More Icosahedral Fulleroids

The discovery of the famous fullerene has raised an interest in the study of other candidates for a modeling of carbon molecules. Motivated by a P. Fowler's question Delgado Friedrichs and Deza defined I(a,b)-fulleroids as cubic convex polyhedra having only a-gonal and b-gonal faces and the symmetry groups isomorphic with the rotation group of the regular icosahedron. In this note we prove that for every n8 there exist infinitely many I(5,n)-fulleroids. This answers positively questions posed recently by Delgado Friedrichs and Deza.     

On the use of symmetry in the ab initio quantum mechanical simulation of nanotubes and related materials

Nanotubes can be characterized by a very high point symmetry, comparable or even larger than the one of the most symmetric crystalline systems (cubic, 48 point symmetry operators). For example, N = 2n rototranslation symmetry operators connect the atoms of the (n,0) nanotubes. This symmetry is fully exploited in the CRYSTAL code. As a result, ab initio quantum mechanical large basis set calculations of carbon nanotubes containing more than 150 atoms in the unit cell become very cheap, because the irreducible part of the unit cell reduces to two atoms only. The nanotube symmetry is exploited at three levels in the present implementation. First, for the automatic generation of the nanotube structure (and then of the input file for the SCF calculation) starting from a two‐dimensional structure (in the specific case, graphene). Second, the nanotube symmetry is used for the calculation of the mono‐ and bi‐electronic integrals that enter into the Fock (Kohn‐Sham) matrix definition. Only the irreducible wedge of the Fock matrix is computed, with a saving factor close to N. Finally, the symmetry is exploited for the diagonalization, where each irreducible representation is separately treated. When M atomic orbitals per carbon atom are used, the diagonalization computing time is close to Nt, where t is the time required for the diagonalization of each 2M × 2M matrix. The efficiency and accuracy of the computational scheme is documented. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

The Chirality of Icosahedral Fullerenes: a Comparison of the Tripling (leapfrog), Quadrupling (chamfering), and Septupling (capra) Transformations

Fullerene polyhedra of icosahedral symmetry have the midpoints of their 12 pentagonal faces at the vertices of a macroicosahedron and can be characterized by the patterns of their hexagonal faces on the (triangular) macrofaces of this macroicosahedron. The numbers of the vertices in fullerene polyhedra of icosahedral symmetry satisfy the Goldberg equation v=20(h ²+hk+k ²), where h and k are two integers and 0 <h≥ k≥ 0 and define a two-dimensional Goldberg vector G = (h, k). The known tripling (leapfrog), quadrupling (chamfering), and septupling (capra) transformations correspond to the Goldberg vectors (1, 1), (2, 0), and (2, 1), respectively. The tripling and quadrupling transformations applied to the regular dodecahedron generate achiral fullerene polyhedra with the full I _h point group. However, the septupling transformation destroys the reflection operations of the underlying icosahedron to generate chiral fullerene polyhedra having only the I icosahedral rotational point group. Generalization of the quadrupling transformation leads to the fundamental homologous series of achiral fullerene polyhedra having 20 n ² vertices and Goldberg vectors (n, 0). A related homologous series of likewise achiral fullerene polyhedra having 60 n ² vertices and Goldberg vectors (n, n) is obtained by applying the tripling transformation to regular dodecahedral C₂₀ to give truncated icosahedral C₆₀ followed by the generalized operations (as in the case of quadrupling) for obtaining homologous series of fullerenes. Generalization of the septupling (capra) transformation leads to a homologous series of chiral C_20m fullerenes with the I point group and Goldberg vectors G=(h, 1) where m=h ²+h+1.     

Symmetry and selection rules in the Raman spectra of carbon nanotubes

The symmetry of achiral single-walled (n,0) and (n,n) carbon nanotubes (CNTs) was examined and the frequencies and types of vibrations allowed in the Raman spectra were calculated. The vibrational spectrum was evaluated as the eigenvalues of the dynamical matrix at the -point of the Brillouin zone. The selection rules for the Raman active vibrations were estimated by the values of the matrix elements responsible for the intensities of corresponding vibrational transitions. The (n,n)-CNTs with even and odd n values are characterized by five and six allowed Raman active vibrations, respectively. The number of Raman active vibrations for (n,0)-CNTs is five if n is even and eight if n is odd. Detailed analysis of the results obtained is presented for the (10,10)-CNT as an example.     

Order,disorder, and geometrical information indices of molecules

Structural features of various molecular systems with symmetry of point groups ranging fromC ₁ to the icosahedral symmetry are analyzed in the framework of the model suggested previously for the evaluation of order and disorder in the arrangement of atoms in a molecule based on the equationQ = I -P/3n (whereQ is the index of order, andP is the number of independent coordinates needed to fix ann-atomic molecule in the Cartesian coordinate system). TheQ value depends on various structural parameters of the molecule: the number of atoms in it, the symmetry, the dimensionality, and the number of structural degrees of freedom. The disorder indexP/3n = 1 -Q correlates with Shannon's entropy of information, andQ correlates with negentropy or excess information; this makes it possible to useP/3n as a new geometrical information molecular index that is obtained by a nonprobabilistic method. Analysis of the relationship between order and chaos in molecular systems, as well as of the specific order indexq =Q/n, makes it possible to identify both general and specific features of molecules.Translated from Izvestiya Akadernii Nauk. Seriya Khimicheskaya, No. 8, pp. 191219-121927, August, 1996.     

Polymorphism in Multisymmetric Close Packings of Equal Spheres on a Spherical Surface

Locally optimum, high-symmetry solutions are sought to the Tammes problem: to determine the maximum angular diameter of n equal circles, which can be packed on a sphere without overlapping. By using triangular surface lattices, in the range 1 n 500, those values of n are determined, for which the circles can be close packed in arrangements different from each other with tetrahedral, octahedral, and icosahedral symmetry. Tables are given of the results.     

Parallel vector analysis based soft resolution of spectral data from pH‐metric titration using symmetry constraint

In this study, a soft method is proposed to calculate concentration and spectral profiles for the two‐way spectral data from dissociation equilibria of polyprotic acids (H_nA). This method has four main distinct steps: (i) a fixed size moving window evolving factor analysis (FSMWEFA) was used to identify the local rank map, (ii) WFA was applied to calculate the concentration profiles of H_nA and Aⁿ⁻ (selection of the window for application of WFA was performed using EFA), (iii) PVA was used to calculate H_{n − 1}A to HA spectral profiles, and (iv) a symmetry constraint, in addition to the non‐negativity constraint, was utilized to obtain the unique concentration and spectral profiles from different acceptable sets of profiles. In the absence of any selective region in the spectral data, the proposed soft method resulted in unique solution without rotational ambiguity. This study is the first application of symmetry constraint on concentration profiles. The rotational ambiguity drastically decreased on considering the constraint of symmetry of the H_{n − 1}A and HA concentration profiles, in addition to non‐negativity of profiles. Simulated examples were used to confirm these approaches. Effect of closeness of dissociation constants on the estimated values of constants was investigated. The results showed that when the difference between pK_a values is more than 1.2, the obtained errors in the estimation of pK_a values are less than about 6.5%. The considered real data were from pH‐metric titration of fluorescein. The obtained spectral and concentration profiles and the estimated pK_a values for fluorescein were in good agreement with the previously reported data. Copyright © 2009 John Wiley & Sons, Ltd.     

Construction and utilisation of planar graphs of two series of IPR fullerenes through the use of threefold rotational symmetry

Threefold rotational symmetry has been used to develop an algorithm for the construction of planar graphs of IPR fullerenes and to factorize their characteristic polynomials. Two series of fullerenes of the formula C_60+12n and C_60+18n have thus been obtained. The algorithm has been shown to be useful for predicting the nature of variation of the point groups of the fullerenes with increased n, for counting the number of ¹³C nuclear magnetic resonance (NMR) signals (along with their relative intensities), and also for obtaining a large part of their eigenspectra. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Some aspects of the symmetry and topology of possible carbon allotrope structures

Elemental carbon has recently been shown to form molecular polyhedral allotropes known as fullerenes in addition to the familiar graphite and diamond known since antiquity. Such fullerenes contain polyhedral carbon cages in which all vertices have degree 3 and all faces are either pentagons or hexagons. All known fullerenes are found to satisfy the isolated pentagon rule (IPR) in which all pentagonal faces are completely surrounded by hexagons so that no two pentagonal faces share an edge. The smallest fullerene structures satisfying the IPR are the known truncated icosahedral C₆₀ of I _h symmetry and ellipsoidal C₇₀ of D _5h symmetry. The multiple IPR isomers of families of larger fullerenes such as C₇₆, C₇₈, C₈₂ and C₈₄ can be classified into families related by the so-called pyracylene transformation based on the motion of two carbon atoms in a pyracylene unit containing two linked pentagons separated by two hexagons. Larger fullerenes with 3ν vertices can be generated from smaller fullerenes with ν vertices through a so‐called leapfrog transformation consisting of omnicapping followed by dualization. The energy levels of the bonding molecular orbitals of fullerenes having icosahedral symmetry and 60n ² carbon atoms can be approximated by spherical harmonics. If fullerenes are regarded as constructed from carbon networks of positive curvature, the corresponding carbon allotropes constructed from carbon networks of negative curvature are the polymeric schwarzites. The negative curvature in schwarzites is introduced through heptagons or octagons of carbon atoms and the schwarzites are constructed by placing such carbon networks on minimal surfaces with negative Gaussian curvature, particularly the so-called P and D surfaces with local cubic symmetry. The smallest unit cell of a viable schwarzite structure having only hexagons and heptagons contains 168 carbon atoms and is constructed by applying a leapfrog transformation to a genus 3 figure containing 24 heptagons and 56 vertices described by the German mathematician Klein in the 19th century analogous to the construction of the C₆₀ fullerene truncated icosahedron by applying a leapfrog transformation to the regular dodecahedron. Although this C₁₆₈ schwarzite unit cell has local O _h point group symmetry based on the cubic lattice of the D or P surface, its larger permutational symmetry group is the PSL(2,7) group of order 168 analogous to the icosahedral pure rotation group, I, of order 60 of the C₆₀ fullerene considered as the isomorphous PSL(2,5) group. The schwarzites, which are still unknown experimentally, are predicted to be unusually low density forms of elemental carbon because of the pores generated by the infinite periodicity in three dimensions of the underlying minimal surfaces. This revised version was published online in July 2006 with corrections to the Cover Date.     

Features of intermolecular interactions in the crystals of metal acetylacetonates

For 12 acetylacetonates of the composition M(acac) _n (n = 2, 3, or 4) and M(acac)(C₂H₄)₂ (M is a metal) the total area (⁰ S) of the faces of Voronoi-Dirichlet polyhedra (VDP) corresponding to all intermolecular contacts of one molecule in the crystal structure and the total volume of pyramids (⁰ V), whose bases are formed of such faces and the vertices are occupied by the nuclei of atoms participating in intermolecular contacts, are determined. The key features of non-bonded interactions are considered. The existence of a linear dependence of the sublimation enthalpy of acetylacetonates on the ⁰ S or ⁰ V parameters of their molecular VDP is revealed. It is shown that the sublimation enthalpy of Ga(acac)₃ requires the refinement and theoretically should be 124 kJ/mol.     

Characterization of Supramolecular Hidden Chirality of Hydrogen‐Bonded Networks by Advanced Graph Set Analysis

Supramolecular hidden chirality of hydrogen‐bonded (HB) networks of primary ammonium carboxylates was exposed by advanced graph set analysis from a symmetric viewpoint in topology. The ring‐type HB (R‐HB) networks are topologically regarded as faces, and therefore exhibit prochirality and positional isomerism due to substituents attached on the faces. To describe the symmetric properties of the faces, additional symbols, Re (right‐handed or clockwise), Si (left‐handed or anticlockwise), and m (mirror), were proposed. According to the symbols, various kinds of faces were classified based on the symmetry. This symmetry consideration of the faces enables us to precisely evaluate supramolecular chirality, especially its handedness, of 0D‐cubic, 1D‐ladder and 2D‐sheet HB networks that are composed of the faces. The 1D‐ladder and 2D‐sheet HB networks generate chirality by accumulating the chiral faces in 1D and 2D manners, respectively, whereas 0D‐cubic HB networks generate chirality based on combinations of eight kinds of faces, similar to the chirality of dice.     

Conformational study of C24 cyclic polyyne clusters

Polyynes were first synthesized before the year 1900, and isolated and characterized after 2000. Cyclic polyynes are of particular interest since possess a high order of symmetry. Furthermore, some studies reported special mechanical properties of the condensed polyyne bulks. The optimal size of polyynes to form rings has been previously investigated and was found to be 24 with a stable cluster of crossing four C₂₄ cyclic polyynes. We investigated in this study the conformation of clusters of polyynes (nC₂₄) by the pattern previously identified to stabilize the cluster. Clusters of 4C₂₄, 10C₂₄, 22C₂₄, 46C₂₄, and 94C₂₄ were designed and subjected to energy minimization. The main finding is the preservation of the symmetry for the nC₂₄ cluster with the increase of its size. The study revealed that 4C₂₄, 10C₂₄, and 22C₂₄ preserve a high symmetry and the calculations suggest an excellent increasing of the cluster stability with the increase of the number of polyyne rings. A 22C₂₄ derived cluster namely 28C₂₄ was found as the one likely to limit the growth of the polyyne clusters.     

Die trikline Kristallstruktur von Ag2Bi2S3Cl2: Pseudosymmetrie und Zwillingsbildung

Ruby‐red crystals of Ag₂Bi₂S₃Cl₂ were synthesized from AgCl and Bi₂S₃ by cooling a melt from 770 K to room temperature. X‐ray diffraction on powders and single‐crystals revealed a triclinic crystal structure with special lattice constants (P &1macr; (No. 2), a = 1085.0(2), b = 717.2(1), c = 1137.6(1) pm, α = 89.80(1)?, β = 74.80(1)?, γ = 87.81(1)?). In the structure [Bi^IIIS₃Cl₄] polyhedra form 2[BiS_3/2Cl_4/4]^‐ double‐layers by sharing common faces and edges. The silver(I) cations between the layers are coordinated either octahedrally by sulfide ions or tetrahedrally by sulfide and chloride ions. The deviations from the monoclinic space group P 1 2₁/c 1 are small and induce twinning along [010]. Further pseudosymmetry is based on the stacking of layer packages with the symmetry of the layer group P (2/c) 2₁/c 2/b.     

Violations of the noncrossing rule due to special symmetry and alternance

This paper reports a study on which behavior of the Hamiltonian gives rise to violation of the noncrossing rule. In principle, the noncrossing rule may be violated when a special symmetry other than spatial and spin symmetries is present or there exists the so-called alternance, which corresponds to a Hamiltonian in a real vector space anticommuting with a Hermitian operator. In the HMO models for pericyclic reactions, violations due to special symmetry or alternance have been found. The [m,n] supra–antara cycloadditions have no symmetry in the traditional sense, but have special symmetry leading to the existence of crossings in the correlation diagram. Alternance results in one crossing in the middle of the correlation diagram of a forbidden pericyclic reaction with intermediate states in the form of even alternant hydrocarbon. For the reactions with intermediate states in the form of odd alternant hydrocarbon such as [2,4]-cycloaddition of an allyl cation or an allyl anion to butadiene, there should be no crossing in the correlation diagrams, and both the supra–supra and the supra–antara processes are predicted to be allowed. Such a prediction is beyond the Woodward–Hoffmann rule.     

Prevalence and Significance of Approximate Symmetry in Organic Pc Structures

A survey has been made of the well determined (R≤0.050) organic crystal structures that have only a single glide plane (e. g., are in space group Pc) and that have more than one crystallographically independent formula unit (Z′>1); the goal was to discover what fraction have additional approximate symmetry. Of the 377 unique structures examined 8 % should almost certainly been refined in a higher-symmetry unit cell; 86 % of the remaining 347 have approximate symmetry that is periodic in at least one dimension. While these percentages are similar to those found for P1 structures (C. P. Brock, Acta Crystallogr., Sect. B 2022 , 78, 576–588), the types of approximate symmetry differ because 89 % of the P1, Z′>1 structures were composed of enantiopure material. The 347 reliable Pc structures include 118 that are slightly distorted or mimicked Cc and P2₁/c structures, 15 of which were reported as having been determined at room temperature. The distortions in another 72 are so large that the approximate symmetry must be seen as periodic in two dimensions only. These results suggest that symmetry lowering may accompany transformation of a crystal nucleus to a macroscopic crystal.     

Pseudosymmetry analysis of molecular orbitals

We introduce a pseudosymmetry analysis of molecular orbitals by means of the newly proposed irreducible representation measures. To do that we define first what we consider as molecular pseudosymmetry and the relationships of this concept with those of approximate symmetry and quasisymmetry. We develop a general algorithm to quantify the pseudosymmetry content of a given object within the framework of the finite group algebra. The obtained mathematical expressions are able to decompose molecular orbitals by means of the irreducible representations of any reference symmetry point group. The implementation and usefulness of the pseudosymmetry analysis of molecular orbitals is demonstrated in the study of σ and π orbitals in planar and nonplanar polycyclic aromatic hydrocarbons and the t_2g and e_g character of the d‐orbitals in the [FeH₆]^3? anion in its high spin state along the Bailar twist pathway. © 2013 Wiley Periodicals, Inc.     

Symmetry calculation for molecules and transition states

The symmetry of molecules and transition states of elementary reactions is an essential property with important implications for computational chemistry. The automated identification of symmetry by computers is a very useful tool for many applications, but often relies on the availability of three‐dimensional coordinates of the atoms in the molecule and hence becomes less useful when these coordinates are a priori unavailable. This article presents a new algorithm that identifies symmetry of molecules and transition states based on an augmented graph representation of the corresponding structures, in which both topology and the presence of stereocenters are accounted for. The automorphism group order of the graph associated with the molecule or transition state is used as a starting point. A novel concept of label‐stereoisomers, that is, stereoisomers that arise after labeling homomorph substituents in the original molecule so that they become distinguishable, is introduced and used to obtain the symmetry number. The algorithm is characterized by its generic nature and avoids the use of heuristic rules that would limit the applicability. The calculated symmetry numbers are in agreement with expected values for a large and diverse set of structures, ranging from asymmetric, small molecules such as fluorochlorobromomethane to highly symmetric structures found in drug discovery assays. The new algorithm opens up new possibilities for the fast screening of the degree of symmetry of large sets of molecules. © 2014 Wiley Periodicals, Inc.     

Pressure Effects on 3dn (n=4, 9) Insulating Compounds: Long Axis Switch in Na3MnF6 not Due to the Jahn-Teller Effect

The pressure-induced switch of the long axis of MnF₆³⁻ units in the monoclinic Na₃MnF₆ compound and Mn³⁺-doped Na₃FeF₆ is explored with the help of first principles calculations. Although the switch phenomenon is usually related to the Jahn-Teller effect, we show that, due to symmetry reasons, it cannot take place in 3dⁿ (n=4, 9) systems displaying a static Jahn-Teller effect. By contrast, we prove that in Na₃MnF₆ the switch arises from the anisotropic response of the low symmetry lattice to hydrostatic pressure. Indeed, while the long axis of a MnF₆³⁻ unit at ambient pressure corresponds to the Mn³⁺−F₃⁻ direction, close to the crystal c axis, at 2.79 GPa the c axis is reduced by 0.29 Å while b is unmodified. This fact is shown to force a change of the HOMO wavefunction favoring that the long axis becomes the Mn³⁺−F₂⁻ direction, not far from crystal b axis, after the subsequent relaxation process. The origin of the different d-d transitions observed for Na₃MnF₆ and CrF₂ at ambient pressure is also discussed together with changes induced by pressure in Na₃MnF₆. The present work opens a window for understanding the pressure effects upon low symmetry insulating compounds containing d⁴ or d⁹ ions.