litosphereЛитосфераLITHOSPHERE (Russia)1681-90042500-302XA.N. Zavaritsky Institute of Geology and Geochemistry10.24930/1681-9004-2024-24-2-240-253litosphere-2058Research ArticleСтатьиArticlesКомбинаторная и алгоритмическая сложность кристаллических структурCombinatorial and algorithmic complexity of crystal structuresБанаруД. А.BanaruD. A.

119334, г. Москва, ул. Косыгина, 19

Darya A. Banaru

19 Kosygin st., Moscow 119334

banaru@geokhi.ruАксеновС. М.AksenovS. M.

184209, Мурманская обл., г. Апатиты, ул. Ферсмана, 14

Sergey M. Aksenov

14 Fersman st., Apatity, Murmansk region 184209

Институт геохимии и аналитической химии им. В.И. Вернадского РАНРоссияV.I. Vernadsky Institute of Geochemistry and Analytical Chemistry, RASRussian FederationКольский научный центр РАНРоссияKola Science Centre, RASRussian Federation202403052024242240253Copyright © Банару Д.А., Аксенов С.М., 20242024Банару Д.А., Аксенов С.М.Banaru D.A., Aksenov S.M.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.lithosphere.ru/jour/article/view/2058

Объект исследования. Числовые индексы, описывающие сложность системы контактов между структурными единицами в кристаллических структурах. Цель исследования. Разработка индекса сложности для системы контактов между периодическими структурными единицами на основании индексов, имеющихся для таковой между структурными единицами в островных (молекулярных) структурах. Материал и методы. Структурные данные были отобраны из кристаллографических баз данных COD, AMCSD и CSD. Анализ системы контактов в структурах проводили методом полиэдров Вороного – Дирихле (ПВД) в пакете программ ToposPro. Результаты. Метод топологического анализа системы контактов в молекулярных кристаллах адаптирован ко всем гетеродесмическим кристаллическим структурам и опробован на структурах соединений нескольких классов. Разработаны целевые индексы сложности. Выводы. Сети контактов между периодическими структурными единицами являются низкоразмерными. Обобщенный структурный класс для таких сетей может быть выведен из исходных кристаллоструктурных данных. Алгоритмическая сложность гетеродесмических структур субаддитивна в отличие от супераддитивной комбинаторной сложности. Впервые рассчитано число опорных контактов, отражающее алгоритмическую сложность структуры на соответствующем уровне структурного описания, между периодическими структурными единицами.

Research subject. Numeral indexes describing the complexity of the system of contacts between structural units in crystal structures. Aim. Development of a complexity index for the system of contacts between periodic structural units based on the indices available for those between structural units in island (molecular) structures. Materials and methods. Structural data were selected from the COD, AMCSD, and CSD crystallographic databases. The system of contacts in the structures was analyzed by the Voronoi–Dirichlet polyhedra (VDP) method in the ToposPro software package. Results. The method of topological analysis of the system of contacts in molecular crystals was adapted to all heterodesmic crystal structures and tested on the structures of compounds of several classes. Target complexity indices were developed. Conclusions. Networks of contacts between periodic structural units are low-dimensional. A generalized structural class for such networks can be derived from the original crystal structure data. The algorithmic complexity of heterodesmic structures is subadditive, in contrast to superadditive combinatorial complexity. For the first time, the number of bearing contacts was calculated between periodic structural units, reflecting the algorithmic complexity of the structure at the appropriate level of structural description.

кристаллическая структуракристаллографическая орбитапозиция Уайкоффакомбинаторная сложностьсильная аддитивностьcrystal structurecrystallographic orbitWyckoff positioncombinatorial complexitystrong additivityРазработка теоретической модели выполнена в рамках госзадания ГЕОХИ РАН. Топологические расчеты поддержаны грантом Российского научного фонда № 20-77-10065. Авторы выражают благодарность чл.-корр. РАН проф. Н.Н. Еремину за ценные советы, а также доценту А.М. Банару за помощь в подготовке рукописи.The development of the theoretical model was carried out within the framework of the state task of GEOKHI RAS. Topologic calculations were supported by the Russian Science Foundation grant No. 20-77-10065. The authors express their gratitude to Prof. N.N. Eremin, a corresponding member of RAS, for his valuable advice, as well as to associate prof. A.M. Banaru for his help in preparing the manuscript.ReferencesУрусов В.С. 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The authors declare that there are no conflicts of interest present.