In a LOCALIZE calculation, localized M.O.'s (LMO's) in double and triple bonds are split into one LMO that is mainly sigma and one or two LMO's that are mainly pi.  For example, a triple bond would consist of a sigma LMO and two pi LMO's.

When BANANA is present, a LOCAL calculation will be done, and these double and triple bond LMO's will be replaced with hybrid LMO's that have equal amounts of sigma character in each M.O.  The resulting LMO's are commonly referred to as banana-bonds because of their shape, although they really look more like éclairs than bananas.  Other LMO's that cannot be mixed without changing their localization properties, such as sigma LMO's in single bonds are not affected when BANANA is present.   

All three sets of M.O.'s, canonical, localized, and localized with BANANA, are equivalent, in that the sets are related to each other by unitary transforms. Use whichever set best first the current needs.

See also LOCAL and RABBIT