Computation of crystal Green's functions in the complex-energy plane with the use of the analytical tetrahedron method

The analytical tetrahedron method (ATM) for evaluating perfect-crystal Green's functions is reviewed. It is shown that the ATM allows for computing matrix elements of the resolvent operator in the entire complex-energy plane. These elements are written as a scalar product involving weighting functions of the complex energy, which are computed on a mesh of $\overrightarrow{\mathrm{k}}$ points in the Brillouin zone. When the usual approximations are made within each tetrahedron, namely linear interpolations for the dispersion relations as well as for the numerator matrix elements, the weighting functions only depend on the perfect-crystal dispersion relations. In addition, the analytical expression obtained for a tetrahedral contribution to the weighting functions is simpler than what is usually expected. Analytical properties of our expressions are discussed and all the limiting forms are worked out. Special attention is paid to the numerical stability of the algorithm producing the Green's-function imaginary part on the real energy axis. Expressions which have been published earlier are subject to computational problems, which are solved in the new formulas reported here.