Helices are a key folding motif in protein structure. The question which factors determine helix stability for a given polypeptide or protein is an ongoing challenge. Here we use van der Waals corrected density-functional theory to address a part of this question in a bottom-up approach. We show how intrinsic helical structure is stabilized with length and temperature for a series of experimentally well studied unsolvated alanine based polypeptides, Ac-Alan-LysH+. By exploring extensively the conformational space of these molecules, we find that helices emerge as the preferred structure in the length range n=4-8 not just due to enthalpic factors (hydrogen bonds and their cooperativity, van der Waals dispersion interactions, electrostatics), but importantly also by a vibrational entropic stabilization over competing conformers at room temperature. The stabilization is shown to be due to softer low-frequency vibrational modes in helical conformers than in more compact ones. This observation is corroborated by including anharmonic effects explicitly through emph{ab initio} molecular dynamics, and generalized by testing different terminations and considering larger helical peptide models.
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