We develop an efficient numerical algorithm for the identification of a large number of saddle points of the potential energy function of Lennard- Jones clusters. Knowledge of the saddle points allows us to find many thousand adjacent minima of clusters containing up to 80 argon atoms and to locate many pairs of minima with the right characteristics to form two-level systems (TLS). The true TLS are singled out by calculating the ground-state tunneling splitting. The entropic contribution to all barriers is evaluated and discussed.
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