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We study the distribution of the minimum free energy (MFE) for the Turner model of pseudoknot free RNA secondary structures over ensembles of random RNA sequences. In particular, we are interested in those rare and intermediate events of unexpected low MFEs. Generalized ensemble Markov-chain Monte Carlo methods allow us to explore the rare-event tail of the MFE distribution down to probabilities like $10^{-70}$ and to study the relationship between the sequence entropy and structural properties for sequence ensembles with fixed MFEs. Entropic and structural properties of those ensembles are compared with natural RNA of the same reduced MFE (z-score).
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