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The single particle energies obtained in a Kohn--Sham density functionaltheory (DFT) calculation are generally known to be poor approximations toelectron excitation energies that are measured in transport, tunneling andspectroscopic experiments such as photo-emission spectroscopy. The correctionto these energies can be obtained from the poles of a single particle Green'sfunction derived from a many-body perturbation theory. From a computationalperspective, the accuracy and efficiency of such an approach depends on how aself energy term that properly accounts for dynamic screening of electrons isapproximated. The $G_0W_0$ approximation is a widely used technique in whichthe self energy is expressed as the convolution of a non-interacting Green'sfunction ($G_0$) and a screened Coulomb interaction ($W_0$) in the frequencydomain. The computational cost associated with such a convolution is high dueto the high complexity of evaluating $W_0$ at multiple frequencies. In thispaper, we discuss how the cost of $G_0W_0$ calculation can be reduced byconstructing a low rank approximation to the frequency dependent part of $W_0$.In particular, we examine the effect of such a low rank approximation on theaccuracy of the $G_0W_0$ approximation. We also discuss how the numericalconvolution of $G_0$ and $W_0$ can be evaluated efficiently and accurately byusing a contour deformation technique with an appropriate choice of thecontour.

Low Rank Approximation in $G_0W_0$ Approximation