Crystal Orbitals Research Articles

The kagomé net: Band theoretical and topological aspects

The Kagomé net (a two-dimensional hexagonal network comprising triangles and hexagons) is a building block of a number of intermetallic compounds. In order to understand the electronic characteristics of such an array of atoms, extended Hückel tight binding calculations have been performed on a Kagomé net of boron atoms. Comparisons have been drawn with other hexagonal nets, namely the graphite net and the close packed triangular net, and topological relationships between these nets are used to explain similarities in their band spectra. It is shown that simple Hückel theory can be used to obtain a rough estimate for the energies of the bands at specific high symmetry k-points, and a way of representing and evaluating energies of crystal orbitals with complex coefficients is discussed. A moments-based analysis of the relative energies of the three nets as a function of electron count is also presented. The effect on its band structure of trigonally distorting the Kagomé net is investigated. Finally, substituted ABC and AB 2 derivatives of the Kagomé net are discussed.

A stability condition for the Hartree–Fock solution of the infinite one-dimensional system

A simplified stability condition for the Hartree–Fock (HF) solution giving the self-consistent field crystal orbitals (SCF-CO) of the infinite one-dimensional system is derived. Since the present formulation, particularly for the systems having nearly or entirely degenerated highest occupied and the lowest unoccupied COs, contains only two physical parameters, that is, the density of states and the Coulomb repulsion integrals both at the Fermi level, it is tractable to examine the stability of the HF solutions of such polymers as well as the ordinary molecular systems. An example of its application to metallic trans-type polyacetylene is also shown.

Structure of the K/Si(100)-(2 x 1) surface: Semiempirical self-consistent-field crystal orbital analysis.

The electronic structure of the K/Si(100)-(2\ifmmode\times\else\texttimes\fi{}1) surface at a K coverage \ensuremath{\Theta}=0.5 has been studied by a semiempirical self-consistent-field Hartree-Fock crystal orbital (CO) formalism supplemented by an INDO (intermediate neglect of differential overlap) Hamiltonian. A symmetric dimer model has been used to represent the structure of the Si(100)-(2\ifmmode\times\else\texttimes\fi{}1) surface. Four possible adsorption sites for the K atoms have been investigated. The relative stabilities of the adsorption sites have been determined by total energy calculations. The Si(100)-(2\ifmmode\times\else\texttimes\fi{}1) surface covered with chemisorbed K atoms is stabilized by multicenter Si-K bonding, which favors K adsorption at surface sites where the coordination to the Si atoms is maximized. The charge transfer from the K overlayer to the Si substrate indicates that the nature of the Si-K bonding is partially ionic. At a K coverage \ensuremath{\Theta}=0.5 the K/Si(100)-(2\ifmmode\times\else\texttimes\fi{}1) system is metallic. The charge carriers at the Fermi level are located at the outermost Si layer. The semiconducting Si(100)-(2\ifmmode\times\else\texttimes\fi{}1) surface is, therefore, metallized upon deposition of K atoms.

The electronic structure and magnetic properties of poly( m-phenylcarbene)

The electronic structure and the magnetic properties of the ferromagnetic organic polymer poly ( m-phenylcarbene ) was studied by application of the unrestricted Hartree-Fock (UHF) crystal orbital (CO) method. In comparison with the restricted Hartree-Fock (RHF) result, it was revealed that the ferromagnetic state is more stable than the non-magnetic state. According to a detailed energy analysis, the stability originates from both the triplet spin configuration at the carbene centre and the delocalized π spins in an antiferromagnetic fashion over the phenyl ring.

On the self-consistent-field linear combination of atomic orbitals for bounded crystal orbitals (SCF-LCAO-BCO) method

The self-consistent-field linear combination of atomic orbitais for bounded crystal orbitais (SCF-LCAO-BCO) method can be used to study the electronic structure of bounded polymers and crystals if they have a significant amount of internal translational symmetry, at the ab initio Hartree-Fock level. The direct recursion (transfer matrix) method (DRM) is applied to its derivation, here only for certain simpler models. It is shown that the application of the free boundary condition instead of the periodic (Born-von Kármán) one halves the conventional one-dimensional Brillouin zone and resolves the double degeneracy of the Bloch states. A longitudinal bulk-state distortion (LBSD) is demonstrated for a linear bounded polymer model possessing perfect translational symmetry and symmetrical intercell interaction matrices. The block diagonalizability is discussed when free boundary conditions are applied. The non-existence of limit wave functions in the overlapping regions of bands is shown for the infinite size limit. The use of certain averaged wave functions is proposed for the calculation of the charge-bond order matrices in order to resolve the above difficulty.

A Theoretical Study of the Ionized State of Polymers with the Localized Molecular Orbital Method

Abstract A localized molecular orbital method for the investigation of the ionized state of polymers is presented. It is based on the Crystal Orbital Method(CO). By performing a super-cell method and Edmiston and Ruedenberg’s LMO method, the Bloch function was transformed into a nearly completely localized Wannier function. Furthermore, the calculation of the configuration interaction was introduced to obtain more reasonable results. The present method was applied to polyacetylene (PA), polyethylene (PE), and polypropylene (PP). Ionization potentials of these polymers were calculated and compared with experimental and other theoretical results. All the calculations were performed using the CNDO/S approximation.

First-principles calculational methods for surface-vacancy formation energies, heats of segregation, and surface core-level shifts.

The matrix Green's-function version of the self-consistent scattering theory of surface-point-defect energetics is extended to the cases of vacancy formation, substitutional adsorption, and surface core-level shifts. Vacancies are treated by zeroing Hamiltonian and overlap matrix elements involving orbitals of atoms being removed and crystal orbitals or the crystal potential. Substitutional adsorption is treated as ordinary adsorption in a vacancy, a treatment which permits use of an arbitrary number of substitutional-atom basis orbitals. Surface core-level shifts, including final-state relaxation effects, are obtained from substitutional adsorption calculations via the Born-Haber-cycle argument of Rosengren and Johansson. In a first comparison with experiment, the surface core-level shift for Al(001) is predicted to be -97 meV, modeling this system as a five-layer film, while different experimental groups report values ranging from 0 to -120 meV, respectively, for the Al(2p) level. Earlier calculations, which neglected final-state relaxation, predict a shift of -94 meV for the Al(1s) and about -120 meV for the Al(2s) and Al(2p) core levels. Comparison with the present result indicates that relaxation effects are small in the Al(001) surface core-level shift.

Solid-state electronic structures of intercalation compounds: The system Mg(Ti 3S 4) 2

The electronic band structure of the intercalation system Mg(Ti 3S 4) 2 is investigated by a three-dimensional (3D) self-consistent-field (SCF) Hartree-Fock (HF) crystal orbital (CO) approach based on an INDO (intermediate neglect of differential overlap) all-valence Hamiltonian. The band-structure properties of Mg(Ti 3S 4) 2 are correlated with those of the binary matrix as well as the dichalcogenide TiS 2 where the transition-metal is in a higher oxidation-state. The calculated electronic-structure data of Mg(Ti 3S 4) 2 and TiS 2 are compared with available results of experimental measurements. For TiS 2 the present computational findings are furthermore compared with previous band-structure calculations. The formation of charge-density-wave (CDW) solutions in Mg (Ti 3S 4) 2 and Ti 3S 4 are analyzed. The CDW condensation is caused by fluctuating valences (i.e. deviations of an orbital occupancy around its average value) in the Ti sublattice. Some associated many-body effects are discussed. Mg(Ti 3S 4) 2 is a system where electronic and ionic conductivity are both operative. A first-order phase-transition of the insulator-semimetal type is predicted for the ternary system as a function of the spatial localization of the intercalant-atoms in the 1D van der Waals channels. Electronic reorganization-processes between the intercalant and the Ti 3S 4 matrix accompanying this phase-transition are analyzed. It is shown, that the rigid band-model is nonvalid for the interpretation of the band-structure properties of the intercalation compound. The results of frequently used methods in band-structure theory of intercalated transition metal chalcogenides are critically reviewed. The present findings render possible a reinterpretation of some electronic consequences of intercalation. Variations of the one-electron energies in intercalated materials were incorrectly predicted by methods employed conventionally in solid-state theory.

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

AbstractA 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.

The two‐dimensional band structure of (polyphthalocyaninato)Ni(II)

AbstractThe two‐dimensional (2D) band structure of (polyphthalocyaninato)Ni(II), Ni(ppc), has been analyzed by a self‐consistent field (SCF) Hartree–Fock (HF) crystal orbital (CO) formalism based on an INDO (intermediate neglect of differential overlap) type Hamiltonian. The calculated HF band gap of Ni(ppc) amounts to 0.24 eV. The highest filled band is a ringlike a1u combination (D4h symmetry label) localized at the carbon sites of the organic fragment. Remarkable hybridization in the valence band leads to the considerable band width Δϵv of 2.92 eV. This value is close to the Δϵv numbers which are conventionally encountered in one‐dimensional metallomacrocycles. The effective width of the states in Ni(ppc) is 13.8 eV. In graphite a net π interval of 13.0 eV is predicted by the present CO formalism; i.e., the energetic distribution of the π electrons is roughly comparable in both 2D solids. The Ni 3d states in Ni(ppc) are far below the Fermi level which is calculated at −4.9 eV; they are predicted between −12.2 and −16.4 eV in the mean‐field approximation. Quasi‐particle corrections lead to a significant shift of these strongly metal‐centered states. Important electronic structure properties of Ni(ppc) are compared with those of 1D metallomacrocycles with similar molecular stoichiometry. The total density of states distribution of Ni(ppc) has been fragmented into projected (ligand π and σ, Ni 3d) contributions in order to allow for a transparent interpretation of the 2D band structure.

Orbital interactions in phyllosilicates: Perturbations of an idealized two-dimensional, infinite silicate frame

The ∞ 2 [Si2O 5 2− ] frame in phyllosilicate minerals is distorted through the rotation and tilting of the silicate tetrahedra and interacts with octahedral cations through its apical oxygens. Qualitative perturbation theory and extended Huckel band structure calculations demonstrate that rotation and tilting distortions of the ∞ 2 [Si2O 5 2− ] frame have little influence on orbital interactions within the frame. The effects which are observed can be traced to next-nearestneighbor, oxygen-oxygen interactions. Analysis of band widths and crystal-orbital-overlap-populations demonstrate the importance of O(2s) orbitals in the silicate bond. Interactions between Si(3s, 3p) and O(2s) atomic orbitals account for about half of the bonding overlap in the Si-O bond. Crystal orbitals within the ∞ 2 [Si2O 5 2− ] frame are perturbed in kaolinite, lizardite, pyrophyllite and talc through interactions of the apical oxygens with octahedrally coordinated Al(III) and Mg(II). These interactions appear to involve states that are non-bonding in an isolated frame, having little effect on the Si-Oapical bond while significantly reducing the apical-oxygen atomic population.

Graphite as an aromatic system

Graphite is a stable structure which does not possess an energy gap at the Fermi level. However, the crystal orbitals of each monolayer remain similar to the benzene molecular orbitals. Different parts of the Brillouin zone take into account either the stability or the reactivity of the compound. The ..pi.. system would be stabilized by a Kekule distortion which would open a gap. The sigma system forbids this distortion.

Band Structures of Intercalation Compounds. The System (NH4)(NH3)3(TiS2)4

The electronic band structure of the intercalation system ammonia TiS2 has been investigated by a semiempirical self-consistent-field (SCF) Hartree-Fock (HF) crystal orbital (CO) formalism supplemented by an INDO (intermediate neglect of differential overlap) Hamiltonian. A two-dimensional (2D) model for the title system with stoichiometry (NH4)(NH3)3(TiS2)4 has been selected on the basis of available experimental data. The model is defined via a TiS2 monolayer coupled to the intercalant monolayer. The corresponding band-structure properties are compared with bandstructure calculations of monolayered TiS2 and bulk TiS2 . For TiS2 available experimental data and numerical results of conventional band-structure approaches are reported. The interaction between the guest-molecules and the host lattice has the character of a redox-process; i.e. one electron per formula unit has been transferred from the intercalant to the TiS2 layer. One consequence of this transfer is a semiconductor-to-metal transition upon intercalation; an additional consequence is a remarkable electronic reorganization in the TiS2 host. The surplus of electronic charge is predominantly localized at the S centers. The electronic states at the Fermi-level are of Ti 3d character. Two electronic configurations of the title system have been investigated. The mean-field ground state is of a charge density wave type with respect to the TiS2 sublattice. A “symmetry adapted” (SA) configuration is predicted at higher energy.

Band Structure Calculations of Two-Dimensional (2 D) Models of the Phases LiBC and CaAl2Si2

The electronic structures of the phases LiBC and CaAl2Si2 have been investigated by two dimensional crystal orbital calculations. The corresponding solid-state ensembles have been separated into anionic (e.g. [BC]-, [AlSi]- , [Al2Si2]2-) and cationic substructures. The employed fragmentation is topologically related to the electron counting schemes used in the classical Zintl approach. The band structures of the anionic fragments have been determined by a semiempirical Hartree-Fock crystal orbital (CO ) Hamiltonian on the basis of the tight-binding approximation. The adopted self-consistent-field variant is an intermediate neglect of differential overlap (INDO ) scheme. The influence of the cationic substructure has been simulated by an electrostatic point-charge model. The computational formalism allows for a suitable explanation of the conformations of the anionic subunits of the studied solids. In the CaAl2Si2 model the site symmetry in the anionic substructure has been studied in larger detail. The Al atoms show a tetrahedral coordination while an inverted tetrahedron (i.e. umbrella-like structure) is realized at the more electronegative Si centers. This local geometry is stabilized by the Coulomb interaction between the anionic [Al2Si2]2- layer and the cationic substructure. The energetic sources leading to the different conformations in LiBC and CaAl2Si2 are also quantified. The observed geometric preferences are rationalized by means of an energy fragmentation into covalent resonance (kinetic energy of the electrons) and classical Coulomb elements. This energy decomposition in the planar and chair-like conformations of LiBC and CaAl2Si2 shows that the kinetic energy favours the former, the electrostatic part the latter conformation. The dimerization of two anionic substructures (CaAl2Si2 structure) is also studied by the tight-binding formalism . The observed geometric differences between intra- and interlayer bonds are satisfactorily reproduced by the semiempirical CO model.

The One‐Dimensional Band Structure of (Tetrazaporphyrinato)iron(II)

AbstractThe one‐dimensional (1 D) band structure of (tetrazaporphyrinato)Fe(II) (Fe(tp)) has been investigated in the framework of a semiempirical Hartree‐Fock (HF) self‐consistent‐field (SCF) crystal orbital (CO) approach. The calculated ground state of this 1D solid is of spatial B1g symmetry. This configuration in the 3d6 material is determined by two half‐filled bands of the insulating Mott‐Hubbard‐type where all CO microstates in the first Brillouin zone are singly occupied. Both Mott‐Hubbard dispersions are strongly metal‐centered, i.e. 3dz2 and 3dxy. The width of the σ‐type (3dz2) band amounts to ca. 0.21 eV while a nearly dispersionless band is predicted for the δ interaction (3dxy). These two half‐filled 3d bands lie below the Fermi energy, while the highest occupied states are of ligand π character. The band structure properties of the Mott‐Hubbard insulator have been determined by means of a grand canonical (GC) averaging procedure which corresponds to an approximate spin‐restricted mean‐field description. The HF SCF energy of the “spin‐paired” 3d6 configuration is predicted to have a higher energy. Therefore, an “aufbau” principle is not valid in the ground state of this 1D material. The implications of this ground state on the magnetic properties of Fe(tp) are also discussed.

Electronic properties of one-dimensional graphite family

Two kinds of one-dimensional graphite series starting from polyacene and polyphenanthrene have been studied on the basis of the tight-binding crystal orbital (CO) calculations. Emphasis has been put on decrease of the band gap, according to the development of the polymer skeleton toward graphite, and distribution of the CO patterns being important to the path of charge carriers.

ChemInform Abstract: Chemisorption Patterns as Predicted by Band Calculations, by Frontier Crystal Orbitals, and by Cluster Calculations: Hydrogen Atoms on Graphite

Chemischer InformationsdienstVolume 17, Issue 35 Physical Inorganic Chemistry ChemInform Abstract: Chemisorption Patterns as Predicted by Band Calculations, by Frontier Crystal Orbitals, and by Cluster Calculations: Hydrogen Atoms on Graphite J. P. LAFEMINA, J. P. LAFEMINASearch for more papers by this authorJ. P. LOWE, J. P. LOWESearch for more papers by this author J. P. LAFEMINA, J. P. LAFEMINASearch for more papers by this authorJ. P. LOWE, J. P. LOWESearch for more papers by this author First published: September 2, 1986 https://doi.org/10.1002/chin.198635023AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume17, Issue35September 2, 1986 RelatedInformation

The band structure of Ni(H 5C 3B 2). An example for energetic stabilization due to dimerization

The band structure of Ni(H 5C 3B 2) the first experimentally verified one-dimensional (1D) polydecker system, has been investigated by means of a semi-empirical crystal orbital (CO) formalism based on an improved INDO (intermediate neglect of differential overlap) hamiltonian. In order to allow for an analysis of the electronic structure of a 1D arrangement composed by unit cells with an ungerade number of electrons a grand canonical (GC) averaging scheme has been used for the definition of the crystal hamiltonian. The synthesized 1D material with 13 valence electrons per simplest stoichiometric unit is a semiconductor and shows nuclear distortions into the direction of the stacking axis (i.e. formation of non-equivalent metal-ligand contacts). The tight-binding calculations lead to a transparent theoretical explanation of this formal dimerization. The 1D arrangement built up by one Ni(H 5C 3B 2) half-sandwich per unit cell reproducing the full translational symmetry is unstable towards a dimerization that leads to a unit which is formed by two half-sandwiches. The energy of the 1D column with the doubled cell is stabilized due to a symmetry-violation of the spatial wavefunction (i.e. symmetry-breaking for a symmetry atomic arrangement). A nuclear distortion (formation of non-equivalent metal-ligand distances) causes an additional energy lowering of the 1D system. The band structure properties in the outer valence region are analyzed. The calculated band gap is in line with he experimentally observed semiconducting properties of the 1D chain. The microstates of the valence band contain both admixtures from the cyclic organic π ligand (leading contribution) and from the transition metal centers (3d xz or 3d yz orbitals).

An efficient technique for the evaluation of lattice sums in crystal orbital (CO) calculations

AbstractAn approximate approach for the determination of lattice sums is presented in two‐ or three‐dimensional solids within the crystal orbital (CO) formalism as derived in the tight‐binding approximation. The solid is divided into two domain types (I and 11) where the interatomic interaction energies are determined in different degrees of sophistication. The matrix elements of the Fock operator are treated exactly within the (inner) spherical domains of type I. An electrostatic model is employed to calculate the coupling terms to atoms localized in the outer domain 11. In this case only the diagonal elements of the mean‐field Hamiltonian are taken into account. Numerical examples to estimate the convergence properties of this approximation are presented for the graphite layer, diamond, two‐ and three‐dimensional BN in the framework of an improved INDO CO approach.

Chemisorption patterns as predicted by band calculations, by frontier crystal orbitals, and by cluster calculations: hydrogen atoms on graphite

Chemisorption patterns as predicted by band calculations, by frontier crystal orbitals, and by cluster calculations: hydrogen atoms on graphite