Once again, into the breach…
Ess, Liu, and De Proft offer another analysis of the protobranching effect.1 As a reminder, Schleyer, Mo and Houk and coworkers argue that the reason why branched alkanes are more stable than linear ones is a stabilizing 1,3-interaction that they call protobranching.2 This proposal has been met with both supporters and vigorous attacks – see these posts.
What is new here is a partitioning of the total DFT energy into three terms. The critical term is one based on the Weizäcker kinetic energy, which is defined as the integral of the gradient of the density squared divided by the density. They call this a “steric energy term”. The second term is the standard electrostatic term, and the last term, which really just picks up the slack, is a “fermionic quantum term”.
Using this partition, they examine a series of bond separation reactions involving alkanes with differing degrees of “protobranches”. The upshot is that the steric energy, which is destabilizing, is less in branched alkanes that linear ones. However, the fermionic quantum term essentially cancels this out, as branched alkanes, being more compact, are more destabilized by this fermionic effect than are linear alkanes. So, the only remaining term, electrostatics is responsible for the branched alkanes being more stable than linear alkanes.
This does not ultimately resolve the issue of whether the protobranching effect, as defined by Schleyer, Mo and Houk, is real, but these authors purposely chose to avoid that question.
(1) Ess, D. H.; Liu, S.; De Proft, F., "Density Functional Steric Analysis of Linear and Branched Alkanes," J. Phys. Chem. A, 2010, ASAP, DOI: 10.1021/jp108577g
(2) Wodrich, M. D.; Wannere, C. S.; Mo, Y.; Jarowski, P. D.; Houk, K. N.; Schleyer, P. v. R., "The Concept of Protobranching and Its Many Paradigm Shifting Implications for Energy Evaluations," Chem. Eur. J. 2007, 13, 7731-7744, DOI: 10.1002/chem.200700602