The Stress Localization Function─A Stress-Derived Real-Space Descriptor for Chemical Bonding Analysis

Abstract Existing quantum-chemical topology descriptors, including the Laplacian of the electron density, the Electron Localization Function (ELF), and the Localized Orbital Locator (LOL), characterize bonding through the density and its low-order derivatives, but they trace the localization of electrons rather than the local balance between the kinetic and potential energy densities and sometimes fail in cases such as heavy atoms, hypervalent centers, and bonds of intermediate character. To address these deficiencies, we introduce the Stress Localization Function (SLF), a real-space bonding descriptor constructed from the quantum-mechanical stress tensor. The trace of the stress tensor is the local virial field, which connects the kinetic and potential energy at each point in space. The SLF is obtained by taking its magnitude, comparing it to the value for a uniform electron gas of the same local density, and mapping the result onto a bounded scale, in analogy to the ELF or LOL. Applied to atoms across the periodic table, the SLF resolves the complete electronic shell structure, including the outermost shells of heavy atoms that the Laplacian fails to recover and that stress-based indicators have not resolved with any single choice of parameter. For the 4d elements, the SLF recovers the full complement of shells with no adjustable parameter. It registers the number of occupied shells rather than the nominal period, as shown by palladium’s anomalous 4d105s0 configuration. The core value is essentially universal (≈0.578), transferable from free atoms to bonded environments. Across a benchmark set of single, double, and triple bonds, the SLF decreases monotonically with bond order while simultaneously tracking bond polarity through the electronegativity of the bonding partner, separating two distinct chemical effects within a single scalar field. The descriptor further characterizes nonprototypical bonding, including three-center bonds, as in the textbook example of H3+ and boranes, the hypervalent center in BrF3, and the σ-donation and π-back-donation of metal carbonyls. The SLF thus provides a stress-derived, energetic view of electron organization that complements existing density-based descriptors.