How many isomers do metallic clusters have?

Stanislav K. Ignatov,1 Sergey N. Belyaev,1 Sergey V. Panteleev,1 Art¸m E. Masunov2,3,4

1 Lobachevsky State University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod 603950, Russia 2 NanoScience Technology Center, University of Central Florida, Orlando, FL 32826, USA 3 South Ural State University, Lenin pr. 76, Chelyabinsk 454080, Russia 4 National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoye shosse 31, Moscow, 115409, Russia


Metallic clusters are the physicochemical objects which have no simple and reliable theory of their chemical structure. In contrast with the organic molecules, which structure can be predicted easily on the basis of several simple rules and restricted set of experimental properties, the structure of metallic clusters both mono- or poly-elemental ones frequently demonstrates a surprising structural diversity depending on their energy, spin multiplicity, and the metal nature. The question arises: how many isomers can be formed in a cluster of a given size n (cluster nuclearity)? How fast does this number grow with increasing n? Can we identify all isomeric structures and evaluate their properties? To clarify the above questions, it would be useful to obtain the entire set of isomers that can be formed for a given type of metal and a given cluster size n. In this work, we try to do this for clusters of magnesium using several approaches to form a complete set of their isomeric structures: (1) generation of new structures based on the complete set of non-isomorphic connected graphs available for the given number of vertices; (2) long-term global optimization, continuing until the limit is reached, at which the optimization of generated structures ceases to lead to new structural classes. (3) additional manual construction of symmetric structures. Thus, the structures of clusters Mgn (n = 2–13) were optimized at several DFT levels using the extensive set of starting geometries (~9000 starting structures, ~820000 explored PES points). It was found that the number of unique stable isomers N quickly grows with n; however, it is significantly lower than the number of possible non-isomorphic graph structures which can be drawn for the given n. Surprisingly, we located only 544 stable isomers among all clusters investigated. It was difficult to distinguish unambiguously between the exponential and power growth models of N(n) although the power model was slightly better. This number of isomers grows with n approximately as n4, whereas the extrapolated N values grow as n8. The clusters geometries obtained at the DFT level were used to adjust two empirical potentials of Gupta type (GP) and modified Sutton-Chen type (SCG3) describing the interactions between the magnesium atoms. Using these potentials, the extensive sets of structures (up to 30000 clusters for each n) were optimized to obtain the dependence of clusters isomer count on n in the continuous range of n=2–30 and for selected n up to 55. It was found that the SCG3 potential, which is closer to the DFT results, gives the number of possible isomers growing as approximately n8.9 whereas GP potential results in the n4.3 dependence. This work was supported by the RFBR (projects No. 20-03-00282 and 18-43-520012)