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.

Exact Excited-State Functionals of the Asymmetric Hubbard Dimer.

The exact functionals associated with the (singlet) ground state and the two singlet excited states of the asymmetric Hubbard dimer at half-filling are calculated using both Levy's constrained search and Lieb's convex formulation. While the ground-state functional is, as is commonly known, a convex function with respect to the density, the functional associated with the doubly excited state is found to be concave. Also, because the density-potential mapping associated with the first excited state is noninvertible, its "functional" is a partial, multivalued function composed of one concave and one convex branch that correspond to two separate domains of the external potential. Remarkably, it is found that, although the one-to-one mapping between density and external potential may not apply (as in the case of the first excited state), each state-specific energy and corresponding universal functional are "functions" whose derivatives are each other's inverse, just as in the ground state formalism.

Møller-Plesset and Density-Fixed Adiabatic Connections for a Model Diatomic System at Different Correlation Regimes.

In recent years, adiabatic connection (AC) interpolations developed within density functional theory (DFT) have been found to provide good performances in the calculation of interaction energies when used with Hartree-Fock (HF) ingredients. The physical and mathematical reasons for such unanticipated performance have been clarified, to some extent, by studying the strong-interaction limit of the Møller-Plesset (MP) AC. In this work, we calculate both the MP and the DFT AC integrand for the asymmetric Hubbard dimer, which allows for a systematic investigation of different correlation regimes by varying two simple parameters in the Hamiltonian: the external potential, Δv, and the interaction strength, U. Notably, we find that, while the DFT AC integrand appears to be convex in the full parameter space, the MP integrand may change curvature twice. Furthermore, we discuss different aspects of the second-order expansion of the correlation energy in each AC, and we demonstrate why the derivative of the λ-dependent density in the MP AC at λ = 0 (i.e., at the HF density) is zero in the model. Concerning the strong-interaction limit of both ACs in the Hubbard dimer setting, we show that the asymptotic value of the MP AC, W∞HF, is lower than (or equal to) its DFT analogue, W∞KS, if the two are compared at a given density, just like in real space. However, we also show that this is not always the case if the two quantities are compared at a given external potential.

The AuSc, AuTi, and AuFe molecules: Determination of the bond energies by Knudsen effusion mass spectrometry experiments combined with ab initio calculations.

The AuTi gaseous molecule was for the first time identified in vapors produced at high temperature from a gold-titanium alloy. The homogeneous equilibria AuTi(g) = Au(g) + Ti(g) (direct dissociation) and AuTi(g) + Au(g) = Au2(g) + Ti(g) (isomolecular exchange) were studied by Knudsen effusion mass spectrometry in the temperature range 2111-2229 K. The so determined equilibrium constants were treated by the "third-law method" of thermodynamic analysis, integrated with theoretical calculations, and the dissociation energy at 0 K was derived as D0K ° (AuTi) = 241.0 ± 5.2 kJ/mol. A similar investigation was carried out for the AuSc and AuFe species, whose dissociation energies were previously reported with large uncertainties. The direct dissociation and the isomolecular exchange with the Au2 dimer were studied in the 1969-2274 and 1842-2092 K ranges for AuSc and AuFe, respectively, and the dissociation energies derived as D0K ° (AuSc) = 240.4 ± 6.0 and D0K ° (AuFe) = 186.2 ± 4.2 kJ/mol. The experimental bond energies are compared with those calculated here by coupled cluster with single, double, and perturbative triple excitations with the correlation-consistent basis sets cc-pVXZ(-PP) and cc-pwCVXZ(-PP) (with X = T, Q, 5), also in the limit of complete basis set, and with those from complete active space self-consistent field-multi-reference configuration interaction calculations, recently available in the literature. The stronger bond of AuTi compared to AuFe parallels the trend observed in monochlorides. This analogy is shown to be more generally observed in the AuM and MCl diatomic series (with M = first row transition metal), in accordance with a picture of "pseudo-halogen" bonding behavior of gold.

Comparing correlation components and approximations in Hartree-Fock and Kohn-Sham theories via an analytical test case study.

The asymmetric Hubbard dimer is a model that allows for explicit expressions of the Hartree-Fock (HF) and Kohn-Sham (KS) states as analytical functions of the external potential, Δv, and of the interaction strength, U. We use this unique circumstance to establish a rigorous comparison between the individual contributions to the correlation energies stemming from the two theories in the {U, Δv} parameter space. Within this analysis of the Hubbard dimer, we observe a change in the sign of the HF kinetic correlation energy, compare the indirect repulsion energies, and derive an expression for the "traditional" correlation energy, i.e., the one that corrects the HF estimate, in a pure site-occupation function theory spirit [Eq. (45)]. Next, we test the performances of the Liu-Burke and the Seidl-Perdew-Levy functionals, which model the correlation energy based on its weak- and strong-interaction limit expansions and can be used for both the traditional and the KS correlation energies. Our results show that, in the Hubbard dimer setting, they typically work better for the HF reference, despite having been originally devised for KS. These conclusions are somewhat in line with prior assessments of these functionals on various chemical datasets. However, the Hubbard dimer model allows us to show the extent of the error that may occur in using the strong-interaction ingredient for the KS reference in place of the one for the HF reference, as has been carried out in most of the prior assessments.

Secondary Kinetic Peak in the Kohn-Sham Potential and Its Connection to the Response Step.

We consider a prototypical 1D model Hamiltonian for a stretched heteronuclear molecule and construct individual components of the corresponding KS potential, namely, the kinetic, the N - 1, and the conditional potentials. These components show very special features, such as peaks and steps, in regions where the density is drastically low. Some of these features are quite well-known, whereas others, such as a secondary peak in the kinetic potential or a second bump in the conditional potential, are less or not known at all. We discuss these features building on the analytical model treated in Giarrusso et al. J. Chem. Theory Comput. 2018, 14, 4151. In particular, we provide an explanation for the underlying mechanism which determines the appearance of both peaks in the kinetic potential and elucidate why these peaks delineate the region over which the plateau structure, due to the N - 1 potential, stretches. We assess the validity of the Heitler-London Ansatz at large but finite internuclear distance, showing that, if optimal orbitals are used, this model is an excellent approximation to the exact wave function. Notably, we find that the second natural orbital presents an extra node very far out on the side of the more electronegative atom.

Exchange-Correlation Energy Densities and Response Potentials: Connection between Two Definitions and Analytical Model for the Strong-Coupling Limit of a Stretched Bond.

We analyze in depth two widely used definitions (from the theory of conditional probability amplitudes and from the adiabatic connection formalism) of the exchange-correlation energy density and of the response potential of Kohn-Sham density functional theory. We introduce a local form of the coupling-constant-dependent Hohenberg-Kohn functional, showing that the difference between the two definitions is due to a corresponding local first-order term in the coupling constant, which disappears globally (when integrated over all space), but not locally. We also design an analytic representation for the response potential in the strong-coupling limit of density functional theory for a model single stretched bond.

Investigation of the Exchange-Correlation Potentials of Functionals Based on the Adiabatic Connection Interpolation.

We have studied the correlation potentials produced by various adiabatic connection models (ACMs) for several atoms and molecules. The results have been compared to accurate reference potentials (coupled cluster and quantum Monte Carlo results) as well as to state-of-the-art ab initio DFT approaches. We have found that all the ACMs yield correlation potentials that exhibit a correct behavior, quite resembling scaled second-order Görling-Levy (GL2) potentials and including most of the physically meaningful features of the accurate reference data. The behavior and contribution of the strong-interaction limit potentials have also been investigated and discussed.

Response Potential in the Strong-Interaction Limit of Density Functional Theory: Analysis and Comparison with the Coupling-Constant Average.

Using the formalism of the conditional amplitude, we study the response part of the exchange-correlation potential in the strong-coupling limit of density functional theory, analyzing its peculiar features and comparing it with the response potential averaged over the coupling constant for small atoms and for the hydrogen molecule. We also use a simple one-dimensional model of a stretched heteronuclear molecule to derive exact properties of the response potential in the strong-coupling limit. The simplicity of the model allows us to unveil relevant features also of the exact Kohn-Sham potential and its different components, namely the appearance of a second peak in the correlation kinetic potential on the side of the most electronegative atom.

Assessment of interaction-strength interpolation formulas for gold and silver clusters.

The performance of functionals based on the idea of interpolating between the weak- and the strong-interaction limits the global adiabatic-connection integrand is carefully studied for the challenging case of noble-metal clusters. Different interpolation formulas are considered and various features of this approach are analyzed. It is found that these functionals, when used as a correlation correction to Hartree-Fock, are quite robust for the description of atomization energies, while performing less well for ionization potentials. Future directions that can be envisaged from this study and a previous one on main group chemistry are discussed.

Communication: Strong-interaction limit of an adiabatic connection in Hartree-Fock theory.

We show that the leading term in the strong-interaction limit of the adiabatic connection that has as weak-interaction expansion the Møller-Plesset perturbation theory can be fully determined from a functional of the Hartree-Fock density. We analyze this functional and highlight similarities and differences with the strong-interaction limit of the density-fixed adiabatic connection case of Kohn-Sham density functional theory.