Slater's nonlocal exchange potential and beyond

The local density approximation (LDA) to the exchange potential V_x( r ), namely the ρ^1/3 electron gas form, was already transcended in Slater's 1951 paper. Here, using Dirac's 1930 form for the exchange energy density ? _x( r ), the Slater (Sl) nonlocal exchange potential V( r ) is defined by 2? _x( r )/ρ( r ). In spherical atomic ions, say the Be or Ne‐like series, this form V( r ) already has the correct behavior in both r → 0 and r → ∞ limits when known properties of the exchange energy density ? _x( r ) and the ground‐state electron density ρ( r ) are invoked. As examples, some emphasis will first be given to the use of the so‐called 1/Z expansion in such spherical atomic ions, for which analytic results can be obtained for both ? _x( r ) and ρ( r ) as the atomic number Z becomes large. The usefulness of the 1/Z expansion is directly demonstrated for the U atomic ion with 18 electrons by comparison with the optimized effective potential prediction. A rather general integral equation for the exchange potential is then proposed. Finally, without appeal to large Z, two‐level systems are considered, with specific reference to the Be atom and to the LiH molecule. In all cases treated, the Slater potential V( r ) is a valuable starting point, even though it needs appreciable quantitative corrections reflecting directly atomic shell structure. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

On the inner and outer radial density functions

Starting from the two-electron radial density D ₂(r ₁,r ₂), a generalized partitioning of the one-electron radial density function D(r) into two component densities D _a (r) and D _b (r) is discussed for many-electron systems. The literature partitioning (Koga and Matsuyama Theor Chem Acc 115:59, 2006) of D(r) into the inner D _<(r) and outer D _>(r) radial densities is shown to minimize the average variance of the two component density functions D _a (r) and D _b (r). It is also found that the average radial separation halved, , constitutes a lower bound to the standard deviation σ of D(r).     

Density functionals in the presence of magnetic field

In this article, density functionals for Coulomb systems subjected to electric and magnetic fields are developed. The density functionals depend on the particle density ρ and paramagnetic current density j^p. This approach is motivated by an adapted version of the Vignale and Rasolt formulation of current density functional theory, which establishes a one‐to‐one correspondence between the nondegenerate ground‐state and the particle and paramagnetic current density. Definition of N‐representable density pairs (ρ,j^p) is given and it is proven that the set of v‐representable densities constitutes a proper subset of the set of N‐representable densities. For a Levy–Lieb‐type functional Q(ρ,j^p), it is demonstrated that (i) it is a proper extension of the universal Hohenberg–Kohn functional to N‐representable densities, (ii) there exists a wavefunction ψ₀ such that , where H₀ is the Hamiltonian without external potential terms, and (iii) it is not convex. Furthermore, a convex and universal functional F(ρ,j^p) is studied and proven to be equal the convex envelope of Q(ρ,j^p). For both Q and F, we give upper and lower bounds. © 2014 Wiley Periodicals, Inc.     

The Formation and Stability of the [FePy3Cl3]· Py Clathrate in the Pyridine–Iron(III) Chloride System: Phase Diagram and Solid–Gas Equilibria Study

The phase diagram of the pyridine–iron(III) chloride system has been studied for the 223–423 K temperature and 0–56 mass-% concentration ranges using differential thermal analysis (DTA) and solubility techniques. A solid with the highest pyridine content formed in the system was found to be an already known clathrate compound, [FePy₃Cl₃]·Py. The clathrate melts incongruently at 346.9 ± 0.3 K with the destruction of the host complex: [FePy₃Cl₃]·Py_(solid)=[FePy₂Cl₃]_(solid) + liquor. The thermal dissociation of the clathrate with the release of pyridine into the gaseous phase (TGA) occurs in a similar way: [FePy₃Cl₃]·Py_(solid)=[FePy2Cl₃]_(solid) + 2 Py_(gas). Thermodynamic parameters of the clathrate dissociation have been determined from the dependence of the pyridine vapour pressure over the clathrate samples versus temperature (tensimetric method). The dependence experiences a change at 327 K indicating a polymorphous transformation occurring at this temperature. For the process in the range 292–327 K, ΔH =70.8 ± 0.8 kJ/mol, ΔS =197 ± 3 J/(mol K), ΔG =12.2 ± 0.1 kJ/mol; in the range 327–368 K, ΔH =44.4 ± 1.3 kJ/mol, ΔS =116 ± 4 J/(mol K), ΔG =9.9 ± 0.3 kJ/mol.     

The Influence of Ionic Strength on Yttrium and Rare Earth Element Complexation by Fluoride Ions in NaClO<Subscript>4</Subscript>, NaNO<Subscript>3</Subscript> and NaCl Solutions at 25 °C

Stability constants of the form _F β ₁(M)=[MF²⁺][M³⁺]⁻¹[F⁻]⁻¹ (where [MF²⁺] represents the concentration of a yttrium or a rare earth element (YREE) complex, [M³⁺] is the free YREE ion concentration, and [F⁻] is the free fluoride ion concentration) were determined by direct potentiometry in NaNO₃ and NaCl solutions. The patterns of log_{10  F} β ₁(M) in NaNO₃ and NaCl solutions very closely resemble stability constant patterns obtained previously in NaClO₄. For a given YREE, stability constants obtained in NaClO₄ were similar to, but consistently larger than _F β ₁(M) values obtained in NaNO₃ which, in turn, were larger than formation constants obtained in NaCl. Stability constants for formation of nitrate and chloride complexes ( and _Cl β ₁(M)=[MCl²⁺][M³⁺]⁻¹[Cl⁻]⁻¹) derived from _F β ₁(M) data exhibited ionic strength dependencies generally similar to those of _F β ₁(M). However, in contrast to the somewhat complex pattern obtained for _F β ₁(M) across the fifteen member YREE series, no patterns were observed for nitrate and chloride complexation constants: neither nor _Cl β ₁(M) showed discernable variations across the suite of YREEs. Nitrate and chloride formation constants at 25 °C and zero ionic strength were estimated as log₁₀  and log_{10  Cl} β ₁^o(M)=0.71±0.05. Although these constants are identical within experimental uncertainty, the distinct ionic strength dependencies of and _Cl β ₁(M) produced larger differences in the two stability constants with increasing ionic strength whereby _Cl β ₁(M) was uniformly larger than .     

Theoretical interpretation of the linear relationship between the hyperfine-structure constants in molecules and their radicals

The hyperfine-interaction constants of a number of radicals (A _H and ) and molecules (J _HH and {ie590-02}) were calculated within the framework of the semiempirical MO LCAO method. It is shown that there is a linear correlation between the atom-atom polarizabilities _rs which determine the values of the J constants and the spin densities in the corresponding radicals.     

Study of the Volumetric Properties of the Aqueous Ionic Liquid 1-Methyl-3-pentylimidazolium Tetrafluoroborate

This paper reports the densities of aqueous solutions of the ionic liquid (IL) 1-methyl-3-pentylimidazolium tetrafluoroborate ([pmim][BF₄]) that were measured from 278.15 to 343.15 K, at intervals of 5 K, using an Anton Parr model DMA 4500 oscillating U-tube densitometer. The apparent molar volume, ^φ V _B, and the partial molar volume of [pmim][BF₄], , were calculated. The values of the apparent molar volume, ^φ V _B, were fitted to Pitzer’s model for volumetric properties by the method of least-squares, which allowed the partial molar volume of the IL at infinite dilution, , and Pitzer’s parameters, β _M,X ^(0)V and β _M,X ^(1)V , to be obtained. The small standard deviations of the fits show that Pitzer’s model is also appropriate for representing the volumetric properties of aqueous solutions of the ionic liquid [pmim][BF₄].     

The Nature of Halogen⋅⋅⋅Halogen Interactions: A Model Derived from Experimental Charge‐Density Analysis

Slightly attractive : The attractive and anisotropic nature of the Cl???Cl interaction in C₆Cl₆ is experimentally demonstrated from an expansion of the electron density ρ( r ) around the chlorine nuclei. The interaction is explained in a model in which there is a bonding attraction involving electron‐deficient (see picture, blue) and electron‐rich (red) regions of adjacent Cl atoms.

     

Are Triphenylamine‐Functionalized or Carbazole‐Functionalized Iridium Complexes the More Effective Phosphorescent Materials? A Theoretical Perspective

The ground and excited states, charge injection/transport, and phosphorescence properties of eleven carbazole‐ and triphenylamine‐functionalized Ir^III complexes were investigated by using the DFT method. By analyzing the spin–orbit coupling (SOC) matrix elements, radiative decay rate constants k_r, and the electronic structures and energies at the ${{\rm{S}}_{\rm{0}}^{{\rm{opt}}} }$ and ${{\rm{T}}_{\rm{1}}^{{\rm{opt}}} }$ states, it was possible to rationalize the order of the experimental phosphorescence quantum yields of a series of Ir^III complexes and to predict that [Ir(Nph‐2‐Cz‐tz)₃] has a higher phosphorescence quantum yield than [Ir(TPA‐tz)₃] (TPA=triphenylamine, tz=thiazolyl, Cz=carbazole, Nph=N‐phenyl). Carbazole‐functionalized Ir^III complexes were shown to be efficient phosphorescent materials that have not only fast but also balanced electron/hole‐transport performance as well as high phosphorescence quantum yields. The phosphorescence emission spectra can be modulated by modifying or replacing a pyridyl substituent.     

Synthesis and Magnetism of Copper(II)-lanthanide(III)-copper(II) Heterotrinuclear Complexes with <Emphasis Type="Italic">N</Emphasis>,<Emphasis Type="Italic">N</Emphasis>′-bis [2-(dimethylamino)ethyl]oxamidocopper(II)

Nine new μ-oxamido-bridged copper(II)-lanthanide(III)-copper(II) heterotrinuclear complexes described by the overall formula Cu₂(dmoxae)₂Ln(NO₃)₃ {Ln = Ce, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er; dmoxae = N,N′-bis[2-(dimethylamino)ethyl]oxamido dianions} have been synthesized by the strategy of ‘complex as ligand’, and characterized by elemental analyses, molar conductivity measurements, i.r. and electronic spectral studies. The variable-temperature susceptibility (2–300 K), e.s.r. measurements, and studies of the Cu₂(dmoxae)₂Gd(NO₃)₃ complex have revealed that the central gadolinium(III) and terminal copper(II) ions are ferromagnetically coupled with the exchange integral J_(Cu-Gd) = +2.1 cm⁻¹, while an antiferromagnetic coupling is detected between the terminal copper(II) ions with J′_(Cu-Cu)=−0.36 cm⁻¹, on the basis of the spin Hamiltonian operator . A plausible mechanism for the ferromagnetic coupling between copper(II) and gadolinium(III) is discussed in terms of spin polarization.     

The Hydrolysis of Protactinium(V) Studied by Capillary Diffusion

The mean diffusion coefficient of ²³³Pa has been measured simultaneously with those of ²²Na and ¹⁵²Eu in 0.5 M (Na, H)ClO₄ solutions with the pH ranging from 0.3 to 13, by the open-end capillary method optimized in order to obtain reproducible and reliable D values at T = 25°C. In the case of Eu(III), the results tend to give higher ₁₃ and ₁₄ hydrolysis constants than the values generally acccepted, but these data are probably affected by the formation of polynuclear or colloidal species as soon as the hydrolysis process is involved. For Pa(V), results are in agreement with the existence of the following two equilibria (I = 0.5 M, T = 25°C):

Hosoya index of unicyclic graphs with prescribed pendent vertices

The Hosoya index z(G) of a (molecular) graph G is defined as the total number of subsets of the edge set, in which any two edges are mutually independent, i.e., the total number of independent-edge sets of G. By G(n, l, k) we denote the set of unicyclic graphs on n vertices with girth and pendent vertices being resp. l and k. Let be the graph obtained by identifying the center of the star S _n-l+1 with any vertex of C _l . By we denote the graph obtained by identifying one pendent vertex of the path P _n-l-k+1 with one pendent vertex of . In this paper, we show that is the unique unicyclic graph with minimal Hosoya index among all graphs in G(n, l, k).      

A Unified Treatment of the Relationship Between Ligand Substituents and Spin State in a Family of Iron(II) Complexes

The influence of ligands on the spin state of a metal ion is of central importance for bioinorganic chemistry, and the production of base‐metal catalysts for synthesis applications. Complexes derived from [Fe(bpp)₂]²⁺ (bpp=2,6‐di{pyrazol‐1‐yl}pyridine) can be high‐spin, low‐spin, or spin‐crossover (SCO) active depending on the ligand substituents. Plots of the SCO midpoint temperature (T ) in solution vs. the relevant Hammett parameter show that the low‐spin state of the complex is stabilized by electron‐withdrawing pyridyl (“X”) substituents, but also by electron‐donating pyrazolyl (“Y”) substituents. Moreover, when a subset of complexes with halogeno X or Y substituents is considered, the two sets of compounds instead show identical trends of a small reduction in T for increasing substituent electronegativity. DFT calculations reproduce these disparate trends, which arise from competing influences of pyridyl and pyrazolyl ligand substituents on Fe‐L σ and π bonding.     

Synthesis and magnetic properties of copper(II)-lanthanoid(III)-copper(II) trinuclear complexes with N,N′-bis(2-aminoethyl)-oxamido ligand

Summary Ten novel -oxamido trinuclear complexes, namely Cu₂–(oxae)₂Ln(ClO₄)₃ (Ln = Y, Pr, Sm, Eu, Gd, Tb, Dy, Ho, Er and Yb), where oxae donotes the N,N-bis(2-aminoethyl)-oxamido dianion, were prepared and characterized. The magnetic susceptibility of Cu₂(oxae)₂Gd(ClO₄)₃ was measured over the 4–300 K range and the observed data were successfully simulated by an equation based on the spin Hamiltonian operator . The exchange integrals J _(Gd-Cu) and J _Gd-Cu were found to be 2.37 and –0.71cm^-1, respectively, indicating that very weakly ferromagnetic spin-exchange interaction operates between copper(II) and gadolinium(III) ions.Visiting scholar: Qufu Normal University.     

The gas-phase pyrolysis of alkyl nitrites. III. Isopropyl nitrite

The rate of decomposition of isopropyl nitrite (IPN) has been studied in a static system over the temperature range of 130–160°C. For low concentrations of IPN (1–5 × 10^?5M), but with a high total pressure of CF₄ (～0.9 atm) and small extents of reaction (～1%), the first-order rates of acetaldehyde (AcH) formation are a direct measure of reaction (1), since k₃ » k₂(NO): \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ {\rm IPN}\begin{array}{rcl} 1 \\ {\rightleftarrows} \\ 2 \\ \end{array}i - \Pr \mathop {\rm O}\limits^. + {\rm NO},i - \Pr \mathop {\rm O}\limits^. \stackrel{3}{\longrightarrow} {\rm AcH} + {\rm Me}. $\end{document} Addition of large amounts of NO (～0.9 atm) in place of CF₄ almost completely suppressed AcH formation. Addition of large amounts of isobutane – t-BuH – (～0.9 atm) in place of CF₄ at 160°C resulted in decreasing the AcH by 25%. Thus 25% of \documentclass{article}\pagestyle{empty}\begin{document}$ i - \Pr \mathop {\rm O}\limits^{\rm .} $\end{document} were trapped by the t-BuH (4): \documentclass{article}\pagestyle{empty}\begin{document}$ i - \Pr \mathop {\rm O}\limits^. + t - {\rm BuH} \stackrel{4}{\longrightarrow} i - \Pr {\rm OH} + (t - {\rm Bu}). $\end{document} The result of adding either NO or t-BuH shows that reaction (1) is the only route for the production of AcH. The rate constant for reaction (1) is given by k₁ = 10^{16.2±0.4–41.0±0.8}/θ sec^?1. Since (E₁ + RT) and ΔH°₁ are identical, within experimental error, both may be equated with D(i-PrO-NO) = 41.6 ± 0.8 kcal/mol and E₂ = 0 ± 0.8 kcal/mol. The thermochemistry leads to the result that \documentclass{article}\pagestyle{empty}\begin{document}$ \Delta H_f^\circ (i - {\rm Pr}\mathop {\rm O}\limits^{\rm .} ) = - 11.9 \pm 0.8{\rm kcal}/{\rm mol}. $\end{document} From ΔS°₁ and A₁, k₂ is calculated to be 10^10.5±0.4M^?1·sec^?1. From an independent observation that k₆/k₂ = 0.19 ± 0.03 independent of temperature we find E₆ = 0 ± 1 kcal/mol and k₆ = 10^9.8+0.4M^?;1·sec^?1: \documentclass{article}\pagestyle{empty}\begin{document}$ i - \Pr \mathop {\rm O}\limits^. + {\rm NO} \stackrel{6}{\longrightarrow} {\rm M}_2 {\rm K} + {\rm HNO}. $\end{document} In addition to AcH, acetone (M₂K) and isopropyl alcohol (IPA) are produced in approximately equal amounts. The rate of M₂K formation is markedly affected by the ratio S/V of different reaction vessels. It is concluded that the M₂K arises as the result of a heterogeneous elimination of HNO from IPN. In a spherical reaction vessel the first-order rate of M₂K formation is given by k₅ = 10^9.4–27.0/θ sec^?1: \documentclass{article}\pagestyle{empty}\begin{document}$ {\rm IPN} \stackrel{5}{\longrightarrow} {\rm M}_2 {\rm K} + {\rm HNO}. $\end{document} IPA is thought to arise via the hydrolysis of IPN, the water being formed from HNO. This elimination process explains previous erroneous results for IPN.     

Some methods and applications of electron density distribution analysis

Accurate and efficient integration of the electron density function over arbitrary regions has been previously achieved by exploiting a separation of variables. Recently, a computer program has been written that calculates ρ, \documentclass{article}\pagestyle{empty}\begin{document}$ \buildrel{\rightharpoonup}\over{\nabla} $\end{document}ρ, and ?²ρ in an expeditious fashion, taking advantage of the separation of variables in the electron density function. Accurate integrations of ?²ρ over arbitrary regions can also be accomplished. The structure of the program is suited especially to vector processors. As a result of the efficiencies of these programs, functions of the electron density, such as the density itself, the surrounding electrostatic potential, ?ρ, and ?²ρ have been calculated in three dimensions. Results of calculations for nitrated cubanes are presented illustrating how the effects of the nitro groups are manifested in the electron density and associated properties.     

Exact solution of the relativistic dynamics of a spin‐½ particle moving in a homogeneous magnetic field

The relativistic dynamics of one spin‐½ particle moving in a uniform magnetic field is described by the Hamiltonian $\mathbf{h}^{0}_{D}(\pi)=c\alpha\cdot\pi+\beta mc^{2}$. The discrete (and semidiscrete) eigenvalues and the corresponding eigenspinors are in principle known from the work of Dirac, Rabi, and Bloch. These are extensively reviewed here. Next, exact solutions are worked out for the recoil dynamics in relative coordinates, which involves the Hamiltonian $\mathbf{h}^{0}_{D}(-\mathbf{k})=-c\alpha\cdot\mathbf{k}+\beta mc^{2}$. Exact solutions are also explicitly calculated in the case where the spin‐½ particle has an anomalous magnetic moment such that its Hamiltonian is given by $\mathbf{h}_{D}(\pi)=\mathbf{h}^{0}_{D}(\pi)-\beta\mu_{\mathrm{ano}}\sigma\cdot\mathbf{B}$. Similar exact solutions are derived here when the recoiling particle has an anomalous magnetic moment, that is, the eigenvalues and eigenspinors of the Hamiltonian $\mathbf{h}_{D}(-\mathbf{k})=\mathbf{h}^{0}_{D}(-\mathbf{k})-\beta\mu_{\mathrm{ano}}\sigma\cdot\mathbf{B}$ are explicitly obtained. The diagonalized and separable form of the Hamiltonian h _D(π), written as $\tilde{\mathbf{h}}_{D}(\pi)$, has exceedingly simple forms of eigenspinors. Similarly, the diagonalized and separable form of the operator h _D(? k ), written as $\tilde{\mathbf{h}}_{D}(-\mathbf{k})$, has very simple eigenspinors. The importance of these exact solutions is that the eigenspinors can be used as bases in a calculation involving many spin‐½ particles placed in a uniform magnetic field. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 209–217, 2001     

Separation of dextrans by gel permeation chromatography

The role of the intrinsic viscosity [η] as separation parameter in gel permeation chromatography (GPC) was studied for dextrans (from Leuconostoc mesenteroids B512) dissolved in water with deactivated silicagel (Porasil) as the column-filling material. For that purpose specific viscosities of dextran fractions eluted by GPC were measured as a function of the elution volume v. Provided that the elution volumes are corrected for zonal spreading, they are related to the intrinsic viscosities in an unambiguous way, probably reflecting a unique relationship between degree of branching and molecular weights. This was further investigated by developing an iteration method to prepare two calibration curves γ(v) and g(v), respectively, relating ln[\documentclass{article}\pagestyle{empty}\begin{document}$\left[ {\bar \eta } \right]$\end {document}] and InM (M is the molecular weight) to v. It required that the weight-average molecular weight M _w, the number-average molecular weight M _n, and the average intrinsic viscosity [\documentclass{article}\pagestyle{empty}\begin{document}$\left[ {\bar \eta } \right]$\end {document}] for a number of dextran samples (broad distributions) be previously known. The calibration curves found lead to consistent values of the above-mentioned averages. Moreover, they allow-establishment of the [\documentclass{article}\pagestyle{empty}\begin{document}$\left[ {\bar \eta } \right]$\end {document}]-M relationship over the range 5000 < M < 500,000.     

Synthesis and Magnetism of Ferromagnetically Coupled [Cu 3 II CrIII] Heterotetranuclear Complexes Bridged by Oxalate Groups

Three novel -oxalato-bridged Cu ₃ ^II Cr^III-type heterotetranuclear complexes described by the overall formula [Cu₃Cr(ox)₃L₃](ClO₄)₃, where ox represents the oxalato dianions, L stands for diaminoethane (en), 1,3-diaminopropane (pn), and 1,2-diaminopropane (ap) respectively, have been synthesized and characterized by elemental analyses, molar conductivity and magnetic moment (room-temperature) measurements, i.r., e.s.r. and electronic spectral studies. It is proposed that these complexes have oxalato-bridged structures consisting of three copper(II) ions and a chromium(III) ion, in which the chromium(III) ion has an octahedral environment and the three copper(II) ions have square-planar environments. Variable temperature magnetic susceptibility (4.2–300 K) measurements and studies of complexes [Cu₃Cr(ox)₃(en)₃](ClO₄)₃ (1) and [Cu₃Cr(ox)₃(pn)₃](ClO₄)₃ (2) revealed the occurrence of an intramolecular ferromagnetic interaction between the copper(II) and chromium(III) ions through the oxalato-bridge within each molecule. The magnetic data have been used also to deduce the indicated -oxalato-bridges [Cu ₃ ^II Cr^III] heterotetranuclear structure. On the basis of the spin Hamiltonian operator, , the magnetic analyses were carried out for the two Cu^II—Cr^III heterotetranuclear complexes and the spin-coupling constants (J) were evaluated as +6.36 cm^–1 for (1) and +7.02 cm^–1 for (2). The results indicate that the bridging oxalato entity should be able to transmit ferromagnetic interactions in the strict orthogonality [Cu ₃ ^II Cr^III] system.