Continued fraction representation of the Fermi-Dirac function for large-scale electronic structure calculations

An efficient and accurate contour integration method is presented for large-scale electronic structure calculations based on the Green function. By introducing a continued fraction representation of the Fermi-Dirac function derived from a hypergeometric function, the Matsubara summation is generalized with respect to distribution of poles so that the integration of the Green function can converge rapidly. Numerical illustrations, evaluation of the density matrix for a simple model Green function and a total energy calculation for aluminum bulk within density functional theory, clearly show that the method provides remarkable convergence with a small number of poles, indicating that the method can be applied to not only the electronic structure calculations, but also a wide variety of problems.

Figure 1

(Color online) The Fermi-Dirac function and its approximants as a function of the real axis $x$. For substantial comparison all the approximants possess the same number of poles, 120.

Figure 2

(Color online) The path of the contour integration associated with Eq. (1). The inset shows the distribution of zero points of Eq. (1).

Figure 3

(Color online) The interval between neighboring poles of Eq. (1) terminated at $N=100$ as a function of an index of poles, where the index of poles is determined in ascending order with respect to the absolute value of poles. The interval for the Matsubara poles is $2\pi$.