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In recent years, the $GW$ method has emerged as a reliable tool for computingcore-level binding energies. The contour deformation (CD) technique has beenestablished as an efficient, scalable, and numerically stable approach tocompute the $GW$ self-energy for deep core excitations. However, core-level$GW$ calculations with CD face the challenge of higher scaling with respect tosystem size $N$ compared to the conventional quartic scaling in valence statealgorithms. In this work, we present the CD-WAC method (CD with $W$ AnalyticContinuation), 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 screenedCoulomb interaction $W$. Our proposed method retains the numerical accuracy ofCD for the computationally challenging deep core case, yielding mean absoluteerrors $<5$ meV for well-established benchmark sets, such as CORE65, forsingle-shot $GW$ calculations. More extensive testing for different $GW$flavors prove the reliability of the method. We have confirmed the theoreticalscaling by performing scaling experiments on large acene chains and amorphouscarbon clusters, achieving speedups of up to 10x for structures of only 116atoms. This improvement in computational efficiency paves the way for moreaccurate and efficient core-level $GW$ calculations on larger and more complexsystems.

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