Gaussian mathematical description

The basis set is needed for a mathematical description of the orbitals within a system and a good basis set is necessary to obtain a good energy quality. The orbitals are built up through linear combination of gaussian functions, which are... 

Gaussian peak a standard bell-shaped curve based on the mathematical description of a random distribution of, in the case of chromatography, component molecules in a band moving through the column. 

For systems involving more particles (more electrons and more nuclear protons and neutrons), the number of variables and other factors immediately exceeds any ability to be calculated precisely. A solution is found, however, in a method that uses an approximation of the orbital description, known as a Slater-type orbital approximation, rather than a precise mathematical description. A third-level Gaussian treatment of the Slater-type orbitals, or STO-3G... 

Gaussian-type function A mathematical description involving an exponential quadratic function which, when... 

The mathematical description of the model is out of the scope of this paper. Briefly, in this model, each reactant beam density is fitted to gaussian radial and temporal distribution functions, the spread in relative translational energy is neglected and the densities are assumed to be constant within the probed volume, which is smaller than the reaction zone. These assumptions result in a simple analytic expression of the overlap integral. Calculations are carried out for each rovibrational state of the outcoming molecule and for extreme velocity vector orientations, i.e, forwards and backwards. An example of the correction function, F, obtained for the A1 + O2 reaction at = 0.49 eV is displayed on Fig. 1, together with the... 

Many optical particle sizing instruments and particle characterization methods are based on scattering by particles illuminated with laser beams. A laser beam has a Gaussian intensity distribution and the often used appellation Gaussian beam appears justified. A mathematical description of a Gaussian beam relies on Davis approximations [45]. An nth Davis beam corresponds to the first n terms in the series expansion of the exact solution to the Maxwell equations in power of the beam parameter s,... 

It should be emphasized that for Markovian copolymers a knowledge of the values of structural parameters of such a kind will suffice to find the probability of any sequence Uk, i.e. for an exhaustive description of the microstructure of the chains of these copolymers with a given average composition. As for the composition distribution of Markovian copolymers, this obeys for any fraction of Z-mers the Gaussian formula whose covariance matrix elements are Dap/l where Dap depend solely on the values of structural parameters [2]. The calculation of their dependence on time, and the stoichiometric and kinetic parameters of the reaction system permits a complete statistical description of the chemical structure of Markovian copolymers to be accomplished. The above reasoning reveals to which extent the mathematical modeling of the processes of the copolymer synthesis is easier to perform provided the alternation of units in macromolecules is known to obey Markovian statistics. 

The evaluation of matrix elements for exphcitly correlated Gaussians (46) and (49) can be done in a very elegant and relatively simple way using matrix differential calculus. A systematic description of this very powerful mathematical tool is given in the book by Magnus and Neudecker [105]. The use of matrix differential calculus allows one to obtain compact expressions for matrix elements in the matrix form, which is very suitable for numerical computations [116,118] and perhaps facilitates a new theoretical insight. The present section is written in the spirit of Refs. 116 and 118, following most of the notation conventions therein. Thus, the reader can look for information about some basic ideas presented in these references if needed. 

The dynamics of a generic linear, ideal Gaussian chain - as described in the Rouse model [38] - is the starting point and standard description for the Brownian dynamics in polymer melts. In this model the conformational entropy of a chain acts as a resource for restoring forces for chain conformations deviating from thermal equilibrium. First, we attempt to exemphfy the mathematical treatment of chain dynamics problems. Therefore, we have detailed the description such that it may be followed in all steps. In the discussion of further models we have given references to the relevant literature. 

The guiding principle in writing down the self-repelling Gaussian chain model is mathematical simplicity, not microscopic faithfulness. Do we have, any idea why such a primitive model properly can explain the experiments To investigate this question we consider how a realistic microscopic description could be reduced to our model. 

It should be emphasized that for the Markovian copolymers, the knowledge of these structure parameters will suffice for finding the probabilities of any sequences LZ, i.e., for a comprehensive description of the structure of the chains of such copolymers at their given average composition. As for the CD of the Markovian copolymers, for any fraction of Z-mers it is described at Z 1 by the normal Gaussian distribution with covariance matrix, which is controlled along with Z only by the values of structure parameters (Lowry, 1970). The calculation of their dependence on time and on the kinetic parameters of a reaction system enables a complete statistical description of the chemical structure of a Markovian copolymer. It is obvious therewith to which extent a mathematical modeling of the processes of the synthesis of linear copolymers becomes simpler when the sequence of units in their macromolecules is known to obey Markov statistics. 

The shape fimctions most encountered in the analysis of absorption profiles are Gaussian fimction, Lorentzian function, Voigt function and the damped oscillator model. The mathematical expressions of these fimctions can be described in fimction of the variable v and in function of two parameters, one characterizing the band maximum vq and one describing the bandwidth. For the latter, different descriptions can be used ... 

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

A polymer is a collection of molecules, called monomers, which interact with each other to form a long flexible chain. For example, a typical polymer like polyethylene consists of a chain of roughly 10 CH2 molecules. The large number of monomers allows for a statistical description of macroscopic properties which are independent of details on the monomer scale. The simplest mathematical model of a polymer chain is referred to as the Gaussian chain [6]. In this model the polymer is described by a position d-dimensional... 

The concrete visualization of macromolecules necessary to formulate detailed predictions of properties provides the fundamental conceptual world that defines this book. Macromolecules are constrained by the same forces that define any molecule, and the atomic level of description is essential for many problems. The inspiration for the presentation found here for the rotational isomeric state model is the classic book by Flory,i Statistical Mechanics of Chain Molecules. When more-global properties are considered, smoothed models such as the Gaussian chain or wormlike chain model are more tractable. The importance of selecting a model that is both empirically adequate and mathematically solvable is stressed. 

The starting point is the probability of finding a conformation i (s) for the linear chain. This problem is discussed in detail in the chapter by Edwards and Muthukumar, and we give a brief description in Section 8.3.2 within this chapter, because of use in rubber theory, and we refer the mathematically interested reader to these parts of this series (Volume 2, Chapter 9). Assuming Gaussian conformations, it is given by the Wiener measure " ... 

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