How good is a Broyden–Fletcher–Goldfarb–Shanno-like update Hessian formula to locate transition structures? Specific reformulation of Broyden–Fletcher–Goldfarb–Shanno for optimizing saddle points

Based on a study of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) update Hessian formula to locate minima on a hypersurface potential energy, we present an updated Hessian formula to locate and optimize saddle points of any order that in some sense preserves the initial structure of the BFGS update formula. The performance and efficiency of this new formula is compared with the previous updated Hessian formulae such as the Powell and MSP ones. We conclude that the proposed update is quite competitive but no more efficient than the normal updates normally used in any optimization of saddle points. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 349–362, 1998     

Broder and Karlin's formula for hitting times and the Kirchhoff Index

We give an elementary proof of an extension of Broder and Karlin's formula for the hitting times of an arbitrary ergodic Markov chain. Using this formula in the particular case of random walks on graphs, we give upper and tight lower bounds for the Kirchhoff index of any N‐ vertex graph in terms of N and its maximal and minimal degrees. We also apply the formula to a closely related index that takes into account the degrees of the vertices between which the effective resistances are computed. We give an upper bound for this alternative index and show that the bound is attained—up to a constant—for the barbell graph. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Accurate and stable numerical Hartree–Fock calculations for atoms. I. The 1s2 ground state of H−, He,Li+, and Be++

New techniques have been developed for atomic self-consistent-field calculations by numerical integration. For the origin and tail regions we present analytical expansions which can represent the solutions to high accuracy. For the numerical integration in the central region a five-point generalization of the Numerov formula is used; the error term is of the order h¹⁰. While this formula is unstable if used in the customary way, stability is achieved by using a Gaussian elimination technique. The new procedures are tested on the ground state of the helium isoelectronic series; with 251 integration points all quantities are calculated with an inherent accuracy of better than 10^–11.     

Titanium (IV) chloride complexes with salen ligands supported on magnesium carrier: Synthesis and use in ethylene polymerization

The magnesium support with the formula MgCl₂(THF)_0.32(Et₂AlCl)_0.36 was used for immobilization of salen complexes of titanium [Ti(salen)Cl₂, Ti(salen(OMe)₂)Cl₂]. The effects of the catalyst composition (i.e. type of titanium complex and type of activator), polymerization temperature, polymerization time, and the effect of comonomer (1‐octene) on the activity of the obtained supported catalysts, on the polymer characteristics (molecular weight, molecular weight distribution, melting point), and on the polymer morphology were studied. The findings were compared to those obtained for corresponding unsupported systems. Catalysts immobilization results in considerable changes in catalysts activity and in properties of resultant polymers. The studied supported catalysts are highly active in ethylene polymerization, their activity increases with increasing temperature and lasts at least 2 hours. Their copolymerizing ability towards 1‐octene is rather low. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 6693–6703, 2009     

Interfacial properties of compressible polymer solutions

A chemomechanical model for the interfacial concentration and density in compressible polymer solutions is formulated using variational principles. The nonlinear model with boundary conditions obtained from phase equilibrium calculations gives the coupled concentration and density profiles. The couplings between chemical and mechanical balances are identified and efficient ways to calculate the interfacial structure is identified. A specific model appropriate to high‐pressure processing of the polyolefins is developed using the modified Sanchez Lacombe equation of state. Bakker's formula for the interfacial tension is adapted to compressible polymer solutions. The structure and tension of a flat interface is characterized using the developed model and material properties of three molecular weight hydrogenated polybutadiene; the main variables of interest were the pressure, polymer molecular weight, and temperature. The relation between the pressure profile across the interface and the interfacial tension is characterized. Scaling power laws for interfacial tension and interfacial thickness as a function of pressure are obtained and contrasted with the corresponding laws observed and predicted for incompressible polymer solutions. It is found that the modified Sanchez Lacombe‐based power law prediction predictions for compressible solutions in terms of pressure quenches are similar to those from those obtained by the Flory‐Huggins incompressible model for temperature quenches. The present results provide the basis for the future study of the kinetics of pressure‐induced phase separation in compressible polymer solutions. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 640–654, 2009     

Evaluation of molecular integral of Cartesian Gaussian type basis function with complex‐valued center coordinates and exponent via the McMurchie–Davidson recursion formula,and its application to electron dynamics

McMurchie–Davidson recursion formula is extended to derive the ab initio molecular integrals with higher angular quantum number complex Gaussian type basis function which has complex‐valued center coordinates and a complex‐valued exponent. Using the analytical recursion formulae, some calculations of electronic dynamics after beta decay of tritium hydride molecular ion HT⁺ are performed by a quantum wave packet method with thawed Gaussian basis functions of s‐ and p‐type. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Scaling law for the radius of gyration of proteins and its dependence on hydrophobicity

The scaling law between the radius of gyration and the length of a polymer chain has long been an interesting topic since the Flory theory. In this article, we seek to derive a unified formula for the scaling exponent of proteins under different solvent conditions. The formula is obtained by considering the balance between the excluded volume effect and elastic interactions among monomers. Our results show that the scaling exponent is closely related to the fractal dimension of a protein's structure at the equilibrium state. Applying this formula to natural proteins yields a 2/5 law with fractal dimension 2 at the native state, which is in good agreement with other studies based on Protein Data Bank analysis. We also study the dependence of the scaling exponent on the hydrophobicity of a protein chain through a simple two‐letters HP model. The results provides a way to estimate the globular structure of a protein, and could be helpful for the investigation of the mechanisms of protein folding. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 207–214, 2009     

Preparation and crystal structures of two salts with the 5‐nitrotetrazolate anion

Tetraphenylphosphonium 5‐nitrotetrazolate ( 2 ) was prepared by metathesis of sodium 5‐nitrotetrazolate dihydrate ( 1 ; NaNT) with tetraphenylphosphonium chloride in acetone. The new compound was fully characterized by vibrational (IR, Raman) and NMR (¹H, ¹³C, and ¹⁴N) spectroscopies, elemental analysis, and mass spectrometry. Attempted synthesis of 2‐methyl‐5‐nitrotetrazole (2‐MeNT) by methylation of 1 with dimethylsulfate at reflux from acetonitrile failed, and crystals of an explosive compound with the formula (NaNT)₂(H₂O)₂CH₃CN ( 3 ), NT = 5‐nitrotetrazolate, formed. X‐ray diffraction techniques were used to determine the crystal structure of 2 and 3 . Compound 2 crystallizes in the orthorhombic space group P2₁2₁2₁ with four molecules in the unit cell and unit cell parameters a = 7.7413(4) Å, b = 13.624(1) Å, c = 21.252(1) Å, and V = 2241.5(2) ų, whereas 3 crystallizes in the orthorhombic space group Ama2 with four formula unit in the unit cell and unit cell parameters a = 14.805(6) Å, b = 9.908(4) Å, c = 8.940(3) Å, and V = 1311.4(1) ų. © 2009 Wiley Periodicals, Inc. Heteroatom Chem 20:35–44, 2009; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/hc.20509     

A fitting formula for the falloff curves of unimolecular reactions,II: Tunneling effects

By taking into account effects of tunneling on the shape of the falloff curves of unimolecular reaction rate constants based on Troe's weak collision integral, a fitting formula for these curves is developed and then examined for several tunneling‐affected reactions. It is demonstrated that, compared to the widely used Troe's formula, the present expression not only substantially enhances the accuracy in fitting for tunneling‐affected reactions, but it is also computationally more efficient in its evaluation. © 2010 Wiley Periodicals, Inc. Int J Chem Kinet 43: 31–42, 2011     

Inactivation of Enterobacter sakazakii of dehydrated infant formula by gamma-irradiation

Enterobacter sakazakii has been implicated as a causal organism in a severe form of neonatal meningitis, with reported mortality rates of 20%. The population at greatest risk is immunocompromised infants of any age. Dried infant formula has been identified as a potential source of the organism in both outbreaks and sporadic cases. The objective of this study was to investigate theirradiation effect of the inactivation on E. sakazakii (ATCC 29544) of a dehydrated infant formula. The D₁₀-values were 0.22–0.27 and 0.76 kGy for broth and dehydrated infant formula, respectively. The irradiation at 5.0 kGy was able to completely eliminate the E. sakazakii inoculated at 8.0 to 9.0 log CFU g⁻¹ onto a dehydrated infant formula. There was no regrowth for all samples during the time they were stored at 10 °C for 6 h after rehydration. The present results indicated that a gamma-irradiation could potentially be used to inactivate E. sakazakii in a dehydrated powdered infant formula.     

Influences of molecular packing on the charge mobility of organic semiconductors: from quantum charge transfer rate theory beyond the first-order perturbation

The electronic coupling between adjacent molecules is an important parameter for the charge transport properties of organic semiconductors. In a previous paper, a semiclassical generalized nonadiabatic transition state theory was used to investigate the nonperturbative effect of the electronic coupling on the charge transport properties, but it is not applicable at low temperatures due to the presence of high-frequency modes from the intramolecular conjugated carbon-carbon stretching vibrations [G. J. Nan et al., J. Chem. Phys., 2009, 130, 024704]. In the present paper, we apply a quantum charge transfer rate formula based on the imaginary-time flux-flux correlation function without the weak electronic coupling approximation. The imaginary-time flux-flux correlation function is then expressed in terms of the vibrational-mode path average and is evaluated by the path integral approach. All parameters are computed by quantum chemical approaches, and the mobility is obtained by kinetic Monte-Carlo simulation. We evaluate the intra-layer mobility of sexithiophene crystal structures in high- and low-temperature phases for a wide range of temperatures. In the case of strong coupling, the quantum charge transfer rates were found to be significantly smaller than those calculated using the weak electronic coupling approximation, which leads to reduced mobility especially at low temperatures. As a consequence, the mobility becomes less dependent on temperature when the molecular packing leads to strong electronic coupling in some charge transport directions. The temperature-independent charge mobility in organic thin-film transistors from experimental measurements may be explained from the present model with the grain boundaries considered. In addition, we point out that the widely used Marcus equation is invalid in calculating charge carrier transfer rates in sexithiophene crystals.     

Remarks on the updated Hessian matrix methods

Optimizing a function with respect to a set of variables using the quasi‐Newton–Raphson method implies updating the Hessian matrix at each iteration. The Broyden–Fletcher–Goldfarb–Shanno update formula is used for minimization and the Murtagh–Sargent–Powell update formula for optimization of first‐order saddle points. Two new formulae are proposed to update the Hessian matrix. One of these formulae is derived using exponential weights and should be used to locate first‐order saddle points. The second formula is a modification of the TS–Broyden–Fletcher–Goldfarb–Shanno update and could used for both minimum and first‐order saddle point optimizations. These two update Hessian matrix formulae present a performance that is the same and in many cases better that the Broyden–Fletcher–Goldfarb–Shanno and Murtagh–Sargent–Powell formulae. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 324–332, 2003     

Energy-density relations in momentum space

The integrated Hellmann-Feynman theorem is used to derive a rigorous relation between the energy and the electron density in momentum space. Choosing the electron mass as a differential parameter, we obtain a formula corresponding to the Wilson-Frost formula in coordinate space. Analysing the mass-dependence of momentum density, we then show that the present formula is equivalent to one of the previous results deduced from the virial theorem. Use of the integral Hellmann-Feynman theorem is also discussed. Several illustrative examples are given for the calculation of energy from momentum density.     

Synthesis and characterization of fluid 1,3‐benzoxazine monomers and their thermally activated curing

Novel mono‐ and difunctional aliphatic oxyalcohol‐based benzoxazines have been synthesized and characterized in detail. Molecular structures of the monomers were investigated by spectral analysis. The obtained benzoxazine monomers exhibit fluidic behavior, which makes them particularly useful for many applications compared to other traditional benzoxazines. Differential scanning calorimetry was used to monitor the thermal crosslinking behavior of synthesized monomers. Mono‐ and bifunctional benzoxazine monomers exhibited low curing exhothermic peak with the onset around 173 and 180 °C, respectively. Relatively, low ring‐opening polymerization temperature was due to the hydroxyl groups present in the structure of the monomers. The hydrogen bonding of hydroxyl groups may cause alignment of the monomers in the liquid state. Thermal stabilty of the polybenzoxazines was studied by thermogravimetric analysis. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2009     

A Lie group method for molecular rovibrational spectra via the broken symmetry of U(r)(2)⊗U(υ)(4)

A special group chain that is appropriate for describing the rovibrational spectra of linear triatomic molecules with respect to both the vibration and the rotation of molecules as harmonic oscillators is given, and the corresponding Hamiltonian is constructed. The eigenvalue expression of the Hamiltonian is similar to the formula commonly used to calculate the rovibrational spectra of linear triatomic molecules. This method eliminates the physical uncertainty brought about by a variety of group chains. The relationships between parameters in the present expression and those in the commonly used expressions are given. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 53–59, 2001     

Removal of Pb and Cd from aqueous media and fish liver using novel polyurethane foam functionalized with pyrazolone as a new metal ion collector

In the present paper, an off-line preconcentration procedure for the determination of cadmium and lead by flame atomic absorption spectrometry (FAAS) is proposed. Polyurethane foam (PUF) functionalized with o-aminophenol (o-AP) followed by Pyrazolone (Pyr) packed in a minicolumn was used as a sorbent material. The metals were retained on the modified PUF, from which it could be eluted and effectively preconcentrated. The detection limits were 0.072 and 0.016 μg L⁻¹ for Pb and Cd respectively. Enrichment factors were 250 and 319 for lead and cadmium respectively. The procedure has been applied successfully to metal determination in water samples, fish liver and reference material.      

On the Kirchhoff index of regular graphs

Using probabilistic tools, we give a compact formula for the Kirchhoff index of any d‐regular N‐vertex graph in terms of d, N, and Kemeny's constant, as well as general upper and lower bounds in terms of d and N. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Semiempirical evaluation of the global hardness of the atoms of 103 elements of the periodic table using the most probable radii as their size descriptors

Relying upon the fact that the density functional computation of the global hardness of the atoms of the elements are still at large and there is some mathematical in congruency between the theory and operational formula of finite difference approximation, we have suggested a radial‐dependent ansatz for evaluating global hardness of atoms as: η=a(7.2/r)+b (in eV), where, “a” and “b” are the constants and r is the absolute radius of atoms in angstrom unit. The ansatz is invoked to evaluate the global hardness of atoms of 103 element of the periodic table. The evaluated new set of global hardness is found to satisfy the sine qua non of a reasonable scale of hardness by exhibiting perfect periodicity of periods and groups and correlating the gross physicochemical properties of elements. The inertness of Hg and extreme reactivity Cs atoms are nicely correlated. The chemical reactivity and its variation in small steps in the series of lanthanide elements are also nicely reproduced. The results of the present semiempirical calculation also have strong correlation with the result of some sophisticated DFT calculation for a set of atoms. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Azine bridge formation during the electrografting of acrylonitrile

We report the electrografting of polyacrylonitrile (PAN) films onto evaporated gold, silver, and copper substrates, and onto platinum foil. The FTIR spectra of the films are virtually identical to library spectra of PAN and show no evidence of the cyclization observed by some authors. Raman spectra of the same films, however, display a broad absorption band centered at ～1520 cm^?1 and a second band at 1096 cm^?1, which are neither present in spin‐coated films of commercially obtained PAN nor are they present in the FTIR spectra. These previously unreported Raman bands are dependent on a number of factors in the electrografting process; the working electrode metal, the monomer concentration, the solvent used, and the electrochemical protocol used for cleaning the working electrode. The relative intensity of the bands was found to be strongest in films grown on platinum and weakest in those grown on gold, and stronger in thin films (<50 nm) than in thick films (>100 nm). We suggest that these new spectral features are associated with azine‐linked polymer chains in a ladder‐like structure close to the electrode surface as a result of the predominantly isotactic brush‐like conformation of the polymer during the early stages of electrografting. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 1685–1695, 2009     

Reconsidering an analytical gradient expression within a divide-and-conquer self-consistent field approach: exact formula and its approximate treatment

An analytical energy gradient formula for the density-matrix-based linear-scaling divide-and-conquer (DC) self-consistent field (SCF) method was proposed in a previous paper by Yang and Lee (YL) [J. Chem. Phys. 103, 5674 (1995)]. Since the formula by YL does not correspond to the exact gradient of the DC-SCF energy, we derive the exact formula by direct differentiation, which requires solving the coupled-perturbed equations while including the inter-subsystem coupling terms. Next, we present an alternative formula for approximately evaluating the DC-SCF energy gradient, assuming the variational condition for the subsystem density matrices. Numerical assessments confirmed that the DC-SCF energy gradient values obtained by the present formula are in reasonable agreement with the conventional SCF values when adopting a reliable buffer region. Furthermore, the performance of the present method was found to be better than that of the YL method.