Bond rotation in buta-1,3-diene
1,3-Butadiene (Preferred IUPAC name: Buta-1,3-diene) is the organic compound, which is industrially important as a monomer in the production of synthetic rubber. It is also the simplest conjugated diene among organic compounds. Since all carbon atoms are $\mathrm{sp^2}$ hybridized, orbital overlap between double bonds is possible and, theoretically, two different conformer shows maximum overlapping, which are s-trans conformers with $\ce{C2-C3}$ dihedral angle of $180^\mathrm{o}$ and s-cis conformer with $\ce{C2-C3}$ dihedral angle of $0^\mathrm{o}$. However, due to steric hindrance, this s-cis conformer is approximately $\pu{16.5 kJ/mol}$ higher in energy than its s-trans counterpart, making this geometry a local energy maximum. Thus, in contrast to the s-trans geometry, which is global minimum, s-cis geometry is not a conformer (Wikipedia).

The double bonds of the s-cis geometry are twisted to give a dihedral angle of around $38^\mathrm{o}$, is considered to be a second conformer (called gauche-rotamer) that is around $\pu{12.0 kJ/mol}$ higher in energy than the s-trans conformer. The high level ab Initio calculations, utilizing coupled cluster theory with quasi-perturbative triple excitations and augmented quadruple $\zeta$ level basis sets, done by Feller and Craig in 2009 showed that the transition state separating the trans- and gauche-rotamers lies $\pu{26.8 kJ/mol}$ above the trans global minimum (Ref.1). These calculations further showed that the gauche rotamer lies $\pu{12.6 kJ/mol}$ above the trans-rotamer and the s-cis-form is a transition state $\pu{2.0 kJ/mol}$ higher than the gauche-rotamer. Wikipedia state that:


  This increased rotational barrier and strong overall preference for a near-planar geometry is evidence for a delocalized π system and a small degree of partial double bond character in the C-C single bond, in accord with resonance theory.


This is illustrated in the following scheme with the most stable conformer of 1,3-butadiene, which has the s-trans conformation (Ref.2):



Quantum chemical calculations on Π-Electron Delocalization in Butadiene has also done and published in 2006. The resultant bond lengths were illustrated in the following diagram and the total abstract of the paper was reproduced here for readers with interest:


  Abstract: Equilibrium structures have been determined for s-trans-1,3-butadiene and ethylene after adjusting the rotational constants obtained from rotational spectroscopy by vibration-rotation constants calculated from the results of quantum chemical calculations. For butadiene, the formal $\ce{C=C}$ bond length is $\pu{1.338 Å}$, and the formal $\ce{C−C}$ bond length is $\pu{1.454 Å}$. For ethylene, the $\ce{C=C}$ bond length is $\pu{1.3305 Å}$. These values appear to be good to $\pu{0.001 Å}$. It is shown for the first time that π-electron delocalization has the structural consequences of increasing the length of the formal double bond by $\pu{0.007 Å}$ and decreasing the length of the formal single bond by $\pu{0.016 Å}$. Comparisons are made with structures computed with several quantum chemical models. The MP2/cc-pVTZ results agree best with the new $r_e$ structure.




References:


D. Feller, N. C. Craig, “High Level ab Initio Energies and Structures for the Rotamers of 1,3-Butadiene,” J. Phys. Chem. A 2009, 113(8),  1601–1607 (DOI: 10.1021/jp8095709).
N. C. Craig, P. Groner, D. C. McKean, “Equilibrium Structures for Butadiene and Ethylene:  Compelling Evidence for Π-Electron Delocalization in Butadiene,” J. Phys. Chem. A 2006, 110(23), 7461–7469 (DOI: 10.1021/jp060695b).