Chemistry - Are there any molecules with delta bonds in their ground states?
Solution 1:
Chromium(II) acetate has a Cr-Cr quadruple bond - and hence a $\delta$ bond - in its ground state. This sort of "paddlewheel" configuration, with four carboxylic acids coordinating a pair of metal ions, is reasonably common (see also copper(II) acetate, for instance), although the extent to which the metals can be considered covalently bonded varies.
Solution 2:
Metal-metal quadruple bonds have seemingly been reported for several cases, quite often by F. A. Cotton. A short search revealed a more recent report on a species with a Cr-Cr quintuple bond: Synthesis of a Stable Compound with Fivefold Bonding Between Two Chromium(I) Centers.
Admittedly, I didn't bother to read the full article, it's just not the kind of chemistry I'm interested in. However, YMMV and you might find it helpful.
Solution 3:
I am a bit late to the party, but I just found this question. When I did research on the φ bond for What would follow in the series sigma, pi and delta bonds? I came across some predictions about possible φ bond in the ground state.
According to Gagliardi and Roos, there should be a φ bond in the ground state of the $\ce{U2}$ molecule. That being said, it is probably incredibly tough to synthesise this molecule.
Metal-metal multiple bonds have become fairly common as my earlier answer demonstrates.
It should be clarified that a quadruple bond does not necessarily mean that a δ bond is involved. Often enough the combination of the dz² metal orbitals lead to the formation of a second σ bond first.
It should further be explained, that a δ bond can be present, but the overall bond order could be lower than four.
Laura Gagliardi, Björn O. Roos, Nature, 2005, 433, 848-851. Available free at Archive ouverte UNIGE.