关于空间群判断失误的札记I.最小素单胞的选取

导出反映晶体结构周期的最小素单胞是正确判断空间群的第一步,其中以还原单胞的选择最为理想。它不仅具有选择的唯一性,且便于确定布拉维格子的类型。本文还讨论单胞选取过小和过大的实例,并给出正确的单胞和空间群以及原子坐标。     

Exploiting space‐group symmetry in fragment‐based molecular crystal calculations

Recent developments in fragment‐based methods make it increasingly feasible to use high‐level ab initio electronic structure techniques to molecular crystals. Such studies remain computationally demanding, however. Here, we describe a straightforward algorithm for exploiting space‐group symmetry in fragment‐based methods which often provides computational speed‐ups of several fold or more. This algorithm does not require a priori specification of the space group or symmetry operators. Rather, the symmetrically equivalent fragments are identified automatically by aligning the individual fragments along their principle axes of inertia and testing for equivalence with other fragments. The symmetry operators relating equivalent fragments can then be worked out easily. Implementation of this algorithm for computing energies, nuclear gradients with respect to both atomic coordinates and lattice parameters, and the nuclear hessian is described. © 2014 Wiley Periodicals, Inc.     

Lattice harmonics for graphite

A general theory for obtaining lattice harmonics of non-symmorphic space groups is presented, and the representation theory for these groups is briefly reviewed, with particular reference to graphite (space groupD _6h ⁴ [P6₃/mmc]). The irreducible matrix representations, compatibilities and lattice harmonics for alll are listed for all symmetry points, lines and planes in the representation domain of the Brillouin zone. The extra degeneracies introduced by time reversal are also considered. An appendix gives full details of the angular momentum conventions used in this paper.     

Energy-band structure for triplet excitons in space group c 52h

The triplet exciton energ-band structure for organic molecular solids belonging to space group C_2h⁵ containing in general four molecules at arbitrary positions in the unit cell is derived in the restricted Frenkel limit for the principal symmetry directions in the zone. The dispersion relations can be obtained without a priori assumptions on the relative strengths of the transfer matrix elements between the different sublattices. The band structure of crystals of current experimental interest like pyrene and 1,4-dibromonaphthalene is discussed.     

Room temperature molecular and lattice structures of a homologous series of anhydrous zinc(II) n-alkanoate

The room temperature structures and lattice arrangements of a homologous series of zinc(II) n-alkanoates from chain length, n(C) = 4-20, inclusive, have been studied using infrared spectroscopy, X-ray diffraction and polarizing light microscopy. Lattice parameters from single crystal and powder diffraction data, for zinc(II) hexanoate, are compared to validate the use of the powder method. Since they are in excellent agreement, the powder data are analyzed by a software programme to determine lattice parameters for all the homologues. These are used, in conjunction with infrared, X-ray, density and molecular model calculations to determine molecular and lattice structures. The compounds are isostructural, in that, each zinc atom is tetrahedrally coordinated to oxygen atoms from four different carboxylate groups and each ligand forms a Z,E-type bidentate bridge with two tetrahedral zinc atoms resulting in a syn-anti arrangement. The hydrocarbon chains are in the fully extended all-trans configuration and are tilted at an average angle of 60 degrees to the zinc basal plane. For the short chain length compounds with n(C) < or =8, a double bilayer in-plane-perpendicular-perpendicular-in-plane arrangement of hydrocarbon chains, with two molecules per unit cell, is indicated. For the others, an interdigitating in-plane-in-plane bilayer with head-to-tail interactions, with one molecule per unit cell, is proposed. A geometric model is presented to account for odd-even chain effects and to explain the differences in melting points and densities between these adducts. All the compounds crystallize in the monoclinic space group with P symmetry and are arranged in a two-dimensional network along the ac plane within the unit cell.     

Fuzzy space periodic symmetries for polyynes and their cyano-compounds

In our previous papers on the molecular fuzzy symmetry, we analyzed the basic characterization in connection with the fuzzy point group symmetry. In this paper, polyynes and their cyano-derivatives are chosen as a prototype of linear molecules to probe the one-dimensional fuzzy space group of parallel translation. It is notable that the space group is an infinite group whereas the point group is a finite group. For the fuzzy point group, we focus on considering the fuzzy characterization introduced due to the difference of atomic types in the monomer through point symmetry transformation in the beginning; and then we consider the difference between the infinity of space group and the finite size of real molecules. The difference between the point group and the space group lies in the translation symmetry transformation. This is the theme of this work. Starting with a simple case, we will only analyze the one-dimensional translation transformation and space fuzzy inversion symmetry transformation in this paper. The theory of the space group is often used in solid state physics; and some of its conclusions will be referred to. More complicated fuzzy space groups will be discussed in our future papers.     

Symmetry and stability of nanotubes based on titanium dioxide

The process of rolling a monolayer of bulk crystal with biperiodical planar lattice to the nanotube was analyzed. It was shown by an example of the carbon nanotubes how the tube symmetry can be revealed through the analysis of symmetry of graphene layers (the layer group with a hexagonal planar lattice) and its changes at the rolling to form the tube. The developed approach can be used to analyze the symmetry of any nanotube. A computer program we developed is discussed that allows to determine the nanotube symmetry using the data on the symmetry and coordinates of the atoms in the nanolayer and get the coordinates of the atoms in the unit cell of the nanotube which can be used for the further quantum-chemical calculations. The method and results of ab initio calculations of the titanium dioxide monolayer stability in the LCAO basis optimized for the bulk crystal, using the hybrid exchange-correlation potential PBE0 are presented. Symmetry properties of nanotubes obtained by rolling the three- and six-plane monolayers (101) and (001) of anatase are discussed. Atomic and electronic structure of TiO₂ nanotubes found by geometry optimization is analyzed. It is shown that titanium dioxide nanotubes based on the three-plane monolayers with hexagonal and square lattice are approximately of the same stability. The data on the stability of nanotubes are essential for the synthesis of new nanomaterials based on titanium dioxide.     

Solid-state polymerization of oxetanes. I. Lattice parameters and packing properties of the monomers

The crystallographic unit cells of melt-crystallized 3,3-bischloromethyloxetane and 3,3-bisbromomethyloxetane were determined by the Weissenberg method. The two isomorphous lattices are triclinic with two molecules in the unit cell. 3,3-Bisfluoromethyloxetane forms plastic crystals in the temperature range between ?36°C and +22°C, as shown by differential calorimetry and NMR broad-line spectroscopy. The Debye-Scherrer diagram and the general physical properties indicate the formation of a face-centered cubic lattice. No correlation between the lattice parameters of the monomer and polymer can be found On the basis of these results, the question is raised as to whether a topochemical polymerization of bishalomethloxetanes, i.e., a solid-state polymerization without destruction of the crystal lattice, can take place at all. The halomethyl side groups of the oxetanes can be shown to possess different conformations in monomer and polymer crystals, so that a conformational change of the groups and rearrangement of the molecules must take place during polymerization. Therefore, a topochemical mechanism for the solid-state polymerization of bishalomethyloxetanes seems to be impossible.     

Crystal structure predictions for acetic acid

Possible crystal structures of acetic acid were generated, considering eight space groups and assuming one molecule in the asymmetric unit. Our grid-search method was compared with a Monte Carlo approach as implemented in the Biosym/MSI Polymorph Predictor. This revealed no sampling deficiencies. A large number of possible crystal structures were found (∼100 within only 5 kJ/mol), including the experimental structure. Energy minimizations were done with a united-atoms force field (GROMOS), an all-atoms force field (AMBER), and a potential that describes the electrostatic interactions with distributed multipoles (DMA). In all cases, the experimental structure had a low lattice energy. The number of hypothetical crystal structures was reduced considerably by removing space-group symmetry constraints, or by a primitive molecular dynamics shake-up. Nevertheless, sufficient structures of equal or lower energy compared with the experimental structure remained to suggest that other factors need to be considered for genuine structure prediction. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 459–474, 1998     

Multiconfigurational SCF approach for high-symmetry molecules and its applications to fullerene trianion

A symmetry-adapted multiconfiguration self-consistent field (MC SCF) approach aimed at calculations of high-symmetry molecules is proposed. The self-consistency procedure applicable to the molecular terms of any symmetry and multiplicity is developed. It holds the symmetry transformation properties of varied molecular orbitals, thus taking advantage of the relationships within the set of two-electron integrals through molecular invariants. For orbital optimization, a unified coupling operator is constructed on the basis of the pseudosecular method providing for efficient convergence to energy minimum. Based on the group-theory technique, computer codes have been developed for straightforward determination of the invariant expansions for two-electron integrals and configuration interaction (CI) matrix elements. Calculated in this way, the expansion coefficients are presented for the three-electron states that originate from joint t_1u and t_1g shells of an icosahedral fullerene C₆₀, the case important for the calculations of anion C₆₀³⁻ representing the charge state of the fullerene molecule in the superconducting ionic solids K₃C₆₀ or Rb₃C₆₀. The results of MC SCF calculations for lowest quasi-π-electronic states of C₆₀³⁻ are discussed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 293–304, 1998     

Efficient algorithm for the spin-free valence bond theory. I. New strategy and primary expressions

A new algorithm for nonorthogonal valence bond (VB) method is presented by using symmetric group approach. In the present algorithm, a new function, called paired-permanent-determinant (PPD), is defined, which is an algebrant and has the same symmetry of a corresponding VB structure. The evaluation of a PPD is carried out by using a recursion formula similar to the Laplace expansion method for determinants. An overlap matrix element in the spin-free VB method may be obtained by evaluating a corresponding PPD, while the Hamiltonian matrix element is expressed in terms of the products of electronic integrals and sub-PPDs. In the present work, some important properties of PPDs are discussed, and the primary procedure for the evaluation of PPD is deduced. Furthermore, the expressions for evaluating both the overlap and Hamiltonian matrix elements are also given in details, which are essential to develop an efficient algorithm for nonorthogonal VB calculations. In the present study, some further effective technical considerations will be adopted, and a new ab initio VB program will be introduced. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 287–297, 1998     

Symmetrizer: Algorithmic determination of point groups in nearly symmetric molecules

Symmetry is an extremely useful and powerful tool in computational chemistry, both for predicting the properties of molecules and for simplifying calculations. Although methods for determining the point groups of perfectly symmetric molecules are well‐known, finding the closest point group for a “nearly” symmetric molecule is far less studied, although it presents many useful applications. For this reason, we introduce Symmetrizer, an algorithm designed to determine a molecule's symmetry elements and closest matching point groups based on a user‐adjustable tolerance, and then to symmetrize that molecule to a given point group geometry. In contrast to conventional methods, Symmetrizer takes a bottom‐up approach to symmetry detection by locating all possible symmetry elements and uses this set to deduce the most probable point groups. We explain this approach in detail, and assess the flexibility, robustness, and efficiency of the algorithm with respect to various input parameters on several test molecules. We also demonstrate an application of Symmetrizer by interfacing it with the WebMO web‐based interface to computational chemistry packages as a showcase of its ease of integration. © 2012 Wiley Periodicals, Inc.     

Symmetry simplifications of space types in configuration interaction induced by orbital degeneracy

Symmetry simplifications are introduced in configuration interaction (CI ) by reducing the number of symmetry-allowed space types if there is degeneracy in some of the molecular orbitals by constructing the unique space types. A new symmetry group which we call the configuration symmetry group is defined and is shown to be expressible as a generalized wreath product group. Generating functions are derived for enumerating the equivalence classes of space types. A double coset method is expounded which constructs the representatives of all equivalence classes of space types using the cycle index of generalized wreath product and the double cosets of label subgroup with generalized wreath product in the symmetric group S_n, if n is twice the number of occupied and virtual orbitals. Method is illustrated with CI using the localized orbitals of polyenes, CI in benzene, and atomic CI for several reference states.     

Derivation of atomistic models for lattices consisting of weakly overlapping structural elements

A theoretical approach based on the arrow diagram technique within the group function theory is applied to an infinite crystal consisting of weakly overlapping structural elements (SE: ions, atoms, and molecules). A perturbation method is used to derive the lattice energy in terms of two‐body, three‐body, etc. types of interactions accompanied by electronic polarization of SEs for the system which is slightly away from some reference configuration. It is demonstrated how an atomistic model for the particular material can be derived from the general theory. A general approach to obtain a parametrization scheme is suggested which is based on a representation of transition and ground‐state reduced density matrices (RDM) of various orders via a finite set of basis functions. Using group theory and necessary conditions for N‐representability of the RDMs, the representation containing the least possible number of the fitting parameters is worked out. The method is illustrated on a special case of spherical SEs (e.g., atoms/ions in atomic/ionic solids) and it is discussed whether existing models are in accord with the ideas developed. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 306–330, 2000     

Crystal calculation using the PCILO method

The PCILO method is used for the study of molecular systems with translational symmetry in a two-dimensional lattice. The calculation for the whole crystal is reduced to the calculation of unit cell pairs. By using the translational symmetry and the first-neighbour approximation, one shows that only four unit cell pairs have to be considered. The procedure described yields the ground-state energy and the charge distribution of the unit cell.     

Low temperature structural transformation in T[Ni(CN)4]·xpyz with x=1,2; T=Mn,Co,Ni,Zn,Cd; pyz=pyrazine

The materials under study are pillared solids T[Ni(CN)₄]·xpyz with one and two (x=1,2) pyrazine (pyz) molecules and where T=Mn, Co, Ni, Zn, Cd. Stimulated by their structural features and potential role as prototype of porous solids for hydrogen storage, the structural stability under cryogenic conditions for this series of pillared solids was studied. At low temperature, in the 100-200 K range, the occurrence of a reversible structural transformation was found. For T=Mn, Co, Zn, Cd, with x=2, the structural transformation was observed to occur around 185 K, and the low temperature phase crystallizes with a monoclinic unit cell (space group Pc). This structure change results from certain charge redistribution on cooling within the involved ligands. For T=Ni with x=1, both the low and high temperature phases crystallize with unit cells of tetragonal symmetry, within the same space group but with a different unit cell volume. In this case the structure change is observed around 120 K. Above that temperature the rotational states for the pyrazine molecule are thermally excited and all the pyrazine molecules in the structure become equivalent. Under this condition the material structure is described using a smaller structural unit. The structural study using X-ray powder diffraction data was complemented with calorimetric and Raman spectroscopy measurements. For the low temperature phases the crystal structures were solved from Patterson methods and then refined using the Rietveld method.     

Chemical applications of topology and group theory. 17. An information theoretical approach to polyhedral symmetry [1]

Information theoretic parameters are described which measure the asymmetry of polyhedra based on partitions of their vertices, faces, and edges into orbits under action of their symmetry point groups. Such asymmetry parameters are all zero only for the five regular polyhedra and are all unity for polyhedra having no symmetry at all, i.e. belonging to the C ₁ symmetry point group. In all other cases such asymmetry parameters have values between zero and unity. Values for such asymmetry parameters are given for all topologically distinct polyhedra having five, six, and seven vertices; all topologically distinct eight-vertex polyhedra having at least six symmetry elements; and selected polyhedra having from nine to twelve vertices. Effects of polyhedral distortions on these asymmetry parameters are examined for the tetrahedron, trigonal bipyramid, square pyramid, and octahedron. Such information theoretic asymmetry parameters can be used to order site partitions which are incomparable by the chirality algebra methods of Ruch and co-workers.     

Ewald summations in systems with two-dimensional periodicity

This study presents formulas for the electrostatic energy of lattices with two-dimensional periodicity, based on Fourier representations and alternatively on the Ewald procedure for convergence acceleration. The work extends the contributions of previous investigators by taking full advantage of plane-group symmetry and by providing analytical formulas for all derivatives of the energy through second order. The derivatives considered include those with respect to the positions of all charges within the unit cell, those with respect to the lattice vectors (cell deformations), and those involving both types of variables. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 385–404, 1998     

A Crystal Orbital approach for two- and three-dimensional solids on the basis of CNDO/INDO Hamiltonians. Basis equations

A Hartree–Fock (HF ) self-consistent field (SCF ) crystal orbital (CO ) formalism for two- and three-dimensional (2D/3D) solids on the basis of semiempirical CNDO /INDO (complete neglect of differential overlap; intermediate neglect of differential overlap) Hamiltonians is presented. The employed SCF variants allow for the treatment of atomic species up to bromine under the inclusion of the first (i.e., 3d) transition metal series. Band structure investigations of 2D and 3D materials containing more than 30 atoms per unit cell are feasible by the present SCF HF CO formalism. The theoretical background of the computational scheme is given in this contribution. Special emphasis is placed on physically reliable truncation criteria for the lattice sums, the adaptation of the crystal symmetry in k space, as well as the suitable choice of domains in Brillouin zone (BZ ) integrations required in the determination of charge-density matrices. The capability and limitations of the semiempirical SCF HF CO approach is demonstrated for some simpler solids by comparing the present computational results with those of ab initio CO schemes as well as conventional numerical methods in soid-state theory. The employed model solids are graphite and BN (2D and 3D networks for both solids) as well as diamond, silicon, germanium, and TiS₂.