Systematic study of elementary models of analytical signals in the form of peaks and waves

An analytical signal represented as a symmetrical peak or a corresponding integral curve (wave) was described using three elementary functions: Gaussian function, derivative of a logistic function, and Cauchy function. The shape and geometric properties of such an analytic peak were characterized by a triangular frame formed by the tangents at the inflection points and the asymptotes to peak branches. In the case of a wave, a frame formed by the tangent at the inflection point of the wave and the asymptotes to its lower and upper branches was used for the same purpose. The use of the shape of differential curves as increments for physicochemical calculations was discussed.     

Modeling of analytical peaks: peaks properties and basic peak functions

The general approach to the detailed characterizing of peak properties based on the use of characteristic points on a peak contour located at the certain levels is proposed. Three basic functions of peaks have been chosen which are submitted in normalized form on height and width. The ability to characterize a peaks shape of a certain analytical series is shown by an example of series of analytical peaks of thallium obtained by stripping voltammetry at linear and step potential sweep. Also the proposed shape parameters are studied for groups of stripping voltammetry and chromatographic peaks.     

The fitting of depth distributions of ion implanted analytes by split student-t functions: The physical basis and a recommended procedure

The non-standardized fitting of depth distributions of ion implanted analytes in solid matrices by arbitrarily chosen functions, as practiced at present, is unsatisfactory for the communication of results and is unacceptable for certification of ion implanted reference materials. Sampling theory applied to the nuclear slowing-down process of swift ions suggests that split Student-t functions should provide the mathematical representation sought. The exact shape and mathematical form of these functions are fully described and readily communicated by the use of five parameters (mode, two shape parameters and two scale parameters). A simple and fast fitting procedure is proposed.Dedicated to Professor Günther Tölg on the occasion of his 60th birthday     

A Mathematical Discussion on Density and Shape Functions, Vector Semispaces and Related Questions

Density and shape functions are studied from the point of view of vector semispace structure and properties. Useful characteristics based on the shell structure of vector semispaces are used to analyze some properties of both functions. Fukui functions and quantum similarity indices are also studied when basic applications of the theory are discussed. Construction of approximate density functions and pseudo wave functions is also outlined. Finally, the original DFT variational theorem is reformulated within the frame of the shape function.     

Rotational symmetry of the molecular potential energy in the Cartesian coordinates

We consider the molecular Born-Oppenheimer potential energy as a function of atomic Cartesian coordinates and discuss the non-stationary Hessian properties arising due to rotational symmetry. A connection with the extended Hessian theory is included. New applications of Cartesian representation for examining and correcting raw numerical Hessian data and a simple formulation of harmonic vibrational analysis of partially optimized systems are proposed. Exemplary calculations for the porphyrin molecule with an internal proton transfer are reported. We also develop the normal transformation method to incorporate the rotational symmetry into the approximate analytical potentials, which are parametrized in the Cartesian coordinates. The transformation converts the coordinates from the space fixed frame to the frame which translates and rotates with the molecule and is determined by the Eckart conditions. New simple analytical formulas for the first and second derivatives of the transformed potential are derived. This fast method can be used to calculate the potential and its derivatives in the simulations of chemical reaction dynamics in the space fixed Cartesian frame without the need to constrain the molecular rotation or to define the local non-redundant internal coordinates.     

Dependence of approximate ab initio molecular loge sizes on the quality of basis functions

Size and shape parameters for the core, bonding, and lone electron pairs of the ten-electron hydrides (CH₄, NH₃, H₂O, HF) were determined from ab initio MO wave functions using various Gaussian basis sets. The fundamental features of approximate electron pair loge representation are somewhat more sensitive to the quality of the basis functions than the molecular total energy. The total size of the molecular electron distribution is less affected by basis set variations than its components: the core, bonding, and lone pair sizes. There is an apparent tendency to “preserve” the total size of molecular distribution.     

An empirically validated analytical model of droplet dynamics in electrowetting on dielectric devices

Explicit analytical models that describe the capillary force on confined droplets actuated in electrowetting on dielectric devices and the reduction in that force by contact angle hysteresis as a function of the three-dimensional shape of the droplet interface are presented. These models are used to develop an analytical model for the transient position and velocity of the droplet. An order of magnitude analysis showed that droplet motion could be modeled using the driving capillary force opposed by contact angle hysteresis, wall shear, and contact line friction. Droplet dynamics were found to be a function of gap height, droplet radius, surface tension, fluid density, the initial and deformed contact angles, contact angle hysteresis, and friction coefficients pertaining to viscous wall friction and contact line friction. The first four parameters describe the device geometry and fluid properties; the remaining parameters were determined experimentally. Images of the droplet during motion were used to determine the evolution of the shape, position, and velocity of the droplet with time. Comparisons between the measured and predicted results show that the proposed model provides good accuracy over a range of practical voltages and droplet aspect ratios.     

A rigorous comparison of theoretical autocorrelation functions with dielectric relaxation parameters for the alpha process of polymers

Recently, tables of parameters used to represent experimental dielectric relaxation data as well as autocorrelation functions have become available. The experimental and autocorrelation function data were represented with the Havriliak–Negami function using rigorous statistical techniques. These tables include not only parameters and their temperature dependencies, but also the confidence intervals for all of the parameters. The important parameters for this work are the two shape parameters, α and β, which represent the width and skewness of the relaxation process, respectively. A comparison is made between the experimental data and several autocorrelation functions by calculating the minimum distance (in units of standard deviations) between the experimental values of αβ parameters for a specific polymer and the αβ parameters corresponding to the autocorrelation functions reported in these tables. Quantities derived from these minimum distances (a distribution function and an error function) are reported for each of the autocorrelation functions. These results are discussed in terms of the basic assumptions of the mode coupling theory. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35: 1887–1897, 1997     

Sum and density of states of polyatomic systems with hindered rotors

The density or sum of states for a collection of independent oscillators, free rotors, and one-dimensional hindered rotors is obtained with good accuracy by numerical inversion of the corresponding total partition function by the method of steepest descents. The hindered-rotor partition functions are used in both classical and quantum forms, the latter in the approximation proposed by Truhlar [J. Comput. Chem., 12, 266 (1991)]. The numerical inversion compares well with analytical results obtained in a simple artificial case and also with an exact count of states in a large ethane-like system. Inversion of the hindered-rotor classical partition function is shown to lead to a somewhat different energy dependence of the sum or density of states, relative to the quantum counterpart, which is considered to be a more realistic representation. The routines presented are simple and fast enough to be of use in microcanonical rate calculations. © 1996 by John Wiley & Sons, Inc.     

Peak recognition imitating human judgement

Summary Peak integration is still a major source of error in analytical techniques such as chromatography (LC and GC), aapillary electrophoresis (CE), spectrosocpy, and electrochemistry. If the baseline is complex, e.g. because of matrix effects, or if the peak shape is irregular, e.g. because of peak tailing, the results are often not satisfactory when classical procedures are used. These shortcomings arise because of the stepwise appearance of the chromatogram. An algorithm that copies the human method of considering baseline and peaks as a whole has already been introduced. Here the use of a straight line as a baseline model led to an improvement in several instances. The baseline is, however, usually not exactly straight and rigid. A baseline model with flexible properties is more advantageous. Thus the smoothing cubic spline function is applied in this work. Here the rigidity can be controlled by use of a parameterp _k. The prediction interval of the spline is used for iterative distinction between baseline and peak regions. Afterwards straightforward optimization of the peak boundaries is applied. More than 50 series of consecutive injections of the same sample (n=40 on average) were used to test the performance of this procedure. The same raw data have been integrated by means of the algorithm described here and by use of commercially available software. The reproducibility of the main component peak are within the series was taken as a measure of integration quality. Typically the new procedure reducesRSD % by approximately 33% (e.g. from 1.5% to 1.0%). The improvement is even more impressive for difficult samples with complex matrices, e.g. blood plasma or polymer excipients. for such samples improvements of up to a factor of 6 are obtained.     

Mathematical functions for the representation of chromatographic peaks.

About ninety empirical functions for the representation of chromatographic peaks have been collected and tabulated. The table, based on almost 200 references, reports for every function: (1) the most used name; (2) the most convenient equation, with the existence intervals for the adjustable parameters and for the independent variable; (3) the applications; (4) the mathematical properties, in relation to the possible applications. The list includes also equations originally proposed to represent peaks obtained in other analytical techniques (e.g. in spectroscopy), which in many instances have proved useful in representing chromatographic peaks as well; the built-in functions employed in some commercial peak-fitting software packages were included, too. Some of the most important chromatographic functions, i.e. the Exponentially Modified Gaussian, the Poisson, the Log-normal, the Edgeworth/Cramér series and the Gram/Charlier series, have been reviewed and commented in more detail.     

Determination of nucleation rate in polymers using isothermal crystallization experiments and computer simulation

Within the frame of overall kinetics of crystallization, polymer crystallization is characterized by only three parameters: the initial density of potential nucleiN ₀, their activation frequencyq and the growth rateG. The growth rateG is rather easy to measure, whereas the nucleation parameters are generally unknown. Our purpose is to determine bothN ₀ andq using experimental isothermal two-dimensional crystallizations and computer simulation. Both these parameters are deduced from the rate of appearance of spherulites expressed as a function of the transformed surface fraction. The activation times of the spherulites are deduced from the shape of the boundaries between spherulites at the end of the transformation. When the growth rateG is known, the evolution of the transformed surface fraction is rebuilt using a computer simulation, so that only one observation of the final stage of the crystallization is needed.     

A formulation of the quantum-mechanical many-body problem in terms of hyperspherical coordinates

Hyperspherical coordinates are used to construct a matrix representation of a general N-particle Hamiltonian in the case where the interaction is electrostatic. The Yukawa interaction can be treated similarly, as is shown in an appendix. The basis functions used to construct the matrix representation of H are mononomials inthe 3N coordinates of the particles, multiplied by functions of the hyperradius. Methods for transforming from this representation to a symmetry-adapted representation are also discussed.     

Evaluation and prediction of the shape of gas chromatographic peaks

The evaluation and prediction of the shape of asymmetric gas chromatographic peaks is important as the knowledge of the amount of tailing permits to foresee the resolution between closely eluting peaks and to select the best analytical conditions for an efficient and rapid separation. A model function was tested in order to approximate the true peak shape obtained on non-polar column by injecting different compounds. The trend of the parameters involved in the used equation has been investigated as a function of column temperature and inlet pressure. The reproduction of the symmetrical or asymmetrical shape of gas chromatographic peaks is satisfactory and the method also permits to predict the shape of peaks obtained in different conditions of temperature and pressure.     

Semi-differential analysis of irreversible voltammetric peaks

Voltammetric peaks obtained by simulation of electrochemical reactions under conditions of linear semi-infinite diffusion with an irreversible electron transfer process are analysed using a semi-differentiation procedure. Obtained semi-derivative peaks, separated or overlapped, are fitted with appropriate mathematical functions. The functions used for data fitting include a function describing symmetrical peaks, proposed by several authors for fitting irreversible semi-derivative peaks, and two alternative functions that express asymmetric shape of the irreversible semi-derivative signals. When applied to the overlapped irreversible semi-derivative peaks, the latter two functions allow calculating certain electrochemical parameters with a better accuracy as compared with the function derived for the symmetrical peaks.