Comprehensive benchmarking of density matrix functional approximations

All-Purpose Measure of Electron Correlation for Multireference Diagnostics

We present an analytical relationship between two natural orbital occupancy-based indices, I N D ¯ and INDmax, and two established electron correlation metrics: the leading term of a configuration interaction expansion, c0, and the D2 diagnostic. Numerical validation revealed that I N D ¯ and INDmax can effectively substitute for c0 and D2, respectively. These indices offer three distinct advantages: (i) they are universally applicable across all electronic structure methods, (ii) their interpretation is more intuitive, and (iii) they can be readily incorporated into the development of hybrid electronic structure methods. Additionally, we draw a distinction between correlation measures and correlation diagnostics, establishing MP2 and CCSD numerical thresholds for INDmax, which are to be used as a multireference diagnostic. Our findings further demonstrate that establishing thresholds for other electronic structure methods can be easily accomplished using small data sets.

Contactium: A strongly correlated model system

At the limit of an infinite confinement strength ω, the ground state of a system that comprises two fermions or bosons in harmonic confinement interacting through the Fermi-Huang pseudopotential remains strongly correlated. A detailed analysis of the one-particle description of this "contactium" reveals several peculiarities that are not encountered in conventional model systems (such as the two-electron harmonium atom, ballium, and spherium) involving Coulombic interparticle interactions. First of all, none of the natural orbitals (NOs) {ψn(ω;r)} of the contactium is unoccupied, which implies nonzero collective occupancies for all the angular momenta. Second, the NOs and their non-ascendingly ordered occupation numbers {νn} turn out to be related to the eigenfunctions and eigenvalues of a zero-energy Schrödinger equation with an attractive Gaussian potential. This observation enables the derivation of their properties, such as the n-4/3 asymptotic decay of νn at the n→∞ limit (which differs from that of n-8/3 in the Coulombic systems), the independence of the confinement energy vn=⟨ψn(ω;r)|12ω2r2|ψn(ω;r)⟩ of n, and the n-2/3 asymptotic decay of the respective contribution νntn to the kinetic energy. Upon suitable scaling, the weakly occupied NOs of the contactium turn out to be virtually identical to those of the two-electron harmonium atom at the ω → ∞ limit, despite the entirely different interparticle interactions in these systems.

Coupling Natural Orbital Functional Theory and Many-Body Perturbation Theory by Using Nondynamically Correlated Canonical Orbitals

We develop a new family of electronic structure methods for capturing at the same time the dynamic and nondynamic correlation effects. We combine the natural orbital functional theory (NOFT) and many-body perturbation theory (MBPT) through a canonicalization procedure applied to the natural orbitals to gain access to any MBPT approximation. We study three different scenarios: corrections based on second-order Møller-Plesset (MP2), random-phase approximation (RPA), and coupled-cluster singles doubles (CCSD). Several chemical problems involving different types of electron correlation in singlet and multiplet spin states have been considered. Our numerical tests reveal that RPA-based and CCSD-based corrections provide similar relative errors in molecular dissociation energies (D_e) to the results obtained using a MP2 correction. With respect to the MP2 case, the CCSD-based correction improves the prediction, while the RPA-based correction reduces the computational cost.

Global Natural Orbital Functional: Towards the Complete Description of the Electron Correlation

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 intrapair electron correlation, and the correlation between orbitals that make up both the pairs and the individual electrons. The interorbital correlation is composed of static and dynamic terms. The concept of dynamic part of the occupation numbers is introduced. To evaluate the accuracy achieved with the GNOF, calculation of a variety of properties is presented. They include the total energies and energy differences between the ground state and the lowest-lying excited state with different spin of atoms from H to Ne, ionization potentials of the first-row transition-metal atoms (Sc-Zn), and the total energies of a selected set of 55 molecular systems in different spin states. The GNOF is also applied to the homolytic dissociation of selected diatomic molecules in different spin states and to the rotation barrier of ethylene, both paradigmatic cases of systems with significant multiconfigurational character. The values obtained agree with those reported at high level of theory and experimental data.

How Many Electrons Does a Molecular Electride Hold?

Electrides are very peculiar ionic compounds where electrons occupy the anionic positions. In a crystal lattice, these isolated electrons often form channels or surfaces, furnishing electrides with many traits with promising technological applications. Despite their huge potential, thus far, only a few stable electrides have been produced because of the intricate synthesis they entail. Due to the difficulty in assessing the presence of isolated electrons, the characterization of electrides also poses some serious challenges. In fact, their properties are expected to depend on the arrangement of these electrons in the molecule. Among the criteria that we can use to characterize electrides, the presence of a non-nuclear attractor (NNA) of the electron density is both the rarest and the most salient feature. Therefore, a correct description of the NNA is crucial to determine the properties of electrides. In this paper, we analyze the NNA and the surrounding region of nine molecular electrides to determine the number of isolated electrons held in the electride. We have seen that the correct description of a molecular electride hinges on the electronic structure method employed for the analyses. In particular, one should employ a basis set with sufficient flexibility to describe the region close to the NNA and a density functional approximation that does not suffer from large delocalization errors. Finally, we have classified these nine molecular electrides according to the most likely number of electrons that we can find in the NNA. We believe this classification highlights the strength of the electride character and will prove useful in designing new electrides.

Singling Out Dynamic and Nondynamic Correlation

The correlation part of the pair density is separated into two components, one of them being predominant at short electronic ranges and the other at long ranges. The analysis of the intracular part of these components permits to classify molecular systems according to the prevailing correlation: dynamic or nondynamic. The study of the long-range asymptotics reveals the key component of the pair density that is responsible for the description of London dispersion forces and a universal decay with the interelectronic distance. The natural range-separation, the identification of the dispersion forces, and the kind of predominant correlation type that arise from this analysis are expected to be important assets in the development of new electronic structure methods in wave function, density, and reduced density-matrix functional theories.

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.

Ozonolysis of 3-carene in the atmosphere. Formation mechanism of hydroxyl radical and secondary ozonides

The gas-phase ozonolysis mechanism of 3-carene is investigated using high level quantum chemistry and kinetic calculations. The reaction follows the Criegee mechanism with an initial addition of O₃ to the [double bond splayed left]C[double bond, length as m-dash]C[double bond splayed right] bond, followed by a chain of unimolecular isomerizations, as 3-carene + O₃→ POZs (primary ozonides) → CIs (Criegee intermediates, 4 conformers) → Ps (products). In the course of the reaction, a large excess of energy retained in the POZs* lead to the prompt unimolecular processes in POZs*, CIs*, and Ps*, and only ∼4% of CIs* could be stabilized by collision at 298 K and 760 Torr. From RRKM-ME calculations, the VHPs* could further dissociate to vinoxy-type radical and OH radical, the SOZs* could isomerize to 3-caronic acid, and DIOs* could be stabilized via collision. The fractional yield of OH radical, in the range of 0.56 to 0.59, agrees reasonably well with the previously measured value of 1.06 (with an uncertainty factor of 1.5).

λ-Density Functional Valence Bond: A Valence Bond-Based Multiconfigurational Density Functional Theory With a Single Variable Hybrid Parameter

A new valence bond (VB)-based multireference density functional theory (MRDFT) method, named λ-DFVB, is presented in this paper. The method follows the idea of the hybrid multireference density functional method theory proposed by Sharkas et al. (2012). λ-DFVB combines the valence bond self-consistent field (VBSCF) method with Kohn–Sham density functional theory (KS-DFT) by decomposing the electron–electron interactions with a hybrid parameter λ. Different from the Toulouse's scheme, the hybrid parameter λ in λ-DFVB is variable, defined as a function of a multireference character of a molecular system. Furthermore, the E_C correlation energy of a leading determinant is introduced to ensure size consistency at the dissociation limit. Satisfactory results of test calculations, including potential energy surfaces, bond dissociation energies, reaction barriers, and singlet–triplet energy gaps, show the potential capability of λ-DFVB for molecular systems with strong correlation.

The Coulomb Hole of the Ne Atom

We analyze the Coulomb hole of Ne from highly-accurate CISD wave functions obtained from optimized even-tempered basis sets. Using a two-fold extrapolation procedure we obtain highly accurate results that recover 97 % of the correlation energy. We confirm the existence of a shoulder in the short-range region of the Coulomb hole of the Ne atom, which is due to an internal reorganization of the K-shell caused by electron correlation of the core electrons. The feature is very sensitive to the quality of the basis set in the core region and it is not exclusive to Ne, being also present in most of second-row atoms, thus confirming that it is due to K-shell correlation effects.

Revitalizing the concept of bond order through delocalization measures in real space

Ab initio quantum chemistry is an independent source of information supplying an ever widening group of experimental chemists. However, bridging the gap between these ab initio data and chemical insight remains a challenge. In particular, there is a need for a bond order index that characterizes novel bonding patterns in a reliable manner, while recovering the familiar effects occurring in well-known bonds. In this article, through a large body of calculations, we show how the delocalization index derived from Quantum Chemical Topology (QCT) serves as such a bond order. This index is defined in a parameter-free, intuitive and consistent manner, and with little qualitative dependency on the level of theory used. The delocalization index is also able to detect the subtler bonding effects that underpin most practical organic and inorganic chemistry. We explore and connect the properties of this index and open the door for its extensive usage in the understanding and discovery of novel chemistry.

Electron-Pair Distribution in Chemical Bond Formation

The chemical formation process has been studied from relaxation holes, Δh(u), resulting from the difference between the radial intracule density and the nonrelaxed counterpart, which is obtained from atomic radial intracule densities and the pair density constructed from the overlap of the atomic densities. Δh(u) plots show that the internal reorganization of electron pairs prior to bond formation and the covalent bond formation from electrons in separate atoms are completely recognizable processes from the shape of the relaxation hole, Δh(u). The magnitude of Δh(u), the shape of Δh(u) ∀ u < R_eq, and the distance between the minimum and the maximum in Δh(u) provide further information about the nature of the chemical bond formed. A computational affordable approach to calculate the radial intracule density from approximate pair densities has been also suggested, paving the way to study electron-pair distributions in larger systems.

The role of topology in organic molecules: origin and comparison of the radical character in linear and cyclic oligoacenes and related oligomers

We discuss the nature of electron-correlation effects in carbon nanorings and nanobelts by a combined approach based on FT-DFT and RAS-SF methods.