The Lanczos tridiagonalization method orthogonally transforms a real symmetric matrix A to symmetric tridiagonal form. Traditionally, this very simple algorithm is suitable when one needs only a few of the lower eigenvalues and the corresponding eigenvectors of very large Hermitian matrices, whose full diagonalization is technically impossible. We introduce here the basic ingredients of the recursion method based on the Lanczos tridiagonalization, and explain how calculation of the DOS as well as the dynamics of wavepackets (and related conductivity) can be performed efficiently.

Lanczos method for the density of states

The Lanczos method is a highly efficient recursive approach for calculation of the electronic structure (Lanczos, 1950). This method, first developed by Haydock, Heine, and Kelly (Haydock, Heine & Kelly, 1972, 1975), is based on an eigenvalue approach due to Lanczos. It relies on computation of Green functions matrix elements by continued fraction expansion, which can be implemented either in real or reciprocal space. These techniques are particularly well suited for treating disorder and defect-related problems, and were successfully implemented to tackle impurity-level calculations in semiconductors using a tight-binding approximation (Lohrmann, 1989), and for electronic structure investigations for amorphous semiconductors, transition metals, and metallic glasses based on linear-muffin-tin orbitals (Bose, Winer & Andersen, 1988). Recent developments include the exploration of a degenerated orbital extended Hubbard Hamiltonian of system size up to ten millions atoms, with the Krylov subspace method (Takayama, Hoshi & Fujiwara, 2004, Hoshi et al., 2012).