How does Lewis acid-base theory explain acid-base catalysis?

According to Lewis theory, an acid is a species capable of accepting an electron pair from a base to form a covalent bond. In the context of catalysis, Lewis acids exert their effect through two primary mechanisms:12

• Assisting Proton Release: Lewis acids can interact with hydrogen-containing compounds present in a system to facilitate the release of a proton to the substrate. For instance, the polymerization of olefins by Lewis acids like boron trifluoride (BF${}_3$) or aluminum chloride (AlCl${}_3$) is often triggered by their interaction with trace amounts of water, generating a carbonium ion that drives the reaction.1
• Direct Substrate Activation: A Lewis acid may act directly on a substrate by withdrawing electrons, thereby converting the substrate into a more reactive form. A classic example is the action of aluminum chloride on alkyl halides, which produces a carbonium ion that can subsequently react with other substances, such as aromatic hydrocarbons.1

These processes demonstrate how Lewis acids catalyze reactions by manipulating electron density or proton availability, even in the absence of traditional proton-based acid catalysts.12