Lanczos and modified Lanczos procedures for the Jahn-Teller systems

The analysis of the dynamical properties of the Jahn-Teller systems requires the computation of eigenstates of large and sparse matrices, for which the use of traditional computational technique is in general precluded. Among the workable methods developed to handle these very large matrices, the Lanczos method and the related recursion method have emerged as the most simple and efficient computational tools for a large variety of applications. The merits of the Lanczos recursion method are particularly evident when a few extreme eigenvalues are desired or when the recursion coefficients can be put in analytic form, as is the case of E ⊗ ε and T ⊗ ε Jahn-Teller systems. In more general situations the Lanczos method is still extremely useful, but at the same time must be used with extreme caution and appropriate implementations. Its main difficulty is related to the finite precision arithmetic of the computers which causes a loss of orthogonality among the states generated by the Lanczos procedure: instabilities in the recursion coefficients can occur, producing the so called "Lanczos phenomena" (ghost states or spurious states). However a precious tool, to identify unambiguously the good eigenvalues from the fake ones, is offered by the following implementation, developed by our group. The Lanczos scheme is applied not to H, but rather to the auxiliary operator A = (H - Et)2, whose ground state is determined through an iterative process which alternates the diagonalization of 2 × 2 Lanczos matrices to a two-pass Lanczos procedure of suitable small dimension. So the eigenstates of a system can be obtained, one at a time, within any desired energy range and with any desired precision. As an exemplification of this modified Lanczos procedure, the T ⊗ τ Jahn-Teller system and the absorption spectrum of ZnS:Fe2+ are considered in detail.