  In solids, crystal field splitting refers to the lifting of atomic orbital
degeneracy by the surrounding ions through the static electric field.
Similarly, we show that the degenerated $d$ orbitals, which were derived in the
harmonic oscillator approximation, are split into a low-lying $d_{x^2+y^2}$
singlet and a $d_{x^2-y^2/xy}$ doublet by the high-order Taylor polynomials of
triangular optical potential. The low-energy effective theory of the orbital
Mott insulator at $2/3$ filling is generically described by the
Heisenberg-Compass model, where the antiferro-orbital exchange interactions of
compass type depend on the bond orientation and are geometrically frustrated in
the triangular lattice. While, for the square optical lattice, the degenerated
$d$ orbitals are split into a different multiplet structure, i.e. a low-lying
$d_{x^2\pm y^2}$ doublet and a $d_{xy}$ singlet, which has its physical origin
in the $C_{4v}$ point group symmetry of square optical potential. Our results
build a novel bridge between ultracold atom systems and solid-state systems for
the investigation of $d$-orbital physics.
