Analysis of critical points on the potential energy surface

A simple classification scheme is proposed for critical points, based only on rankr and signatures of the (n,n)-matrixG of harmonic force constants. The determination ofr ands, e.g. by the well-known factorizationG=L ^T gL (L: triangular matrix,g: diagonal matrix), has several theoretical as well as practical (computational) advantages over the inspection of eigenvalues ofG, so far used in quantum chemistry. The eigenvalues are sufficient butnot necessary for a classification whereas rank and signature are the only necessary and sufficient prerequisites for solving the task. For the purpose of presenting a working example, by calculating only a 2×2 torque constant matrix, it is shown that the coplanar ethylbenzene is unstable in the CNDO/2 picture.     

Quantitative Correlations for the Estimation of Thermodynamic and Molecular Properties of Pyridines and 2,2′-Bipyridyines

Standard heats of formation, entropies, ionization potentials, and molecular dipole moments of a series of pyridines have been calculated by MNDO, AM1 and PM3 methods. Linear relationship have been established which permit a priori estimation of thermodynamic and molecular characteristics of pyridines. Correlation have been found between the values of pK_a for 2,2′-bipyridines for aqueous solutions and their gas phase proton affinities.__________Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 3, 391–402, March, 2005.     

Structural disorder in Ba0.6Sr0.4Al2O4

The structural disorder in Ba_0.6Sr_0.4Al₂O₄ (space group P6₃22) was investigated by X-ray powder diffraction and selected-area electron diffraction (SAED). The initial structural model was determined using direct methods, and it was further modified by the combined use of Rietveld method and maximum-entropy method (MEM). MEM-based pattern fitting method was subsequently applied, resulting in the final reliability indices of R_wp=9.61%, R_p=6.96%, R_B=1.40% and S=1.25. The electron density distribution was satisfactorily expressed by the split-atom model in which the strontium/barium and oxygen atoms were split to occupy the lower symmetry sites. The diffuse scattering in SAED was mainly attributable to the positional disorder of oxygen atoms.     

Syntheses and Characterization of Polymetallosiloxanes from Silicic Acid and Metal Chlorides

Syntheses and characterization of polymetallosiloxanes by the non-hydrolysis sol-gel process using no metal alkoxides were investigated. The reaction of silicic acid (SA) with MCl₄ (M = Ti, Zr) in the molar ratios SA/MCl₄ = 0.5–3.0 using a tetrahydrofuran-methanol solvent formed polymetallosiloxane (PMS), which was insoluble in organic solvents regardless of the molar ratio. The PMS was isolated as esterified polymetallosiloxane by esterification with isopropyl alcohol for various periods, which were soluble in methanol, acetone, and tetrahydrofuran. The number average molecular weight was 1000–3200 for esterified polytitanosiloxane and 3400–11000 for esterified polyzirconosiloxane. Esterified polymetallosiloxanes had no melting point but decomposition point. The results of analytical data indicated that esterified polymetallosiloxane and/or polymetallosiloxane consisted of the main chain of Si–O–Si and Si–O–M linkage with the pendants of alkoxy, silanol, and chloro group.     

Tetraalkylammonium Picrates in the Dichloromethane–Water System: Ion-Pair Formation and Liquid–Liquid Distribution of Free Ions and Ion Pairs

Equilibria concerning picrates of tetraalkylammonium ions (Me₄N⁺, Et₄N⁺, Pr₄N⁺, Bu₄N⁺, Bu₃MeN⁺) in a dichloromethane−water system have been investigated at 25 ^∘C. The 1:1 ion-pair formation constants (K _IP,o ^o) in dichloromethane at infinite dilution were conductometrically determined. The distribution constants (K _D ^o) of the ion pairs and the free cations between the solvents were determined by a batch-extraction method. The K _IP,o ^o value varies in the cation sequence, Bu₄N⁺ ≈ Pr₄N⁺ ≈ Et₄N⁺ < Bu₃MeN⁺ < < Me₄N⁺; this trend is explained by the electrostatic cation−anion interaction taking into account the structures of the ion pairs determined by density functional theory calculations. For the ion pairs of the symmetric R₄N⁺ cations, there is a linear positive relationship between log₁₀ K _D ^o and the number of methylene groups in the cation (N _{CH 2}). The ion pair of asymmetric Bu₃MeN⁺ has a higher distribution constant than that expected from the above log₁₀ K _D ^o versus N _{CH 2} relationship. These cation dependencies of log₁₀ K _D ^o for the ion pairs are explained theoretically by using the Hildebrand-Scatchard equation. For all the cations, the log₁₀ K _D ^o value of the free cation increases linearly with N _{CH 2}; the variation of log₁₀ K _D ^o is discussed by decomposing the distribution constant into the Born-type electrostatic contribution and the non-Born one, and attributed to the latter that is governed by the differences in the molar volumes of the cations. The cation dependencies of the ion-pair extractability and ion pairing in water are also discussed. An erratum to this article can be found at     

Quantum-chemical analysis of the CuCl2 molecule

This paper reports on quantum-chemical analysis of the linear structure of CuCl₂ by Hartree-Fock (HF) and density functional theory (DFT) methods and also by time-dependent HF (TD HF) and DFT (TD DFT) techniques. Using pure DFT exchange correlation functional (B3LYP) yields the best agreement with the experimental electronic spectra of CuCl₂. In this case, the odd electron is delocalized over the molecule, spin density on copper being 0.27. The ground state of the CuCl₂ molecule is ²Π_g with linear geometry.     

A comparison of two algorithms for warping of analytical signals

The alignment of analytical signals is an important preprocessing step when further analysis (e.g. PCA) requires the same lengths of all of them. Two techniques for alignment of profiles, namely dynamic time warping (DTW) and correlation optimized warping (COW) were tested and compared. The attention was focused on chromatographic and spectroscopic profiles. Simulated and two sets of real data were studied in this study.     

Bonferroni-type inequalities; Chebyshev-type inequalities for the distributions on [0, n]

An elementary majorant-minorant method to construct the most stringent Bonferroni-type inequalities is presented. These are essentially Chebyshev-type inequalities for discrete probability distributions on the set {0, 1,..., n}, where n is the number of concerned events, and polynomials with specific properties on the set lead to the inequalities. All the known results are proved easily by this method. Further, the inequalities in terms of all the lower moments are completely solved by the method. As examples, the most stringent new inequalities of degrees three and four are obtained. Simpler expressions of Mrgritescu's inequality (1987, Stud. Cerc. Mat., 39, 246–251), improving Galambos' inequality, are given.     

A modified steepest descent method with applications to maximizing likelihood functions

In maximizing a non-linear function G(), it is well known that the steepest descent method has a slow convergence rate. Here we propose a systematic procedure to obtain a 1–1 transformation on the variables , so that in the space of the transformed variables, the steepest descent method produces the solution faster. The final solution in the original space is obtained by taking the inverse transformation. We apply the procedure in maximizing the likelihood functions of some generalized distributions which are widely used in modeling count data. It was shown that for these distributions, the steepest descent method via transformations produced the solutions very fast. It is also observed that the proposed procedure can be used to expedite the convergence rate of the first derivative based algorithms, such as Polak-Ribiere, Fletcher and Reeves conjugate gradient methods as well.     

Grouped coordinate minimization using Newton's method for inexact minimization in one vector coordinate

Letf(x,y) be a function of the vector variablesx R ⁿ andy R ^m. The grouped (variable) coordinate minimization (GCM) method for minimizingf consists of alternating exact minimizations in either of the two vector variables, while holding the other fixed at the most recent value. This scheme is known to be locally,q-linearly convergent, and is most useful in certain types of statistical and pattern recognition problems where the necessary coordinate minimizers are available explicitly. In some important cases, the exact minimizer in one of the vector variables is not explicitly available, so that an iterative technique such as Newton's method must be employed. The main result proved here shows that a single iteration of Newton's method solves the coordinate minimization problem sufficiently well to preserve the overall rate of convergence of the GCM sequence.The authors are indebted to Professor R. A. Tapia for his help in improving this paper.