Gaussian probability density, molecular

In gas kinetic theory, the probability density for a component of the molecular velocity is a Gaussian distribution. The normalized probability distribution for Vx, the X component of the velocity, is given by... 

It is often desired to compare the distribution of observations in a population with a theoretical distribution. The normal distribution (also called the Gaussian distribution) is a particularly important theoretical distribution in molecular modelling. The probability density function for a general normal distribution is ... 

Log normal Distribution (logarithmic normal distribution) A statistical probability-density function, characterized by two parameters, that can sometimes provide a faithful representation of a polymer s molecular-weight distribution or the distribution of particle sizes in ground, brittle materials. It is a variant of the familiar normal or Gaussian distribution in which the logarithm of the measured quantity replaces the quantity itself. It s mathematical for is... 

Each HE molecular orbital is written as a linear combination of functions describing atomic orbitals. The entire set of such equations for the atomic orbitals in a molecule is called a basis set, and each equation is called a basis set function. Ideally, these basis set fimctions would have the properties of hydrogenic wave functions, particularly with regard to the radial dependence of electron density probability as a function of distance of the electron from the nucleus, r. A type of basis set function proposed by Slater uses a radial component incorporating and such functions are called Slater t)rpe orbitals (STOs). Gaussian type orbitals (GTOs) have radial dependence and are easier to solve analytically, but they do not describe the radial dependence of electron density as well as do the STOs. 

In most electron-diffraction studies both the electron beam and the molecular beam are assumed to have vanishing cross-sections. This approximation is probably satisfactory for the electron beam. The effect of a non-zero cross-section for the molecular beam varies with the density distribution and with the nozzle-to-plate distance. - Rundgren has found that for a Gaussian density distribution with standard deviation, am, the observed values for will be too large by approximately... 

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