Quotients of group algebrae in the calculation of intermediate ligand field matrix elements

The structure of the classes of symmetry elements excluded during the subduction of the representations of SU(2) onto the finite group 0* is shown to quantitatively define the relationship of the coupling algebrae of these two groups. This relationship is formalized as a quotient algebra. This quotient algebra is realized as 3-like symbols which exist whether or not the quotient can be defined as a group. These symbols distribute the value of a reduced matrix element of SU(2) onto the subduced reduced matrix elements of O* and are termed Partition Coefficients. Since the structure of the excluded symmetry classes is independent of the quantization of O*, these Partition Coefficients can be used to define the values of the matrix elements of O* without reference to the form of its basis set. Thus, the choice of physical interpretation of the ligand field is unimportant. The strong field, weak field, Russell-Saunders and j-j coupling models are all unified in terms of the Partition Coefficients and the 3 symbols which are appropriate to the quantization.     

The non-additive ligand field

The semi-empirical ligand field is a perturbation operator whose consequences are taken to first order using a basis set ofl functions. Since the basis spans an irreducible representation of the 3-dimensional rotation-inversion groupR _3i it is useful to express the operator as a sum of components of irreducible tensor operators with respect to this group. IfR _3i is reduced with respect to the molecular subgroup the electronic factor of each term in the sum must be totally symmetrical within this group. This choice of operator leads to thecrystal field parameterization without implying an electrostatic model. Alternatively a shift operator withinl space may be chosen as the essential part of the perturbation operator. This leads to theligand field parameterization. Between the two parameterizations there exists a one to one relationship, whose coefficients are proportional to 3l symbols. This relationship is given together with a brief discussion of the reasons for the proposed parameterizations.     

Clebsch–Gordan coefficients for chains of groups of interest in quantum chemistry. III. The point symmetry groups

This work amalgamates some basic elements defined in the first paper of this series and in related papers with the theory of coupling coefficients for an arbitrary group with the view of generating the Clebsch–Gordan coefficients and V symbols for point symmetry groups. The connection between Clebsch–Gordan coefficients and V symbols is established for an arbitrary group in a form that reduces to the one known for the chain SU(2) ? U(1). The Clebsch–Gordan coefficients and V symbols of any point symmetry group G are shown to be obtainable from Clebsch–Gordan coefficients and \documentclass{article}\pagestyle{empty}\begin{document}$ {\bar f} $\end{document} symbols of the chain SU(2) ? G through the resolving of a system of nonlinear equations.     

The effects of titanium or chromium doping on the crystal structure of V2O3

The crystal structures of five samples of (Ti_xV_1?x)₂O₃ (0.011 ≤ x ≤ 0.077) and seven samples of (Cr_xV_1?x)₂O₃ (both metallic and insulating phases, 0 ≤ x ≤ 0.05) were determined from X-ray diffraction data collected from single crystals. These compounds are isomorphous with α-alumina. The cell dimensions change such that the a axes increase and the c axes decrease with increasing Ti or Cr. In the CrV₂O₃ system, from 0 to 1.25% Cr doping, changes in structure parallel those observed in the TiV₂O₃ system. These changes are consistent with a slight weakening of the bonding metal-metal interactions in the basal plane, leading to an increase in the metal-metal distances coupled with changes which maintain constant metal-oxygen distances. A discontinuity appears at about 1.25% Cr as the transition from metal to insulating behavior occurs with increasing Cr content. No change in crystal symmetry accompanies this transformation. It appears that the metal-metal bonding interactions are retained even in the insulating phase of Cr-doped V₂O₃. A comparison of the structural variation in the Cr- and Ti-doped systems suggests that the change from metallic to insulating behavior cannot be a structure effect. These changes are, however, consistent with the band model proposed by others for these systems.     

Metal-insulator transition in VnO2n−1

Research on phase relationships and structure studies by electron diffraction confirm V_nO_2n?1 (n = 3–9) phases between V₂O₃ and VO₂. Metal-insulator phase transitions have been found in all phases but V₃O₅ and V₇O₁₃. Electrical, magnetic and thermodynamic properties associated with the transitions are reported for sintered samples or for single crystals prepared by a vapor-transport method. The results are collated and reviewed in summarized form.     

Symmetrization of eigenfunctions of angular momentum in point groups

The eigenfunctions |jm〉 of angular momentum can combine linearly to make basis functions of irreducible representations of point groups. We surmount the projection operator and find a new method to calculate the combination coefficients. It is proven that these coefficients are components of eigenvectors of some hermitian matrices, and that for all pure rotation point groups, the coefficients can be made real numbers by properly choosing the azimuth angles of symmetry elements of point groups in the coordinate system. We apply the coupling theory of angular momentum to obtain the general formulas of the basis functions of point groups. By use of our formulas, we have calculated the basis functions with half‐integers j from 1/2 to 13/2 of double‐valued irreducible representations for the icosahedral group. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 286–302, 2001     

Time-dependent perturbation: a recursive generation of the transition amplitudes between bound states

We present a new expansion of the solution to the time-dependent Schrödinger equation it ∂U/∂t = {H₀ + V(t)}U. A complete set of eigenvectors of H₀ spanning the Hilbert space in which H₀ and V operate is partitioned in two subsets. Transition amplitudes from the first subspace to the second one are calculated by building an adequate series of intermediate representations with respect to the couplings which produce the transition from the model space into its orthogonal complement. These expansions yield a coupling matrix series V⁽ⁿ⁾ for which an iterative solution V⁽ⁿ⁾ → V⁽ⁿ⁺¹⁾ is derived. This solution leads to a recursive numerical treatment of the calculation of transition amplitudes. A simple example, concerning a harmonic oscillator under an intense laser field, is considered using a Fourier analysis of the perturbation.     

Calculation of j and jm symbols for arbitrary compact groups. I. Methodology

A systematic method is established for computing the coupling coefficients associated with arbitrary compact groups using only the general properties of the coefficients and the specific character theory of the relevant groups. The basic character theory is first outlined. Then a primitive set of 6j symbols is defined and their computation sketched with careful attention being given to matters of phase fixing. A recursive method is developed for calculating arbitrary 6j symbols using the primitive set. Finally, a primitive set of 3jm factors is set up and then the general 3jm factors computed recursively.     

Phase-fixed double-group 3-Γ symbols. V. 3-Γ symbols and coupling coefficients for all the octahedral double group-subgroup hierarchies

For all the subgroup hierarchies descending from the octahedral double group O ^*, we have obtained sets of 3- symbols and discuss here their properties. We have entirely real sets of 3- symbols for the tetrahedral and tetragonal hierarchies as well as for O ^* C ₃ ^* . For the latter hierarchy and the tetragonal ones, formalisms almost as powerful as the classical one for the rotation group may be established. We also discuss results obtained for cases with strict adaption to D ₃ ^* where it is now known that non-real 3- symbols are unavoidable.The 3- symbols are phase-fixed by the specification of basis functions (or, equivalently, subduction coefficients) generating them.The significance of the concept of associated representations of O* is discussed. The problems raised by the two multiplicity triples UT₁U and UT₂U in O* are given particular attention.     

Use of the Tóth and Unilan Equations for Chromatographic Testing of the Adsorption Properties of Active Carbon

The adsorption and chromatographic properties of active carbon samples obtained by adsorptive/desorptive modification have been studied by inverse gas chromatography to assess their suitability for adsorption of analytes from gas and liquid phases. This paper may help in the choice of an appropriate method for characterizing the surface of an active carbon with regard to particular analytes. Therefore, the specific, V _{g (T )}, and net, V _{S ( T )}, retention volumes were determined from the retention times of the peak maxima and centres of gravity of the peaks. A mathematical link between retention volume data and the monolayer capacity, a _m , was derived by use of a hyperbolic-type equation which enables the values of the terms of the Tóth and Unilan equations to be found. Adsorption second virial coefficients, B _{2 S} , calculated from the values of the terms of the Tóth and Unilan equations were used as a sensitive test of the accuracy of the calculated data. There were no significant differences between the isosteric enthalpies obtained by use of virial coefficients or retention volumes. It was shown experimentally there is substantial parallelism between the magnitudes of the differential isosteric enthalpy and the differential entropy for propane chromatographed on the adsorption systems tested. The cause of non-linear behaviour of plots of ln V _{g ( T )} against T ^?1 for different dynamic adsorption systems is discussed.     

Calculations of Jahn-Teller coupling constants in the weak-field basis via the angular overlap model: a simple operator equivalent technique

Within the framework of the angular overlap model the matrix elements of the linear Jahn-Teller operator may easily be calculated in the weak-field basis in terms of simple operator equivalents. The method is applied to the calculation of the 〈M_J|?V/?Q|M_j〉 matrix elements for the |LSJMj〉 ground states of t^x species (x = 1) ? 13) in )^* symmetry.     

Phase-fixed double-group 3-Γ symbols. VI. Real 3-Γ symbols and coupling coefficients for the group hierarchy I *⊃ C 5 *

It is demonstrated that for the group-subgroup hierarchy I ^* C ₅ ^* , one may choose standard irreducible matrix representations and corresponding all-real sets of 3- symbols which obey a formalism just as elegant as the classical one for the 3-j symbols of the rotation double group. The 3- symbols are phase-fixed by the specification of basis functions (or, equivalently, subduction coefficients) generating them and based on functions first given by McLellan.Other icosahedral double-group hierarchies are also briefly discussed.     

A visual approach to stationary phase selectivity classification based on the Snyder–Dolan Hydrophobic-Subtraction Model

A novel type of stationary phase selectivity classification “triangle” has been developed based on the Snyder–Dolan (S–D) Hydrophobic-Subtraction Model, wherein the apices of a set of four triangles represent the relative contributions of steric hindrance (χ_S), hydrogen-bonding acidity (χ_A), hydrogen-bonding basicity (χ_B), cation-exchange capacity (χ_C) to selectivity. We found that “effective selectivity” of a stationary phase is mathematically given by the ratio of system dependent interaction coefficients but not their absolute values. Thus by normalizing the S*, A, B and C terms of the S–D model by H, we were able to obtain four parameters which fully define the chromatographic selectivity of the stationary phases. By examining the parameters in groups of three, we can represent all the result in a set of four “selectivity triangles”. The distinctive feature of this approach compared to the S–D phase classification scheme is that it allows the visualization of column selectivity by plotting three-dimensional data in a two-dimensional space. Moreover, it very clearly shows that the RPLC columns thus far characterized cover only a small fraction of separation selectivity space leaving a great deal of room for researchers to develop novel RPC materials. Various applications of these “selectivity triangles” will be discussed in this paper.     

Characterization of 1,3-alternate calix[4]arene-silica bonded stationary phases and their comparison to selected commercial columns by using principal component analysis

Twelve calix[4]arene stationary phases in 1,3-alternate conformation, synthesized in the authors’ laboratory, were characterized in terms of their surface coverage, hydrophobic selectivity, aromatic selectivity, shape selectivity, hydrogen bonding capacity and ion-exchange capacity. The set of tests commonly used for evaluation of commercially available stationary phases was applied to assess fundamental chromatographic properties of the calixarene phases. The new calixarene phases were compared to each other, to Caltrex and LiChrosorb C-18 columns. Principal component analysis has been used to provide comparison between 1,3-alternate calix[4]arene phases and commercially available phenyl, fluorophenyl and fluoroalkyl columns.     

Potential functions of internal rotation in gauche-isomers of n-mononitroalkanes

Internal rotation in gauche-mononitroalkanes CH₃(CH₂) _n NO₂, n ≤ 7 is studied by means B3LYP/6-311++g(3df,3pd) and MP2/6-311++g(3df,3pd). Thirty six potential functions (V(φ)) are found for all rotations. It is shown that starting from a particular n value, the potential functions associated with certain molecular fragments change slightly during the lengthening of a hydrocarbon chain. Similar dependences V(φ) are presented in the form of generalized functions with averaged coefficients (V _av(φ)) that are transferable and recommended for use in molecular simulations. The rotation of tops without symmetry elements is considered.     

Molar excess volumes and molar excess enthalpies of methylenebromide + aromatic hydrocarbon mixtures

Molar excess volumes V^e and molar excess enthalpies H^e of binary methylenebromide (i) +benzene. +toluene, and + o^?, + m^? and + p-xylene (j) mixtures have been determined at 298.15 and 308.15 K. The data have been analysed in terms of recent approaches for solutions of nonelectrolytes, and the results suggest that these mixtures are characterised by specific interactions between the components. Self-volume interaction coefficients V_iiV_jj have also been evaluated.     

Studies of xLiFePO4·yLi3V2(PO4)3/C composite cathode materials with high tap density and high performance prepared by sol spray drying method

The xLiFePO₄·yLi₃V₂(PO₄)₃/C cathode materials are synthesized by a sol spray drying method. X-ray diffraction results reveal that the xLiFePO₄·yLi₃V₂(PO₄)₃/C (x,y?≠?0) composites are composed of LiFePO₄ and Li₃V₂(PO₄)₃ phases, and no impurities are detected. The samples show spherical particles with the size of 0.5–5 μm, and the tap densities of all the samples are higher than 1.5 g cm^?3. Electrochemical tests show that the xLiFePO₄·Li₃V₂(PO₄)₃/C (x,y?≠?0) composites exhibit much better performance than the single LiFePO₄/C or Li₃V₂(PO₄)₃/C. Among all the samples, 3LiFePO₄·Li₃V₂(PO₄)₃/C possesses the best comprehensive performance in terms of the discharge capacity, average working voltage, and rate capability. At 1, 5, and 10 C rates, the sample shows first discharge capacities of 152.0, 134.3, and 116.8 mAh g^?1 and capacity retentions of 99.2, 98.2, and 97.7 % after 100 cycles, respectively. The excellent electrochemical performance of micron-sized xLiFePO₄·Li₃V₂(PO₄)₃/C (x,y?≠?0) powders is owing to the homogeneous mixing of reactants at a molecular level by sol spray drying, the incorporation of fast ion conductor Li₃V₂(PO₄)₃, and the mutual doping in LiFePO₄ and Li₃V₂(PO₄)₃.     

Solvatochromic relationship: Prediction of distribution of ionic solutes in aqueous two-phase systems

Partition ratios of several ionic compounds in 20 different polymer/polymer aqueous two-phase systems (ATPS) containing 0.15 M NaCl in 0.01 M phosphate buffer, pH 7.4, were determined. The differences between the electrostatic properties of the phases in all the ATPS were estimated from partitioning of the homologous series of dinitrophenylated-amino acids. Also the solvatochromic solvent parameters characterizing the solvent dipolarity/polarizability (π*), solvent hydrogen-bond donor acidity (α), and solvent hydrogen-bond acceptor basicity (β) of aqueous media were measured in the coexisting phases of the ATPS. The solute-specific coefficients for the compounds examined were determined by the multiple linear regression analysis using the modified linear solvation energy relationship equation. The minimal number of ATPS necessary for determination of the coefficients was established and 10 ATPS were selected as a reference ATPS set. The solute-specific coefficients values obtained with this reference set of ATPS were used to predict the partition ratios for the compounds in 10 ATPS not included in the reference set. The predicted partition ratios values were compared to those determined experimentally and found to be in good agreement. It is concluded that the presented model of solute–solvent interactions as the driving force for solute partitioning in polymer/polymer ATPS describes experimental observations with 90–95% accuracy.