  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.
