Heat capacity and density of solutions of calcium and cadmium nitrates in N-methylpyrrolidone at 298.15 K

The heat capacity and density of solutions of calcium and cadmium nitrates in N-methylpyrrolidone (MP) at 298.15 K are studied by calorimetry and densimetry. The obtained data are discussed in relation to certain features of solvation and complex formation in solutions of these salts. The standard partial molar heat capacities and volumes ( $\overline {C_{p^2 }^0 }$ and $\overline {V_2^0 }$ ) of the electrolytes in MP are calculated. The standard heat capacities $\overline {C_{p^i }^0 }$ and volumes $\overline {V_i^0 }$ of Ca²⁺ and Cd²⁺ ions in MP at 298.15 K were determined, along with the contribution from specific interactions to the values of $\overline {C_{p^i }^0 }$ and $\overline {V_i^0 }$ of Cd²⁺ ions in MP solution.     

Thermodynamic Analysis of Homologous α-Amino Acids in Aqueous Potassium Fluoride Solutions at Different Temperatures

Partial molal volumes ( $V_{\phi} ^{0}$ ) and partial molal compressibilities ( $K_{\phi} ^{0}$ ) for glycine, L-alanine, L-valine and L-leucine in aqueous potassium fluoride solutions (0.1 to 0.5?mol?kg^?1) have been measured at T=(303.15,308.15,313.15 and 318.15) K from precise density and ultrasonic speed measurements. Using these data, Hepler coefficients ( $\partial^{2}V_{\phi} ^{0}/\partial T^{2}$ ), transfer volumes ( $\Delta V_{\phi} ^{0}$ ), transfer compressibilities ( $\Delta K_{\phi} ^{0}$ ) and hydration number (n _H) have been calculated. Pair and triplet interaction coefficients have been obtained from the transfer parameters. The values of $V_{\phi} ^{0}$ and $K_{\phi} ^{0}$ vary linearly with increasing number of carbon atoms in the alkyl chain of the amino acids. The contributions of charged end groups ( $\mathrm{NH}_{3}^{+}$ , COO^?), CH₂ group and other alkyl chains of the amino acids have also been estimated. The results are discussed in terms of the solute?Ccosolute interactions and the dehydration effect of potassium fluoride on the amino acids.     

Synergistic extraction of europium and americium into nitrobenzene by using hydrogen dicarbollylcobaltate and bis(diphenylphosphino)methane dioxide

Extraction of microamounts of europium and americium by a nitrobenzene solution of hydrogen dicarbollylcobaltate (H⁺B^?) in the presence of bis(diphenylphosphino)methane dioxide (DPPMDO, L) has been investigated. The equilibrium data have been explained assuming that the species $ {\text{HL}}^{ + } $ , $ {\text{HL}}_{2}^{ + } $ , $ {\text{ML}}_{2}^{3 + } $ , $ {\text{ML}}_{3}^{3 + } $ and $ {\text{ML}}_{4}^{3 + } $ (M³⁺ = Eu³⁺, Am³⁺) are extracted into the organic phase. The values of extraction and stability constants of the species in nitrobenzene saturated with water have been determined. It was found that the stability constants of the corresponding complexes $ {\text{EuL}}_{n}^{3 + } $ and $ {\text{AmL}}_{n}^{3 + } $ , where n = 2, 3 and L is DPPMDO, in water–saturated nitrobenzene are comparable, whereas in this medium the stability of the cationic species $ {\text{AmL}}_{4}^{3 + } $ (L = DPPMDO) is somewhat higher than that of $ {\text{EuL}}_{4}^{3 + } $ with the same ligand L.     

Transport of Water by Group 1 and 2 Ions with t-Butyl Alcohol as Reference Substance: Comparison with Raffinose and Dioxan

Measurement of the transport of water with respect to the second solvent component in a binary aqueous mixture gives the Washburn number, $ w_{\text{W}} = (n_{\text{W}} )_{ + } t_{ + } - (n_{\text{W}} )_{ - } t_{ - } $ , in a transport number determination, where the ions move in opposite directions, and give the Erdey–Grúz number, $ \Upsigma n_{\text{W}} = (n_{\text{W}} )_{ + } + (n_{\text{W}} )_{ - } $ , in a diffusion experiment, where the ions move in the same direction. Here n _W and t are the number of water molecules and transport number, respectively, of the anion or cation. Combination of the results of these two experiments allows unambiguous determination of values for the solvent transport numbers, $ n_{\text{W}} $ , of the individual ions. While the values of $ n_{\text{W}} $ depend on the cosolvent, at high dilutions of the second component the highest value of $ n_{\text{W}} $ found, $ N_{\text{W}} $ , should approach the number of water molecules transported by the ion in pure water, $ N_{\text{W}}^{0} $ . New data for alkali-metal, alkaline-earth metal, hydrogen and halide ions in dilute mixtures of t-butyl alcohol with water are presented. Values of $ N_{\text{W}} $ rounded to whole numbers thus found are: 12 (Li⁺), 10 (Na⁺), 6 (K⁺), 5 (Rb⁺), 5 (Cs⁺), 1 (H⁺), 13 (Ca²⁺), 16 (Sr²⁺) and 15 (Ba²⁺). Factors influencing preferential solvation are briefly discussed. Detailed recalculations of $ n_{\text{W}} $ in the raffinose–water system from literature data also allows resolution of a problem with the Onsager Relations.     

Correlation consistent, Douglas–Kroll–Hess relativistic basis sets for the 5p and 6p elements

The diatomic carbon molecule has a complex electronic structure with a large number of low-lying electronic excited states. In this work, the potential energy curves (PECs) of the four lowest lying singlet states ( $X^{1} \Sigma^{ + }_{g}$ , $A^{1} \Pi_{u}$ , $B^{1} \Delta_{g}$ , and $B^{\prime1} \Sigma^{ + }_{g}$ ) were obtained by high-level ab initio calculations. Valence electron correlation was accounted for by the correlation energy extrapolation by intrinsic scaling (CEEIS) method. Additional corrections to the PECs included core–valence correlation and relativistic effects. Spin–orbit corrections were found to be insignificant. The impact of using dynamically weighted reference wave functions in conjunction with CEEIS was examined and found to give indistinguishable results from the even weighted method. The PECs showed multiple curve crossings due to the $B^{1} \Delta_{g}$ state as well as an avoided crossing between the two $^{1} \Sigma^{ + }_{g}$ states. Vibrational energy levels were computed for each of the four electronic states, as well as rotational constants and spectroscopic parameters. Comparison between the theoretical and experimental results showed excellent agreement overall. Equilibrium bond distances are reproduced to within 0.05 %. The dissociation energies of the states agree with experiment to within ~0.5 kcal/mol, achieving “chemical accuracy.” Vibrational energy levels show average deviations of ~20 cm^?1 or less. The $B^{1} \Delta_{g}$ state shows the best agreement with a mean absolute deviation of 2.41 cm^?1. Calculated rotational constants exhibit very good agreement with experiment, as do the spectroscopic constants.     

Free ions Pr IV (4\text{ f }^{2}) and Tm IV (4\text{ f }^{12}): intermediate versus LS coupling scheme

The intermediate and LS-coupling schemes for the free lanthanide ions $\text{ Pr }^{3+}$ Pr 3 + and $\text{ Tm }^{3+}$ Tm 3 + have been compared by the matrix elements of the tensor operator ${{\varvec{U}}}^{({\varvec{k}})}, \text{ k } = 2, 4, 6$ U ( k ) , k = 2 , 4 , 6 . The necessary eigenvectors and eigenvalues have been computed with the aid of four parameters, $\text{ F }_{2}, \text{ F }_{4}, \text{ F }_{6}$ F 2 , F 4 , F 6 , and $\zeta _{4\mathrm{f}}$ ζ 4 f , known from free-ion spectra of the same ions. It has been found that both coupling types for each ion lead to close values of ${\vert }{{\varvec{U}}}^{({\varvec{k}})}{\vert }^{2}$ | U ( k ) | 2 only for transitions from the ground level to certain lower-lying energy levels within the $4\text{ f }^\mathrm{N}$ 4 f N configuration.     

A mathematical approach to chemical equilibrium theory for gaseous systems—III: \hbox {Q}_\mathrm{p}, \hbox {Q}_\mathrm{c}, and \hbox {Q}_{\mathrm{x}}

The reaction quotient Q can be expressed in partial pressures as $\hbox {Q}_\mathrm{P}$ or in mole fractions as $\hbox {Q}_{\mathrm{x}}$ . $\hbox {Q}_\mathrm{P}$ is ostensibly more useful than $\hbox {Q}_{\mathrm{x}}$ because the related $\hbox {K}_{\mathrm{x}}$ is a constant for a chemical equilibrium in which T and P are kept constant while $\hbox {K}_{\mathrm{P}}$ is an equilibrium constant under more general conditions in which only T is constant. However, as demonstrated in this work, $\hbox {Q}_{\mathrm{x}}$ is in fact more important both theoretically and technically. The relationships between $\hbox {Q}_{\mathrm{x}}$ , $\hbox {Q}_\mathrm{P}$ , and $\hbox {Q}_{\mathrm{C}}$ are discussed. Four examples of applications are given in detail.     

A CASPT2//CASSCF study of the quartet excited state \tilde{a}^{4}A^{\prime\prime} of the HCNN radical

Complete active space self-consistent field and second-order multiconfigurational perturbation theory methods have been performed to investigate the quartet excited state ${\tilde{a}}^{4}{A^{\prime\prime}}$ potential energy surface of HCNN radical. Two located minima with respective cis and trans structures could easily dissociate to CH $({\tilde{a}}^{4}\Sigma^{-})$ and $N_{2} ({\tilde{X}}^{1}\Sigma_{\rm g}^{+})$ products with similar barrier of about 16.0 kcal/mol. In addition, four minimum energy crossing points on a surface of intersection between ${\tilde{a}}^{4}A^{\prime\prime}$ and X ( $X={\tilde{X}}^{2}A^{\prime\prime}$ and ${\tilde{A}}^{2}A^{\prime}$ ) states are located near to the minima. However, the intersystem crossing ${\tilde{a}}^{4}A^{\prime\prime} \rightarrow X$ is weak due to the vanishingly small spin–orbit interactions. It further indicates that the direct dissociation on the ${\tilde{a}}^{4}{A^{\prime\prime}}$ state is more favored. This information combined with the comparison with isoelectronic HCCO provides an indirect support to the recent experimental proposal of photodissociation mechanism of HCNN.     

Acid–Base Properties of the \mathrm{Fe}(\mathrm{CN})_{6}^{3-}/\mathrm{Fe}(\mathrm{CN})_{6}^{4-} Redox Couple in the Presence of Various Background Mineral Acids and Salts

The acid?Cbase behavior of $\mathrm{Fe}(\mathrm{CN})_{6}^{4-}$ was investigated by measuring the formal potentials of the $\mathrm{Fe}(\mathrm{CN})_{6}^{3-}$ / $\mathrm{Fe}(\mathrm{CN})_{6}^{4-}$ couple over a wide range of acidic and neutral solution compositions. The experimental data were fitted to a model taking into account the protonated forms of $\mathrm{Fe}(\mathrm{CN})_{6}^{4-}$ and using values of the activities of species in solution, calculated with a simple solution model and a series of binary data available in the literature. The fitting needed to take account of the protonated species $\mathrm{HFe}(\mathrm{CN})_{6}^{3-}$ and $\mathrm{H}_{2}\mathrm{Fe}(\mathrm{CN})_{6}^{2-}$ , already described in the literature, but also the species $\mathrm{H}_{3}\mathrm{Fe}(\mathrm{CN})_{6}^{-}$ (associated with the acid?Cbase equilibrium $\mathrm{H}_{3}\mathrm{Fe}(\mathrm{CN})_{6}^{-}\rightleftharpoons \mathrm{H}_{2}\mathrm{Fe}(\mathrm{CN})_{6}^{2-} + \mathrm{H}^{+}$ ). The acidic dissociation constants of $\mathrm{HFe}(\mathrm{CN})_{6}^{3-}$ , $\mathrm{H}_{2}\mathrm{Fe}(\mathrm{CN})_{6}^{2-}$ and $\mathrm{H}_{3}\mathrm{Fe}(\mathrm{CN})_{6}^{-}$ were found to be $\mathrm{p}K^{\mathrm{II}}_{1}= 3.9\pm0.1$ , $\mathrm{p}K^{\mathrm{II}}_{2} = 2.0\pm0.1$ , and $\mathrm{p}K^{\mathrm{II}}_{3} = 0.0\pm0.1$ , respectively. These constants were determined by taking into account that the activities of the species are independent of the ionic strength.     

Study of Malachite Green Fading in Water–Ethanol–Ethylene Glycol Ternary Mixtures

The rate constant of malachite green (MG⁺) alkaline fading was measured in water–ethanol–ethylene glycol ternary mixtures. This reaction was studied under pseudo-first-order conditions at 283–303 K. In each series of experiments, the concentration of ethanol was kept constant and the concentration of ethylene glycol was changed. It was shown that due to hydrogen bonding and hydrophobic interaction between MG⁺ and alcohol molecules the observed reaction rate constant, $ k_{\text{obs}} $ , increased in the water–ethanol–ethylene glycol ternary mixtures. The fundamental rate constants of MG⁺ fading in these solutions ( $ k_{1} $ , $ k_{ - 1} $ and $ k_{2} $ ) were obtained by the SESMORTAC model. Analysis of $ k_{1} $ and $ k_{2} $ values in solutions containing constant ethanol concentrations show that in low concentrations of ethylene glycol, hydrogen bonding formed between ethanol and ethylene glycol molecules and in high concentrations of ethylene glycol, ethanol as a solvent for ethylene glycol affected the reaction rate.     

Volumetric and Viscometric Studies in Binary Liquid Mixtures of 2-Methyl-2-propanol with Some Ketones at Different Temperatures

Densities, ??, and viscosities, ??, of binary mixtures of 2-methyl-2-propanol with acetone (AC), ethyl methyl ketone (EMK) and acetophenone (AP), including those of the pure liquids, were measured over the entire composition range at 298.15, 303.15 and 308.15?K. From these experimental data, the excess molar volume $V_{\mathrm{m}}^{\mathrm{E}}$ , deviation in viscosity ????, partial and apparent molar volumes ( $\overline{V}_{\mathrm{m},1}^{\,\circ }$ , $\overline{V}_{\mathrm{m},2}^{\,\circ }$ , $\overline{V}_{\phi ,1}^{\,\circ}$ and $\overline{V}_{\phi,2}^{\,\circ} $ ), and their excess values ( $\overline{V}_{\mathrm{m},1}^{\,\circ \mathrm{E}}$ , $\overline{V}_{\mathrm{m,2}}^{\,\circ \mathrm{ E}}$ , $\overline {V}_{\phi \mathrm{,1}}^{\,\circ \mathrm{ E}}$ and $\overline{V}_{\phi \mathrm{,2}}^{\,\circ \mathrm{ E}}$ ) of the components at infinite dilution were calculated. The interaction between the component molecules follows the order of AP > AC > EMK.     

Optimal descriptors as a tool to predict the thermal decomposition of polymers

Quantitative structure-property relationship for the thermal decomposition of polymers is suggested. The data on architecture of monomers is used to represent polymers. The structures of monomers are represented by simplified molecular input-line entry system. The average statistical quality of the suggested quantitative structure-property relationships for prediction of molar thermal decomposition function $\hbox {Y}_{\mathrm{d},1/2}$ is the following: $\hbox {r}^{2}=0.970 \pm 0.01$ and $\hbox {RMSE}=4.71\pm 1.01\,(\hbox {K}\times \hbox {kg}\times \hbox {mol}^{-1})$ .     

A thermodynamic study of complexation of 18-crown-6 with Zn2+, Tl+, Hg2+ and {\text{UO}}^{{{\text{2 + }}}}_{{\text{2}}} cations in acetonitrile-dimethylformamide binary media and study the effect of anion on the stability constant of (18C6-Na+) complex in methanol solutions

The equilibrium constants and thermodynamic parameters for complex formation of 18-crown-6(18C6) with Zn²⁺, Tl⁺, Hg²⁺ and $ {\text{UO}}^{{{\text{2 + }}}}_{{\text{2}}} $ cations have been determined by conductivity measurements in acetonitrile(AN)-dimethylformamide(DMF) binary solutions. 18-crown-6 forms 1:1 complexes [M:L] with Zn²⁺, Hg²⁺ and $ {\text{UO}}^{{{\text{2 + }}}}_{{\text{2}}} $ cations, but in the case of Tl⁺ cation, a 1:2 [M:L₂] complex is formed in most binary solutions. The thermodynamic parameters ( $ \Delta {\text{H}}^{ \circ }_{{\text{c}}} $ and $ \Delta {\text{S}}^{ \circ }_{{\text{c}}} $ ) which were obtained from temperature dependence of the equilibrium constants show that in most cases, the complexes are enthalpy destabilized but entropy stabilized and a non-monotonic behaviour is observed for variations of standard enthalpy and entropy changes versus the composition of AN/DMF binary mixed solvents. The obtained results show that the order of selectivity of 18C6 ligand for these cations changes with the composition of the mixed solvent. A non-linear relationship was observed between the stability constants (logK_f) of these complexes with the composition of AN/DMF binary solutions. The influence of the $ {\text{ClO}}^{ - }_{{\text{4}}} $ , $ {\text{NO}}^{ - }_{{\text{3}}} $ and $ {\text{Cl}}^{ - } $ anions on the stability constant of (18C6-Na⁺) complex in methanol (MeOH) solutions was also studied by potentiometry method. The results show that the stability of (18C6-Na⁺) complex in the presence of the anions increases in order: $ {\text{ClO}}^{ - }_{{\text{4}}} $  >  $ {\text{NO}}^{ - }_{{\text{3}}} $  >  $ {\text{Cl}}^{ - } $ .     

The principal measure and distributional (p, q)-chaos of a coupled lattice system with coupling constant \varepsilon =1 related with Belusov–Zhabotinskii reaction

We consider the following system coming from a lattice dynamical system stated by Kaneko (Phys Rev Lett, 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction: $$\begin{aligned} x_{n}^{m+1}=(1-\varepsilon )f\left( x_{n}^{m}\right) +\frac{1}{2}\varepsilon \left[ f(x_{n-1}^{m})+f\left( x_{n+1}^{m}\right) \right] , \end{aligned}$$ where $m$ is discrete time index, $n$ is lattice side index with system size $L$ (i.e., $n=1, 2, \ldots , L$ ), $\varepsilon \ge 0$ is coupling constant, and $f(x)$ is the unimodal map on $I$ (i.e., $f(0)=f(1)=0$ , and $f$ has unique critical point $c$ with $0<c<1$ and $f(c)=1$ ). In this paper, we prove that for coupling constant $\varepsilon =1$ , this CML (Coupled Map Lattice) system is distributionally $(p, q)$ -chaotic for any $p, q\in [0, 1]$ with $p\le q$ , and that its principal measure is not less than $\mu _{p}(f)$ . Consequently, the principal measure of this system is not less than $\frac{2}{3}+\sum _{n=2}^{\infty }\frac{1}{n}\frac{2^{n-1}}{(2^{n}+1) (2^{n-1}+1)}$ for coupling constant $\varepsilon =1$ and the tent map $\Lambda $ defined by $\Lambda (x)=1-|1-2x|, x\in [0, 1]$ . So, our results complement the results of Wu and Zhu (J Math Chem, 50:2439–2445, 2012).     

Compound-specific isotope analysis of benzotriazole and its derivatives

Compound-specific isotope analysis (CSIA) is an important tool for the identification of contaminant sources and transformation pathways, but it is rarely applied to emerging aquatic micropollutants owing to a series of instrumental challenges. Using four different benzotriazole corrosion inhibitors and its derivatives as examples, we obtained evidence that formation of organometallic complexes of benzotriazoles with parts of the instrumentation impedes isotope analysis. Therefore, we propose two strategies for accurate $\delta^{13}$ C and $\delta^{15}$ N measurements of polar organic micropollutants by gas chromatography isotope ratio mass spectrometry (GC/IRMS). Our first approach avoids metallic components and uses a Ni/Pt reactor for benzotriazole combustion while the second is based on the coupling of online methylation to the established GC/IRMS setup. Method detection limits for on-column injection of benzotriazole, as well as its 1-CH $_{3}$ -, 4-CH $_{3}$ -, and 5-CH $_{3}$ -substituted species were 0.1–0.3 mM and 0.1–1.0 mM for δ¹³C and δ¹⁵N analysis respectively, corresponding to injected masses of 0.7–1.8 nmol C and 0.4–3.0 nmol N, respectively. The Ni/Pt reactor showed good precision and was very long-lived ( $>$ 1000 successful measurements). Coupling isotopic analysis to offline solid-phase extraction enabled benzotriazole-CSIA in tap water, wastewater treatment effluent, activated sludge, and in commercial dishwashing products. A comparison of $\delta ^{13}$ C and $\delta ^{15}$ N values from different benzotriazoles and benzotriazole derivatives, both from commercial standards and in dishwashing detergents, reveals the potential application of the proposed method for source apportionment.     

Diamond hydrocarbons revisited: partitioned formula tables of diamondoids

Isomeric diamondoids with the same number $n$ of adamantane units (or cells), which share the same molecular formula $\text{ C}_\mathrm{Q}(\text{ CH})_\mathrm{T}(\text{ CH}_{2})_\mathrm{S}$ , can be divided into valence isomers by partitioning the number $C $ of their carbon atoms according to whether they are Quaternary, Tertiary, or Secondary: $C =Q +T +S$ . Each [ $n$ ]diamondoid has a dualist (or inner dual) with $n$ vertices (situated at centers of adamantane units), and edges connecting vertices of adjacent adamantane units sharing a chair-shaped hexagon of carbon atoms. Such a dualist is characterized by a quadruplet of indices (denoted as p, s, t, q for primary, secondary, tertiary, or quaternary) specifying again the connectivity of each vertex by assimilating it with a virtual carbon atom. The diamond lattice is self-dual. Dualists help in classifying diamondoids as catamantanes with acyclic dualists, perimantanes with dualists having chair-shaped six-membered rings, or coronamantanes with dualists having only higher-membered rings. In turn, catamantanes can be either regular when they have formulas $\text{ C}_{4n+6}\text{ H}_{4n+12}$ , or irregular when the numbers of carbon and hydrogen atoms are lower than the above values for the given numbers $n$ of adamantane units. Regular catamantanes can have branched or non-branched dualists and they are isomeric when having the same $n$ . Partitioned formulas reflect the branching patterns, encoded in their dualists. Partition formulas and codes are presented for all possible diamondoids with up to 7 adamantane units. A remarkable symmetry is observed for the table of partition periodic table of regular catamantanes with up to 7 adamantane units. Isomeric irregular catamantanes with six or more adamantane units may be valence-isomeric (or homomeric, sharing both the molecular and the partitioned formulas), or heteromeric when they have different partitioned formulas.     

Thermodynamic properties of polymorphic forms of theophylline. Part II: The enthalpies of solution in water at 298.15 K

The enthalpies of solution $ \Updelta_{sol}^{{}} H_{m}^{{}} $ of polymorphic forms I and II of theophylline in water at 298.15 K using the isoperibol solution calorimeter have been determined in the range of concentration (0.311–1.547) · 10^?3 /mol · kg^?1. The enthalpies of hydration $ \Updelta_{hyd}^{{}} H_{m}^{o} $ were determined from the experimentally obtained the enthalpies of solution for aqueous solutions and previously determined enthalpies of sublimation $ \Updelta_{s}^{g} H_{m}^{o} . $      

Thermochemical Properties of n-Undecylammonium Bromide Monohydrate C11H28BrNO(s)

The crystal structure of n-undecylammonium bromide monohydrate was determined by X-ray crystallography. The crystal system of the compound is monoclinic, and the space group is P2₁/c. Molar enthalpies of dissolution of the compound at different concentrations m/(mol·kg^?1) were measured with an isoperibol solution–reaction calorimeter at T = 298.15 K. According to the Pitzer’s electrolyte solution model, the molar enthalpy of dissolution of the compound at infinite dilution ( $ \Updelta_{\text{sol}} H_{\text{m}}^{\infty } $ ) and Pitzer parameters ( $ \beta_{\text{MX}}^{(0)L} $ and $ \beta_{\text{MX}}^{(1)L} $ ) were obtained. Values of the apparent relative molar enthalpies ( $ {}^{\Upphi }L $ ) of the title compound and relative partial molar enthalpies ( $ \bar{L}_{2} $ and $ \bar{L}_{1} $ ) of the solute and the solvent at different concentrations were derived from experimental values of the enthalpies of dissolution.     

The state of Cu2+ ions in concentrated aqueous ammonia solutions of copper nitrate

Stabilization of Cu²⁺ ions in concentrated aqueous ammonia solutions of copper nitrate in a wide range of ammonium ion concentrations has been studied by EPR and electronic absorption spectroscopy. Three types of Cu²⁺ associates with different types of orbital ordering have been identified. The ammonium ion concentration in a solution has a decisive effect on the type of orbital ordering of Cu²⁺ ions in associates. In all cases, Cu²⁺ ordering in associates is caused by the existence of bridging OH groups in the axial and equatorial positions of [Cu(NH₃) _n (H₂O)_{6 ? n} ]²⁺ complexes (n < 6). At a high concentration of ammonium ions, weakly bound associates of tetramminecopper with the $d_{x^2 - y^2 }$ ground state are formed. In solutions with low ammonium concentrations, bulky associates with the $d_{y^2 }$ and $d_{x^2 - z^2 }$ ground states and associates of Cu²⁺ ions with the $d_{x^2 - y^2 }$ ground state with hydroxyl groups in the equatorial plane and axial water molecules are formed.     

A proposed family of variationally correlated first-order density matrices for spin-polarized three-electron model atoms

In early work of March and Young (Phil Mag 4:384, 1959), it was pointed out for spin-free fermions that a first-order density matrix (1DM) for $N-1$ particles could be constructed from a 2DM ( $\Gamma $ ) for $N$ fermions divided by the diagonal of the 1DM, the density $n(\mathbf{r}_1)$ , as $2\Gamma (\mathbf{r}_1,\mathbf{r}^{\prime }_2;\mathbf{r}_1,\mathbf{r}_2)/n(\mathbf{r}_1)$ for any arbitrary fixed $\mathbf{r}_1$ . Here, we thereby set up a family of variationally valid 1DMS constructed via the above proposal, from an exact 2DM we have recently obtained for four electrons in a quintet state without confining potential, but with pairwise interparticle interactions which are harmonic. As an indication of the utility of this proposal, we apply it first to the two-electron (but spin-compensated) Moshinsky atom, for which the exact 1DM can be calculated. Then the 1DM is found for spin-polarized three-electron model atoms. The equation of motion of this correlated 1DM is exhibited and discussed, together with the correlated kinetic energy density, which is shown explicitly to be determined by the electron density.