Rigid-unit modes and the quantitative determination of the flexibility possessed by zeolite frameworks

This paper is concerned with the way in which the flexibility of zeolite frameworks can be calculated and expressed in an exact and quantitative manner. It is found that zeolites modeled as frameworks of tetrahedra are extremely flexible but only as a few very specific modes in specific places. These modes give specificity in the action of a zeolite: in lining up of Bronstedt acid sites to fit reacting molecules, in adapting the shapes of channel windows to the shapes of certain diffusing molecules, and in attaching catalytically active cations to the framework. These floppy modes or "rigid-unit modes" (RUMs) can be very localized, and we show how local RUMs may be calculated precisely as wave packets of standard RUM phonon modes for any given zeolite, Applications to six zeolites are given. Local modes may be static deformations costing virtually no energy as in holding cations, or fluctuating as in facilitating the diffusion of certain molecules, or quasi-static as in catalyzing a reaction with acidic sites. In addition, it is possible to attempt to create local RUMs with eigenvectors that have a desired form. This means that the presence of any kind of specific flexibility within a zeolite can be tested for. Thus it becomes possible to determine the exact nature of the flexibility present within any aluminosilicate zeolite. In particular, local RUMs enable one to determine which parts of a framework are most flexible and hence more likely to be catalytically active.