Electronic Vector Potential from the Exact Factorization of a Complex Wavefunction.

We generalize the definitions of local scalar potentials named vkin and vN-1, which are relevant to properly describe phenomena such as molecular dissociation with density-functional theory, to the case in which the electronic wavefunction corresponds to a complex current-carrying state. In such a case, an extra term in the form of a vector potential appears which cannot be gauged away. Both scalar and vector potentials are introduced via the exact factorization formalism which allows us to express the given Schrödinger equation as two coupled equations, one for the marginal and one for the conditional amplitude. The electronic vector potential is directly related to the paramagnetic current density carried by the total wavefunction and to the diamagnetic current density in the equation for the marginal amplitude. An explicit example of this vector potential in a triplet state of two non-interacting electrons is showcased together with its associated circulation, giving rise to a non-vanishing geometric phase. Some connections with the exact factorization for the full molecular wavefunction beyond the Born-Oppenheimer approximation are also discussed.

Optimal Transport Distances to Characterize Electronic Excitations.

Understanding the character of electronic excitations is important in computational and reaction mechanistic studies, but their classification from simulations remains an open problem. Distances based on optimal transport have proven very useful in a plethora of classification problems and, therefore, seem a natural tool to try to tackle this challenge. We propose and investigate a new diagnostic Θ based on the Sinkhorn divergence from optimal transport. We evaluate a k-NN classification algorithm on Θ, the popular Λ diagnostic, and their combination, and assess their performance in labeling excitations, finding that (i) the combination only slightly improves the classification, (ii) Rydberg excitations are not separated well in any setting, and (iii) Θ breaks down for charge transfer in small molecules. We then define a length-scale-normalized version of Θ and show that the result correlates closely with Λ for results obtained with Gaussian basis functions. Finally, we discuss the orbital dependence of our approach and explore an orbital-independent version. Using an optimized combination of the optimal transport and overlap diagnostics together with a different metric is in our opinion the most promising for future classification studies.

Møller-Plesset Adiabatic Connection at Large Coupling Strengths for Open-Shell Systems.

We study the adiabatic connection that has as weak-coupling expansion the Møller-Plesset perturbation series, generalizing to the open-shell case previous closed-shell results for the large-coupling limit. We first focus on the hydrogen atom with fractional spins, providing results along the adiabatic connection from small to large coupling strengths. We reveal an intriguing phase diagram and an equation for the large-coupling leading order that has closed-form solutions for specific choices of its relevant quantum numbers. We then show that the hydrogen atom results provide variational estimates for the large-coupling leading terms for the general many-electron open-shell case in terms of functionals of the Hartree-Fock α-spin and ß-spin densities.

Large-Z atoms in the strong-interaction limit of DFT: Implications for gradient expansions and for the Lieb-Oxford bound.

We numerically study the strong-interaction limit of the exchange-correlation functional for neutral atoms and Bohr atoms as the number of electrons increases. Using a compact representation, we analyze the second-order gradient expansion, comparing it with the one for exchange (weak interaction limit). The two gradient expansions, at strong and weak interaction, turn out to be very similar in magnitude but with opposite signs. We find that the point-charge plus continuum model is surprisingly accurate for the gradient expansion coefficient at strong coupling, while generalized gradient approximations, such as Perdew-Burke-Ernzerhof (PBE) and PBEsol, severely underestimate it. We then use our results to analyze the Lieb-Oxford bound from the point of view of slowly varying densities, clarifying some aspects on the bound at a fixed number of electrons.

Regularized and Opposite Spin-Scaled Functionals from Møller-Plesset Adiabatic Connection─Higher Accuracy at Lower Cost.

Noncovalent interactions (NCIs) play a crucial role in biology, chemistry, material science, and everything in between. To improve pure quantum-chemical simulations of NCIs, we propose a methodology for constructing approximate correlation energies by combining an interpolation along the Møller-Plesset adiabatic connection (MP AC) with a regularization and spin-scaling strategy applied to MP2 correlation energies. This combination yields cosκos-SPL2, which exhibits superior accuracy for NCIs compared to any of the individual strategies. With the N4 formal scaling, cosκos-SPL2 is competitive or often outperforms more expensive dispersion-corrected double hybrids for NCIs. The accuracy of cosκos-SPL2 particularly shines for anionic halogen bonded complexes, where it surpasses standard dispersion-corrected DFT by a factor of 3 to 5.

Electronic Excited States in Extreme Limits via Ensemble Density Functionals.

Density functional theory (DFT) has greatly expanded our ability to affordably compute and understand electronic ground states, by replacing intractable ab initio calculations by models based on paradigmatic physics from high- and low-density limits. But, a comparable treatment of excited states lags behind. Here, we solve this outstanding problem by employing a generalization of density functional theory to ensemble states (EDFT). We thus address important paradigmatic cases of all electronic systems in strongly (low-density) and weakly (high-density) correlated regimes. We show that the high-density limit connects to recent, exactly solvable EDFT results. The low-density limit reveals an unnoticed and most unexpected result-density functionals for strictly correlated ground states can be reused directly for excited states. Nontrivial dependence on excitation structure only shows up at third leading order. Overall, our results provide foundations for effective models of excited states that interpolate between exact low- and high-density limits, which we illustrate on the cases of singlet-singlet excitations in H_{2} and a ring of quantum wells.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Self-Consistent Implementation of Kohn-Sham Adiabatic Connection Models with Improved Treatment of the Strong-Interaction Limit.

Adiabatic connection models (ACMs), which interpolate between the limits of weak and strong interaction, are powerful tools to build accurate exchange-correlation functionals. If the exact weak-interaction expansion from the second-order perturbation theory is included, a self-consistent implementation of these functionals is challenging and still absent in the literature. In this work, we fill this gap by presenting a fully self-consistent-field (SCF) implementation of some popular ACM functionals. While using second-order perturbation theory at weak interactions, we have also introduced new generalized gradient approximations (GGAs), beyond the usual point-charge-plus-continuum model, for the first two leading terms at strong interactions, which are crucial to ensure robustness and reliability. We then assess the SCF-ACM functionals for molecular systems and for prototypical strong-correlation problems. We find that they perform well for both the total energy and the electronic density and that the impact of SCF orbitals is directly connected to the accuracy of the ACM functional form. For the H2 dissociation, the SCF-ACM functionals yield significant improvements with respect to standard functionals also thanks to the use of the new GGAs for the strong-coupling functionals.

Gradient Expansions for the Large-Coupling Strength Limit of the Møller-Plesset Adiabatic Connection.

The adiabatic connection that has, as weak-interaction expansion, the Møller-Plesset perturbation series has been recently shown to have a large coupling-strength expansion, in terms of functionals of the Hartree-Fock density with a clear physical meaning. In this work, we accurately evaluate these density functionals and we extract second-order gradient coefficients from the data for neutral atoms, following ideas similar to the ones used in the literature for exchange, with some modifications. These new gradient expansions will be the key ingredient for performing interpolations that have already been shown to reduce dramatically MP2 errors for large noncovalent complexes. As a byproduct, our investigation of neutral atoms with large number of electrons N indicates that the second-order gradient expansion for exchange grows as N log(N) rather than as N, as often reported in the literature.

Variations of the Hartree-Fock fractional-spin error for one electron.

Fractional-spin errors are inherent in all current approximate density functionals, including Hartree-Fock theory, and their origin has been related to strong static correlation effects. The conventional way to encode fractional-spin calculations is to construct an ensemble density that scales between the high-spin and low-spin densities. In this article, we explore the variation of the Hartree-Fock fractional-spin (or ghost-interaction) error in one-electron systems using restricted and unrestricted ensemble densities and the exact generalized Hartree-Fock representation. By considering the hydrogen atom and H+ 2 cation, we analyze how the unrestricted and generalized Hartree-Fock schemes minimize this error by localizing the electrons or rotating the spin coordinates. We also reveal a clear similarity between the Coulomb hole of He-like ions and the density depletion near the nucleus induced by the fractional-spin error in the unpolarized hydrogen atom. Finally, we analyze the effect of the fractional-spin error on the Møller-Plesset adiabatic connection, excited states, and functional- and density-driven errors.

Noncovalent Interactions from Models for the Møller-Plesset Adiabatic Connection.

Given the omnipresence of noncovalent interactions (NCIs), their accurate simulations are of crucial importance across various scientific disciplines. Here we construct accurate models for the description of NCIs by an interpolation along the Møller-Plesset adiabatic connection (MP AC). Our interpolation approximates the correlation energy, by recovering MP2 at small coupling strengths and the correct large-coupling strength expansion of the MP AC, recently shown to be a functional of the Hartree-Fock density. Our models are size consistent for fragments with nondegenerate ground states, have the same cost as double hybrids, and require no dispersion corrections to capture NCIs accurately. These interpolations greatly reduce large MP2 errors for typical π-stacking complexes (e.g., benzene-pyridine dimers) and for the L7 data set. They are also competitive with state-of-the-art dispersion enhanced functionals and can even significantly outperform them for a variety of data sets, such as CT7 and L7.

Dispersion without Many-Body Density Distortion: Assessment on Atoms and Small Molecules.

The "fixed diagonal matrices" (FDM) dispersion formalism [Kooi, D. P.; et al. J. Phys. Chem. Lett. 2019, 10, 1537] is based on a supramolecular wave function constrained to leave the diagonal of the many-body density matrix of each monomer unchanged, reducing dispersion to a balance between kinetic energy and monomer-monomer interaction. The corresponding variational optimization leads to expressions for the dispersion energy in terms of the ground-state pair densities of the isolated monomers only, providing a framework to build new approximations without the need for polarizabilities or virtual orbitals. Despite the underlying microscopic real space mechanism being incorrect, as in the exact case there is density relaxation, the formalism has been shown to give extremely accurate (or even exact) dispersion coefficients for H and He. The question we answer in this work is how accurate the FDM expressions can be for isotropic and anisotropic C6 dispersion coefficients when monomer pair densities are used from different levels of theory, namely Hartree-Fock, MP2, and CCSD. For closed-shell systems, FDM with CCSD monomer pair densities yield a mean average percent error for isotropic C6 dispersion coefficients of about 7% and a maximum absolute error within 18%, with a similar accuracy for anisotropies. The performance for open-shell systems is less satisfactory, with CCSD pair densities performing sometimes worse than Hartree-Fock or MP2. In the present implementation, the computational cost on top of the monomer's ground-state calculations is O(N4). The results show little sensitivity to the basis set used in the monomer's calculations.

Large coupling-strength expansion of the Møller-Plesset adiabatic connection: From paradigmatic cases to variational expressions for the leading terms.

We study in detail the first three leading terms of the large coupling-strength limit of the adiabatic connection that has as weak-interaction expansion the Møller-Plesset perturbation theory. We first focus on the H atom, both in the spin-polarized and the spin-unpolarized cases, reporting numerical and analytical results. In particular, we derive an asymptotic equation that turns out to have simple analytical solutions for certain channels. The asymptotic H atom solution for the spin-unpolarized case is then shown to be variationally optimal for the many-electron spin-restricted closed-shell case, providing expressions for the large coupling-strength density functionals up to the third leading order. We also analyze the H2 molecule and the uniform electron gas.

London dispersion forces without density distortion: a path to first principles inclusion in density functional theory.

We analyse a path to construct density functionals for the dispersion interaction energy from an expression in terms of the ground state densities and exchange-correlation holes of the isolated fragments. The expression is based on a constrained search formalism for a supramolecular wavefunction that is forced to leave the diagonal of the many-body density matrix of each fragment unchanged, and is exact for the interaction between one-electron densities. We discuss several aspects: the necessary features of a density functional approximation for the exchange-correlation holes of the monomers, the optimal choice of the one-electron basis (named "dispersals"), and the functional derivative with respect to monomer density variations.