... vanish.1
The term adiabatic does not actually distinguish between eigenfunctions of ${\hat H}_{el}$ and ${\hat H}_0$. Eigenfunctions of ${\hat H}_{el}$ will still have nonzero matrix elements $U^{(so)}_{kk'}$ and hence nonzero $U_{kk'}$ if spin-orbit coupling is considered.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.