  The order-disorder type phase transition caused by the vibronic interaction
(collective Jahn-Teller effect) in a monoclinic CuInP$_2$S$_6$ crystal is
analyzed. For this purpose, a trigonal protostructure model of CuInP$_2$S$_6$
is created, through a slight change in the crystal lattice parameters of the
CuInP$_2$S$_6$ paraelectric phase. In parallel to the group-theoretical
analysis, the DFT-based {\it ab initio} band structure calculations of the
CuInP$_2$S$_6$ protostructure, para-, and ferriphases are performed. Using the
elementary energy bands concept, a part of the band structure from the vicinity
of the forbidden energy gap, which is created by the $d$-electron states of
copper, has been related with a certain Wyckoff position where the
Jahn-Teller's centers are localized. A construction procedure of the vibronic
potential energy matrix is generalized for the case of crystal using the
elementary energy bands concept and the group theoretical method of invariants.
The procedure is illustrated by the creation of the adiabatic potentials of the
$\Gamma_5$-$\Gamma_5$ vibronic coupling for the protostructure and paraphase of
the CuInP$_2$S$_6$ crystal. A structure of the obtained adiabatic potentials is
analyzed, followed by conclusions on their transformation under a phase
transition and the discussion on the possibility for the spontaneous
polarization to arise in this crystal.
