We argue that any general mathematical measure of density error, no matter how reasonable, is too arbitrary to be of universal use. However the energy functional itself provides a universal relevant measure of density errors. For the self-consistent density of any Kohn-Sham calculation with an approximate functional, the theory of density-corrected density functional theory (DC-DFT) provides an accurate, practical estimate of this ideal measure. We show how to estimate the significance of the density-driven error even when exact densities are unavailable. In cases with large density errors, the amount of exchange-mixing is often adjusted, but we show this is unnecessary. Many chemically relevant examples are given.

All-electron fixed-node diffusion Monte Carlo provides benchmark spin gaps for four Fe(II) octahedral complexes. Standard quantum chemical methods (semilocal DFT and CCSD(T)) fail badly for the energy difference between their high- and low-spin states. Density-corrected DFT is both significantly more accurate and reliable and yields a consistent prediction for the Fe-Porphyrin complex.

Density-corrected density functional theory (DC-DFT) is enjoying substantial success in improving semilocal DFT calculations in a wide variety of chemical problems. This paper provides the formal theoretical framework and assumptions for the analysis of any functional minimization with an approximate functional. We generalize DC-DFT to allow comparison of any two functionals, not just comparison with the exact functional. We introduce a linear interpolation between any two approximations, and use the results to analyze global hybrid density functionals. We define the basins of density-space in which this analysis should apply, and give quantitative criteria for when DC-DFT should apply. We also discuss the effects of strong correlation on density-driven error, utilizing the restricted HF Hubbard dimer as an illustrative example. arXiv:1908.05721v1 [physics.chem-ph]

Empirical fitting of parameters in approximate density functionals is common. Such fits conflate errors in the self-consistent density with errors in the energy functional, but density-corrected DFT (DC-DFT) separates these two. We illustrate with catastrophic failures of a toy functional applied to H 2 + at varying bond lengths, where the standard fitting procedure misses the exact functional; Grimme's D3 fit to noncovalent interactions, which can be contaminated by large density errors such as in the WATER27 and B30 data sets; and doublehybrids trained on self-consistent densities, which can perform poorly on systems with densitydriven errors. In these cases, more accurate results are found at no additional cost by using Hartree−Fock (HF) densities instead of self-consistent densities. For binding energies of small water clusters, errors are greatly reduced. Range-separated hybrids with 100% HF at large distances suffer much less from this effect.

Dispersion corrections of various kinds usually improve DFT energetics of weak noncovalent interactions. However, in some cases involving molecules or halides, especially those with σ-hole interactions, the density-driven errors of uncorrected DFT are larger than the dispersion corrections. In these abnormal situations, HF-DFT (using Hartree−Fock densities instead of self-consistent densities) greatly improves bond energies, while dispersion corrections can even worsen the results. On the other hand, pnictogen bonds and the S22 data set are normal and are not improved by this procedure. Such effects should be accounted for when parametrizing dispersion interactions.

Kohn−Sham (KS) inversion, that is, the finding of the exact KS potential for a given density, is difficult in localized basis sets. We study the precision and reliability of several inversion schemes, finding estimates of densitydriven errors at a useful level of accuracy. In typical cases of substantial densitydriven errors, Hartree−Fock density functional theory (HF-DFT) is almost as accurate as DFT evaluated on CCSD(T) densities. A simple approximation in practical HF-DFT also makes errors much smaller than the density-driven errors being calculated. Two paradigm examples, stretched NaCl and the HO•Cl − radical, illustrate just how accurate HF-DFT is.

Recently, various energy transducers driven by the relative motion of solids and liquids have been demonstrated. However, in relation to the energy transducer, a proper understanding of the dynamic behavior of ions remains unclear. Moreover, the energy density is low for practical usage mainly due to structural limitations, a lack of material development stemming from the currently poor understanding of the mechanisms, and the intermittently generated electricity given the characteristics of the water motion (pulsed signals). Here, we verify a hypothesis pertaining to the ion dynamics which govern the operation mechanism of the transducer. In addition, we demonstrate enhanced energy transducer to convert the mechanical energy of flowing water droplets into continuous electrical energy using an electrolyte-insulator-semiconductor structure as a device structure. The output power per droplet mass and the ratio of generated electric energy to the kinetic energy of water drops are 0.149v mW·g·m·s and 29.8%, respectively, where v is the speed of the water droplet.

HF-DFT, the practice of evaluating approximate density functionals on Hartree−Fock densities, has long been used in testing density functional approximations. Density-corrected DFT (DC-DFT) is a general theoretical framework for identifying failures of density functional approximations by separating errors in a functional from errors in its selfconsistent (SC) density. Most modern DFT calculations yield highly accurate densities, but important characteristic classes of calculation have large density-driven errors, including reaction barrier heights, electron affinities, radicals and anions in solution, dissociation of heterodimers, and even some torsional barriers. Here, the HF density (if not spincontaminated) usually yields more accurate and consistent energies than those of the SC density. We use the term DC(HF)-DFT to indicate DC-DFT using HF densities only in such cases. A recent comprehensive study (J. Chem. Theory Comput. 2021, 17, 1368−1379) of HF-DFT led to many unfavorable conclusions. A reanalysis using DC-DFT shows that DC(HF)-DFT substantially improves DFT results precisely when SC densities are flawed.