The electronic structure and geometry of polymethylaluminoxane (MAO) [ 1Al(CH 3 )O1 ] n with different size (n = 4 -12) have been studied using quantum-chemical DFT (density functional theory) calculations. It has been found: 1) Starting from n = 6, the three-dimensional oxo-bridged (cage) structure of MAO is more stable than the cyclic structure. 2) Both for cage structure and for cyclic structure the Lewis acidity of Al atoms characterized by their net positive charge amplifies with increasing size of MAO (n). 3) Trimethylaluminium (AlMe 3 ) reacts with the cage structure of MAO with cleavage of an Al-O dative bond and formation of acidic tri-coordinated Al v and basic di-coordinated O v atoms in the MAO molecule. Two molecules AlMe 3 are associated with acidic Al v and basic O v centers. As the MAO increases in size, the acidity of Al v centers amplifies and the distance Al v -(AlMe 3 ) shortens; on the contrary, interaction of AlMe 3 with O v centers weakens and the distance O v -(AlMe 3 ) increases with increasing n value. The total heat of Al 2 Me 6 interaction with MAO (sum interaction of Al v -(AlMe 3 ) and O v -(AlMe 3 )) noticeably decreases as the size of MAO increases (from 50.9 kcal/mol for n = 4 to 20.2 kcal/mol for n = 12). It is proposed that acidic Al v and basic O v centers formed in the cage structure of MAO interact with zirconocene yielding 'cation-like' zirconium active centers.