nondynamic correlation effects. The resurgence of interest in density matrix functional methods has brought several new methods including the family of Piris Natural Orbital Functionals (PNOF). In this work, we compare PNOF5 and PNOF6, which include nondynamic electron correlation effects to some extent, with other standard ab initio methods in the H4 D(4h)/D(2h) potential energy surface (PES). Thus far, the wrongful behavior of single-reference methods at the D(2h)-D(4h) transition of H4 has been attributed to wrong account of nondynamic correlation effects, whereas in geminal-based approaches, it has been assigned to a wrong coupling of spins and the localized nature of the orbitals. We will show that actually interpair nondynamic correlation is the key to a cusp-free qualitatively correct description of H4 PES. By introducing interpair nondynamic correlation, PNOF6 is shown to avoid cusps and provide the correct smooth PES features at distances close to the equilibrium, total and local spin properties along with the correct electron delocalization, as reflected by natural orbitals and multicenter delocalization indices.

Natural occupation numbers in two-electron quantum rings

Natural orbitals (NOs) are central constituents for evaluating correlation energies through efficient approximations. Here, we report the closed-form expression of the NOs of two-electron quantum rings, which are prototypical finite-extension systems and new starting points for the development of exchange-correlation functionals in density functional theory. We also show that the natural occupation numbers for these two-electron paradigms are in general non-vanishing and follow the same power law decay as atomic and molecular two-electron systems.