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Reduced density matrix functional theory (RDMFT) and coupled cluster theory restricted to paired double excitations (pCCD) are emerging as efficient methodologies for accounting for the so-called non-dynamic electronic correlation effects. Up to now, molecular calculations have been performed with real-valued orbitals. However, before extending the applicability of these methodologies to extended systems, where Bloch states are employed, the subtleties of working with complex-valued orbitals and the consequences of imposing time-reversal symmetry must be carefully addressed. In this work, we describe the theoretical and practical implications of adopting time-reversal symmetry in RDMFT and pCCD when allowing for complex-valued orbital coefficients. The theoretical considerations primarily affect the optimization algorithms, while the practical implications raise fundamental questions about the stability of solutions. Specifically, we find that complex solutions lower the energy when non-dynamic electronic correlation effects are pronounced. We present numerical examples to illustrate and discuss these instabilities and possible problems introduced by N-representability violations.

We present an original multi-state projective diabatization scheme based on the Green's function formalism that allows the systematic mapping of many-body ab initio calculations onto effective excitonic models. This method inherits the ability of the Bethe-Salpeter equation to describe Frenkel molecular excitons and intermolecular charge-transfer states equally well, as well as the possibility for an effective description of environmental effects in a QM/MM framework. The latter is found to be a crucial element in order to obtain accurate model parameters for condensed phases and to ensure their transferability to excitonic models for extended systems. The method is presented through a series of examples illustrating its quality, robustness, and internal consistency.

Due to the infinite summation of bubble diagrams, the $GW$ approximation of Green's function perturbation theory has proven particularly effective in the weak correlation regime, where this family of Feynman diagrams is important. However, the performance of $GW$ in multireference molecular systems, characterized by strong electron correlation, remains relatively unexplored. In the present study, we investigate the ability of $GW$ to handle closed-shell multireference systems in their singlet ground state by examining four paradigmatic scenarios. Firstly, we analyze a prototypical example of a chemical reaction involving strong correlation: the potential energy curve of \ce{BeH2} during the insertion of a beryllium atom into a hydrogen molecule. Secondly, we compute the electron detachment and attachment energies of a set of molecules that exhibit a variable degree of multireference character at their respective equilibrium geometries: \ce{LiF}, \ce{BeO}, \ce{BN}, \ce{C2}, \ce{B2}, and \ce{O3}. Thirdly, we consider a \ce{H6} cluster with a triangular arrangement, which features a notable degree of spin frustration. Finally, the dissociation curve of the \ce{HF} molecule is studied as an example of single bond breaking. These investigations highlight a nuanced perspective on the performance of $GW$ for strong correlation, depending on the level of self-consistency, the choice of initial guess, and the presence of spin-symmetry breaking at the Hartree-Fock level., Comment: 11 pages, 4 figures

The accurate computation of non-linear optical properties (NLOPs) in large polymers requires accounting for electronic correlation effects with a reasonable computational cost. The Random Phase Approximation (RPA) used in the adiabatic connection fluctuation theorem is known to be a reliable and cost-effective method to render electronic correlation effects when combined with the density-fitting techniques and the integration over imaginary frequencies. We explore the ability of the RPA energy expression to predict NLOPs by evaluating RPA electronic energies in the presence of finite electric fields to obtain (using the finite difference method) static polarizabilities and hyper-polarizabilities. We show that RPA based on hybrid functional self-consistent field calculations yields as accurate NLOPs as the best-tuned double-hybrid functionals developed today, with the additional advantage that RPA avoids any system-specific adjustment.

The $GW$ approximation is a well-established method for calculating ionization potentials and electron affinities in solids and molecules. For numerous years, obtaining self-consistent $GW$ total energies in solids has been a challenging objective that is not accomplished yet. However, it was shown recently that the linearized $GW$ density matrix permits a reliable prediction of the self-consistent $GW$ total energy for molecules [F. Bruneval et. al. J. Chem. Theory Comput. 17, 2126 (2021)] for which self-consistent $GW$ energies are available. Here we implement, test, and benchmark the linearized $GW$ density matrix for several solids. We focus on the total energy, lattice constant, and bulk modulus obtained from the $GW$ density matrix and compare our findings to more traditional results obtained within the random phase approximation (RPA). We conclude on the improved stability of the total energy obtained from the linearized $GW$ density matrix with respect to the mean-field starting point. We bring compelling clues that the RPA and the $GW$ density matrix total energies are certainly close to the self-consistent $GW$ total energy in solids if we use hybrid functionals with enriched exchange as a starting point., Comment: 16 pages, 8 figures

While the effect of relativity in the electronic density has been widely studied, the effect on the pair probability, intracule, and extracule densities has not been studied before. Thus, in this work, we unveil new insights related to changes on the electronic structure caused by relativistic effects. Our numerical results suggest that the mean inter-electronic distance is reduced (mostly) due to scalar-relativistic effects. As a consequence, an increase of the electron-electron repulsion energy is observed. Preliminary results suggest that this observation is also valid when electronic correlation effects are considered.

A range-separation of the Coulomb hole into two components, one of them being predominant at long interelectronic separations (hcI ) and the other at short distances (hcII ), is exhaustively analyzed throughout various examples that put forward the most relevant features of this approach and how they can be used to develop efficient ways to capture electron correlation. We show that hcI , which only depends on the first-order reduced density matrix, can be used to identify molecules with a predominant nondynamic correlation regime and differentiate between two types of nondynamic correlation, types A and B. Through the asymptotic properties of the hole components, we explain how hcI can retrieve the long-range part of electron correlation. We perform an exhaustive analysis of the hydrogen molecule in a minimal basis set, dissecting the hole contributions into spin components. We also analyze the simplest molecule presenting a dispersion interaction and how hcII helps identify it. The study of several atoms in different spin states reveals that the Coulomb hole components distinguish correlation regimes that are not apparent from the entire hole. The results of this work hold the promise to aid in developing new electronic structure methods that efficiently capture electron correlation.

Nonlinear optical properties (NLOPs) play a major role in photonics, electro-optics and optoelectronics, and other fields of modern optics. The design of new NLO molecules and materials has benefited from the development of computational tools to analyze the relationship between the electronic structure of molecules and its optical response. In this paper, we present a new means to analyze the response property through the partition of NLOPs in terms of orbital contributions (PNOC). This tool can be used to obtain a local representation of the NLOPs, providing a powerful visualization aid to connect the magnitude of the optical property with some parts of the molecule. Unlike other methods to analyze NLOPs, the PNOC decomposes the optical property into orbitals of the unperturbed system, furnishing this method with the ability to assess the performance of single- and multi-determinant electronic structure methods. PNOC can be also used to design small basis sets for an accurate description of large systems, saving a substantial amount of computer time for the calculation of optical properties.

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., Comment: This document is the unedited Author's version of a Submitted Work that was subsequently accepted for publication in J. Phys. Chem. Lett., copyright American Chemical Society after peer review. To access the final edited and published work see http://doi.org/10.1021/acs.jpclett.9b01376

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 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., Comment: 9 pages, 6 figures

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 wavefunction into a functional of the 1RDM, known as the top-down method. Here, we show that the phase dilemma also appears in the bottom-up method, in which the functional E[1RDM] is generated by progressive inclusion of N-representability conditions on the reconstructed two-particle reduced density matrix. It is shown that an adequate choice of signs is essential to accurately describe model systems with strong non-dynamic (static) electron correlation, specifically, the one-dimensional Hubbard model with periodic boundary conditions and hydrogen rings. For the latter, the Piris natural orbital functional 7 (PNOF7), with phases equal to -1 for the inter-pair energy terms containing the exchange-time-inversion integrals, agrees with exact diagonalization results., Comment: 7 pages, 3 Figures, Special issue in honor of Hardy Gross

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., Comment: 7 pages, 6 figures, 1 table

van der Waals interactions govern the physics of a plethora of molecular structures. It is well known that the leading term in the distance-based London expansion of the van der Waals energy for atomic and molecular dimers decays as $1/R^6$, where $R$ is the dimer distance. Using perturbation theory, we find the leading term in the distance-based expansion of the intracule pair density at the interatomic distance. Our results unveil a universal $1/R^3$ decay, which is less prone to numerical errors than the $1/R^6$ dependency, and it is confirmed numerically in H$_2$ and He$_2$ molecules. This \textit{signature} of van der Waals interactions can be directly used in the construction of approximate pair density and energy functionals including vdW corrections., Comment: 4 pages, 3 figures

The lowest-lying states of LiH have been widely used to develop and calibrate many different methods in quantum mechanics. In this paper we show that the electron-transfer processes occurring in these two states are a difficult test for chemical bonding descriptors and can be used to assess new bonding descriptors on its ability to recognize the harpoon mechanism. To this aim, we study the bond formation mechanism in a series of diatomic molecules. In all studied electron-reorganization mechanisms, the maximal electron-transfer variation point along the bond formation path occurs when about half electron has been transferred from one atom to another. If the process takes places through a harpoon mechanism, this point of the reaction path coincides with the avoided crossing. The electron sharing indices and one-dimensional plots of the electron localization function and the Laplacian of the electron density along the molecular axis can be used to monitor the bond formation in diatomics and provide a distinction between the harpoon mechanism and a regular electron-reorganization process., Comment: 21 pages, 9 figures