Vibronic reduction factors in E ⊗ (β₁ + β₂) and their Berry's phase manifestations

In a linearly coupled dihedral E ⊗ (β₁ + β₂) Jahn-Teller system one finds two different q-factors (say, q₁ and q₂) and a p-factor. For stronger β₁- than β₂-coupling, one obtains q₁ > q₂ and in the strong linear vibronic coupling limit, q₁ → 0.5, q₂ → 0. At all strengths, the historical relation "p = 2q - 1" turns into p = q₁ + q₂ - 1. A slowly rotating external field induces in the electronic wave-functions, upon a full turn, a Berry's phase equal to the solid angle subtended by the field's vector at the origin. We have calculated (by employing a quasi-classical approximation to solve the dynamic problem for the ground vibronic state) the corresponding phase change, now diminished by the vibronic reduction factors.