The theorems at the core of density functional theory (DFT) state that the energy of a many-electron system in its ground state is fully defined by its electron density distribution. This connection is made via the exact functional for the energy, which minimizes at the exact density. For years, DFT development focused on energies, implicitly assuming that functionals producing better energies become better approximations of the exact functional. We examined the other side of the coin: the energy-minimizing electron densities for atomic species, as produced by 128 historical and modern DFT functionals. We found that these densities became closer to the exact ones, reflecting theoretical advances, until the early 2000s, when this trend was reversed by unconstrained functionals sacrificing physical rigor for the flexibility of empirical fitting.

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Kepp argues in his Comment, among other concerns, that the atomic densities we have considered are not relevant to molecular bonding. However, this does not change the main conclusion of our study, that unconstrained fitting of flexible functional forms can make a density functional more interpolative but less widely predictive.K epp argues in his Comment (1) that our Report (2) does not properly assess the accuracy of approximate functionals for molecules. Many studies have been published to assess the accuracy of such functionals for various applications. We have, instead, investigated the ability of these functionals to make predictions for electron densities-the property employed in computations of all other properties but almost never used to fit or assess functionals. We deliberately chose atomic systems because they are somewhat different from molecules and because they should be well described by the tested functional forms. We found that nonempirical or few-parameter empirical functionals were much more predictive than flexible (highly unconstrained) multiparameter empirical functionals. We will respond in detail to the Comment after clarifying the methodological issue in density functional theory (DFT) and more generally.There is a systematic, nonempirical way to improve approximations to the exact density functional for the exchange-correlation energy (3, 4): (i) Prove the existence of the exact functional and derive exact formal expressions for it. (ii) Discover mathematical properties of the exact functional (exact constraints), which include limits, scaling relations, equalities, and bounds. (iii) Develop approximate but computationally tractable forms for the approximations at various levels of flexibility, including the local spin density approximation (LSDA), generalized gradient approximations (GGAs), meta-GGAs, and hybrids tested in our paper. (iv) Impose the "exact constraints" from step ii on each form, as appropriate. (v) If a form still retains some flexibility, fit it to energies/densities of appropriate norms (4), systems for which the form can be expected to be highly accurate. (Bonded systems are not appropriate norms, because in them the standard approximations display an understood but uncontrollable error cancellation between exchange and correlation. Fitting to bonded systems could make a functional more interpolative but less widely predictive, in the sense of the next paragraph.) The period 1965 to 1979 focused on step i, the 1980s on step ii, 1965 to 1998 on step iii, and the 1970s to the present on step iv, whereas the first appropriate norm for step v was the uniform electron gas (1965). A recent example of steps i to v is the SCAN (strongly constrained and appropriately normed) meta-GGA (4), which respects all 17 exact constraints that a meta-GGA can and works well for diversely bonded molecules and materials [e.g., reference 5 in (2)]. Because SCAN was fitted only to energies of nonbonded systems, every bonding description from it is a genuine prediction. On the meta-GGA lev...

The energy of stereoelectronic interactions in N-C-S and N-N-C systems in tetrahydro[1,3,4]thiadiazolo[3,4- c][1,3,4]thiadiazole was estimated by means of R. W. Bader's quantum theory of "atoms in molecules" (AIM) and natural bond orbital analysis (NBO). The results were compared with those obtained by analysis of rho( r) derived from high-resolution X-ray diffraction data. The analysis of the data obtained allowed one to find a correlation between geometric characteristics of the stereoelectronic interactions, NBO mixing energies and the AIM properties of atoms. Significant variations of nitrogen atom atomic basin populations in different conformers were explained by sterical interactions between their electron lone pairs.

The three-dimensional atomic structure of MoS2–organic layered systems is obtained for the first time, providing insight into the surface chemistry of charged MoS2 sheets.