Long-Range Configuration Interaction with an Short-Range Correction and an Asymptotic Lower Bound†.

Short-range corrections to long-range selected configuration interaction calculations are derived from perturbation theory considerations and applied to harmonium (with two to six electrons for some low-lying states). No fitting to reference data is used, and the method is applicable to ground and excited states. The formulas derived are rigorous when the physical interaction is approached.

Modified Expression for the Hamiltonian Expectation Value Exploiting the Short-Range Behavior of the Wave Function.

The expectation value of the Hamiltonian using a model wave function is widely used to estimate the eigenvalues of electronic Hamiltonians. We explore here a modified formula for models based on a long-range interaction. It scales differently the singlet and triplet components of the repulsion between electrons not present in the model (its short-range part).

Exploring the role of mean-field potentials and short-range wave function behavior in the adiabatic connection.

In this article, we explore the construction of Hamiltonians with long-range interactions and their corrections using the short-range behavior of the wave function. A key aspect of our investigation is the examination of the one-particle potential, kept constant in our previous work, and the effects of its optimization on the adiabatic connection. Our methodology involves the use of a parameter-dependent potential dependent on a single parameter to facilitate practical computations.

Second-order adiabatic connection: The theory and application to two electrons in a parabolic confinement.

The adiabatic connection formalism, usually based on the first-order perturbation theory, has been generalized to an arbitrary order. The generalization stems from the observation that the formalism can be derived from a properly arranged Taylor expansion. The second-order theory is developed in detail and applied to the description of two electrons in a parabolic confinement (harmonium).

Correcting Models with Long-Range Electron Interaction Using Generalized Cusp Conditions.

Sources of energy errors resulting from the replacement of the physical Coulomb interaction by its long-range erfc(μ)/ approximation are explored. It is demonstrated that the results can be dramatically improved and the range of μ giving energies within chemical accuracy limits significantly extended if the generalized cusp conditions are used to represent the wave function at small . The numerical results for two-electron harmonium are presented and discussed.

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.

The effect of uncertainty on building blocks in molecules.

Probabilities to find a chosen number of electrons in flexible domains of space are calculated for highly correlated wave functions. Quantum mechanics can produce higher probabilities for chemically relevant arrangements of electrons in these regions. However, the probability to have a given arrangement, e.

Was Pauling Mistaken about Metals?

Pauling described metallic bonds using resonance. The maximum probability domains in the Kronig-Penney model can show a picture of it. When the walls are opaque (and the band gap is large) the maximum probability domain for an electron pair essentially corresponds to the region between the walls: the electron pairs are localized within two consecutive walls.

Models and corrections: Range separation for electronic interaction-Lessons from density functional theory.

Model Hamiltonians with long-range interaction yield energies are corrected taking into account the universal behavior of the electron-electron interaction at a short range. Although the intention of this paper is to explore the foundations of using density functionals combined with range separation, the approximations presented can be used without them, as illustrated by a calculation on harmonium. In the regime, when the model system approaches the Coulomb system, they allow the calculation of ground states, excited states, and properties, without making use of the Hohenberg-Kohn theorem.

Concluding remarks for the new horizons in density functional theory Faraday Discussion.

The present contribution tries to succinctly review the progress presented during the Faraday Discussions New horizons in density functional theory that have taken place online, 2-4 September 2020.

Probabilistic performance estimators for computational chemistry methods: Systematic improvement probability and ranking probability matrix. II. Applications.

In Paper I [P. Pernot and A. Savin, J.

Probabilistic performance estimators for computational chemistry methods: Systematic improvement probability and ranking probability matrix. I. Theory.

The comparison of benchmark error sets is an essential tool for the evaluation of theories in computational chemistry. The standard ranking of methods by their mean unsigned error is unsatisfactory for several reasons linked to the non-normality of the error distributions and the presence of underlying trends. Complementary statistics have recently been proposed to palliate such deficiencies, such as quantiles of the absolute error distribution or the mean prediction uncertainty.

Curing basis-set convergence of wave-function theory using density-functional theory: A systematically improvable approach.

The present work proposes to use density-functional theory (DFT) to correct for the basis-set error of wave-function theory (WFT). One of the key ideas developed here is to define a range-separation parameter which automatically adapts to a given basis set. The derivation of the exact equations are based on the Levy-Lieb formulation of DFT, which helps us to define a complementary functional which corrects uniquely for the basis-set error of WFT.

Probabilistic performance estimators for computational chemistry methods: The empirical cumulative distribution function of absolute errors.

Benchmarking studies in computational chemistry use reference datasets to assess the accuracy of a method through error statistics. The commonly used error statistics, such as the mean signed and mean unsigned errors, do not inform end-users on the expected amplitude of prediction errors attached to these methods. We show that, the distributions of model errors being neither normal nor zero-centered, these error statistics cannot be used to infer prediction error probabilities.

When does a functional correctly describe both the structure and the energy of the transition state?

Requiring that several properties are well reproduced is a severe test on density functional approximations. This can be assessed through the estimation of joint and conditional success probabilities. An example is provided for a small set of molecules, for properties characterizing the transition states (geometries and energies).

Spooky correlations and unusual van der Waals forces between gapless and near-gapless molecules.

We consider the zero-temperature van der Waals (vdW) interaction between two molecules, each of which has a zero or near-zero electronic gap between a ground state and the first excited state, using a toy model molecule (equilateral H) as an example. We show that the van der Waals energy between two ground state molecules falls off as D instead of the usual D dependence, when the molecules are separated by distance D. We show that this is caused by a perfect "spooky" correlation between the two fluctuating electric dipoles.

Exchange-Correlation Functionals via Local Interpolation along the Adiabatic Connection.

The construction of density-functional approximations is explored by modeling the adiabatic connection locally, using energy densities defined in terms of the electrostatic potential of the exchange-correlation hole. These local models are more amenable to the construction of size-consistent approximations than their global counterparts. In this work we use accurate input local ingredients to assess the accuracy of a range of local interpolation models against accurate exchange-correlation energy densities.

Maximum probability domains for the analysis of the microscopic structure of liquids.

We introduce the concept of maximum probability domains (MPDs), developed in the context of the analysis of electronic densities, in the study of the microscopic spatial structures of liquids. The idea of locating a particle in a three dimensional region, by determining the domain where the probability of finding that, and only that, particle is maximum, gives an interesting characterization of the local structure of the liquid. The optimization procedure, required for the search of the domain of maximum probability, is carried out by the implementation of the level set method.

Prediction uncertainty of density functional approximations for properties of crystals with cubic symmetry.

The performance of a method is generally measured by an assessment of the errors between the method's results and a set of reference data. The prediction uncertainty is a measure of the confidence that can be attached to a method's prediction. Its estimation is based on the random part of the errors not explained by reference data uncertainty, which implies an evaluation of the systematic component(s) of the errors.