  Recent state-of-the-art crystallographic investigations of transition metal
spinel compounds have revealed that the d- orbital charge carriers undergo
ordering transitions with the formation of local "molecular bonding" units such
as dimers in MgTi2O4, octomers in CuIr2S4, and heptamers in AlV2O4. Herein, we
provide a unifying scheme involving one- dimensional orbital wave ordering
applicable to all of these spinels. The relative phase of the orbitals in the
chains is shown to be crucial to the formation of different local units, and
thus both the amplitude and phase of the orbital wave play important roles.
Examination of Horibe et al.'s [1] structure for AlV2O4 serves as the vehicle
for developing the general behavior for such orbital wave ordering. Ordered
AlV2O4 will be seen to organize into three equivalent chains in 2D Kagome
planes coupled so as to form units of three dimer bonds. Three additional
equivalent chains manifest a more complex tetramerization with three different
charge states and two different bonding schemes. The orbital wave ordering
scheme developed is extended to other spinel and related compounds with local
triangular transition metal coordination and partial filling of the t2g-d
orbitals.
