  The selection rule on vibronic angular momentum of $t_{1u}^n \otimes h_g$
Jahn-Teller problem ($n = $ 1-5) is reinvestigated. It is shown that among
three adiabatic orbitals only two have nonzero Berry phase. Thus, the Berry
phase of adiabatic electronic configurations depends on the spin multiplicity
as well as the number of electrons. On this basis, the general relation between
the Berry phase and the angular momentum is described. It allows, in
particular, to clarify the nature of vibronic states arising from high spin
configurations. In comparison with the previous solution for the low-lying
vibronic states for bimodal systems, the present solutions correctly fulfill
all the symmetry requirement.
