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.     

六核钽铑二元混合簇合物的几何和电子结构

采用密度泛函理论对六核钽、铑八面体纯簇及其混合簇的几何结构和电子性质进行了研究.计算结果表明:大部分钽铑混合簇稳定构型的对称性均较低,为C1或Cs点群,只有[Ta2Rh4Cl4H8(CN)6]4-团簇的稳定构型对称性较高,为C2h或C4v点群;混合簇的最高占据分子轨道(HOMO)与最低未占据分子轨道(LUMO)能隙(ΔEH-L)均较小,介于0.52-1.00eV之间;混合簇的前线轨道主要由骨架金属原子的d电子贡献,随着Rh原子替代Ta原子个数的递增,Ta—Rh键对混合簇稳定构型所起作用逐渐增加,Ta—Ta键所起作用减小,而Rh—Rh键为非键或反键性质.     

Generation of carbon-cage polyhedra

The problem of the generation of polyhedra with degree-3 vertices, and faces each of which is pentagonal or hexagonal, is addressed in order to characterize carbon in order to characterize clusters of ca. 30 or more atoms. Following Eulerian type arguments such polyhedra are subcategorized in terms of numbers of different types of local structures. An algorithm to generate such polyhedra is developed, and its computer implementation is described. Results for smaller than 80-vertex cages of subcategories anticipated to be more chemically relevant are reported. The singular position of the truncated-icosahedron (buckminsterfullerene) structure is noted.     

Polyhedral structures with three-, four-, and five fold symmetry in metal-centered ten-vertex germanium clusters

Studies using density functional theory (DFT) at the hybrid B3LYP level indicate that the relative energies of structures with three-fold, four-fold, and five-fold symmetry for centered 10-vertex bare germanium clusters of the general type M@Ge(10) (z) depend on the central metal atom M and the skeletal electron count. For M@Ge(10) clusters with 20 skeletal electrons the DFT results agree with experimental data on the isoelectronic centered 10-vertex bare metal clusters. Thus the lowest energy structure for Ni@Ge(10), isoelectronic with the known Ni@In(10) (10-), is a C(3v) polyhedron derived from the tetracapped trigonal prism. However, Zn@Ge(10) (2+) is isoelectronic with the known cluster Zn@In(10) (8-), which has the lowest energy structure, a D(4d) bicapped square antiprism. For the clusters Ni@Ge(10) (2-), Cu@Ge(10) (-), and Zn@Ge(10) that have 22 skeletal electrons the lowest energy structures are the D(4d) bicapped square antiprism predicted by the Wade-Mingos rules. For the clusters Ni@Ge(10) (4-), Cu@Ge(10) (3-), and Zn@Ge(10) (2-) that have 24 skeletal electrons the lowest energy structures are C(3v) polyhedra with 10 triangular faces and 3 quadrilateral faces derived from a tetracapped trigonal prism by extreme lengthening of the edges of the capped triangular face of the underlying trigonal prism. For the clusters Cu@Ge(10) (5-) and Zn@Ge(10) (4-) that have 26 skeletal electrons the lowest energy structures are the D(5d) pentagonal antiprisms predicted by the Wade-Mingos rules and the C(3v) tetracapped trigonal prism as a somewhat higher energy structure. However, for the isoelectronic Ni@Ge(10) (6-) the relative energies of these two structure types are reversed so that the C(3v) tetracapped trigonal prism becomes the global minimum. The effects of electron count on the geometries of the D(5d) pentagonal prism and D(4d) bicapped square antiprism centered metal cluster structures are consistent with the bonding/antibonding characteristics of the corresponding HOMO and LUMO frontier molecular orbitals.     

Growth and Formation of Fullerene Clusters

Theoretical studies on the stability and electronic structure of small carbon clusters assuming chain, ring, bowl, and fullerene structures have been carried out using a linear combination of atomic orbitals molecular orbital approach within a density functional formalism. Our studies on clusters containing between 12 and 60 atoms indicate three regimes for the growth and formation of carbon clusters. In clusters containing less than 20 atoms, the most stable geometry is the ring arrangements. Between 20 and 28 atoms, clusters with very different geometry have comparable energies. For clusters with larger than 30 atoms, the fullerene structures are the most stable structures. An analysis of the electronic structure shows a distinct correlation between the geometry and the nature of electronic states.     

Self-assembly of nanoscale,porous T-symmetric molecular adamantanoids

Treatments of lanthanide nitrate or perchlorate and C2-symmetric 2,2'-hydroxy-1,1'-binaphthalene-6,6'-dicarboxylic acid (6,6'-H2BDA) led to diastereoselective self-assembly of nanoscale, porous molecular adamantanoids [Ln4(BDA)6(H2O)12]*12DMF (Ln = Gd, La, 1a,b). These adamantanoid clusters possess perfect T symmetry as a result of the C2-symmetric nature of BDA bridging ligands and C3-symmetric nature of lanthanide ions. Face-to-face intercluster hydrogen bonds formed between 2,2'-dihydroxyl groups of BDA ligands and carboxylate oxygen atoms direct the assembly of 3D polycages based on chiral molecular adamantanoid building blocks which possess two different types of open channels.     

Molecules and crystals with both icosahedral and cubic symmetry

Notwithstanding the apparent incompatibility between octahedral and icosahedral symmetries, fragments with the two types of symmetry coexist in many molecules and crystals, as evidenced by continuous shape and symmetry measures. A geometric analysis of Platonic and Archimedean polyhedra and of a variety of molecular and crystal structures strongly suggests that octahedral symmetry is latent in icosahedral polyhedra and vice versa. In this Feature Article, new concepts and structural data from the literature combine to offer a perspective view of complex molecular and extended structures. Its influence on the common cubic packing of icosahedral molecules is discussed for a variety of examples, including water clathrates, dodecahedrane, Buckminsterfullerene, the Pd145 and Mo132 clusters and several intermetallic phases.     

Core-shell photoabsorption and photoelectron spectra of gas-phase pentacene: experiment and theory

The C K-edge photoabsorption and 1s core-level photoemission of pentacene (C22H14) free molecules are experimentally measured, and calculated by self-consistent-field and static-exchange approximation ab initio methods. Six nonequivalent C atoms present in the molecule contribute to the C 1s photoemission spectrum. The complex near-edge structures of the carbon K-edge absorption spectrum present two main groups of discrete transitions between 283 and 288 eV photon energy, due to absorption to pi* virtual orbitals, and broader structures at higher energy, involving sigma* virtual orbitals. The sharp absorption structures to the pi* empty orbitals lay well below the thresholds for the C 1s ionizations, caused by strong excitonic and localization effects. We can definitely explain the C K-edge absorption spectrum as due to both final (virtual) and initial (core) orbital effects, mainly involving excitations to the two lowest-unoccupied molecular orbitals of pi* symmetry, from the six chemically shifted C 1s core orbitals.     

Special stability of cationic MPb12+ clusters and superalkali character of neutral MPb12 clusters (M = B, Al, Ga, In, and Tl)

The electronic structures and stabilities of cationic MPb12+ clusters (M = B, Al, Ga, In, and Tl) with 50 valence electrons are investigated within density functional theory. It is shown that, at the B3LYP/cc-pVDZ(-PP) and BPW91/cc-pVDZ(-PP) levels of theory, the structures of MPb12+ with icosahedra (I(h)) symmetry are energetically favorable, and their high stabilities may arise from the closed-shell nature of the pi subsystems which are subject to the 2(N(pi + 1)2 rule with N(pi = 1). In addition, the possessing of large nucleus-independent chemical shifts of the five kinds of clusters reflects the common aromatic character of these clusters. From the comparison of our studies on the binding energies and the highest occupied molecular orbital and the lowest unoccupied molecular orbital energy gaps, the cluster AlPb12+ has higher stability than the others and this is consistent with the recent mass-spectrometric discovery of Al-doped Pb(n)+ clusters, in which AlPb12+ is highly abundant. The same methods are used to search for the structures of the neutral MPb12 clusters. The calculations reveal that the most stable geometries of the BPb12 and GaPb12 clusters have I(h) symmetry, the AlPb12 and InPb12 clusters have T(h) symmetry, and the TlPb12 cluster has C5v symmetry. Furthermore, the vertical ionization potentials of the neutral MPb12 clusters are smaller than that of some alkali atoms, indicating that the neutral MPb12 clusters possess superalkali character.     

Spin-orbit effects on a gold-based superatom: a relativistic Jellium model

The inclusion of relativistic effects always brings to the scientific community great and stimulating surprises. To consider the spin-orbit term, which accounts for the interaction between the spatial and spin coordinates, requires the use of double point groups of symmetry in order to solve the Dirac equation or the two component approximation to it, leading to total angular momenta (j) functions, atomic or molecular spinors, instead of pure orbital angular momenta (l), atomic or molecular orbitals. Large and small components, derived from the Dirac treatment, depict wavefunctions corresponding to fermions, electrons, which are described for the first time for a superatom case. In addition, their behavior is revisited in order to clarify the effects of the inclusion of the spin-orbit coupling into the electronic structure calculations, which can be extended to other superatoms, clusters, molecules and atoms.     

Density functional study of methyl chemisorption on polycyclic aromatic hydrocarbons.

The reactions of methyl radicals with large (up to C(96)H(24)) polycyclic aromatic hydrocarbons (PAHs) are studied by density functional calculations to shed light on the experimentally observed deposition of carbon on highly oriented pyrolytic graphite (HOPG), which occurs when hot HOPG (decorated by nanometre-sized defects) is exposed to methyl radicals. The equilibrium structures of the reaction products, together with transition structures for PAHs up to the size of phenanthroperylene, are determined using the density functionals B3LYP, TPSSh, BP86 and TPSS. The structures are analysed by computing the pi orbital axis vector (POAV) and the altitude of the reactive carbon above the molecular plane of the PAH. The strongest C-CH(3) bonds are found at the edges of the PAHs, where the s character of the C orbital involved in the bond is roughly 25 % (sp(3) hybrid orbital). Carbon atoms inside the PAH form bonds with the methyl radical through atomic orbitals with about 16 % s character in the POAV analysis. These bonds are much weaker than those at the edges of the PAH, while the reactive carbon has moved about 40 pm above the molecular plane. At the edges, the PAH carbon atoms do not leave the molecular plane to this extent. The computed barrier heights and geometrical parameters of the transition structures are in agreement with Hammond's postulate, and the relative energies of all of the equilibrium structures can be rationalized by Hückel molecular orbital (HMO) theory.     

Au12M (M=Na,Mg, Al,Si, P,S, Cl)团簇的结构、稳定性和电子性质

采用基于密度泛函理论的第一性原理方法系统地研究了Au12M(M=Na,Mg,Al,Si,P,S,Cl)团簇的结构、稳定性和电子性质.对团簇的平均结合能、镶嵌能、垂直离化势、最高占据分子轨道(HOMO)和最低未占据分子轨道(LUMO)的能级差、电荷布居分析、自然键轨道(NBO)进行了计算和讨论.对于Au12M(M=Na,Mg,Al)团簇,它们形成了内含M原子的最稳定的笼状结构.然而对于Au12M(M=Si,P,S,Cl)团簇,它们却形成了以M元素为顶点的稳定锥形结构.在这些团簇中发现Au12S团簇相对是最稳定的,这是由于Au12S团簇形成了稳定的满壳层的电子结构.自然电荷布居分析表明:对于所有的Au12M(M=Na,Mg,Al,Si,P,S,Cl)团簇电荷总是从Au原子转向M原子.自然键轨道和HOMO分析表明Au12M团簇中发生了Au原子的s-d轨道和M原子的p轨道间的杂化现象.     

Waugh-型镍钼杂多酸的量子化学计算

采用半经验INDO法，完成了Waugh-型镍钼杂多酸（NH44）6〔NIMO9O32〕·6H2O的量子化学计算，得到146个成键轨道和78个反键轨道，轨道能级，电荷和键序等数据。通过对这些数据的分析，表明杂多酸分子中端氧和Ni都可能为化学活性中心，各原子轨道在分子轨道中都占有一定比例，其HOMO和LUMO能级皆为负值，其体系具有进一步接受电子的能力。这些结论与实验事实一致。     

Quantum-chemical calculations of molybdenum chloride clusters Mo13C24, Mo13Cl26, and Mo13Cl30

Quantum-chemical calculations of the structures and thermodynamic stabilities of isomers of molecular clusters of lower Mo chlorides Mo₁₃Cl₂₄, Mo₁₃Cl₂₆, and Mo₁₃Cl₃₀, as well as of the Mo₁₃Cl ₂₆ ^? anion of cuboctahedral or icosahedral structure, have been performed by the DFT B3LYP method with the use of the Lanl2dz basis set for the Mo atoms and the 631G(d) basis set for the Cl atoms. The highest symmetry polyhedra are considered: Mo₁₃Cl₃₀ and Mo₁₃Cl₂₄ have a structure of centered icosahedron and cuboctahedron with bridging Cl atoms sitting atop the midpoints of 30 and 24 edges, respectively. In both cases, calculations lead to a rapid increase in the Mo-Mo distance to the values that are evidence of the absence of metal-metal bonds; hence, the cluster structure, in its classical meaning, is not retained. Both the neutral cluster Mo₁₃Cl₂₆ and the Mo₁₃Cl ₂₆ ^? anion—a symmetric icosahedron, in which Cl ions sit atop the centers of 20 faces and 6 Cl atoms are terminal, and a cuboctahedron, in which 14 Cl atoms are located on the three- and fourfold axes over the centers of molybdenum faces and 12 terminal Cl atoms are located on twofold axes. For both the neutral cluster and the anion, stable structures have been obtained. In both cases, the total energies are lower for the initial icosahedral configuration.     

On the use of spatial symmetry in ab initio calculations. Transformation of the two-electron integrals from atomic orbital to localized molecular orbital basis

Anm ⁵-dependent integral transformation procedure from atomic orbital basis to localized molecular orbitals is described for spatially extended systems with some Abelian symmetry groups. It is shown that exploiting spatial symmetry, the number of non-redundant integrals for normal saturated hydrocarbons can be reduced by a factor of 2.5-3.5, depending on the size of the system and on the basis. Starting from a list of integrals over basis functions in canonical order, the number of multiplications of the four-index transformation is reduced by a factor of 2.8-3.5 as compared to that of Diercksen's algorithm. It is pointed out that even larger reduction can be achieved if negligible integrals over localized molecular orbitals are omitted from the transformation in advance.     

Potential magnetic properties of nanotubes (n, 0) with Klein and Fujita edges

Analytical solutions for localized states of zigzag-type nanotube (NT) fragments with various combinations of Klein and Fujita borders are considered using the Hückel approach. It is shown that the equations for determining molecular orbitals (MOs) in systems with two Klein edges are similar to equations for systems with two Fujita edges. An analytical formula for the energies of all ?? MOs is obtained for systems that have a Klein edge on one side and a Fujita edge on the other. It is established that these systems have n orbitals with energy ?? that are localized on the Fujita and Klein edges in dependence on the MO symmetry. The degeneracy of edge orbitals indicates that there is a tendency toward single occupancy of them and to the appearance of spin (magnetic) properties. In addition, the energies of the states of different multiplicity for NT fragments (8, 0) are calculated using the CASSCF approach. It is shown that the ground state has a multiplicity of 9, as was also indicated by estimates obtained using the density functional method (B3LYP). It is concluded that zigzag-type NTs with asymmetric edges have a tendency to exhibit spin properties. It is noted that the construction of nanoscale magnetic materials based on them is very promising.     

The nature of hydrogen bonds and conjugation in H-bonded systems

Analysis of the electron density distributions in the spatial regions of hydrogen bonds reveals the existence of both ?? and ?? binding. The ??-type electron density distribution is shown to be determined by occupied orbitals of both bonding and antibonding character. The ?? system can be conjugated and provides the predominant binding of molecules in nonplanar structures. The mutual orientation of the molecules involved in a hydrogen bond is found to be optimal when the X-H bond in the proton donor and the orbital of the acceptor, the main contributor to ?? binding, have a common axis of symmetry. The conjugated ?? electron system in molecular clusters determines the so-called cooperativity that is manifested in the deviation of the properties of hydrogen-bonded systems from pairwise additivity.     

Endohedral nickel and palladium atoms in metal clusters: analogy to endohedral noble gas atoms in fullerenes in polyhedra with five-fold symmetry

Nickel and palladium atoms with their closed-shell d(10) electronic configurations are encapsulated in the icosahedral clusters [Ni@Ni(10)E(2)(CO)(18)](4-)(E = Sb, Bi, Sb[rightward arrow]Ni(CO)(3), CH(3)Sn and n-C(4)H(9)Sn) and the geometrically related pentagonal antiprismatic cluster Pd@Bi(10)(4+) found in Bi(14)PdBr(16). Such endohedral d(10) atoms in pentagonal antiprismatic clusters are donors of zero skeletal electrons and interact only weakly with the atoms in the surrounding polyhedron so that they may be regarded as analogous to endohedral noble gases in fullerenes such as He@C(60). On the other hand, endohedral nickel and palladium atoms in 10- and 11-vertex flattened deltahedral bare metal clusters of group 13 metals without five-fold symmetry, such as Ni@E(10)(10-) found in Na(10)NiE(10)(E = Ga, In) and Pd@Tl(11)(7-) found in A(8)Tl(11)Pd (A = Cs, Rb, K), interact significantly with the cluster atoms, particularly those at the flattened vertices of the deltahedron. The role of endohedral d(10) atoms Ni and Pd in polyhedra with five-fold symmetry as "pseudo-noble-gases" can be related to their positions at the "composite divide" of the "Metallurgists' Periodic Table" proposed by H. E. N. Stone on the basis of alloy systematics as well as the equivalence of the five d orbitals in polyhedra with five-fold symmetry.     

Relativistic computational investigation: the geometries and electronic properties of TaSi(n)+ (n = 1-13, 16) clusters

The equilibrium geometries, stabilities, and electronic properties of the TaSi(n)+ (n = 1-13, 16) clusters are investigated systematically by using the relativistic density functional method with generalized gradient approximation. The small-sized TaSi(n)+ clusters with slight geometrical adjustments basically keep the frameworks that are analogous to the neutrals while the medium-sized charged clusters significantly deform the neutral geometries, which are confirmed by the calculated AIP and VIP values. Furthermore, the optimized geometries of the charged clusters agree with the experimental results of Hiura and co-workers (Hiura, H.; Miyazaki, T.; Kanayama, T. Phys. Rev. Lett. 2001, 86, 1733). The highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) gaps of the charged clusters are generally increased as the cluster size goes from n = 1 to 13; and the large HOMO-LUMO gaps of charged clusters resulting from the positive charge indicate that their chemical stabilities are stronger than their neutral counterparts, especially for n = 4, 6, and 7 clusters. Additionally, the contributions of the d orbitals of the Ta atom to the HOMO and LUMO reveal that the chemical activity of the d orbitals of the Ta atom decreases gradually as the number of silicon atoms increases. This interesting finding is in good agreement with the recent experimental results on the reactive activities of the H2O and transition-metal silicon clusters (Koyasu, K.; Akutsu, M.; Mitsui, M.; Nakajima, A. J. Am. Chem. Soc. 2005, 127, 4998). Generally, the positive charge significantly influences the electronic and geometric structures of the charged clusters. Finally, the most stable neutral and charged TaSi16 clusters are found to be fullerene-like structures and the HOMO-LUMO gap in charged form is detectable experimentally.     

Computer modeling of the structure of large molecules: II. Cluster C1500

The structure of a C₁₅₀₀ cluster modeling a “planar” polymeric buckminsterfullerene(VIII) is determined by the PM3 quantum-chemical method. The cluster consists of 25 oblate octa-, hexa-, and tetra-valent C₆₀ polyhedra and has D _2h symmetry. The calculated parameters of a rectangular unit cell are as follows: 9.016 Å along the bridging bonds in the polymer plane and 9.084 Å along the bridging bonds over and under the polymer plane. Hybrid σ orbitals of three-coordinate atoms are directed at an angle of 9.7°–9.8° to the polyhedron edges and those of four-coordinate atoms, at an angle of 8.3°–8.6° to the rectangular bridge edges. The s contribution into π orbitals is negligibly small. The GAMESS program with a new algorithm for diagonalization of large dense symmetrical matrices and a new VALENCY program were used for calculations.