In this study, an early-working algorithm is designed to evaluate derivatives of electron repulsion integrals (DERIs) for heavy-element systems. The algorithm is constructed to extend the accompanying coordinate expansion and transferred recurrence relation (ACE-TRR) method, which was developed for rapid evaluation of electron repulsion integrals (ERIs) in our previous article (M. Hayami, J. Seino, and H. Nakai, J. Chem. Phys. 2015, 142, 204110). The algorithm was formulated using the Gaussian derivative rule to decompose a DERI of two ERIs with the same sets of exponents, different sets of contraction coefficients, and different angular momenta. The algorithms designed for segmented and general contraction basis sets are presented as well. Numerical assessments of the central processing unit time of gradients for molecules were conducted to demonstrate the high efficiency of the ACE-TRR method for systems containing heavy elements. These heavy elements may include a metal complex and metal clusters, whose basis sets contain functions with long contractions and high angular momenta.
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