The density of atomic systems is analysed via the Single-Exponential Decay Detector (SEDD). SEDD is a scalar field designed to explore mathematical, rather than physical, properties of electron density. Nevertheless, it has been shown that SEDD can serve as a descriptor of bonding patterns in molecules as well as an indicator of atomic shells [P. de Silva, J. Korchowiec, and T. A. Wesolowski, ChemPhysChem13, 3462 (2012)]. In this work, a more detailed analysis of atomic shells is done for atoms in the Li–Xe series. Shell populations based on SEDD agree with the Aufbau principle even better than those obtained from the Electron Localization Function, which is a popular indicator of electron localization. A link between SEDD and the local wave vector is given, which provides a physical interpretation of SEDD.

The recently introduced molecular descriptor (Single Exponential Decay Detector - SEDD) [P. de Silva, J. Korchowiec, T. A. Wesolowski, ChemPhysChem 2012, 13, 3462] is used to visualize bonding patterns in molecules. In each point of space SEDD is simply related to the electron density:

SEDD(r) = ln[1/Ï�²(∇(∇Ï�/Ï�)²)²}.

Either experimental or computed densities �(r) can be used to evaluate SEDD. Here, maps of SEDD are obtained from theoretical densities and reveal such features as core electrons, chemical bonds, lone pairs and delocalization in aromatic systems. It is shown that SEDD provides fingerprints of aromaticity, which can be separated into geometric and electronic effects.

In methods based on frozen-density embedding theory or subsystem formulation of density functional theory, the non-additive kinetic potential (v_t^nad(r)) needs to be approximated. Since v_t^nad(r) is defined as a bifunctional, the common strategies rely on approximating v_t^nad[�_A,�_B](r). In this work, the exact potentials (not bifunctionals) are constructed for chemically relevant pairs of electron densities (�_A and �_B) representing: dissociating molecules, two parts of a molecule linked by a covalent bond, or valence and core electrons. The method used is applicable only for particular case, where �_A is a one-electron or spin-compensated two-electron density, for which the analytic relation between the density and potential exists. The sum �_A + �_B is, however, not limited to such restrictions. Kohn-Sham molecular densities are used for this purpose. The constructed potentials are analyzed to identify the properties which must be taken into account when constructing approximations to the corresponding bifunctional. It is comprehensively shown that the full von Weizsäcker component is indispensable in order to approximate adequately the non-additive kinetic potential for such pairs of densities.

We introduce a new tool (single exponential decay detector: SEDD) to extract information about bonding and localization in atoms, molecules, or molecular assemblies. The practical evaluation of SEDD does not require any explicit information about the orbitals. The only quantity needed is the electron density (calculated or experimental) and its derivatives up to the second order.

Within the linear combination of atomic orbitals (LCAO) approximation, one can distinguish two different Kohn-Sham potentials. One is the potential available numerically in calculations, and the other is the exact potential corresponding to the LCAO density. The latter is usually not available, but can be obtained from the total density by a numerical inversion procedure or, as is done here, analytically using only one LCAO Kohn-Sham orbital. In the complete basis-set limit, the lowest-lying Kohn-Sham orbital suffices to perform the analytical inversion, and the two potentials differ by no more than a constant. The relation between these two potentials is investigated here for diatomic molecules and several atomic basis sets of increasing size and quality. The differences between the two potentials are usually qualitative (wrong behavior at nuclear cusps and far from the molecule even if Slater-type orbitals are used) and δ-like features at nodal planes of the lowest-lying LCAO Kohn-Sham orbital. Such nodes occur frequently in LCAO calculations and are not physical. Whereas the behavior of the potential can be systematically improved locally by the increase of the basis sets, the occurrence of nodes is not correlated with the size of the basis set. The presence of nodes in the lowest-lying LCAO orbital can be used to monitor whether the effective potential in LCAO Kohn-Sham equations can be interpreted as the potential needed for pure-state noninteracting v-representability of the LCAO density. Squares of such node-containing lowest-lying LCAO Kohn-Sham orbitals are nontrivial examples of two-electron densities which are not pure-state noninteracting v-representable.