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By exploiting an efficient evaluation of the Hessian-vector product, $k$-CIAH achieves $O(N_k^2 n^3)$ scaling in both CPU time and memory, matching that of previously reported first-order $k$-space approaches while improving upon the $O(N_k^3 n^3)$ scaling of $\\Gamma$-point CIAH, where $N_k$ denotes the number of $k$-points sampling the first Brillouin zone and $n$ characterizes the unit-cell size. Benchmark calculat"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"By exploiting an efficient evaluation of the Hessian-vector product, k-CIAH achieves O(N_k^2 n^3) scaling in both CPU time and memory, matching that of previously reported first-order k-space approaches while improving upon the O(N_k^3 n^3) scaling of Γ-point CIAH... yields an overall computational efficiency approximately 2-3-fold higher than first-order k-space methods and orders of magnitude higher than Γ-point CIAH for localizing 1000-5000 orbitals. 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