Three independent methods for intermediate Jahn-Teller coupling

Three different and independent methods are used to find eigenfunctions and eigenvalues for a linear Jahn-Teller Hamiltonian. They are the following: diagonalization on a Born-Oppenheimer basis developed in the adiabatic limit, diagonalization on a Glauber grates basis developed; in the strong-coupling limit, and construction of the eigenfunctions by means of the Lanczos method. We explore the space of interactions aiming at the intermediate-coupling limit, finding the regions of best convergence for each method. Comparison among the three methods in terms of their numerical results for energy and expected optical transitions leads to regions of total and partial agreement, Conditions for several zero-phonon lines are discussed. The dominant line is not always the threshold; line and it is determined by a nontrivial balance involving all interactions. Conclusions are focused toward finding reliable methods for the different regions of the parameter space. Preparation is done for applications of this approach to explaining optical spectra of magnetic impurities in crystals under different coupling regimes.