Ro-vibrational spectroscopy of molecules represented by a Tietz–Hua oscillator potential

Accurate low and high-lying bound states of Tietz–Hua oscillator potential are presented. The radial Schrödinger equation is solved efficiently by means of the generalized pseudospectral method that enables optimal spatial discreti zation. Both $\ell =0$ and rotational states are considered. Ro-vibrational levels of six diatomic molecules viz., H $_2$ , HF, N $_2$ , NO, O $_2$ , O $_2^+$ are obtained with good accuracy. Most of the states are reported here for the first time. A detailed analysis of variation of eigenvalues with $n, \ell $ quantum numbers is made. Results are compared with literature data, wherever possible. These are also briefly contrasted with the Morse potential results.     

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.     

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.     

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).     

Towards a definition of almost–equienergetic graphs

The energy $E(G)$ of a graph $G$ , a quantity closely related to total $\pi $ -electron energy, is equal to the sum of absolute values of the eigenvalues of $G$ . Two graphs $G_a$ and $G_b$ are said to be equienergetic if $E(G_a)=E(G_b)$ . In 2009 it was discovered that there are pairs of graphs for which the difference $E(G_a)-E(G_b)$ is non-zero, but very small. Such pairs of graphs were referred to as almost equienergetic, but a precise criterion for almost–equienergeticity was not given. We now fill this gap.     

Heats of formation of the amino acids re-examined by means of W1-F12 and W2-F12 theories

We have obtained accurate heats of formation for the twenty natural amino acids by means of explicitly correlated high-level thermochemical procedures. Our best theoretical heats of formation, obtained by means of the ab initio W1-F12 and W2-F12 thermochemical protocols, differ significantly (RMSD = 2.3 kcal/mol, maximum deviation 4.6 kcal/mol) from recently reported values using the lower-cost G3(MP2) method. With the more recent G4(MP2) procedure, RMSD drops slightly to 1.8 kcal/mol, while full G4 theory offers a more significant improvement to 0.72 kcal/mol (max. dev. 1.4 kcal/mol for glutamine). The economical G4(MP2)-6X protocol performs equivalently at RMSD = 0.71 kcal/mol (max. dev. 1.6 kcal/mol for arginine and glutamine). Our calculations are in excellent agreement with experiment for glycine, alanine and are in excellent agreement with the recent revised value for methionine, but suggest revisions by several kcal/mol for valine, proline, phenylalanine, and cysteine, in the latter case confirming a recent proposed revision. Our best heats of formation at 298 K ( $\Delta H_{f,298}^{\circ }$ ) are as follows: at the W2-F12 level: glycine ?94.1, alanine $-$ 101.5, serine $-$ 139.2, cysteine $-$ 94.5, and methionine $-$ 102.4  kcal/mol, and at the W1-F12 level: arginine $-$ 98.8, asparagine $-$ 146.5, aspartic acid $-$ 189.6, glutamine $-$ 151.0, glutamic acid $-$ 195.5, histidine $-$ 69.8, isoleucine $-$ 118.3, leucine $-$ 118.8, lysine $-$ 110.0, phenylalanine $-$ 76.9, proline $-$ 92.8, threonine $-$ 149.0, and valine $-$ 113.6 kcal/mol. For the two largest amino acids, an average over G4, G4(MP2)-6X, and CBS-QB3 yields best estimates of $-$ 58.4 kcal/mol for tryptophan, and of $-$ 117.5 kcal/mol for tyrosine. For glycine, we were able to obtain a “quasi-W4” result corresponding to $\hbox {TAE}_e$  = 968.1, $\hbox {TAE}_0$  = 918.6, $\Delta H_{f,298}^{\circ }=-90.0$ , and $\Delta H_{f,298}^{\circ }=-94.0$  kcal/mol.     

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.     

Kekuléan benzenoids

A Kekulé structure for a benzenoid or a fullerene $\Gamma $ is a set of edges $K$ such that each vertex of $\Gamma $ is incident with exactly one edge in $K$ , i.e. a perfect matching. All fullerenes admit a Kekulé structure; however, this is not true for benzenoids. In this paper, we develop methods for deciding whether or not a given benzenoid admits a Kekulé structure by constructing Kekulé structures that have a high density of benzene rings. The benzene rings of the Kekulé structure $K$ are the faces in $\Gamma $ that have exactly three edges in $K$ . The Fries number of $\Gamma $ is the maximum number of benzene rings over all possible Kekulé structures for $\Gamma $ and the set of benzene rings giving the Fries number is called a Fries set. The Clar number is the maximum number of independent benzene rings over all possible Kekulé structures for $\Gamma $ and the set of benzene rings giving the Clar number is called a Clar set. Our method of constructing Kekulé structures for benzenoids generally gives good estimates for the Clar and Fries numbers, often the exact values.     

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.     

Note on the asymptotic isomer count of large fullerenes

The leading term in the large- $N$ asymptotics of the isomer count of fullerenes with $N$ carbon atoms is extracted from the published enumerations for $N\le 400$ with the help of methods of series analysis. The uncovered simple $N^9$ scaling is distinct from isomer counts of most classes of chemical structures that conform to mixed exponential/power-law asymptotics. The second leading asymptotic term is found to be proportional to $N^{25/3}$ . A conjecture concerning isomer counts of the IPR fullerenes is also formulated.     

Bound state solutions of the Deng–Fan molecular potential with the Pekeris-type approximation using the Nikiforov–Uvarov (N–U) method

By employing the Pekeris-type approximation to deal with the centrifugal term, we solve the Schrödinger equation with the Deng–Fan molecular potential for all values of $l$ (orbital angular momentum quantum number). Using the Nikiforov–Uvarov (N–U) method, the approximate analytical bound state energy eigenvalues and the corresponding wave functions are obtained. The results obtained are in good agreement with those ones found in the literature. The bound state energy eigenvalues for a set of diatomic molecules (HCl, LiH, H $_{2}$ , ScH, TiH, VH, CrH, CuLi, TiC, NiC, ScN and ScF) corresponding to the Deng–Fan molecular potential for arbitrary values of n and $l$ quantum numbers are reported.     

A note on Li–Yorke chaos in a coupled lattice system related with Belusov–Zhabotinskii reaction

This paper is concerned with the following system which comes from a lattice dynamical system stated by Kaneko in (Phys Rev Lett 65:1391–1394, 1990) and is related to the Belusov–Zhabotinskii reaction: $$\begin{aligned} x_{n}^{m+1}=(1-\varepsilon )f(x_{n}^{m})+\frac{1}{2}\varepsilon \left[ f(x_{n-1}^{m})+f(x_{n+1}^{m})\right] , \end{aligned}$$ x n m + 1 = ( 1 ? ε ) f ( x n m ) + 1 2 ε [ f ( x n ? 1 m ) + f ( x n + 1 m ) ] , where $m$ m is discrete time index, $n$ n is lattice side index with system size $L$ L (i.e., $n=1, 2, \ldots , L$ n = 1 , 2 , … , L ), $\varepsilon $ ε is coupling constant, and $f(x)$ f ( x ) is the unimodal map on $I$ I (i.e., $f(0)=f(1)=0$ f ( 0 ) = f ( 1 ) = 0 and $f$ f has unique critical point $c$ c with $0<c<1$ 0 < c < 1 and $f(c)=1$ f ( c ) = 1 ). It is proved that for coupling constant $\varepsilon =1$ ε = 1 , this CML (Coupled Map Lattice) system is chaotic in the sense of Li–Yorke for each unimodal selfmap on the interval $I=[0, 1]$ I = [ 0 , 1 ] .     

Electron-pair shell density approximation applied to inner and outer densities of atoms

The shell density approximation to the electron-pair radial density of atoms is applied to the inner $D_< (r)$ and outer $D_> (r)$ densities, which are two components of the single-electron density $D(r)$ . The inner and outer densities are found to be expressed by product sums of shell densities and shell distributions or their complements. The expressions clarify physical meaning of the two densities and give examples for constructing two-electron properties from single-electron properties. Examination of the 53 atoms He through Xe shows that the quantum similarity indices between the original and approximate densities, bounded by 0 (complete dissimilarity) and 1 (complete similarity), are never smaller than 0.99998 and 0.99987 for the inner and outer densities, respectively. The local nature of the shell density and the monotonically increasing property of the shell distribution are used to derive simple shellwise lower and upper bounds to $D_< (r)$ and $D_> (r)$ in terms of $D(r)$ and the numbers of shell electrons. Numerical tests of the bounds demonstrate their utility.     

Clar and Fries numbers for benzenoids

A Kekulé structure of a benzenoid or a fullerene $\Gamma $ Γ is a set of edges $K$ K such that each vertex of $\Gamma $ Γ is incident with exactly one edge in $K$ K . The set of faces in $\Gamma $ Γ that have exactly three edges in $K$ K are called the benzene faces of $K$ K . The Fries number of $\Gamma $ Γ is the maximum number of benzene faces over all possible Kekulé structures for $\Gamma $ Γ . The Clar number is the maximum number of independent benzene faces over all possible Kekulé structures for $\Gamma $ Γ . It is often assumed, but never proved, that some set of independent benzene faces giving the Clar number is a subset of a set of benzene faces giving the Fries number. In Hartung (The Clar structure of fullerenes, Ph.D. Dissertation. Syracuse University, 2012) it is shown that this assumption is false for a large class of fullerenes. In this paper, we prove that this assumption is valid for a large a class of benzenoids.     

Short- and long-range behavior of the inner and outer densities

We report the short- and long-range behavior of the inner $ \rho_<(r) $ and outer $ \rho_>(r) $ densities, which result from a rigorous partitioning of the spherically averaged one-electron density $ \rho(r) $ in many-electron atoms. It is found that for a small r, $ \rho_<(r) $ has one-electron nature but $ \rho_>(r) $ has two-electron nature. For a large r, however, the opposite is true.     

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.     

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}}^{ - } $ .     

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.     

The HOMFLY polynomials of odd polyhedral links

This paper extends the methodology for the construction of odd polyhedral links. Building blocks are odd chain tangles, each of which consists of finitely many $2n+1$ -twist tangles for any nonnegative integer $n$ . For any polyhedral graph $G$ , replacing each edge with an odd chain tangle results in an infinite collection of odd polyhedral links. The relationship between the HOMFLY polynomials of these odd links and the $Q^{d}$ -polynomial of $G$ is established. It leads to the determination of the span of the HOMFLY polynomial, the bound on the braid index and the genus of each odd link. Our results show that these indices depend not only on the building blocks but also on the graph $G.$