  Motivated by the recent activities on the Ni-based diamond lattice
antiferromagnet NiRh$_2$O$_4$, we theoretically explore on a general ground the
unique spin and orbital physics for the Ni$^{2+}$ ions with a $3d^8$ electron
configuration in the tetrahedral crystal field environment and on a diamond
lattice Mott insulator. The superexchange interaction between the local moments
usually favors magnetic orders. Due to the particular electron configuration of
the Ni$^{2+}$ ion with a partially filled upper $t_{2g}$ level and a fully
filled lower $e_g$ level, the atomic spin-orbit coupling becomes active at the
linear order and would favor a spin-orbital-entangled singlet with quenched
local moments in the single-ion limit. Thus, the spin-orbital entanglement
competes with the superexchange and could drive the system to a quantum
critical point that separates the spin-orbital singlet and the magnetic order.
We further explore the effects of magnetic field and uniaxial pressure. The
non-trivial response to the magnetic field is intimately tied to the underlying
spin-orbital structure of the local moments. We discuss the future experiments
such as doping and pressure, and point out the correspondence between different
electron configurations.
