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). A significant improvement relative to the first-order theory has been obtained.

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(µr)/r 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 r. 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.g., that corresponding to chemical formulas (bonds or atoms), is low although being often maximal. Like in valence bond theory, it is useful to consider alternative distributions of electrons. Exchanges of electrons should be considered not only between atoms but also between other types of regions, such as those attributed to lone pairs. It is useful to have definitions flexible enough to allow users to find the most relevant representations. We tentatively suggest a tool (the effective number of parties) to help one make the choice.

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. However, when the walls become transparent (and the band gaps closes), the maximum probability domain can be moved through the system without a significant loss in probability.

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. Asymptotically, the technique is improvable and allows for error estimates that can validate the results. Some considerations for correcting the errors of finite basis sets in this spirit are also presented. Being related to the present understanding of density functional approximations, the results are comparable to those obtained with the latter, as long as these are accurate.

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. 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. We introduce here a new score, the systematic improvement probability, based on the direct system-wise comparison of absolute errors. Independent of the chosen scoring rule, the uncertainty of the statistics due to the incompleteness of the benchmark datasets is also generally overlooked. However, this uncertainty is essential to appreciate the robustness of rankings. In the present article, we develop two indicators based on robust statistics to address this problem: Pinv, the inversion probability between two values of a statistic, and Pr, the ranking probability matrix. We demonstrate also the essential contribution of the correlations between error sets in these scores comparisons.

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. Chem. Phys. 152, 164108 (2020)], we introduced the systematic improvement probability as a tool to assess the level of improvement on absolute errors to be expected when switching between two computational chemistry methods. We also developed two indicators based on robust statistics to address the uncertainty of ranking in computational chemistry benchmarks: Pinv, the inversion probability between two values of a statistic, and Pr, the ranking probability matrix. In this second part, these indicators are applied to nine data sets extracted from the recent benchmarking literature. We also illustrate how the correlation between the error sets might contain useful information on the benchmark dataset quality, notably when experimental data are used as reference.

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. The coupling of DFT and WFT is done through the definition of a real-space representation of the electron-electron Coulomb operator projected on a one-particle basis set. Such an effective interaction has the particularity to coincide with the exact electron-electron interaction in the limit of a complete basis set, and to be finite at the electron-electron coalescence point when the basis set is incomplete. The non-diverging character of the effective interaction allows one to define a mapping with the long-range interaction used in the context of range-separated DFT and to design practical approximations for the unknown complementary functional. Here, a local-density approximation is proposed for both full-configuration-interaction (FCI) and selected configuration-interaction approaches. Our theory is numerically tested to compute total energies and ionization potentials for a series of atomic systems. The results clearly show that the DFT correction drastically improves the basis-set convergence of both the total energies and the energy differences. For instance, a sub kcal/mol accuracy is obtained from the aug-cc-pVTZ basis set with the method proposed here when an aug-cc-pV5Z basis set barely reaches such a level of accuracy at the near FCI level.

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. To overcome this limitation, we advocate for the use of more informative statistics, based on the empirical cumulative distribution function of unsigned errors, namely, (1) the probability for a new calculation to have an absolute error below a chosen threshold and (2) the maximal amplitude of errors one can expect with a chosen high confidence level. Those statistics are also shown to be well suited for benchmarking and ranking studies. Moreover, the standard error on all benchmarking statistics depends on the size of the reference dataset. Systematic publication of these standard errors would be very helpful to assess the statistical reliability of benchmarking conclusions.

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 H3) as an example. We show that the van der Waals energy between two ground state molecules falls off as D-3 instead of the usual D-6 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. The phenomenon is related to, but not the same as, the "resonant" interaction between an electronically excited and a ground state molecule introduced by Eisenschitz and London in 1930. It is also an example of "type C van der Waals non-additivity" recently introduced by one of us [J. F. Dobson, Int. J. Quantum Chem. 114, 1157 (2014)]. Our toy molecule H3 is not stable, but symmetry considerations suggest that a similar vdW phenomenon may be observable, despite Jahn-Teller effects, in molecules with a discrete rotational symmetry and broken inversion symmetry, such as certain metal atom clusters. The motion of the nuclei will need to be included for a definitive analysis of such cases, however.

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. The importance of the strictly correlated electrons (SCE) functional describing the strong coupling limit is emphasized, enabling the corresponding interpolated functionals to treat strong correlation effects. In addition to exploring the performance of such models numerically for the helium and beryllium isoelectronic series and the dissociation of the hydrogen molecule, an approximate analytic model is presented for the initial slope of the local adiabatic connection. Comparisons are made with approaches based on global models, and prospects for future approximations based on the local adiabatic connection are discussed.