Correlation between Fourier series expansion and Hückel orbital theory

The Fourier series can be used to describe periodic phenomena such as the one-dimensional crystal wave function. By the trigonometric treatements in Hückel theory it is shown that Hückel theory is a special case of Fourier series theory. Thus, the conjugated π system is in fact a periodic system. Therefore, it can be explained why such a simple theorem as Hückel theory can be so powerful in organic chemistry. Although it only considers the immediate neighboring interactions, it implicitly takes account of the periodicity in the complete picture where all the interactions are considered. Furthermore, the success of the trigonometric methods in Hückel theory is not accidental, as it based on the fact that Hückel theory is a specific example of the more general method of Fourier series expansion. It is also important for education purposes to expand a specific approach such as Hückel theory into a more general method such as Fourier series expansion.     

Computer-assisted spectrophotometry: multicomponent analysis with a discrete fourier transform

A computer-assisted method for analysis of multicomponent mixtures by use of conventional absorbance as well as discrete Fourier transforin coefficients (combined trigonometric functions) is presented. The program can store absorbance data (A vs. λ), process data by convolution with combined trigonometric functions, apply least-squares analysis and solve the resultant simultaneous linear equations, and display data on screen, printer or plotter.     

新极谱法实验数据的富里叶变换平滑

本文提出一种新的富里叶变换平滑算法并应用于处理阶梯扫描伏安法的实验数据。方法的建立主要是基于平滑原理的缩减步骤、三角函数周期性及和角公式计算正弦、余弦的简化方法。本法对新极谱法实验数据的处理较最小二乘多项式拟合平滑效果更显著。     

Approximation of enthalpy of formation by Fourier series

A possible approximation of the enthalpy of formation of C,H,N,O-containing compounds by trigonometric Fourier series has been shown.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 8, pp. 1389–1390, August, 1994.     

Simplification of gamma-ray spectral data by using Fourier transform

A method is proposed to represent gamma-ray response spectra by Fourier series for the purpose of compressing spectral data. The usefulness of the method was confirmed by applying it to a spectral library of a NaI detector. In the method, a response spectrum as a wave is described by superposition of sine (cosine) waves with low frequencies, whose coefficient parameters can be obtained by a Fast Fourier Transform program. The relation between the number of parameters and the fitting error is discussed, and as the result, it is shown that the number of parameters can be reduced to about a half. The merits and features are presented in practical application of the method to the analysis of gamma-ray spectra.     

Overlap Integrals Between Irregular Solid Harmonics and STOs Via the Fourier Transform Methods

In this paper, a unified analytical and numerical treatment of overlap integrals between Slater type orbitals (STOs) and irregular Solid Harmonics (ISH) with different screening parameters is presented via the Fourier transform method. Fourier transform of STOs is probably the simplest to express of overlap integrals. Consequently, it is relatively easy to express the Fourier integral representations of the overlap integrals as finite sums and infinite series of STOs, ISHs, Gegenbauer, and Gaunt coefficients. The another mathematical tools except for Fourier transform have used partial-fraction decomposition and Taylor expansions of rational functions. Our approach leads to considerable simplification of the derivation of the previously known analytical representations for the overlap integrals between STOs and ISHs with different screening parameters. These overlap integrals have also been calculated for extremely large quantum numbers using Gegenbauer, Clebsch-Gordan and Binomial coefficients. The accuracy of the numerical results is quite high for the quantum numbers of Slater functions, irregular solid harmonic functions and for arbitrary values of internuclear distances and screening parameters of atomic orbitals.     

Mathematical Relationships Between Bond-Bending Force Fields

Molecular mechanics softwares adopt various set of force field functions. In some cases, reliable data from one set of force field parameters cannot be used in a software that adopts another set of force field. Using mathematical approach, exact relationships between parameters from three bond-bending force fields, namely the (i) harmonic cosine angle, (ii) polynomial series, and (iii) Fourier series, are herein developed. Parameters from these three potential functions are further related to the approximate form, the harmonic angle function, which is valid for small change in chemical bond angle.     

Evaluation of some trigonometric series occurring in infinite chain polymer calculations

Closed formulas are presented for a number of trigonometric series whose summands contain inverse powers of one or two quadratic forms. It is shown how to arrange the formulas to remove apparent numerical instabilities, and useful numerical approximations are identified together with the range of their validity.     

Some applications of nonlinear convergence accelerators

A number of established nonlinear convergence acceleration methods are presented. It is discussed how these methods, and also some new algorithms, can be applied successfully to Fourier series. Some of the new accelerators are tailored for these-series. Others are also useful for other types of series and sequences, and not only for convergence acceleration, but also for extrapolation or summation problems. Applications to the numerical inversion of the Laplace transform and to a Fourier series arising in the Matsubara formalism are presented. © 1993 John Wiley & Sons, Inc.     

Rotational mechanism for vibrational relaxation in rigid media. Interaction potentials

Using the assumption of pairwise additive forces we derive the interaction potential acting upon a substantial impurity in a crystalline lattice. The angle dependent parts of the force on the impurity internal vibrations are represented in the form of a Fourier series. Numerical calculations of the Fourier coefficients of the force are presented for a model system employing empirical argon—argon interactions. The higher Foutier coefficients are shown to decrease rapidly in conformity with the simple model potential used to describe the vibrational relaxation in a preceding paper.     

Note on the solution of secular problems with two non-orthogonal basis functions

A convenient trigonometric expression for the eigenfunctions and eigenvalues of 2 x 2 secular problems including overlap is presented.     

Antiradical activity of aryl- and hetarylhydrazones in ethylbenzene oxidation

A relation between the kinetic parameters of the antiradical activity of aryl and hetarylhydrazones and their electronic structure is revealed in the form of the correlation between the reaction rate constants, bond energies of the reaction centers and the antioxidants ionization potentials. The relations obtained can be used to predict the antiradical activity in a series of hydrazones.     

Vibrational assignment, structure and intramolecular hydrogen bond of 4-methylamino-3-penten-2-one

The molecular structure, intramolecular hydrogen and vibrational frequencies of 4-methylamino-3-penten-2-one were investigated by a series of density functional theoretical (DFT) calculations and ab initio calculation at the post-Hartree-Fock (MP2) level. Fourier transform infrared and Fourier transform Raman spectra of this compound and its deuterated analogue were clearly assigned. The calculated geometrical parameters show a strong intramolecular hydrogen bond with a N...O distance of 2.622-2.670 A. This bond length is about 0.02 A shorter than that in its parent, 4-amino-3- penten-2-one which is in agreement with spectroscopic results. Furthermore, the conformations of methyl groups with respect to the plane of the molecule and with respect to each other were investigated.     

A new hybrid double divisor ratio spectra method for the analysis of ternary mixtures

A new spectrophotometric method was developed for the simultaneous determination of ternary mixtures, without prior separation steps. This method is based on convolution of the double divisor ratio spectra, obtained by dividing the absorption spectrum of the ternary mixture by a standard spectrum of two of the three compounds in the mixture, using combined trigonometric Fourier functions. The magnitude of the Fourier function coefficients, at either maximum or minimum points, is related to the concentration of each drug in the mixture. The mathematical explanation of the procedure is illustrated. The method was applied for the assay of a model mixture consisting of isoniazid (ISN), rifampicin (RIF) and pyrazinamide (PYZ) in synthetic mixtures, commercial tablets and human urine samples. The developed method was compared with the double divisor ratio spectra derivative method (DDRD) and derivative ratio spectra-zero-crossing method (DRSZ). Linearity, validation, accuracy, precision, limits of detection, limits of quantitation, and other aspects of analytical validation are included in the text.     

Low-dispersion electrokinetic flows for expanded separation channels in microfluidic systems: multiple faceted interfaces

A novel methodology to design on-chip conduction channels is presented for expansion of low-dispersion separation channels. Designs are examined using two-dimensional numerical solutions of the Laplace equation with a Monte Carlo technique to model diffusion. The design technique relies on trigonometric relations that apply for ideal electrokinetic flows. Flows are rotated and stretched along the abrupt interface between adjacent regions having differing specific permeability. Multiple interfaces can be placed in series along a channel. The resulting channels can be expanded to extreme widths while minimizing dispersion of injected analyte bands. These channels can provide a long path length for line-of-sight optical absorption measurements. Expanded sections can be reduced to enable point detection at the exit section of the channel. Designed to be shallow, these channels have extreme aspect ratios in the wide section, greatly increasing the surface-to-volume ratio to increase heat removal and decrease unwanted pressure-driven flow. The use of multiple interfaces is demonstrated by considering several three-interface designs. Faceted flow splitters can be constructed to divide channels into any number of exit channels while minimizing dispersion. The resulting manifolds can be used to construct medians for structural support in wide, shallow channels.     

General synthesis and aggregation behaviour of a series of single-chain 1,omega-bis(phosphocholines)

The synthesis and physicochemical characterisation of a series of polymethylene-1,omega-bis(phosphocholines) with even-numbered chain lengths between 22 and 32 carbon atoms is described. Two new synthetic strategies for the preparation of long-chain 1,omega-diols as hydrocarbon building blocks are presented. The temperature-dependent self-assembly of the single-chain bolaamphiphiles was investigated by cryo transmission electron microscopy (cryo-TEM), differential scanning calorimetry (DSC), and Fourier transform infrared spectroscopy (FTIR).     

Studies of rearrangement reactions of protonated and lithium cationized 2-pyrimidinyloxy-N-arylbenzylamine derivatives by MALDI-FT-ICR mass spectrometry

The rearrangement reactions of protonated and lithium-cationized 2-pyrimidinyloxy-N-arylbenzylamine derivatives were studied by Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR MS) and infrared multiphoton dissociation mass spectrometry (IRMPD). Our results show that three kinds of rearrangement reactions occur in IRMPD processes. First, nearly all protonated 2-pyrimidinyloxy-N-arylbenzylamine derivatives undergo Pathway A to form the K ion series. It is proposed that this rearrangement (migration of a substituted benzyl group) proceeds by way of a gas-phase intramolecular S(N)2 reaction. Second, a gas phase intramolecular S(N)Ar type rearrangement mechanism is proposed to explain the formation of the F ion series from protonated and lithium-cationized 5 (or 6). This skeletal rearrangement reaction competes with the S(N)2 reaction of the Pathway A, which produces the K ion series, in IRMPD of protonated 5 and 6. Third, the formation pathway of the W ion series is explained by a gas phase Cope type rearrangement mechanism.     

Convolution and deconvolution in the synthesis and analysis of staircase voltammograms

The comparison of staircase voltammograms with linear scan voltammograms is complicated by the former's response being dependent on the measurement time. A method for the prediction and removal of that dependence is given, presenting the data in a form similar to that of semi-differential analysis. An efficient algorithm for simulating the response of a reversible system to repetitive stepwise excitation is developed utilizing the fast Fourier transform convolution procedure. A brief comparison of several convolution procedures is presented.     

DFT studies of the conformational/structural dependencies of geminal 1H,1H scalar coupling 2J(H,H') in substituted methanes

A study is presented of the structural dependencies for scalar, interproton J-coupling across two bonds in a series of substituted methanes. The coupled perturbed, density functional theory method with a B3PW91 functional and aug-cc-pVTZ-J basis sets is used to examine coupling between geminal protons (2)J(H,H') in methane and a series of substituted compounds CH(3)X (X = CH3, CH(2)CH(3), CH=CH2, CH=O, and NH2) as functions of the dihedral angle phi measured about the C1-X2 bonds. All four contributions are obtained but all conformational effects are dominated by the Fermi contact term. Simple linear combination of atomic orbitals (LCAO)-molecular orbital (MO) sum-over-states methods are used to examine the relationships of the coupling constants with dihedral angles as well as internal H-C-H and H-C1-X2 angles. This study explores some novel aspects of geminal H-H coupling including an analysis of the asymmetry in the conformational dependencies arising from non-next-nearest neighbor interactions. For each of the substituted methanes, explicit trigonometric/exponential expressions are given and these accurately reproduce the (2)J(H,H') structural dependencies with standard deviations usually less than 0.03 Hz. The molecular structures for representative bicyclic molecules were fully optimized, and DFT results for (2)J(H,H') reproduce all the trends in the experimental data. A discussion is given on the applicability of the equations for H--H coupling in the substituted methanes to coupling in the bicyclic molecules.     

Irradiation methods for production of high specific activity radionuclides in no carrier added form

High Specific Activity Radio-Nuclides (HSARN) are a powerful tool to label a large range of chemical species at very low concentration levels. In order to obtain these radiotracers in a very high specific activity form, it is necessary to optimize the production methods, to separate and purify them from the irradiated target without the addition of inactive carrier, to carry out a series of analytical and radioanalytical tests to determine their true specific activity and to verify a series of purity parameters. A review of irradiation methods and nuclear parameters adopted in our laboratories for the preparation of some tens of radiotracers are presented.