Probability distribution function, structure calculations

The general structure of the theoretical tool of ST should now be quite clear. For each set of independent variables we define a partition function. This partition function is related to a thermodynamic quantity through one of the fundamental relationships. On the other hand, each of the summands in the PF is proportional to the probability of realizing the specific value of the variable on which the summation is carried out. Having the probability distribution, for each set of independent variables, one can write down various averages over that distribution function. The calculation of such averages consists of the main outcome of ST. 

Figure 7. The probability distribution function of the Z sctn-es computed for the population of false positives. A set of 47 sequences from the 547 set of proteins with known structures without homologs in the HL set is used to sample the distribution of Z scores for false positives. Each of the sequences is aligned to all the structures included in HL set. Hie Z semes are calculated far the 200 best matches (according to energy) using 100 shuffled sequences. The observed distribution of Z scores is represented by +. The dashed line shows the attempted analytical fit to a Gaussian distribution, whereas the solid line the analytical fit to the expected extreme value (double exponential) distribution. Note the significant tail to the right, which is the probability of obtaining a relatively large Z score by chance. See text far mote details.

Let us start with the simplest structural example the degree of polymerization (DP). With the polymerization model just described, and assuming an equal likelihood for the selection of any polymer molecule from the mixture, it is possible to calculate the probability distribution function for the chain length n. The probability distribution function for n is the probability of finding a molecule with a given chain length n in a polymer sample. For experimental purposes, the probability function is the fraction of all polymer molecules that possess the stated chain length, n. 

Structure calculations using the probability distribution function... 

Upon applying the calibration constants obtained from the data of one sample to evaluate the SEC data of the other samples, the calculated Mn and values correspond fairly well (within 10-20%) with the absolute MW parameters of the samples. This agreement also suggests that the samples probably have similar chain structures. The distribution functions for samples PN-1 and IL-22 are plotted in Figures 4 and 5. The molecular weight distributions of both polymers are similar to distribution curves reported for derivatized poly(organo)phosphazenes (4-10). 

The Hartree-Fock or self-consistent-field approximation is a simplification useful in the treatment of systems containing more than one electron. It is motivated partly by the fact that the results of Hartree-Fock calculations are the most precise that still allow the notion of an orbital, or a state of a single electron. The results of a Hartree-Fock calculation are interpretable in terms of individual probability distributions for each electron, distinguished by characteristic sizes, shapes and symmetry properties. This pictorial analysis of atomic and molecular wave functions makes possible the understanding and prediction of structures, spectra and reactivities. 

In the Fiirth hole model for molten salts, the primary attraction is that it allows a rationalization of the empirical expression = 3.741 r p. In this model, fluctuations of the structure allow openings (holes) to occur and to exist for a short time. The mean hole size turns out to be about the size of ions in the molten salt. For the distribution function of the theory (the probability of having a hole of any size), calculate the probability of finding a hole two times the average (thereby allowing paired-vacancy diffusion), compared with that of finding the most probable hole size. 

Other important microscopic structural information that has received much experimental[5,40,49,50] and theoretical[9,36,37] attention is the molecular orientation profile. This is the probability distribution P(6,z) for the angle 9 between a vector fixed in the molecule frame and the normal to the interface, calculated as a function of the distance z from the interface. For water dipoles, this quantity seems to behave quite similarly for a number of water/organic liquid interfaces. An example is shown in Figure 6 for the water/CCU interface. 

To calculate the foam cell size distribution function we consider, following Mihira an isolated cell foam structure model (Fig. 24). Let r be a true cell radius, and s the radius of sectional circles on the cut surface X, f and s their mean values, of and of their mean square deviations, and f(r) and f(s) their distribution functions. We will denote by x the depth of a cell dissected by the plane X (Fig. 24) and calculate the probability P(r,x) of cells having a radius in the range from r to (r + dr) and a depth from x to (x + dx). The probability P(r) for the cells dissected by the plane X to have a radius r is ... 

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