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As 2nd carbon and 3rd carbon of butadiyne are sp hybridized so there are two unhybridised p orbital mutually perpendicular to each other in each of carbon atom also these two orbitals are separated by a sigma bond.
Is There a possibility that it could show conjugation from the both the unhybridised p orbitals as opposed just conjunction takes place in either one plane ?
I'm not able to confirm this hypothesis as I don't have reliable resources.
"For butadiene...the formal (single) C−C bond length is 1.454 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 0.007 Å and decreasing the length of the formal single bond by 0.016 Å." https://pubs.acs.org/doi/pdf/10.1021/jp060695b
"An empirical equilibrium structure based on experimental rotational constants for 13 isotopic species of diacetylene and computed zero-point vibrational corrections is determined: ... r CC (triple bond) = 1.2085 Å, r C-C = 1.3727 Å and in good agreement with the best theoretical structure." https://www.sciencedirect.com/science/article/abs/pii/S0022285208000969
The first reference notes that conjugation in butadiene shortens the middle C-C single bond by 0.016 Å; the second reference notes the middle single C-C bond in diacetylene (butadiyne) is shortened to 1.3727 Å from 1.454 Å in butadiene, a reduction of 0.0813 Å, about five times as much as occurs when going from no conjugation on either side of the single bond to double bonds on both sides.
This bond shortening is a strong case for extra conjugation in diacetylene (butadiyne). The absolute linearity of diacetylene is a big help; in butadiene, the rotation around the C-C single bond would reduce conjugation.
Look at a flat ring; maybe there is more conjugation in cyclopentadiene: "The microwave spectra of the three monosubstituted 13C species of cyclopentadiene have been investigated. The effective rotational constants of these species together with those of the main isotopic species give the structural parameters for the ring. Labeling the methylene carbon as C1 and numbering the other carbons sequentially we obtain: r(C1C2)=1.509 Å, r(C2C3)=1.342 Å, r(C3C4)=1.469 Å, ∠C1C2C3=109.3°, ∠C2C3C4=109.4°, ∠C2C1C5=102.8°." https://aip.scitation.org/doi/10.1063/1.1697207 The C-C single bond between the two double bonds is r(C3C4)=1.469 Å, actually a little longer than in butadiene (1.454 Å)! Steric restrictions in the ring may account for the difference, but it's small, so may be insignificant.
