Effect of nonorthogonality of electronic wave functions on the approximate calculation of transition dipole moments

We discuss a method of taking into account the nonorthogonality of independent approximations for the electronic wave functions of states of a single symmetry type in the calculation of transition dipole moments. The effectiveness of the method is illustrated for the example of the electronic transition C²_r-A²_i of the molecule BO.Translated from Teoreticheskaya i Eksperimental'naya Khimiya, Vol. 26, No. 2, pp. 215–217, March–April, 1990     

A theoretical study on cyclacenes: Analytical tight‐binding approach

We present a theoretical study of cyclacene molecules performed at tight‐binding level. The orbital energies and eigenvectors have been analytically computed, and exact expressions for the axial component of the total position spread and polarizability tensors have been obtained. In absence of dimerization, the system has a D_nh symmetry, where n is the number of hexagonal units. The energy bands present no gap at the Fermi level, and to this fact it corresponds a diverging (per‐electron) polarizability for in the direction of the system symmetry axis. The two (degenerate) components of the polarizability on the σ_h symmetry plane, conversely, remain finite for . The total position spread tensor presents a qualitatively different behavior, since all the three components of the position spread per electron remain finite for . The results are analyzed and discussed for both axial and planar components separately as these are affected differently with respect to the increasing system size. Both dipole polarizability and total position spread have been computed using an ab initio approach for the smallest systems, to compare the analytical tight‐binding expressions with a higher‐level theory.     

Compositions and structures of nanoparticles of pentagonal axial symmetry <Emphasis Type="Italic">D</Emphasis><Subscript>5<Emphasis Type="Italic">d</Emphasis></Subscript>: Nanotubes

The method of determination of the structure and the number of atoms in the shells of nanoparticles as a function of the arrangement of atoms at the symmetry elements of a symmetry group has been developed. The formulas for the calculation of the number of particles with symmetry group D _5d are reported. The number of particles in these shells is determined by three structurally invariant numbers and the “quantum number” of the group order n. The classification of all possible nanostructures with symmetry group D _5d is given: C _θ+10z , z = 0, 1, 2, …, where the basic shells are C _θ = C ₂, C ₁₀, C ₁₂. The sum rule has been obtained for the coordination numbers of shell sites located at symmetry axes. Pentagonal axial nanoparticles are shown to be the initial shells for obtaining (5,5) and (10,10) armchair nanotubes or (5,0) and (10,0) zigzag nanotubes. The general formula of these nanotubes closed with icosahedral and dodecahedral caps is N _20+10p , N _60+10p (p = 1, 2, …). The graphical constructions of all classes of nanoparticles and nanotubes of the pentagonal axial type are reported.     

Synthesis,characterization and function of ternary copper(II) complexes

Ternary complexes of glycine, alanine, -alanine, serine and ethylenediamine with copper iminodiacetate have been prepared and characterized by their i.r., u.v.-vis. and e.s.r. spectra. The data suggest that the complexes possess axial symmetry and a distorted octahedral configuration in aqueous solution. The H2O molecule on the axial copper site can be replaced by O2–. The bonding parameters, which fall between 0.84 and 0.88, indicate that the bonds between copper(II) and the ligands possess more ionic character. The ternary complex containing serine displays the highest catalytic activity in dismutation of the superoxide radical O2–. This activity may be explained on the basis of the stabilization of CuII-O2– by hydrogen bonding.     

Electron paramagnetic resonance study of a Eu related defect in CsBr:Eu needle image plates for computed radiography

The electron paramagnetic resonance (EPR) spectrum of needle image plates of CsBr doped with Eu²⁺, which are proposed as new X-ray storage phosphors for computed radiography, is studied at room temperature and Q-band microwave frequencies (34 GHz). X-ray diffraction analysis demonstrates that the CsBr:Eu²⁺ needles have an 0 0 1 out of plane (perpendicular to the plate) orientation, and contrary to expectation that the in plane orientation is not random. The room temperature EPR spectrum is attributed to a single centre which is related to Eu²⁺ with axial 0 0 1 symmetry. Using the spin Hamiltonian parameters extracted from the spectrum recorded with the magnetic field parallel to the needles’ axes, we convincingly simulate both the spectrum of a powdered image plate and the single crystal like angular dependence of intact pieces of image plate. The knowledge of the symmetry of this centre, which appears to be related with the radiation sensitivity of the plate, presents a first step in finding its model and role in the X-ray storage process.     

E.p.r. spectra and bonding parameters of copper(ii) complexes of 2-methyl-4,6-diacylphenols,of the dibenzoyl analogues and of some of their derivatives

Summary The e.p.r. spectra of mononuclear copper(II) complexes of 2-methyl-4,6-diacylphenols and dibenzoylphenols, and their condensation products with 1,3-diaminopropane have been examined. The MO coefficients evaluated from e.p.r. spectral parameters indicate that the in-plane-, in-plane- and out-of-plane- bondings are significantly covalent. The e.p.r. spectra of the macrocyclic dicopper(II) complexes derived from 2-methyl-4,6-diacyl(benzoyl)phenols and 1,3-diaminopropane measured at 77 K reveal the presence of mononuclear species of axial symmetry.     

Geometric theory for adhering lipid vesicles

This paper aims at constructing a general mathematical model for the equilibrium theory of adhering lipid vesicles from a geometrical point of view. Based on the generalized potential functional, a few differential operators and their integral theorems on curved surfaces, the general normal and tangential equilibrium differential equations and boundary conditions are given at the first time for inhomogeneous lipid vesicles. A general boundary condition is first put forward including line tension. No assumptions are made either on the symmetry of the vesicle or on that of the substrate. The physical and biological meaning of the equilibrium differential equations and the boundary conditions are discussed. Numerical simulation results based on the Helfrich energy for adhering lipid vesicles under the axial symmetric condition show the effectiveness and convenience of the present theory.     

ESR study of Jahn-Teller distortions and the structure of Cu(II) complexes in [Cu(bipy)2X](ClO4) and [Cu(phen)2X](ClO4)

The Cu(II) complexes in [Cu(bipy)₂X](ClO₄) and [Cu(phen)₂X](ClO₄) were studied by ESR spectroscopy at =9.8 GHz and optical absorption at 7000–20,000 cm^–1. The axially symmetrical g factors, corresponding to the g, were established for the distorted trigonal bipyramidal coordination of Cu(II) with C _2v symmetry. Analysis of the possible vibrational distortions of the complex showed that the axial symmetry of the g factor is a consequence of orientational averaging realized for an electronic vibrational state with a two-well adiabatic potential at the final height of the barrier and vibrational frequency greater than the ESR frequency.Slovakian Technical Institute and Institute of Geology and Geophysics, Siberian Branch, Academy of Sciences of the USSR. Translated from Zhurnal Strukturnoi Khimii, Vol. 30, No. 5, pp. 74–77, September–October, 1989.     

Electron spin resonance and electronic spectral investigation on square planar copper(II) complexes ofN-ethyl- andN-propylimidazole

Summary Electron spin resonance studies have been carried out onN-ethylimidazole andN-propylimidazole 4 : 1 complexes of CuX₂ salts (X = ClO ₄ ^– , NO ₃ ^– , , Br^– or Cl^–) in their polycrystalline and undiluted form at 295 K and 77 K. Cupric ion hyperfine structural resolution is observed for all the complexes at 295 K and the spectra are characteristic of a CuN₄ chromophore with axial symmetry. In complexes involving ClO , Br^– and Cl^– the anions are nonbonding, whereas those with the NO ₃ ^– anion are weakly bonding. The electronic and e.s.r. spectral data have been correlated. The resolution of Cu²⁺ ion hyperfine structure in these complexes is attributed to a decrease in the dipolar interaction and has been observed for the first time since the first resolution reported in 1954 for CuN₄ coordination with square planar symmetry for ,,,-tetraphenylporphyrincopper (CuTPP).     

Crystal structure and electronic properties of bis(mepirizole) copper(II) perchlorate. Correlation between the electronic spectrum and CuN4 chromophore distortion from tetrahedral symmetry

Summary The crystal structure of the title compound has been determined from single-crystal x-ray diffraction data. The crystals are tetragonal, space group P4₂2₁2, withz=2 in a unit cell of dimensionsa=b=9.575(2),c=16.135(5) Å. The structure was solved by the usual Patterson and Fourier techniques, and was refined by least-squares analysis to an R value of 0.057 for 545 observed reflexions. Two mepirizole molecules are arranged in a flattened tetrahedral manner (average Cu-N, 1.974 Å) around Cu, and coordinated through N atoms. The dihedral angle between N-Cu-N planes of the two ligands is 53.6+°. This geometry seems to result from steric interaction between the ligands.The e.s.r. spectrum is axial (g g > 2.0). A study of the maximum d-d transition as a function of CuN₄ chromophore distortion from T_d symmetry is reported.4-Methoxy-2-(5-methoxy-3-methyl-pyrazol-1-yl) -6-methylpyrimidine.     

Hypervirial theorems and symmetry

It is pointed out that to ensure that an optimal variational wave function having a certain symmetry satisfies the hypervirial theorem forW, it is sufficient thatiSW, whereS is the projector onto the symmetry type in question, be a possible variation of . Application is made to the tensor hypervirial theorem for atoms.     

Correlation Instability and Phase Transitions in Octahedral Fluoride Systems

An ab initio study of octahedral hexafluoride molecules and ions has shown that innersphere correlation interactions lower the initial O_h symmetry of these molecules and ions to C₁ and that the observed high symmetry is pseudosymmetry. The microscopic mechanism of correlation instability of octahedral molecules and ions is related to the electric dipole nature of dispersion interactions. The anomalously high values of entropy for polymorphic transformations in transition and chalcogenideelement hexafluorides fit the model of order–disorder phase transitions in a system of internally distorted octahedral groups.     

Molecular and crystal structure of the clathrochelate complex FeDm3(BF)2⋅C6H6 and parameters of its 57Fe Mossbauer spectra

The molecular and crystal structure of the macrobicyclic complex FeDm₃(BF)₂C₆H₆ was established by X-ray diffraction analysis at 138, 208, and 291 K. The X-ray diffraction data were compared with the ⁵⁷Fe Mossbauer spectral parameters of this complex in the temperature range of 135-290 K. A substantial disagreement between the experimental values of quadrupole splitting and the corresponding values calculated based on the concept of partial quadrupole splitting was attributed to the displacement of the Fe atom from the center of the cavity of the macrobicyclic ligand and to the distortion of the axial symmetry of the electron density distribution about the metal atom.     

Hydrogen/argon plasma jet with methane addition

Spatial distributions of plasma parameters are presented for a H₂/Ar plasma jet with addition of methane. The plasma has been generated at atmospheric pressure by a 200 A (20 kW) nontransferred do arc. Optical emission spectroscopy has been used for the measurements assuming the plasma jet to be optically thin and to have an axial symmetry. Local spectral ernissivity values have been evaluated using a routine Abel inversion procedure. Half- width and emissivity of H spectral line have been measured to determine the electron density and temperature of the plasma. The densities of excited C, CH radicals have been evaluated from the absolute emissivities of relevant molecular emission bands measured in limited spectral intervals in the visible spectrum. The emissivity ratios have been used to fund rotational and vibrational temperatures. The results supply information on methane decomposition and the behavior of molecular radicals in close-to-thermal plasma jets.     

Theoretical studies of [n]paracyclophanes and their valence isomers

Summary The valence isomerisations of benzene, [6]- and [7]paracyclophane to their Dewar benzene and prismane isomers are studied with the MNDO method using the unrestricted Hartree-Fock (UHF) and the configuration interaction (C.I.) approximations. The enthalpy of the reaction Dewar benzene benzene is H° _r =–68.9 kcal/mol and the activation enthalpy is H°=27.9 kcal/mol (with C.I.). The reaction path hasC _2v symmetry.The determination of several points of the lowest potential energy surface of [6]- and [7]paracyclophanes leads to a minimum reaction path having the same topology as for the potential energy surface of the nonbridged benzene. The only difference is a quantitative change in the energy values of the aromatic isomers due to the deformation introduced by the alkyl chain. For [6]paracyclophane, the activation enthalpy is H°=24.6 kcal/mol and the activation entropy is S ⁰=0.6 cal K^–1 mol^–1 calculated with C.I.The enthalpy of the reaction prismane Dewar benzene is H° _r –32 kcal/mol and the activation enthalpy is H°19 kcal/mol. The highest molecular symmetry group common to both molecules isC _2v , whereas the symmetry group of the reaction path is lowered toC _s . Along this reaction path is located a biradicaloid intermediate, separated by low activation barriers from the products. No significant changes of the potential energy surfaces are found for the bridged [n]prismanes and the [n]Dewar benzenes.All the calculated values, reaction enthalpies, activation enthalpies and entropies, are in a good agreement with literature experimental data.This article is dedicated to Professor A. Pullman     

Some remarks on propagators and resolvents

Summary The background of the propagator methods introduced into quantum chemistry by Linderberg and Öhrn is briefly reviewed. Emphasis is put on its connection with the superresolvent associated with the Liouvillians superoperator as shown by Goscinskiet al. The paper stresses the importance of the effectiveness of the resolvent approach in general, and points out that, for the ordinary Hamiltonian, this method is not only conceptually important as the basis for infinite-order perturbation theory and rational approximations but also as the foundation for a large-scale computational effort still to come. Even if it may be premature to ask for a similar effort related to the superoperator, it would certainly be of value to carry out some test calculations based on the Hilbert-Schmidt binary product, in which the Liouvillian superoperator is automatically self-adjoint, and to compare the results with those obtained from the more conventional propagator methods. The importance of the use of approximations based on inner projections in both these approaches is also mentioned.     

Effect of molecular weight and temperature on the isothermal crystallization of poly(oxetane)

Poly(oxetane) fractions ranging in number-average molecular weights from 7800 to 157000 have been isothermally crystallized in the temperature range from –50 to 19 C, using dilatometric and calorimetric techniques. In both cases, reproducible isotherms were obtained with an Avrami exponent equal to three. The crystallization rate against crystallization temperature presents a maximum at –30 C. The level of crystallinity changes with molecular weight and the influence of this parameter on the rate of crystallization is pronounced. The crystallization temperature coefficient was studied using nucleation theory and it was found an slight increase in the basal interfacial free energy for the lowest molecular weight fraction. For the analysis of the temperature coefficient at the higher undercoolings, different approximations for the free energy of fusion and the transport term have been considered. The conclusion of this analysis is that, independently of these approximations, the obtained temperature coefficients are the same.     

Chemical algebra. I: Fuzzy subgroups

Using the notion of fuzzy subset, the algebraic formulation of the constant of stereogenic pairing equilibria between skeletal analogs (previously disclosed) is connected to symmetry group theory. A distinction is introduced between geometrical (skeletal) symmetry and topographical (numerical parameters) symmetry. In order to describe topographical symmetry, a formal extended definition of a subgroup is proposed. Fuzzy subsets of the skeletal groupG are endowded with a structure which can be defined without referring to the geometrical representation of the abstract group isomorphic toG. The relevance of these propositions is evidenced by their integer interpretation meeting basic definitions of group theory, as well as by their role in expressing chemical pairing constants.     

Symmetry functions adapted to the subgroup chain U(7) ⊃ SO(7) ⊃ G2 ⊃ SO(3) ⊃ G

Summary In this paper the Lie algebra technique is used to construct symmetry functions adapted to the subgroup chain U(7) SO(7) G ₂ SO(3) G, which is one of symmetry group chains appearing in the weak ligand field scheme for f ^N ions. The functions are expressed in terms of the Gelfand states.     

Relativistic Gaussian basis sets for the atoms hydrogen to neon

Gaussian basis sets for use in relativistic molecular calculations are developed for atoms and ions with one to ten electrons. A relativistic radial wavefunction coupled to an angular function of l-symmetry is expanded into a linear combination of spherical Gaussians of the form r ^l exp (–r ²). One set of basis functions is used for all large and small components of the same angular symmetry. The expansion coefficients and the orbital exponents have been determined by minimizing the integral over the weighted square of the deviation between the Dirac or Dirac-Fock radial wavefunctions and their analytical approximations. The basis sets calculated with a weighting function inversely proportional to the radial distance are found to have numerical constants very similar to those of their energy-optimized non-relativistic counterparts. Atomic sets are formed by combining l-subsets. The results of relativistic and non-relativistic calculations based on these sets are analyzed with respect to different criteria, e.g. their ability to reproduce the relativistic total energy contribution and the spin-orbit splitting. Contraction schemes are proposed.Dedicated to Prof. Dr. A. Neckel on occasion of his 60th birthday