A promising trend in plasmonics involves shrinking the size of plasmon-supporting structures down to a few nanometers, thus enabling control over light-matter interaction at extreme-subwavelength scales. In this limit, quantum mechanical effects, such as nonlocal screening and size quantization, strongly affect the plasmonic response, rendering it substantially different from classical predictions. For very small clusters and molecules, collective plasmonic modes are hard to distinguish from other excitations such as single-electron transitions. Using rigorous quantum mechanical computational techniques for a wide variety of physical systems, we describe how an optical resonance of a nanostructure can be classified as either plasmonic or nonplasmonic. More precisely, we define a universal metric for such classification, the generalized plasmonicity index (GPI), which can be straightforwardly implemented in any computational electronic-structure method or classical electromagnetic approach to discriminate plasmons from single-particle excitations and photonic modes. Using the GPI, we investigate the plasmonicity of optical resonances in a wide range of systems including: the emergence of plasmonic behavior in small jellium spheres as the size and the number of electrons increase; atomic-scale metallic clusters as a function of the number of atoms; and nanostructured graphene as a function of size and doping down to the molecular plasmons in polycyclic aromatic hydrocarbons. Our study provides a rigorous foundation for the further development of ultrasmall nanostructures based on molecular plasmonics
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