An IEF-PCM study of solvent effects on the Faraday {\mathcal{B}} term of MCD

We present the first theoretical investigation of solvent effects on the Faraday ${\mathcal{B}}$ term of magnetic circular dichroism (MCD) at the density–functional level of theory. In our model, the solvent is described by the polarizable continuum model in its integral-equation formulation. We present the extensions required for including electron correlation effects using density–functional theory (DFT) as well as the necessary extensions for including the effects of a dielectric continuum. The new code is applied to the study of the Faraday ${\mathcal{B}}$ term of MCD in a series of benzoquinones. It is demonstrated that electron correlation effects, as described by DFT, are essential in order to recover the experimentally observed signs of the ${\mathcal{B}}$ term. Dielectric continuum effects increase, in general, the magnitude of the ${\mathcal{B}}$ term, leading to an overestimation of the experimental observations in most cases.     

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.     

Molecular Interactions in 1-Ethyl-3-Methylimidazolim Tetrafluoroborate + Amide Mixtures: Excess Molar Volumes, Excess Isentropic Compressibilities and Excess Molar Enthalpies

The densities, ρ ₁₂, and speeds of sound, u ₁₂, of 1-ethyl-3-methylimidazolium tetrafluoroborate (1) + N-methylformamide or N,N-dimethylformamide (2) binary mixtures at (293.15. 298.15. 303.15, 308.15 K), and excess molar enthalpies, $ H_{12}^{\text{E}} $ H 12 E , of the same mixtures at 298.15 K have been measured over the entire mole fraction range using a density and sound analyzer (Anton Paar DSA-5000) and a 2-drop microcalorimeter, respectively. Excess molar volume, $ V_{12}^{\text{E}} $ V 12 E , and excess isentropic compressibility, $ \left( {\kappa_{S}^{\text{E}} } \right)_{12} $ ( κ S E ) 12 , values have been calculated by utilizing the measured density and speed of sound data. The observed data have been analyzed in terms of: (i) Graph theory and (ii) the Prigogine–Flory–Patterson theory. Analysis of the $ V_{12}^{\text{E}} $ V 12 E data in terms of Graph theory suggest that: (i) in pure 1-ethyl-3-methylimidazolium tetrafluoroborate, the tetrafluoroborate anion is positioned over the imidazoliun ring and there are interactions between the hydrogen atom of (C–H{edge}) and proton of the –CH₃ group (imidazolium ring) with fluorine atoms of tetrafluoroborate anion, and (ii) (1 + 2) mixtures are characterized by ion–dipole interactions to form a 1:1 molecular complex. Further, the $ V_{12}^{\text{E}} $ V 12 E , $ H_{12}^{\text{E}} $ H 12 E and $ \left( {\kappa_{S}^{\text{E}} } \right)_{12} $ ( κ S E ) 12 values determined from Graph theory compare well with their measured experimental data.     

A note on the principal measure and distributional ({\varvec{p, q}})-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction

García Guirao and Lampart in (J Math Chem 48:159–164, 2010) presented a lattice dynamical system stated by Kaneko in (Phys Rev Lett 65:1391–1394, 1990) which is related to the Belusov–Zhabotinskii reaction. In this paper, we prove that for any non-zero coupling constant $\varepsilon \in (0, 1)$ , this coupled map lattice system is distributionally $(p, q)$ -chaotic for any pair $0\le p\le q\le 1$ , and that its principal measure is not less than $(1-\varepsilon )\mu _{p}(f)$ . Consequently, the principal measure of this system is not less than $$\begin{aligned} (1-\varepsilon )\left( \frac{2}{3}+\sum \limits _{n=2}^{\infty }\frac{1}{n}\frac{2^{n-1}}{(2^{n}+1) (2^{n-1}+1)}\right) \end{aligned}$$ for any non-zero coupling constant $\varepsilon \in (0, 1)$ and the tent map $\Lambda $ defined by $$\begin{aligned} \Lambda (x)=1-|1-2x|,\quad x\in [0, 1]. \end{aligned}$$      

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.     

Explicit form of Pauli potential for direct derivation of pair density from a two-particle differential equation for the quintet state of four electrons with harmonic interparticle interactions

Recently, an analytic two-particle density matrix (2DM) has been derived for the quintet state of four electrons interacting via two-body harmonic forces. Here we use this 2DM to extract the exact pair density $\Gamma (\mathbf{r}_1,\mathbf{r}_2)$ . This is then employed in the known two-particle partial differential equation for the pair density amplitude to extract the Pauli potential $v_P(\mathbf{r}_1,\mathbf{r}_2)$ for this quintet state.     

Thermodynamic investigations of 1-ethyl-3-methylimidazolium tetrafluoroborate and cycloalkanone mixtures

The densities, ρ, speeds of sound, u, and heat capacities, (C _P)_mix, for binary 1-ethyl-3-methylimidazolium tetrafluoroborate (1) + cyclopentanone or cyclohexanone (2) mixtures within temperature range (293.15–308.15 K) and excess molar enthalpies, H ^E, at 298.15 K have been measured over the entire composition range. The excess molar volumes, V ^E, excess isentropic compressibilities, \( \kappa_{\text{S}}^{\text{E}}, \) and excess heat capacities, \( C_{\text{P}}^{\text{E}}, \) have been computed from the experimental results. The V ^E, \( \kappa_{\text{S}}^{\text{E}} \) , H ^E, and \( C_{\text{P}}^{\text{E}} \) values have been calculated and compared with calculated values from Graph theory. It has been observed that V ^E, \( \kappa_{\text{S}}^{\text{E}} \) , H ^E, and \( C_{\text{P}}^{\text{E}} \) values were predicted by Graph theory compare well with their experimental values. The V ^E, \( \kappa_{\text{S}}^{\text{E}}, \) and H ^E thermodynamic properties have also been analyzed in terms of Prigogine–Flory–Patterson theory.     

Use of auxiliary functions {Q_{ns}^q} and {G_{-ns}^q} in evaluation of multicenter integrals over integer and noninteger n-Slater type orbitals arising in Hartree–Fock–Roothaan equations for molecules

With the help of expansion relations for the two-center Slater type orbitals (STOs) charge densities established by the author from the use of complete orthonormal sets of Ψ ^α -exponential type orbitals (Ψ ^α -ETOs), where α = 1, 0, ? 1, ? 2, . . . , a large number of series expansion formulas for the multicenter integrals of integer and noninteger n-STOs (ISTOs and NISTOs) occurring in Hartree–Fock–Roothaan (HFR) equations for molecules is derived through the auxiliary functions ${Q_{ns}^q}$ and ${G_{-ns}^q}$ , and one- and two-center basic integrals of ISTOs. The analytical relations for basic integrals are presented. As an example of application, the calculations have been performed for the ground state of electronic configuration of ${{\it CH}_4((1a_1)^{2}(2a_1)^{2}(1t_{2x})^{2} (1t_{2y})^{2} (1t_{2z})^{2},{}^1A_1)}$ using combined HFR theory suggested by the author.     

The principal measure and distributional (p, q)-chaos of a coupled lattice system related with Belusov–Zhabotinskii reaction

García Guirao and Lampart (J Math Chem 48:66–71, 2010; J Math Chem 2 48:159–164, 2010) said that for non-zero couplings constant, the lattice dynamical system is more complicated. Motivated by this, in this paper, we prove that this coupled lattice system is distributionally (p, q)-chaotic for any pair 0?≤ p?≤ q?≤ 1 and its principal measure 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 ${0 < \epsilon < 1}$ .     

The equilibrium melting temperature and isothermal crystallisation kinetics of cyclic poly(butylene terephthalate) and styrene maleimide (c-PBT/SMI) blends

The equilibrium melting point ( $ T_{\text{m}}^{0} $ ) and isothermal crystallisation kinetics of cyclic poly(butylene terephthalate) (c-PBT) and styrene maleimide (SMI) blends prepared by solid dispersion and in situ polymerisation of cyclic butylene terephthalate oligomers (CBT) within SMI were investigated. This c-PBT/SMI blend is a miscible semicrystalline–amorphous blend system. The $ T_{\text{m}}^{0} $ of c-PBT/SMI blends was determined using the Hoffman and Weeks method, while Avrami crystallisation kinetic model have been applied to study their isothermal crystallisation kinetics. It was found that $ T_{\text{m}}^{0} $ decreased with increasing SMI content in the blend compositions. All the crystallisation exotherms were obtained from differential scanning calorimetry under isothermal experimental conditions. The average value of Avrami exponent, n, is in the range of 2.4–2.8 for the primary crystallisation process for c-PBT and its blends, which suggest that heterogeneous nucleation of spherulites occurred and growth of spherulites was between two-dimensional and three-dimensional.     

Theoretical study of N–H· · ·O hydrogen bonding properties and cooperativity effects in linear acetamide clusters

We investigated geometry, energy, ${\nu_{{\text{N--H}}}}$ harmonic frequencies, ¹⁴N nuclear quadrupole coupling tensors, and ${n_{\rm O}\to \sigma _{{\text{N--H}}}^\ast}$ charge transfer properties of (acetamide) _n clusters, with n = 1 ? 7, by means of second-order Møller-Plesset perturbation theory (MP2) and DFT method. Dependency of dimer stabilization energies and equilibrium geometries on various levels of theory was examined. B3LYP/6-311++G** calculations revealed that for acetamide clusters, the average hydrogen-bonding energy per monomer increases from ?26.85 kJ mol^?1 in dimer to ?35.12 kJ mol^?1 in heptamer; i.e., 31% cooperativity enhancement. The n-dependent trend of ${\nu_{{\text{N--H}}}\,{and}\,^{14}}$ N nuclear quadrupole coupling values were reasonably correlated with cooperative effects in ${r_{{\text{N--H}}}}$ bond distance. It was also found that intermolecular ${n_{\rm O}\to \sigma_{{\text{N--H}}}^\ast}$ charge transfer plays a key role in cooperative changes of geometry, binding energy, ${\nu_{{\text{N--H}}}}$ harmonic frequencies, and ¹⁴N electric field gradient tensors of acetamide clusters. There is a good linear correlation between ¹⁴N quadrupole coupling constants, C _Q (¹⁴N), and the strength of Fock matrix elements (F _ij ). Regarding the ${n_{\rm O}\to \sigma_{{\text{N--H}}}^\ast}$ interaction, the capability of the acetamide clusters for electron localization, at the N–H· · ·O bond critical point, depends on the cluster size and thereby leads to cooperative changes in the N–H· · ·O length and strength, N–H stretching frequencies, and ¹⁴N quadrupole coupling tensors.     

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).     

Volumetric Properties of 1,3-Dioxolane + Toluene + o- or p-Xylene Ternary Mixtures at 25.00 °C and Atmospheric Pressure

Excess molar volumes, $ V_{123}^{\text{E}} $ V 123 E , of 1, 3-dioxolane (1) + toluene (2) + o- or p-xylene (3) ternary mixtures have been determined dilatometrically over the entire composition range at 298.15 K. For thermodynamic consistency the experimental values were fitted to Redlich–Kister Equation. The $ V_{123}^{\text{E}} $ V 123 E values of 1, 3-dioxolane (1) + toluene (2) + o- or p-xylene (3) ternary mixtures have been found to be negative over the whole composition range. It has been observed that $ V_{123}^{\text{E}} $ V 123 E values calculated by graph theory are of the same sign and magnitude with respect to their experimental values.     

Density functional study on structures, stabilities, and electronic properties of size-selected Pd n Si q (n = 1–7 and q = 0, +1, −1) clusters

The density functional theory (DFT) calculations within the framework of generalized gradient approximation have been employed to systematically investigate the geometrical structures, stabilities, and electronic properties of Pd _n Si ^q (n = 1–7 and q = 0, +1, ?1) clusters and compared them with the pure ${\text{Pd}}_{n + 1}^{q}$ (n = 1–7 and q = 0, +1, ?1) clusters for illustrating the effect of doping Si atom into palladium nanoclusters. The most stable configurations adopt a three-dimensional structure for both pure and Si-doped palladium clusters at n = 3–7. As a result of doping, the Pd _n Si clusters adopt different geometries as compared to that of Pd _n+1. A careful analysis of the binding energies per atom, fragmentation energies, second-order difference of energies, and HOMO–LUMO energy gaps as a function of cluster size shows that the clusters ${\text{Pd}}_{4}^{ + }$ , ${\text{Pd}}_{4}$ , ${\text{Pd}}_{8}^{ - }$ , ${\text{Pd}}_{5} {\text{Si}}^{0, + , - }$ , and ${\text{Pd}}_{7} {\text{Si}}^{0, + , - }$ possess relatively higher stability. There is enhancement in the stabilities of palladium frameworks due to doping with an impurity atom. In addition, the charge transfer has been analyzed to understand the effect of doped atom and compared further.     

Symmetry-itemized enumeration of quadruplets of RS-stereoisomers. II. The partial-cycle-index method of the USCI approach combined with the stereoisogram approach

The symmetry-itemized enumeration of quadruplets of stereoisograms is discussed by starting from a tetrahedral skeleton, where the partial-cycle-index (PCI) method of the unit-subduced-cycle-index approach (Fujita in Symmetry and combinatorial enumeration of chemistry. Springer, Berlin, 1991) is combined with the stereoisogram approach (Fujita in J Org Chem 69:3158–3165, 2004, Tetrahedron 60:11629–11638, 2004). Such a tetrahedral skeleton as contained in the quadruplet of a stereoisogram belongs to an RS-stereoisomeric group denoted by $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ , where the four positions of the tetrahedral skeleton accommodate achiral and chiral proligands to give quadruplets belonging to subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ according to the stereoisogram approach. The numbers of quadruplets are calculated as generating functions by applying the PCI method. They are itemized in terms of subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ , which are further categorized into five types. Quadruples for stereoisograms of types I–V are ascribed to subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ , where their features are discussed in comparison between RS-stereoisomeric groups and point groups.     

Natural Bond Orbital (NBO) Population Study of Some Para-Substituted N-Methyl-N-nitrosobenzenesulfonamide Biological Molecules in MeCN Solution

A theoretical study of several para-substituted N-methyl-N-nitrosobenzenesulfonamide biological molecules in MeCN solution has been performed using quantum computational ab initio RHF and density functional B3LYP and B3PW91 methods with the 6-311++G(d,p) basis set. Geometries obtained from DFT calculations were used to perform natural bond orbital analysis. The results show that an intramolecular hydrogen bond exists in the selected molecules, which is confirmed by the NBO analysis. The p characters of the two nitrogen natural hybrid orbitals $ \sigma_{{{\text{N}}3 - {\text{N}}2}} $ increase with increasing $ \sigma_{p} $ values of the para-substituent group on the benzene ring, which results in a lengthening of the N₃–N₂ bond. It is noted that the weakness of the N–N bond is due to $ n_{{{\text{O}}1}} \to \sigma_{{{\text{N}}3 - {\text{N}}2}}^{*} $ delocalization and is responsible for the longer N₃–N₂ bond. In addition, there is a direct correlation between hyperconjugation $ n_{{{\text{O}}1}} \to \sigma_{{{\text{N}}3 - {\text{N}}2}}^{*} $ and the bond dissociation energy in the system, which is confirmed by comparison with isoelectronic isomers.     

CORAL: QSPRs of enthalpies of formation of organometallic compounds

Available on the Internet the CORAL software gives reasonable good prediction for standard enthalpy of formation for selected organometallic compounds (n = 132). The approach is tested using five random splits of the considered data into the sub-training set (n = 32–49), calibration set (n = 36–51), test set (n = 10–29), and the validation set (n = 22–41). Compounds of the validation set are not involved in building up the models. The average statistical quality of prediction is the following: correlation coefficient ( $\overline{R^{2}} )$ R 2 ¯ ) $0.991\pm 0.005$ 0.991 ± 0.005 and standard error of estimation ( $\overline{s} )$ s ¯ ) $22.9 \pm 5.6$ 22.9 ± 5.6  kJ/mol.     

Excess Molar Volumes and Excess Isentropic Compressibilities of o-Toluidine + Tetrahydropyran with Pyridine or Benzene or Toluene Ternary Mixtures at 298.15, 303.15 and 308.15 K

The densities, ρ ₁₂₃, and speeds of sound, u ₁₂₃, of ternary o-toluidine (OT, 1) + tetrahydropyran (THP, 2) + pyridine (Py) or benzene or toluene (3) mixtures have been measured as a function of composition at 298.15, 303.15 and 308.15 K. Values of the excess molar volumes, $ V_{123}^{\text{E}} , $ and excess isentropic compressibilities, $ (\kappa_{\text{S}}^{\text{E}} )_{123} , $ of the studied mixtures have been determined by employing the measured experimental data. The observed thermodynamic properties were fitted with the Redlich–Kister equation to determine adjustable ternary parameters and standard deviations. The $ V_{123}^{\text{E}} $ and $ (\kappa_{\text{S}}^{\text{E}} )_{123} $ values were also analyzed in terms of Graph theory. It was observed that Graph theory correctly predicts the sign as well as magnitude of $ V_{123}^{\text{E}} $ and $ (\kappa_{\text{S}}^{\text{E}} )_{123} $ values of the investigated mixtures. Analysis of the data suggests strong interactions and a more close packed arrangement in OT (1) + THP (2) + Py (3) mixtures as compared to those of the OT (1) + THP (2) + benzene (3) or toluene (3) mixtures. This may be due to the presence of a nitrogen atom in Py which results in stronger interactions for the OT:THP molecular entity as compared to those with benzene or toluene.     

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 ] .