Distribution functions gaussian

The Hartree-Fock limit for the ground state of the hydrogen molecule can be achieved in principle by using a basis set of s-type Gaussian functions distributed along the internuclear axis. Such a basis set will be termed an axis basis set. The atom-centred basis functions, designated (ac), are located on the points... 

The final ingredient in this theory is an assumption permitting averaging with respect to the density fluctuations to eliminate the condition here on the densities. The simplest such assumption is that these coordinates obey a gaussian functional distribution built on the information [py(r) X ) and [8py r)8p r ) X )f, this... 

Initially, the polyatom program was coded to limit the basis set to 50 elements . By 1966, the ibmol program could (in principle) handle a maximum of 800 Gaussian functions distributed on 50 centers. This program also used contracted Gaussian basis functions which had been introduced earlier by dementi. 

As usual there is the question of the initial conditions. In general, more than one frozen Gaussian function will be required in the initial set. In keeping with the frozen Gaussian approximation, these basis functions can be chosen by selecting the Gaussian momenta and positions from a Wigner, or other appropriate phase space, distribution. The initial expansion coefficients are then defined by the equation... 

Mixmre models have come up frequently in Bayesian statistical analysis in molecular and structural biology [16,28] as described below, so a description is useful here. Mixture models can be used when simple forms such as the exponential or Dirichlet function alone do not describe the data well. This is usually the case for a multimodal data distribution (as might be evident from a histogram of the data), when clearly a single Gaussian function will not suffice. A mixture is a sum of simple forms for the likelihood ... 

Applying the TABS model to the stress distribution function f(x), the probability of bond scission was calculated as a function of position along the chain, giving a Gaussian-like distribution function with a standard deviation a 6% for a perfectly extended chain. From the parabolic distribution of stress (Eq. 83), it was inferred that fH < fB near the chain extremities, and therefore, the polymer should remain coiled at its ends. When this fact is included into the calculations of f( [/) (Eq. 70), it was found that a is an increasing function of temperature whereas e( increases with chain flexibility [100],... 

Under specific circumstances, alternative forms for ks j have been proposed like the parabolic or the truncated Gaussian probability distribution function for example [154]. 

In order to render the expression for d AFa) in a usable form, it remains to evaluate pk and pi. We have already pointed out that the average segment density of a molecule will be greatest at the center of gravity and that it will decrease smoothly as the distance 5 (Fig. 114,a) from the center is increased. While the distribution will not be exactly a Gaussian function of s, it may be so represented without introducing an appreciable error in our final result, which can be shown to be insensitive to the exact form assumed for the radial dependence of the segment density. Hence we may let... 

W(Xy yy z)y W r) Density and radial distribution functions for the end-to-end coordinates of a polymer chain (usually Gaussian functions). 

The most intense 826-cm band is broader than the other bands. The broadened band suggests a frequency distribution in the observed portion of the surface. Indeed, the symmetric peak in the imaginary part of the spectrum is fitted with a Gaussian function rather than with a Lorenz function. The bandwidth was estimated to be 56 cm by considering the instrumental resolution, 15 cm in this particular spectrum. This number is larger than the intrinsic bandwidth of the bulk modes [50]. 

Figure 16.6 TEM micrographs of titania-supported Au particles. The nominal thickness of An was (a) 0.13 nm (h) 0.78nm (c) 1.56nm (d) 2.33 nm. The Au deposition rate was 2.6 X 10 nms. Particle size distributions of Au for various deposition times are shown in the plot, with the distrihutions fitted to a normal Gaussian function.

Dependence of results from the prior-prejudice distribution. Non-uniform prior-prejudice distributions (NUP for short in what follows) were initially introduced by Jauch and Palmer by centering 3D Gaussian functions at the nuclear positions [29]. They found that the low-density regions of the crystal changed significantly upon introduction of the NUP, but the uneven distribution of errors persisted. 

Assuming the distribution models are accurate and that they model all the possible behaviors in the data set, Bayes s theorem says that pup2, and p3 are the probabilities that the unknown sample is a member of class 1, 2, or 3, respectively. The distributions are modeled using multivariate Gaussian functions in a method known as expectation maximization. ... 

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