scholarly journals Complexity Parameters for Molecular Solids

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1399
Author(s):  
Alexander M. Banaru ◽  
Sergey M. Aksenov ◽  
Sergey V. Krivovichev
Keyword(s):  
Crystal Structures ◽  
Degrees Of Freedom ◽  
Inorganic Compounds ◽  
Molecular Crystals ◽  
Structural Complexity ◽  
Molecular Solids ◽  
Complexity Measures ◽  
Inorganic Crystal ◽  
Shannon Information Entropy ◽  
Intermolecular Contacts

Structural complexity measures based on Shannon information entropy are widely used for inorganic crystal structures. However, the application of these parameters for molecular crystals requires essential modification since atoms in inorganic compounds usually possess more degrees of freedom. In this work, a novel scheme for the calculation of complexity parameters (HmolNet, HmolNet,tot) for molecular crystals is proposed as a sum of the complexity of each molecule, the complexity of intermolecular contacts, and the combined complexity of both. This scheme is tested for several molecular crystal structures.

scholarly journals Crystal Chemical Relations in the Shchurovskyite Family: Synthesis and Crystal Structures of K2Cu[Cu3O]2(PO4)4 and K2.35Cu0.825[Cu3O]2(PO4)4

Crystals ◽  
2021 ◽  
Vol 11 (7) ◽  
pp. 807
Author(s):  
Ilya V. Kornyakov ◽  
Sergey V. Krivovichev
Keyword(s):  
Crystal Structure ◽  
Crystal Structures ◽  
Structural Complexity ◽  
Crystal Chemical ◽  
X Ray Diffraction ◽  
Complexity Measures ◽  
X Ray ◽  
The Family ◽  
Similarities And Differences ◽  
Related Compounds

Single crystals of two novel shchurovskyite-related compounds, K2Cu[Cu3O]2(PO4)4 (1) and K2.35Cu0.825[Cu3O]2(PO4)4 (2), were synthesized by crystallization from gaseous phase and structurally characterized using single-crystal X-ray diffraction analysis. The crystal structures of both compounds are based upon similar Cu-based layers, formed by rods of the [O2Cu6] dimers of oxocentered (OCu4) tetrahedra. The topologies of the layers show both similarities and differences from the shchurovskyite-type layers. The layers are connected in different fashions via additional Cu atoms located in the interlayer, in contrast to shchurovskyite, where the layers are linked by Ca2+ cations. The structures of the shchurovskyite family are characterized using information-based structural complexity measures, which demonstrate that the crystal structure of 1 is the simplest one, whereas that of 2 is the most complex in the family.

scholarly journals The Principle of Maximal Simplicity for Modular Inorganic Crystal Structures

Crystals ◽  
2021 ◽  
Vol 11 (12) ◽  
pp. 1472
Author(s):  
Sergey V. Krivovichev
Keyword(s):  
Crystal Structures ◽  
Structure Prediction ◽  
Structural Information ◽  
Structural Complexity ◽  
Configurational Entropy ◽  
Sensu Stricto ◽  
Least Action ◽  
Inorganic Crystal ◽  
Principle Of Least Action ◽  
Structure Description

Modularity is an important construction principle of many inorganic crystal structures that has been used for the analysis of structural relations, classification, structure description and structure prediction. The principle of maximal simplicity for modular inorganic crystal structures can be formulated as follows: in a modular series of inorganic crystal structures, the most common and abundant in nature and experiments are those arrangements that possess maximal simplicity and minimal structural information. The latter can be quantitatively estimated using information-based structural complexity parameters. The principle is applied for the modular series based upon 0D (lovozerite family), 1D (biopyriboles) and 2D (spinelloids and kurchatovite family) modules. This principle is empirical and is valid for those cases only, where there are no factors that may lead to the destabilization of simplest structural arrangements. The physical basis of the principle is in the relations between structural complexity and configurational entropy sensu stricto (which should be distinguished from the entropy of mixing). It can also be seen as an analogy of the principle of least action in physics.

Ladders of information: what contributes to the structural complexity of inorganic crystals

2018 ◽  
Vol 233 (3-4) ◽  
pp. 155-161 ◽  
Author(s):  
Sergey V. Krivovichev
Keyword(s):  
Crystal Structures ◽  
Inorganic Compounds ◽  
Structural Complexity ◽  
Unit Cell ◽  
Shannon Information ◽  
Topological Complexity ◽  
Crystal Chemical ◽  
Quantitative Estimates ◽  
Hydration State ◽  
Inorganic Crystals

AbstractComplexity is one of the important characteristics of crystal structures, which can be measured as the amount of Shannon information per atom or per unit cell. Since complexity may arise due to combination of different factors, herein we suggest a method of ladder diagrams for the analysis of contributions to structural complexity from different crystal-chemical phenomena (topological complexity, superstructures, modularity, hydration state, etc.). The group of minerals and inorganic compounds based upon the batagayite-type [M(TO4)ϕ] layers (M=Fe, Mg, Mn, Ni, Zn, Co; T=P, As; ϕ=OH, H2O) is used as an example. It is demonstrated that the method allows for the quantitative estimates of various contributions to the complexity of the whole structure.

Voronoi–Dirichlet tesselation as a tool for investigation of polymorphism in molecular crystals with C w H x N y Oz composition and photochromic properties

2012 ◽  
Vol 68 (3) ◽  
pp. 305-312 ◽  
Author(s):  
Viktor N. Serezhkin ◽  
Larisa B. Serezhkina ◽  
Anna V. Vologzhanina
Keyword(s):  
Hydrogen Bonds ◽  
Crystal Structures ◽  
Nitro Group ◽  
General Formula ◽  
Molecular Crystals ◽  
Photochromic Properties ◽  
Intermolecular Contacts

The non-bonded interactions in five sets of polymorph substances with photochromic properties have been investigated within the Voronoi–Dirichlet approach. Twenty compounds with the general formula C w H x N y O z were analyzed. Among ten possible types of non-bonded interactions at least five types are observed in the crystal structures of compounds under discussion. For all the structures the majority of interactions involve H atoms, namely London forces (H...H and H...C) and hydrogen bonds (H...O and H...N). A conformational polymorph was stated to be characterized by a unique set of inter- and intramolecular non-bonded interactions. It was quantitatively demonstrated that molecules in the same conformation can pack in a different way, and, vice versa, the change in conformation of a molecule does not prevent a substance from realising the same set of intermolecular contacts. In accordance with the data obtained for 2,4-dinitrobenzylpyridine derivatives, only conformational polymorphs with an intramolecular N...N interaction between a nitro group and a pyridine are photochromic.

Dynamical complexity of human responses: a multivariate data-adaptive framework

2012 ◽  
Vol 60 (3) ◽  
pp. 433-445 ◽  
Author(s):  
M.U. Ahmed ◽  
N. Rehman ◽  
D. Looney ◽  
T.M. Rutkowski ◽  
D.P. Mandic
Keyword(s):  
Real World ◽  
Degrees Of Freedom ◽  
Multivariate Data ◽  
Structural Complexity ◽  
Multiscale Entropy ◽  
Real World Data ◽  
Complexity Measures ◽  
Long Range Correlations ◽  
Multivariate Empirical Mode Decomposition ◽  
Mode Decomposition

Abstract Established complexity measures typically operate at a single scale and thus fail to quantify inherent long-range correlations in real-world data, a key feature of complex systems. The recently introduced multiscale entropy (MSE) method has the ability to detect fractal correlations and has been used successfully to assess the complexity of univariate data. However, multivariate observations are common in many real-world scenarios and a simultaneous analysis of their structural complexity is a prerequisite for the understanding of the underlying signal-generating mechanism. For this purpose, based on the notion of multivariate sample entropy, the standard MSE method is extended to the multivariate case, whereby for rigor, the intrinsic multivariate scales of the input data are generated adaptively via the multivariate empirical mode decomposition (MEMD) algorithm. This allows us to gain better understanding of the complexity of the underlying multivariate real-world process, together with more degrees of freedom and physical interpretation in the analysis. Simulations on both synthetic and real-world biological multivariate data sets support the analysis.

Structural complexity of minerals: information storage and processing in the mineral world

2013 ◽  
Vol 77 (3) ◽  
pp. 275-326 ◽  
Author(s):  
S. V. Krivovichev
Keyword(s):  
Crystal Structure ◽  
Crystal Structures ◽  
Information Content ◽  
High Pressures ◽  
Structural Information ◽  
Structural Complexity ◽  
Information Storage ◽  
Inorganic Crystal Structure Database ◽  
Complexity Measures ◽  
Geological Processes

AbstractStructural complexity of minerals is characterized using information contents of their crystal structures calculated according to the modified Shannon formula. The crystal structure is considered as a message consisting of atoms classified into equivalence classes according to their distribution over crystallographic orbits (Wyckoff sites). The proposed complexity measures combine both size- and symmetry-sensitive aspects of crystal structures. Information-based complexity parameters have been calculated for 3949 structure reports on minerals extracted from the Inorganic Crystal Structure Database. According to the total structural information content, IG, total, mineral structures can be classified into very simple (0–20 bits), simple (20–100 bits), intermediate (100–500 bits), complex (500–1000 bits), and very complex (> 1000 bits). The average information content for mineral structures is calculated as 228(6) bits per structure and 3.23(2) bits per atom. Twenty most complex mineral structures are (IG, total in bits): paulingite (6766.998), fantappieite (5948.330), sacrofanite (5317.353), mendeleevite-(Ce) (3398.878), bouazzerite (3035.201), megacyclite (2950.928), vandendriesscheite (2835.307), giuseppetite (2723.097), stilpnomelane (2483.819), stavelotite-(La) (2411.498), rogermitchellite (2320.653), parsettensite (2309.820), apjohnite (2305.361), antigorite (m = 17 polysome) (2250.397), tounkite (2187.799), tschoertnerite (2132.228), farneseite (2094.012), kircherite (2052.539), bannisterite (2031.017), and mutinaite (2025.067). The following complexity-generating mechanisms have been recognized: modularity, misfit relationships between structure elements, and presence of nanoscale units (clusters or tubules). Structural complexity should be distiguished from topological complexity. Structural complexity increases with decreasing temperature and increasing pressure, though at ultra-high pressures, the situation may be different. Quantitative complexity measures can be used to investigate evolution of information in the course of global and local geological processes involving formation and transformation of crystalline phases. The information-based complexity measures can also be used to estimate the 'ease of crystallization' from the viewpoint of simplexity principle proposed by J.R. Goldsmith (1953) for understanding of formation of simple and complex mineral phases under both natural and laboratory conditions. According to the proposed quantitative approach, the crystal structure can be viewed as a reservoir of information encoded in its complexity. Complex structures store more information than simple ones. As erasure of information is always associated with dissipation of energy, information stored in crystal structures of minerals must have an important influence upon natural processes. As every process can be viewed as a communication channel, the mineralogical history of our planet on any scale is a story of accumulation, storage, transmission and processing of structural information.

scholarly journals On an extension of Krivovichev's complexity measures

2020 ◽  
Vol 76 (4) ◽  
pp. 534-548 ◽  
Author(s):  
Wolfgang Hornfeck
Keyword(s):  
Crystal Structure ◽  
Shannon Entropy ◽  
Degrees Of Freedom ◽  
Structural Complexity ◽  
Complexity Measure ◽  
Additional Contribution ◽  
Complexity Measures ◽  
Free Parameters ◽  
Atomic Coordinates ◽  
Shed Light

An extension is proposed of the Shannon entropy-based structural complexity measure introduced by Krivovichev, taking into account the geometric coordinational degrees of freedom a crystal structure has. This allows a discrimination to be made between crystal structures which share the same number of atoms in their reduced cells, yet differ in the number of their free parameters with respect to their fractional atomic coordinates. The strong additivity property of the Shannon entropy is used to shed light on the complexity measure of Krivovichev and how it gains complexity contributions due to single Wyckoff positions. Using the same property allows for combining the proposed coordinational complexity measure with Krivovichev's combinatorial one to give a unique quantitative descriptor of a crystal structure's configurational complexity. An additional contribution of chemical degrees of freedom is discussed, yielding an even more refined scheme of complexity measures which can be obtained from a crystal structure's description: the six C's of complexity.

In-situ high-pressure study of the ordered phase of ethyl propionate

2007 ◽  
Vol 63 (1) ◽  
pp. 111-117 ◽  
Author(s):  
Roman Gajda ◽  
Andrzej Katrusiak
Keyword(s):  
High Pressure ◽  
Degrees Of Freedom ◽  
Molecular Volume ◽  
Ethyl Propionate ◽  
Intermolecular Contacts ◽  
High Pressure Study ◽  
Pressure Phase ◽  
Three Degrees Of Freedom ◽  
Volume Difference

Ethyl propionate, C5H10O2 (m.p. 199 K), has been in-situ pressure-frozen and its structure determined at 1.34, 1.98 and 2.45 GPa. The crystal structure of the new high-pressure phase (denoted β) is different from phase α obtained by lowering the temperature. The freezing pressure of ethyl propionate at 296 K is 1.03 GPa. The molecule assumes an extended chain s-trans–trans–trans conformation, only slightly distorted from planarity. The closest intermolecular contacts in both phases are formed between carbonyl O and methyl H atoms; however, the ethyl-group H atoms in phase β form no contacts shorter than 2.58 Å. A considerable molecular volume difference of 24.2 Å3 between phases α and β can be rationalized in terms of degrees of freedom of molecules arranged into closely packed structures: the three degrees of freedom allowed for rearrangements of molecules confined to planar sheets in phase α, but are not sufficient for obtaining a densely packed pattern.

scholarly journals Correction to Benchmarking Coordination Number Prediction Algorithms on Inorganic Crystal Structures

2021 ◽  
Author(s):  
Hillary Pan ◽  
Alex M. Ganose ◽  
Matthew Horton ◽  
Muratahan Aykol ◽  
Kristin Persson ◽  
...  
Keyword(s):  
Crystal Structures ◽  
Coordination Number ◽  
Prediction Algorithms ◽  
Inorganic Crystal

Influence of 3-{5-[4-(diethylamino)benzylidene]rhodanine}propionic acid on the conformation of 5-(4-chlorobenzylidene)-2-(4-methylpiperazin-1-yl)-3H-imidazol-4(5H)-one

2018 ◽  
Vol 74 (11) ◽  
pp. 1427-1433 ◽  
Author(s):  
Ewa Żesławska ◽  
Wojciech Nitek ◽  
Waldemar Tejchman ◽  
Jadwiga Handzlik
Keyword(s):  
Crystal Structures ◽  
Crystal Packing ◽  
Protonated Form ◽  
Acid Form ◽  
Asymmetric Unit ◽  
X Ray Diffraction ◽  
Basic Form ◽  
Piperazine Ring ◽  
X Ray ◽  
Intermolecular Contacts

The arylidene–imidazolone derivatives are a group of compounds of great interest in medicinal chemistry due to their various pharmacological actions. In order to study the possible conformations of an arylidene–imidazolone derivative, two new crystal structures were determined by X-ray diffraction, namely (Z)-5-(4-chlorobenzylidene)-2-(4-methylpiperazin-1-yl)-3H-imidazol-5(4H)-one, C15H17ClN4O, (6), and its salt 4-[5-(4-chlorobenzylidene)-5-oxo-4,5-dihydro-3H-imidazol-2-yl]-1-methylpiperazin-1-ium 3-{5-[4-(diethylamino)benzylidene]-4-oxo-2-thioxothiazolidin-3-yl}propionate, C15H18ClN4O+·C17H19N2O3S2 −, (7). Both compounds crystallize in the space group P\overline{1}. The basic form (6) crystallizes with two molecules in the asymmetric unit. In the acid form of (6), the N atom of the piperazine ring is protonated by proton transfer from the carboxyl group of the rhodanine acid derivative. The greatest difference in the conformations of (6) and its protonated form, (6c), is observed in the location of the arylidene–imidazolone substituent at the N atom. In the case of (6c), the position of this substituent is close to axial, while for (6), the corresponding position is intermediate between equatorial and axial. The crystal packing is dominated by a network of N—H...O hydrogen bonds. Furthermore, the crystal structures are stabilized by numerous intermolecular contacts of types C—H...N and C—H...Cl in (6), and C—H...O and C—H...S in (7). The geometry with respect to the location of the substituents at the N atoms of the piperazine ring was compared with other crystal structures possessing an N-methylpiperazine moiety.

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