In light of the swift progression and increasing complexity in theories of molecular interactions, this paper aimed to revisit and refine the existing theory for exchange interactions, applying a second quantization approach, and to offer its generalization to a permutation operator of a general order. Engaging with interactions involving excited states and chemical compounds that contain metals, such a generalization becomes critically important in many pivotal applications. This is predominantly because the prevailing density-matrix-based theory from Schaeffer and Jansen 2 (of 2nd order) or Waldrop and Patkowski 4 (pertaining to 3rd order induction) is applicable only to single determinant wave functions, and extending it to more intricate wave functions, which encompass electronic correlation, presents notable challenges.