Second-order nonlinear processes like Sum-Frequency Generation (SFG) are essentially defined in the electric dipolar approximation. However, when dealing with the SFG responses of bulk, big nanoparticles, highly symmetric objects, or chiral species, magnetic and quadrupolar contributions play a significant role in the process too. We extend the diagrammatic theory for linear and nonlinear optics to include these terms for single objects as well as for multipartite systems in interaction. Magnetic and quadrupolar quantities are introduced in the formalism as incoming fields, interaction intermediates, and sources of optical nonlinearity. New response functions and complex nonlinear processes are defined, and their symmetry properties are analyzed. This leads to a focus on several kinds of applications involving nanoscale coupled objects, symmetric molecular systems, and chiral materials, both in line with the existing literature and opening new possibilities for original complex systems.
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