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The Quantum Theory of Atoms in Molecules, due to Bader, is applied to periodic systems. Results for molecular and crystalline urea are presented. Changes in both bond critical points and atomic properties due to changes of chemical environment are described. A rationale for the different lengths of the in-plane and out-of-plane hydrogen bonds and for the lengthening of the CO bond in bulk urea is provided in terms of the properties of the Laplacian of the oxygen atom electron density distribution. An evaluation of molecular and atomic volume changes indicates that the decrease of molecular volume upon change of phase from gas to solid originates primarily from a contraction of the atomic basins directly involved in hydrogen bonds. Other atoms show a small expansion. The considerable decrease of oxygen and hydrogen atomic volumes is related to the mutual penetration of their van der Waals envelopes following hydrogen bond formation. The results confirm that urea is more polar in the solid phase.

Analysis of the chemical bonding in the position space, instead of or besides that in the wave function (Hilbert) orbital space, has become increasingly popular for crystalline systems in the past decade. The two most frequently used investigative tools, the Quantum Theory of Atoms in Molecules and Crystal (QTAIMAC) and the Electron Localization Function (ELF) are thoroughly discussed. The treatment is focussed on the topological peculiarities that necessarily arise from the periodicity of the crystal lattice and on those facets of the two tools that have been more debated, especially when these tools are applied to the condensed phase. In particular, in the case of QTAIMAC, the physical and chemical significance of the bond paths for the very weak or the supposedly repulsive interactions, the distinctive features and the appropriateness of the several schemes that have been proposed to classify chemical bonds, and, finally, the relative importance of the local and integrated electron density properties for describing intermolecular interactions. In the case of the ELF, particular attention is devoted to how this function is formulated and to the related physical meaning, and to how can the ELF be chemically interpreted and properly analysed in crystals. Several examples are reported to illustrate all these points and for critically examine the answers obtained and the problems encountered. The discussed examples encompass the case of molecular crystals, Zintl phases, intermetallic compounds, metals, supported and unsupported metal-metal bonds in organometallics, ionic solids, crystal surfaces, crystal defects, etc. Whenever possible joint ELF and QTAIMAC studies are considered, with particular emphasis on the comparison of the bond description afforded by the ELF and the Laplacian of the electron density. Two recently proposed functions, the Localized Orbital Locator (LOL) and the Source Function in its integrated or local form are also presented, in view of their potential interest for stud ies of chemical bonding in crystals. The use of approximated ELF and LOL, as derived from the density functional form of the positive kinetic energy density, is also discussed.

solids; a "semiconductor" has a narrow bandgap across which electrons can be excited by light; these classifications are therefore based on measurable, macroscopic properties. A fingerprint of five coexisting identifiers was recently used to define a concept termed "metavalent" bonding (MVB): [4,5] metavalent solids show i) moderate electronic conductivity (≈10 2 -10 4 S cm −1 ); ii) increased coordination numbers incompatible with the (8-N) rule for semiconductors; iii) large optical dielectric constants, ε ∞ ; iv) large bond polarizability, as measured by Born effective charges, Z*; and v) large lattice anharmonicity, as measured by the Grüneisen parameter, |γ TO |. In terms of conductivity and coordination numbers, metavalent solids are therefore located between the covalent and metallic regimes-but they are distinctly different from both because they show anomalously large response properties [5] and a unique bond-breaking mechanism [4] not observed in either covalent or metallic solids. This definition based on a set of observable properties directly led to a revision of the "resonant bonding" model (which had previously been widely used to describe the bonding in PCMs [6] ) by showing that the response properties of PCMs are fundamentally different from those of resonantly bonded benzene and graphite. [5] A 2D map is created for solid-state materials based on a quantum-mechanical description of electron sharing and electron transfer. This map intuitively identifies the fundamental nature of ionic, metallic, and covalent bonding in a range of elements and binary compounds; furthermore, it highlights a distinct region for a mechanism recently termed "metavalent" bonding. Then, it is shown how this materials map can be extended in the third dimension by including physical properties of application interest. Finally, it is shown how the map coordinates yield new insight into the nature of the Peierls distortion in phase-change materials and thermoelectrics. These findings and conceptual approaches provide a novel avenue to tailor material properties. Materials DesignThe ORCID identification number(s) for the author(s) of this article can be found under https://doi.