Carl Kemnitz has been paying attention to the Gronert vs Wodrich&Schleyer debate on the origin of stability in branched alkanes and protobranching as reported on in this blog in 2009 (DOI). In a nutshell, branched alkanes are more stable (more favorable heat of formation) than their linear counterparts thanks to protobranches (attractive 1,3 alkyl alkyl interactions) if you believe Wodrich or despite repulsive steric HCH, HCC or CCC interactions if you side with Gronert. No wonder Kemnitz labels the debate as controversial with diametrically opposed explanations. There is obviously work to be done.
No evidence was found for Gronerts physical model based on steric repulsions although his heat of formation group additivity method continues to hold. Kemnitz notes that on going from n-butane (1 PB, 8HCH, 14 HCC, 2CCC) to isobutane (3PB, 9 HCH, 12 HCC, 3 CCC) for every protobranch you add you also gain one HCH interaction, one CCC interaction and lose 2 HCC interactions. All aliphatic hydrocarbons share this property and Kemnitz was able to compute the energies involved arriving at 32 kcal/mol for CCC, 22 kcal/mol for HCC and 15 kcal/mol for HCH interactions. Adding a protobranch then involves an energy gain of 15+32-2*22 = +3 kcal/mol meaning that the contributions are destabilizing after all.
That leaves the protobranching model. Interestingly Wodrich never supplied his theory with a physical model (protobranches stabilize, that's it) but Kemnitz has developed one of its own or rather two. NBO calculations (natural bond orbitals see Goldbook) indicate stabilization through delocalization of electrons more precisely by sigma-sigma(star) excitation which this blog guesses is something like hyperconjugation with electrons in a sigma - carbon-carbon bond also occupy the empty sigma star orbital of the adjacent C-C bond. In another approach similarities are noted between protobranching and the charge-shift (CS) bond concept, the brain child of Shaik & Hiberty (DOI). CS bonds result from fluctuations in charge density and when applied to propane pure covalent bonding only accounts for 49% to the total picture and ionic contributions make up the rest. Specifically two 1,3-ionic structures H3C+ CH2 -H3C and H3C- CH2 + H3C make up 9% , require a protobranch and lower the total energy by 1.6 kcal/mol, a value close to the stabilization energy per protobranch.