Crystal Orbitals Research Articles

Symmetry projection of crystal wave functions by means of a computer

The theory of the irreducible representations of space groups and the symmetry projection of crystal wave functions can be written in a more concise form, when the irreducible representations of arbitrary finite groups can be calculated. Two computer programs have been written. Starting with a minimum of input they produce crystal space orbitals and plane waves respectively, symmetry adapted to the irreducible representations of any one of the 230 space groups. The irreducible representations are generated by the program itself.

CNDO/2 and MINDO/2 energy band structures of some homopolynucleotides

AbstractThe all‐valence electron band structures of the periodic DNA models polyC, polyT, and polyA have been calculated with the aid of the CNDO/2 and MINDO/2 crystal orbital (CO) method.According to the obtained results the valence and conduction bands are always of π type. The widths of the bands are usually larger in the CNDO case, than those obtained for the same systems in the PPP CO approximation. On the other hand the MINDO results show similar widths as the PPP ones. The forbidden bandwidths are by 3–4 eV larger in the CNDO/2 case, than in the PPP one, while the MINDO/2 CO method has resulted in similar gaps like the PPP calculation.

Charge Carrier Mobilities in Periodic DNA Models

The drift mobility of electrons and holes has been calculated for various periodic DNA models using LCAO crystal orbitals and applying the variational method of transport theory. Since the energy bands are narrow, inelastic electron-acoustic phonon scattering has been taken into account for all states of the first Brillouin zone including normal and Umklapp processes. The mean free paths of the charge carriers have been estimated too, to test the consistency of the delocalized description. The mobility values obtained for 300°K lie between 30–100 cm2 V−1·sec−1 for models with bandwidths of 0.1–0.3 eV and between 4–20 cm2 V−1· sec−1 for those with bandwidths of 0.02–0.1 eV, respectively. The temperature dependence follows the T−n law with n = 1.5–2 for the former and with n=2–2.5 for the latter group. The calculated mean free paths show real delocalization for the relatively broader bands, but the application of the band model seems to be questionable for the extremely narrow band cases.

Crystal field surface orbital — Bond-energy bond- order (CFSO-BEBO) model for chemisorption: Application to hydrogen adsorption on a platinum (111) surface

A model describing adsorption which combines features of the bond-energy bond-order relationships of gas phase kinetics and spectroscopy, and the concept of crystal field orbitals has been formulated. This model is thus termed the crystal field surface orbital — bond energy bond order (CFSO-BEBO) model of chemisorption. The calculation is empirical in nature since single bond energies for metal-metal and metal-gas bonds are deduced from available data on bulk compounds or transition metal complexes. Alternatively, the single order bond energies may be estimated from suitable mixing rules or deduced from sublimation energies and spectroscopic gas phase data. Spectroscopic data are also used for the bond energies of the stable gas phase species and for the correlation between bond energy and bond order. The bond-energy bond-order correlation is assumed to be linear for metal-metal and metal-gas bonds.This model is used to describe the interaction of hydrogen with platinum (111) in a detailed way, i.e., the potential energy of interaction along the reaction (or adsorption) coordinate. The trapping probability of hydrogen on platinum is consistent with this CFSO-BEBO model, and the energies of the two predicted chemical adsorption states correspond to the observed experimental heats of adsorption, 17 and 31 kcal/mole, with the low energy state being molecular. LEED and flash desorption experiments also give strong support for the calculated potential energy along the reaction coordinate.

Molecular Orbital Approximation Study of the Co2+ Impurity in ZnSe – Ligand Hyperfine Structure

AbstractFor the complex of tetrahedral symmetry given by a Co2+ ion and its neighbours in a ZnSe crystal molecular orbitals are determined by a semiempirical method, The experimental data used were the cubic field splitting parameter 10 Dq and the spectroscopic splitting parameter g. The molecular orbitals are given as linear combinations of the Hartree‐Fock atomic orbitals with appropriate numerical coefficients. They are used for the calculation of the superhyperfine interaction. The results are compared with results of EPR measurements.

Tetrahedral transition-metal complex spectra

Low-temperature single-crystal absorption, emission and magnetic studies are reported on the tetrahedral complexes [(CH3)4N]2MnCl4 and [(CH3)4N]2MnBr4. All nine d-d bands predicted by crystal field theory are observed in the optical spectra, many with detailed fine structure. The general features of the spectra are analysed using the Tanabe-Sugano strong-field scheme, including the Tree's correction, with excellent results. The finer spectral details are accounted for by spin-orbit interaction via a complete 3d 5 crystal field-spin orbit calculation. The effects of tetragonal distortions are considered and shown to be small. Several ‘mystery’ bands are observed at low temperatures in thick crystals and discussed in terms of possible sextet-doublet transitions.

ESR and ENDOR Investigation of Interstitial Hydrogen Atoms in Alkali Halides. II. Theoretical Interpretation

AbstractA theoretical interpretation of the superhyperfine interactions, the proton hyperfine interaction, and the g‐factor of H centres in KF, KC1, KBr, NaCl, and RbCl is presented. The anisotropic distribution of the hydrogen electron into the lattice is well explained by a wave function which was obtained by orthogonalization of the hydrogen Is function to an internally orthogonal set of crystal ion orbitals. A slight covalency to the nearest halogen neighbours is taken into account. The g‐factor can also be understood on the basis of this wave function, if the electronic as well as the charge transfer excited states are considered. In order to explain the proton hyperfine interaction the influence of van der Waals forces has to be included.

Calculation of the effects of divalent metal ions on nucleotide base pairs. II. Single G-C and A-T and poly(G-C) and (A-T).

SCF calculations have been carried out for the electronic energy levels and wavefunctions of single G–C and A–T and poly(G–C) and (A–T) in the presence of Mg2+ ion attached to the electronegative sites in various ways. The P–P–P approximation was used for the single base pairs, and its equivalent, the semiempirical SCF–LCAO–crystal-orbital (CO) approximation for the polymer. Energy levels and energy bands are presented for all cases that have been investigated. The results supplement those of an earlier paper.

Tetrahedral transition metal complex spectra: The tetrachloroferrate (III) anion

With one exception previous assignments of the electronic spectrum of the tetrahedral tetrachloroferrate(III) anion have relied on spectra taken in solution at room temperature. We report here the low temperature single crystal polarized absorption spectra of tetra(n-butyl) ammonium tetrachloroferrate(III), tetraphenylarsonium tetrachloroferrate(III), and dimethyl-ammonium tetrachloroferrate(III). One new band has been observed and a number of fine structure components have been resolved. The six observed electronic transitions have been fit using three adjustable parameters ( B , C/B , Dq ) by minimizing the r.m.s. deviation between observed and calculated crystal field energies. It was necessary in these calculations to include the Tree's correction factor for a correct assignment of the spectrum. A complete crystal field spin orbit calculation was performed in an effort to determine the origin of the observed fine structure.

Calculation of the effects of hydration and divalent metal ions on nucleotide base pairs: single G-C and poly(G-C).

SCF calculations have been carried out for the electronic energy levels and wavefunctions of single G–C and poly(G–C) in the presence of a neutral water molecule or Mg1+ ion. The Pariser- Parr–Pople (P–P–P) approximation was used for the single base pair, and its equivalent, the semiempirical SCF–LCAO–crystal-orbital (CO) approximation, for the polymer. Energy levels and energy bands are presented for four different cases. According to the results, the presence of the Mg2+ ion very strongly changes the band structure of the poly(G–C). The probable consequences of this on the conductive properties of DNA and the possibility of ab initio calculations on periodic DNA models are discussed.

Spin-Polarized Orbitals for Localized States in Crystals

Using the concept of different orbitals for different spins, in conjunction with the tight-binding approximation, it is possible to construct a set of spin-polarized orbitals (SPO's) in which electrons with different spins are localized on different crystal atoms. Thus, SPO's allow for the correlation between electrons with different spins and, for a pure crystal, are very similar to Löwdin's alternant crystal orbitals. For one-dimensional crystals, an analysis which resembles that for linear mixed crystals reveals that, for α- or β-spin electrons, the energy levels of the nonlocalized SPO's form two bands, while those of the localized SPO's lie either outside these bands or in the gap between them. Finally, Green's matrix method is employed to examine the effect on the SPO's of the perturbation produced by interior impurity atoms in the crystal, or chemisorbed atoms at the surface.

Physical Methods for Treating the π-Electron Systems of Biopolymers such as Polypeptides and Polynucleotides in the Hückel Approximation

Two simple physical methods are presented for calculating the electronic band structures of biopolymers such as hydrogen-bonded polypeptides or homopolynucleotides in the Huckel approximation. One is an extended LCAO MO method, or a method of Linear Combination of Molecular Orbitals, and the other is an extended Bloch wave method where the crystal orbitals are constructed from the molecular orbitals of elementary group with the aid of group theoretical considerations. The extended Bloch orbital method seems to be the most simple and powerful tool for dealing with the electronic structures of the multiple conjugate π-electron systems. The brief applications of these two methods to actual biopolymers are made only for the illustration.

A Self-consistent Calculation of the Graphite Band

The approximate self-consistent field theory, recently developed in calculations of molecular wave functions, is applied to a two-dimensional crystal - the graphite layer. The crystal orbitals for the π electrons are constructed from atomic orbitals, as in the usual tight binding approximation, but the methods of many-electron theory are used in setting up the self-consistent field. Matrix elements of the one-electron Hartree-Fock Hamiltonian (which in the usual tight binding model are identified empirically) are expressed in terms of the π-electron density matrix, whose elements may be evaluated by integration over the occupied region of k space, and are systematically adjusted until Hamiltonian and charge density are self-consistent. The results, besides being of intrinsic interest, suggest that with the further development of automatic computation similar calculations on three-dimensional crystals will become feasible.

SELECTION RULES AND TEMPERATURE DEPENDENCE OF THE FIRST-ORDER RAMAN EFFECT IN CRYSTALS

Starting from the most general scattering formulae, the current theory of the Raman effect in crystals is modified in such a way as to remove the well-known discrepancies between theory and experiment concerning the selection rules for calcite and similar crystals. A distinction is made between electrons in delocalized crystal orbitals and electrons in localized atomic or molecular orbitals and it is shown that only the latter produce a Raman scattering in agreement with the unmodified theory. The general formula for the scattering by delocalized electrons is analyzed and it is found that the magnitude of the components [Formula: see text] of the first-order polarizability (qi normal coordinate of the scattering lattice vibration) depends on the wave vectors Q′ and Q″ of incident and scattered light. The wave vector dependence of the first-order polarizability requires selection rules for the first-order Raman effect which do not correspond to the full symmetry of the scattering crystal but only to the symmetry operations of the group of Q = Q′ – Q″ which leave Q unchanged. These modified selection rules are shown to be compatible with experiment. The influence of mechanical anharmonicity and of polarizability derivatives of higher order on the first-order Raman effect is also discussed. It is found that these non-linear effects offer no satisfactory explanation for the great intensity of forbidden lines in the Raman spectrum of some crystals. Concerning temperature effects the non-linear terms in the intensity formulae are found to be of greater importance and are tentatively suggested as being responsible for the anomalous temperature dependence of low frequency external lattice vibrations.