  Short unbranched alkanes are known to prefer linear conformations, while long
unbranched alkanes are folded. It is not known with certainty at what chain
length the linear conformation is no longer the global minimum. To clarify this
point, we use {\it ab initio} and density functional methods to compute the
relative energies of the linear and hairpin alkane conformers for increasing
chain lengths. Extensive electronic structure calculations are performed to
obtain optimized geometries, harmonic frequencies and accurate single point
energies for the selected alkane conformers from octane through octadecane.
Benchmark CCSD(T)/cc-pVTZ single point calculations are performed for chains
through tetradecane, while approximate methods are required for the longer
chains up to octadecane. Using frozen natural orbitals to unambiguously
truncate the virtual orbital space, we are able to compute composite CCSD
FNO(T) single point energies for all the chain lengths. This approximate
composite method has significant computational savings compared to full CCSD(T)
while retaining $\sim0.15$ kcal/mol accuracy compared to the benchmark results.
More approximate dual-basis resolution-of-the-identity double-hybrid DFT
calculations are also performed and shown to have reasonable $0.2-0.4$ kcal/mol
errors compared with our benchmark values. After including contributions from
temperature dependent internal energy shifts, we find the preference for folded
conformations to lie between hexadecane and octadecane, in excellent agreement
with recent experiments [L\"{u}ttschwager, N. O.; Wassermann, T. N.; Mata, R.
A.; Suhm, M. A. {\it Angew. Chem. Int. Ed.} 2013, 52, 463].
