Six atomic clusters with lower energy than the Cambridge Cluster Database minima (modified Dzugutov and Z1 potentials)

Six cluster configurations with lower energy than the minima listed in the Cambridge Cluster Database (CCD), for two pair potentials: modified Dzugutov (mDz; Doye and Wales, Phys. Rev. Lett. 86, 5719, 2001) and Z1 (Doye, Wales, Zetterling and Dzugutov, J. Chem. Phys. 118, 2792, 2003). They were emailed on 22 October 2026 to Jonathan Doye, author of the table pages (which ask for improvements by email), and to David Wales, who hosts the CCD. No reply as of 29 October 2026. Nothing is claimed about global optimality. Results Table, N CCD listed energy This work mDz 74 -180.982763 -180.987553 mDz 134 -343.386067 -343.390040 mDz 177 -462.579581 -462.582660 mDz 186 -487.78611 -487.857776 Z1 54 -146.333923 -146.350095 Z1 68 -185.275450 -185.363426 Our energies are rounded to six decimals; CCD energies are as listed. Energies are in the reduced units of each potential. CCD values come from the two tables (mDzugutov/table.html and Zetterling/table.html), read on 17 and 28 October 2026. The values are identical on both dates; the Last-Modified header is 25 July 2024. The gap for mDz 186 (0.072) is much larger than for the other mDz sizes (0.003 to 0.005). Method Basin-hopping, and seeding from the author's best configurations at N - 1 (one atom added) and N + 1 (one atom removed), then Newton polishing. The search code was written with Claude Opus 5.5 under the author's direction. It is not included: the deposit documents the results and their verification. The same energy code reproduces the CCD's own coordinate files: within 1e-6 for all 248 sizes of the mDz table, and within 3.4e-6 of the listed value for the six sizes here. For Z1 it agrees within 4.2e-6 with the printed constants, except for N = 13, 25, 33 and 37 (not in this deposit). There the energy recomputed from the CCD file is 7.1 to 7.6 higher than the listed value. The cause is not established. Verification Interval arithmetic: for each size, the energy of our file and the energy recomputed from the CCD's coordinate file are enclosed (120 bits, mpmath 1.3.0, iv module) from the decimal coordinates as written. Our enclosure lies strictly below the CCD one, with gaps from 0.0031 (mDz 177) to 0.088 (Z1 68). This is interval_check.py, included exactly as run (see SHA256SUMS). Second checker, verify_deepseek.py: written on 29 October 2026 by DeepSeek V4.1 Flash (through OpenRouter), from the potential definitions and file formats given in its prompt (the same definitions as in potentials.py). It did not see our code. It is included unchanged. It uses 80-digit decimal arithmetic (high precision, not interval-certified). It requires both gaps (to the recomputed CCD file and to the listed value) to exceed 1e-5, and the largest gradient component to be below 1e-8. It guards against coding errors, not against an error in the potential definitions. On 29 October 2026 it accepted the six configurations. Its energies match the interval enclosures to nine decimals; the largest gradient component is 3.4e-13 to 5.7e-12. It rejected seven corrupted copies (the script that writes them is not included): the CCD coordinates relabelled as ours, with a false printed energy; a missing atom line; a wrong atom count; one atom moved by 1e-3; two coincident atoms; an mDz CCD reference file missing one atom; a Z1 CCD reference file missing one atom. A separate 40-digit program found the same energies within 1e-12. Numerically, each configuration is a local minimum: gradient below 1e-11, six Hessian eigenvalues at zero within tolerance (translations and rotations), smallest non-zero eigenvalue between 3.81 (Z1 68) and 7.88 (mDz 74). That program is not included, and the Hessian check was not repeated independently. These are different structures, not CCD minima polished further. Re-polishing the CCD minima lowers them by only 7e-9 to 6e-8. The largest difference between the sorted interatomic distances of ours and the CCD's ranges from 0.11 to 0.38 across the six sizes. Replay of the README on a fresh copy, 29 October 2026: sha256 6/6, verify_deepseek.py 6/6 PASS, interval_check.py 6/6 certified. Files Six files <table>_<N>.xyz: N, then a comment line with our floating-point energy and an internal working note in French (kept so that the files are byte-identical to those sent to the CCD), then one line X x y z per atom. Lines end with CRLF. CCD-references.tsv: for each size, the CCD listed energy, and the URL and sha256 of the CCD coordinate file used as reference (fetched 17 October 2026). The CCD files are not redistributed here. interval_check.py and potentials.py (Python, numpy, mpmath). verify_deepseek.py (standard library only). README.txt and SHA256SUMS. Limits Z1: the constants V0 = 0.04682632 and rc = 2.64909 are used as printed in the paper. The interval-certified ordering refers to the potential with these constants. Recomputing rc and V0 so that V(rc) = 0 and V'(rc) = 0 hold exactly changes the gaps by less than 1e-7 (numerical, not interval-enclosed). The comparison is with the CCD tables only. mDz: no lower energy was found on the Oxford mirror of the CCD pages, among the titles of works citing the source papers (OpenAlex), or in five web searches. Z1: no systematic literature search was made. The ABCluster paper (Zhang and Dolg, Phys. Chem. Chem. Phys. 17, 24173, 2015) tests a potential called "Z". Its full text was not read, so it is unknown whether that is Z1, for which sizes, and with which energies. Use of AI The search, the computations and interval_check.py were produced with Claude Opus 5.5 (Anthropic) in Claude Code under the author's direction. verify_deepseek.py was written by DeepSeek V4.1 Flash. The author is responsible for the deposit. No AI system is an author.