Generalized quadrature for finite temperature Green’s function methods

In electronic structure and quantum transport calculations, many physical quantities are integrations over the electron energy, weighted by the Fermi–Dirac distribution function. Green’s function based approaches commonly circumvent the numerically difficult real energy integration by extending the integrand analytically into the complex energy plane, and using a Gaussian quadrature integration over a complex energy contour for zero temperature. For finite temperatures, a much slower convergent sum over the Matsubara frequencies is necessary. We present a generalized quadrature method that uses orthogonal polynomials on the manifold of Matsubara frequencies to enable rapid convergence. Both Gaussian quadrature integration and Matsubara frequency summation methods are shown to be limiting cases of the generalized method. Tests on an all-electron ab initio code and total energy calculation of interacting Anderson impurity model show convergence with a small number of energy mesh points.