PM3 and DFT (6-311G/ B3LYP) level calculations were carried out for the 5Radialene molecule, which is exhibit D5h symmetry. The obtained equilibrium geometry was applied for the calculation of all 3N−6 vibration frequencies, and for the analysis of its normal coordinates and symmetry species, in addition to some physical properties such as heat of formation, total energy, dipole moment and energy difference of HOMO and LUMO levels (ΔELUMO-HOMO), using Gaussian-03 program. The so calculated frequencies according to DFT (6-311G/ B3LYP) fall in the ranges;
CH2 str. (3016-3098 cm-1), C=C str. (1662-1709cm-1), ring (C-C str.) (1268-1464 cm-1). δCH2 (890-1317cm-1), (δCCC) (562-631cm-1), γCH2 (738-946cm-1) and γring (γCCC) (14-805cm-1), and according to PM3 fall in the ranges;
CH2 str. (3124-3138cm-1), C=C str. (1873-1939cm-1), ring str. (C-C str.) (1289-1430cm-1). δCH2 (946-1503cm-1), (δCCC) (549-777cm-1), γCH2 (673-1007cm-1) and γring (γCCC) (54-785cm-1).
Other interesting correlations were also be obtained for the frequencies of similar vibrations. Distribution of electronic charge density on atoms of 5Radialene molecule were also calculated and studied.
The substantial key to initiate an explicit statistical formula for a physically specified continua is to consider a derivative expression, in order to identify the definitive configuration of the continua itself. Moreover, this statistical formula is to reflect the whole distribution of the formula of which the considered continua is the most likely to be dependent. However, a somewhat mathematically and physically tedious path to arrive at the required statistical formula is needed. The procedure in the present research is to establish, modify, and implement an optimized amalgamation between Airy stress function for elastically-deformed media and the multi-canonical joint probability density functions for multivariate distribution complet
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This Book is the second edition that intended to be textbook studied for undergraduate/ postgraduate course in mathematical statistics. In order to achieve the goals of the book, it is divided into the following chapters. Chapter One introduces events and probability review. Chapter Two devotes to random variables in their two types: discrete and continuous with definitions of probability mass function, probability density function and cumulative distribution function as well. Chapter Three discusses mathematical expectation with its special types such as: moments, moment generating function and other related topics. Chapter Four deals with some special discrete distributions: (Discrete Uniform, Bernoulli, Binomial, Poisson, Geometric, Neg
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