(15) In the T ≥ 0 case, we can visualize the contour and poles as shown in Figure (adapted from Figure in reference which shows the T = 0 case). The poles of G appear both above and below the real axis, and are weighted by the occupation factor f s (visualized as darker pink representing f s closer to 1). The partial occupations enter into the screened coulomb interaction through the irreducible polarizability. The inclusion of partial occupations using the above method is straightforwardly applicable to the evaluation of the GW electronic structure and RPA correlation energy using the static subspace approximation . Within this approach, the evaluation of the frequency dependence part of the polarizability (as needed for the integration over the imaginary axis of equation 5) is calculated using a compressed basis obtained from the low rank approximation of the static (ω = 0) polarizability. The new basis of size N b is obtained by using a screening threshold to select the eigenpairs of the static symmetryzed polarizability with the greatest absolute eigenvalues. The eigenvectors selected in this way define the subspace basis used to evaluate the χ 0 (ω) for (ω ̸ = 0). this is achieved by projection of the M nn ′ (k, q, G) transition matrix elements of equation 2 from their original plane-wave basis onto the subspace basis. If N G is the original plane-wave basis size used for the expansion of χ 0 , the evaluation of its assiciated frequency dependent part is spedup by a factor proportional to (N G /N b ) 2 .