Accelerating core-level $GW$ calculations by combining the contour deformation approach with the analytic continuation of $W$

In recent years, the $GW$ method has emerged as a reliable tool for computing core-level binding energies. The contour deformation (CD) technique has been established as an efficient, scalable, and numerically stable approach to compute the $GW$ self-energy for deep core excitations. However, core-level $GW$ calculations with CD face the challenge of higher scaling with respect to system size $N$ compared to the conventional quartic scaling in valence state algorithms. In this work, we present the CD-WAC method (CD with $W$ Analytic Continuation), which reduces the scaling of CD applied to the inner shells from $O(N^5)$ to $O(N^4)$ by employing an analytic continuation of the screened Coulomb interaction $W$. Our proposed method retains the numerical accuracy of CD for the computationally challenging deep core case, yielding mean absolute errors $<5$ meV for well-established benchmark sets, such as CORE65, for single-shot $GW$ calculations. More extensive testing for different $GW$ flavors prove the reliability of the method. We have confirmed the theoretical scaling by performing scaling experiments on large acene chains and amorphous carbon clusters, achieving speedups of up to 10x for structures of only 116 atoms. This improvement in computational efficiency paves the way for more accurate and efficient core-level $GW$ calculations on larger and more complex systems.