The coupling symbols and algebrae of subducible,octahedrally projected ligand field eigenvectors

The 3 symbols required for the application of the Wigner-Eckart theorem to strong ligand field matrix elements are derived for complex basis functions quantized on the C ₄ ^Z , C ₃ ^XYZ , C ₂ ^Z and C ₂ ^XY axes of an octahedron. This scheme provides a standardized analysis technique for the matrix elements of subgroups in each of the four physically significant chains of the double group O _h ^* . This standardization yields the minimum necessary number of ligand field parameters in any subgroup and makes possible the direct comparability of equivalent parameters in different symmetries. A unique numerical labelling for both representations and complex components on each axis provides both a simple component selection rule algebra and numerical phase factors governing permutation and conjugation of the 3 symbols.     

The Coordination Number of Lu(III) in a Mixed System of Methanol and Water

The stability constants (_1(F)) of the monofluoro complex of Lu(III) and those (_1(Cl)) of the monochloride solvent-shared ion-pair of Lu(III) have been determined in mixed solvents of methanol and water at 0.10 and 1.00 mol·dm^–3 ionic strengths, respectively. The variation in ln_1(F) with an increase in the mole fraction of methanol (X _s) in the mixed solvent system showed an acute-angled convex inflection point at X _s 0.12, an acute-angled concave inflection point at X _s 0.22, and another acute-angled convex inflection point at X _s 0.27. It was concluded that the first and the second convex inflection points denoted the CN of Lu³⁺ from CN = 8 to a mixture of CN = 8 and 7 and from CN = 8 and 7 to a mixture containing CN = 6, respectively. The concave point is the starting point of a change in the CN of Lu(III) in LuF²⁺ from CN = 8 to a mixture of CN = 8 and 7. The values of two inflection points of the CN around Lu³⁺ are consistent with the inflection points of the variation in the values of ln_1(Cl) versus the dielectric constant of the mixed solvent.     

Analytic matrix elements of the dipole moment and Herman-Wallis coefficients

Perturbation theory proves to be a powerful approach to obtain in analytic form both vibration-rotational energies and matrix elements of the dipole moment of diatomic molecules in terms of the expansion parameter = 2B _e/gw_e,B _e and _e being, respectively, the equilibrium rotational and harmonic vibrational spectral parameters. A systematic and efficient algorithm has been developed to execute such calculations with sufficient accuracy for most physical applications when the potential-energy function is accurately represented in the Dunham form. The method also provides analytic expressions of the Herman-Wallis coefficientsC _v ^v andD _v ^v for the vibration-rotational overtone bandsv ¹v for diatomic molecules in¹ electronic states.     

Statistical branching of heterochains

A general theory for the statistical branching of heterochains is proposed on the basis of the random sampling technique. Consider a polymer mixture that consists of N types of chains whose weight fractions are w_i (i = 1, 2, …, N), and number- and weight-average chain lengths are P̄_np,i and P̄_wp,i, respectively. Suppose the transition probability that a branch point on a chain of type i is connected to a chain end of a type j chain is given by p_ij. When the branching density of chains of type i is ρ_i, the weight-average chain length is given by $\bar P_w = W\sum\nolimits_{m = 0}^\infty T ^m \sum\nolimits_{n = 0}^\infty {SU} ^n 1$, where S is the diagonal matrix whose elements are $S_{ii} = \bar P_{wp,i}$, 1 is the column vector whose elements are all unity, U is the transition matrix whose elements are given by $u_{ij} = \rho _i p_{ij} P_{np,j} ,T$ is another transition matrix whose elements are given by t_ij = (w_j/w_i)U_ji, and W is the row vector whose elements are w_i. Simpler expressions of P̄_w are presented for binary systems. In addition to the multicomponent systems, the present equation could also be used such as for free-radical polymerization with long-chain branching, by considering primary chains formed at different times as different types of polymer chains. For the prediction of the full molecular weight distribution, a Monte Carlo simulation method is used to illustrate the resulting distribution profiles.     

Empirical equations of state for adsorption layer: Alkylthiopolyoxyethylene glycols

Surface tension isotherms ofn-alkylthiopolyoxyerhylene glycols:n-C _x H_2x+1S(CH₂CH₂O) _y H, wherex=5 to 8,y=3 or 4, were approximated with orthogonal polynomials to get good quality values of surface pressure (II) and molar area of the adsorbed layer (). The modified Volmer (*(–₀)=Z*R*T) van der Waals and virial equations of state were used to correlate and in terms of real two-dimensional gas. The combination of Volmer and van der Waals equations of state made it possible to determine the interaction energy, , which was prescribed to cohesion of hydrophobic chains in the adsorption layer. The value of for the amphiphiles in question was in the range 0.97–1.91R*T and the average contribution per methylene group was ca. 0.21R*T.The Lennard-Jones potentials calculated from second virial coefficient were of the same range as , but no clear relation was found between their values and number of structure elements of the alkylthiopolyoxyethlene glycols.Presented during 7th International Conference: Surface and Colloid Science, July 7–13, 1991, Compiegne, France     

Trace elements in the water cycle of the Šalek valley,Slovenia, using INAA

Three watersheds were studied by sampling bulk precipitation deposition, seepage water at 50 cm soil depth and spring water. As the main analytical method for determination of trace elements and heavy metals in water samples, thek ₀-based method of INAA was used. The results showed an increased content and concentration range of trace elements in precipitation, soil water and spring water in the vicinity of the otanj Thermal Power Plant. We demonstrated that thek ₀-based method of INAA as a multielement nondestructive technique is a highly suitable approach to determining some toxic trace elements in environmental studies of the water cycle.     

Dependence of muonic x-ray intensity spectra and bond ionicities

It is shown that the muonicK /K intensity ratios in oxygen of oxides of third-row elements are correlated to the bond ionicities. A similar correlation is found in chlorine of corresponding compounds. These correlations confirm the dominant role played by the outer electrons of the bond partners in the coulomb capture mechanism of negative muons.     

Some comments on the parameter choices for S.C.M.O. calculations on five membered heteroatomic π-electron systems

It has been reported [1] that the usual semi-empirical S.C.M.O. treatments of heteroatomic -electron systems yield unexpected dependencies of charge distribution on the _core terms involving the heteroatoms. Other workers [2] have claimed that the most usual choices of parameters yield chemically unreasonable charge densities in furan and pyrrole. It is shown here that the first effect is related to the relative magnitudes of the diagonal elements of the Fock matrix and that the second depends very strongly on the nonnearest neighbor F matrix elements but can be overcome with certain parameter choices by the inclusion of non-nearest neighbor _core terms in the calculations.

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Zusammenfassung Bei den üblichen semiempirischen SCMO-Rechnungen an Heteroaromaten ergeben sich unerwartete Abhängigkeiten der Ladungsverteilung von den _Rumpf-Termen [1]. Die gebräuchlichste Parameterwahl führt zu chemisch unverständlichen Ladungsverteilungen bei Furan und Pyrrol [2]. Es wird gezeigt, daß der erste Effekt mit der relativen Größe der Diagonalelemente der Fockmatrix zusammenhängt; der zweite Effekt hängt sehr stark von den Nichtnachbarelementen ab: Bei Einschluß von Nichtnachbar- _Rumpf-Termen und entsprechender Parameterwahl erhält man befriedigende Ladungsverteilungen.

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Résumé Il a été indiqué [1] que le traitement semi-empirique S.C.M.O. habituel des systèmes d'électrons dans les hétérocycles comporte des relations in attendues entre la distribution de charge et les termes _coeur impliquant les hétéroatomes. D'autres chercheurs [2] ont affirmé que les choix les plus communs de paramètres fournissent des densités de charge dans le furane et le pyrrole déraisonnables sur le plan chimique. On montre ici que le premier effet est lié aux grandeurs relatives des éléments diagonaux de la matrice de Fock et que le second effet dépend fortement des éléments non immédiatement voisins de cette matrice, mais peut être annulé avec certains choix de paramètres par introduction des termes non voisins _coeur dans les calculs.

     

Predicting 1-octanol-water partition coefficient by high-performance liquid chromatography gradient elution

The method to predict 1-octanol-water partition coefficients (K _ow) from capacity factors (k) obtained by Reversed-Phase High-Performance Liquid Chromatography (RP-HPLC) has been extended to use gradient elution rather than isocratic elution. The mobile phase has been changed either linearly or exponentially with time. The initial composition of the mobile phase and its rate of change affected the log K _ow versus log k relation. This relation was exponential in linear gradient experiments. For non-linear gradient elution in which the water fraction of the mobile phase decreased exponentially from 100% to approach 0% asymptotically, a physically-based equation describing the dependence of log K _ow on log k has been derived. Without any preliminary estimation, RP-HPLC gradient elution allows a precise prediction of log K _ow over a range of nearly six orders of magnitude.     

In-beam determination of k0 factors for short-lived nuclides

The prompt -activation analysis (PGAA) facility at Budapest Research Reactor offers a unique possibility to perform in-beam measurements. Several k ₀ factors for decay -lines of short-lived nuclides have been determined accurately by means of in-beam activation. The present values compare well with literature data. New k ₀ factors are proposed for ^24mNa and ^60mCo.     

Direct evaluation of one-electron properties in coupled cluster methods

Summary An analysis of a method for approximate calculations of expectation values for one-electron operators from available coupled cluster amplitudes is presented and illustrated numerically for the polarizability of the Be atom. The one-particle density matrix resulting from the present approach is accurate through the fourth order in the electron correlation perturbation. It has been found that, in order to obtain quantitative agreement between the energy derivative results and the approximate expectation value formalism, the third orderT ₁ T ₂⁽⁰⁾ wave function term must be included into the calculation of the one-particle density matrix. The present method is also considered as a promising tool for calculations of higher-order atomic and molecular properties from high level correlated wave functions.     

The polyiodide salts: Pyridinium pentaiodide;β-naphthylammonium pentaiodide; andN-methyl-γ-picolinium heptaiodide. Structures with channel inclusion features

The crystal structures of three polyiodode salts are reported (pyridinium pentaiodide, monoclinic,P2 ₁/m,a=9.221(5),b=12.918(5),c=6.026(4) Å, =103.60(7)^o,Z=2,R _F=0.087 for 1187 intensities; -naphthyl-ammonium pentaiodide, triclinic,173-1,a=10.390(5),b=9.502(5),c=4.462(3) Å, =99.19(7), =90.40(7),=108.49(8)^o,Z=2,R _F=0.059 for 1319 intensities;N-methyl--picolinium heptaiodide, monoclinic,C2/c,a=19.315(7),b=12.714(5),c=8.442(4) Å, =107.26(7)^o,Z=4,R _F=0.065 for 1336 intensities). All three structures can be described as having channel inclusion features; the cations are contained in channels in polyiodide frameworks based on different arrangements of I₂ molecules and I ₃ ^– anions. This structural type is the converse of the more widespread kind where polyiodide anions are contained in an organic matrix (e.g., cyclodextrin polyiodides). Supplementary Data relating to this article are deposited with the British Library as Supplementary Publication No. SUP 82016 (30 pages).Part X of Crystal Structures of Polyiodide Salts and Complexes. For Part IX, see F. H. Herbstein, G. M. Reisner, and W. Schwotzer,Acta Crystallogr. C, accepted for publication, 1984     

Entropy in dissimilarity and chirality measures

Within the prospect of quantifying the geometrical dissimilarity of molecular models on the basis of a thermodynamical formalism, the algebra of stereogenic pairing equilibria is reviewed and applied to molecular geometry: developing Rassat's proposition, an interaction energy of two figures F and F is taken as proportional tod _H ^Emphasis>/2 (F, F), whered _H denotes the Hausdorff distance. IfG is a group of rotations in E _n the geometrical version of the general equation (E) of the chemical algebra defines a distance extensionD _p(F,F) ofd _H(F,F), which is independent of the orientations of F and F, and where the coefficientp is interpreted as the reciprocal of a temperature-like parameter:p 1/T. At K (p = ), no formal entropy contributes to the definition of the uniform distanceD . At K (p = 0), the discrimination between homo- and hetero-pairing of figures by the harmonic distance Do is averaged over orientation states. Temperature-dependent chirality measuresc _p are derived fromD _p, andc is analogous to Mislow's chirality measure. If T and oT are normalized enantiomorphic triangles with coincident centroids inE ₂,c _p(T) =D _p (T, T) is calculated forp = 0 andp = , and discussed for 0 <p < . Finally, the Hausdorff interaction model is putatively related to energy profiles versus dihedral angle inmeso- anddl-molecules.     

Molecular Interactions in Formamide + Isomeric Butanols: An Ultrasonic and Volumetric Study

Densities, , ultrasonic speeds, u and viscosities, of the binary mixtures of formamide (FA) with 1-butanol, 2-methyl-1-propanol, and 2-methyl-2-propanol, including those of pure liquids, were measured over the whole composition range at 35°C. Using the experimental values of , u and , the deviations in isentropic compressibility, _s , excess volume, V ^E, viscosity, , and excess Gibbs energy of activation of viscous flow, G* ^E , were calculated from the linear dependence of these parameters on composition of mixtures. The apparent molar isentropic compressibility, K _,2 and apparent molar volume, V ^,2 of alcohols in FA were also calculated. The variations of these parameters with composition are discussed from the point of view of intermolecular interactions in these mixtures. The V ^E data have also been analyzed using Prigogine–Flory–Patterson theory. An analysis of each of the three contributions, viz., interactional, free volume, and P* effect to V ^E shows that P*, the internal pressure parameter of the theory, plays a dominant role in deciding the sign and magnitude of V ^E.     

CH Acidity of Di- and Trisubstituted Methanes

It is shown that the CH acidity of di- and trisubstituted methanes can be studied using the ap- proaches consisting in singling out the contributions of physically significant factors from the overall acidification mechanisms in the gas phase and in solution. This procedure implies formal decomposition of the calculated gas-phase deprotonation energy Edeprot into the following terms: electrostatic energy of proton detachment from the acid CH bond, with the state of the remainder of the molecule absolutely unperturbed (E ₁); electronic relaxation energy of the resulting molecular residue and formation of a virtual carbanion therefrom (E ₂); the Edeprot component due to displacement of the atomic nuclei on changing from the virtual to real carbanion E ₃. Relationships between the energy components E ₁, E ₂, E ₃, and the commonly used characteristics of the molecular structure were investigated. The parameter E ₁ is selectively sensitive to the inductive effect of the substituent. Imperfect correlation between E ₁ and the sum of the _I constants can be due to the fact that the contributions to the _I constants from the effective charge on the hydrogen atom of the CH bond being deprotonated and from the polarizabilities of the substituents are not taken into account. In contrast to monosubstituted methanes, in di- and trisubstituted methanes there is no correlation between the E ₂ component and the ¹ J(^13CH) constants. The linear dependence linking the sums of the components E ₁ + E ₂ and the pyramidal angle in the carbanions is responsible for the relaxation nature of the effects associated with the E ₂ + E ₃ sum. Comparison of the data obtained with the calculated patterns of redistribution of the effective charges on atoms accompanying conversion of CH acids to carbanions enabled elucidation of the relative contribution of each of the components, E ₂ and E ₃, to the deprotonation energy of selected groups of substituted methanes. The previously developed technique of separating the energy of protolytic equilibrium in the gas from that of solvation processes in solution enabled assessment of the contributions from electrostatic solvation to pK _a in DMSO. The same technique of singling out the solvation component due to intermolecular hydrogen bonds from pK _a(H₂O) was used in studying the acid-base equilibria for substituted methanes in aqueous solution. It was shown that the solvation effects manifested in the liquid-phase CH acidity can be modeled by the effects revealed for analogous hydrogen bonds of ion-molecule complexes in the gas phase. The relationships between the strength of hydrogen bonds and the CH-acidic properties of compounds in the gas phase and liquid water are similar.     

Vibrational isotope effect of linear and planar molecules: Deuterated bromoethenes

Linear (planar) molecules A and B which are identical except for isotopic substitutions at the atomic sites are considered. Stretching (bending, out-of-plane) frequencies _k and normal modes _k of the isotopically perturbed molecule B are expressed in terms of stretching (bending, out-of-plane) frequencies _i and the corresponding normal modes _i of the unperturbed molecule A. Complete specification of the unperturbed normal modes is not required. All that is needed are stretching (bending, out-of-plane) amplitudes | _i of the normal modes _i at those sites that are affected by isotopic substitution. The rule which interlaces frequencies _k of molecule A with frequencies _i of molecule B is derived. Given two isotopic molecules A and B that differ by a single isotopic substitution at site , the inversion relation is derived. This relation expresses unperturbed stretching (bending, out-of-plane) amplitudes at the site in terms of stretching (bending, out-of-plane) frequencies of molecules A and B . As an example, out-of-plane vibrations of deuterated bromoethene were considered. In the simplest method 12 out-of-plane frequencies of four polydeuterated bromoethenes were calculated from 12 out-of-plane frequencies of bromoethene and three monodeuterated bromoethenes. Standard deviation of thus calculated frequencies from experimental frequencies is =2.74 cm^–1. In another method, 15 out-of-plane frequencies of four polydeuterated bromoethenes and selected monodeuterated bromoethene are calculated from 9 out-of-plane frequencies of bromoethene and the remaining two monodeuterated bromoethenes. Depending on which monodeuterated bromoethene is selected (1-, cis- or trans-), standard deviation of thus obtained frequencies from experimental frequencies is 1=2.84 cm^–1, c=2.96 cm^–1 and t=2.72 cm^–1.     

The viscosity of aqueous solutions of alkali and tetraalkylammonium halides at 25°C

The viscosities of most alkali and tetraalkylammonium halides have been measured in water at 25°C. The relative viscosities can be fitted, up to 1M, with the relation _r =1+A c^1/2+B c+D ². TheA term depends on long-range coulombic forces, andB is a function of the size and hydration of the solute. When combined with partial-molal-volume data, the difference B –0.0025V° is mostly a measure of the solute-solvent interactions. IonicB are obtained if the tetraethylammonium ion is assumed to obey Einstein's law. TheD parameter depends on higher terms of the long-range coulombic forces, on higher terms of the hydrodynamic effect, and on structural solute-solute interactions. As such, it cannot be interpreted unambiguously.     

Relativistic bond lengthening of UO 2 2+ and UO2

Summary Relativistic calculations on UO₂ [1] have shown that relativity leads to substantial bondlengthening in this compound, in contrast to the bond contraction found almost exclusively for other compounds. The bond lengthening isnot caused by the relativistic expansion of the 5f valence AO of U, which is the primary bond forming orbital on U in UO₂. The origin of the bond lengthening can be traced back to the semi-core resp. subvalence character of the U 6p AO. The valence character of 6p shows up in an increasing depopulation of the 6p upon bond shortening, and hence loss of mass-velocity stabilization. The core character of 6p shows up in large off-diagonal mass-velocity matrix elements 5p|h _MV|6p which are shown to have an overall bond lengthening effect. The larger expansion in UO₂ than in UO ₂ ²⁺ is due to destabilization of U levels in UO₂, caused by repulsion of the two additional 5f electrons.The present analysis corroborates the picture of relativistic bond length effects of Ref. [2].     

Ligand-structure effect on electrochemical reduction of copper(II) complexes

Summary Half-wave potentials for a one-electron reduction of copper(II) complexes containing polydentate ligands can be calculated using the equationE _1/2=E ⁰(Cu^2+/+)+ _{i j} E _i where E _i are contributions related to the electronic and steric properties of the ligands. The values of 18 E _i contributions are presented and explained, and the effect of the solvents on the half-wave potentials is exemplified.Dedicated to Prof. Dr. Viktor Gutmann to his 70th birthday     

Characterization of polymer networks by measurements of the freezing point depression

Based on the phenomenon of freezing point depression of a solvent byT, experimental evidence is presented to show that the distance between the junction points can be calculated fromT. Direct measurements of the temperature-time-curve of the cooling network and the Differential Scanning Calorimetry offer the determination ofT. Except the mean distances ¯d _c in dependence on cross-linking density, swelling degree, and other network parameters, the distribution of the distance between the junction pointsH(d_c) can be determined, which allows conclusions on the course of cross-linking reaction. This paper attempts to give experimental evidence of influences of the breadth ofH(d_c) on application-relevant properties.