among the types of reactions that can be catalyzed by lewis acids, those with carbonyl-containing substrates have received the greatest amount of attention. the first major discovery in this area was in 1960, when yates and eaton reported the significant acceleration of the diels-alder reaction by alcl when maleic anhydride is the dienophile. early theoretical studies that depended on frontier orbital analysis established that lewis acid catalysis operates via lowering of the dienophile's lumo energy, which is still the accepted rationalization. the concept of lowered lumo energy is also used to explain the dramatically enhanced electrophilic reactivity of carbonyl compounds (whose lumo is the c-o π* orbital) towards mild nucleophilic reagents, as in the cases of the mukaiyama aldol reaction and sakurai reaction.