Statistical weight function

It is, of course, not possible to include a statistical weighting function exp — WJkT) directly in the proton wave function, since this must actually be inserted in the density matrix. The total jump rate, however, amounts essentially to a summation of quantum contributions like (9.68) from energy levels below the barrier top together with a classical contribution like (9.59) frorh higher levels. 

We can discuss MD data on a more sophisticated level by introducing the probability distribution P(X), for each variable X. For the energy E, for example, and for a general ensemble whose statistical weight function is W,... 

To construct an ensemble conjugate to the MD ensemble, we have to introduce a second parameter which gives 3 a nonzero average value. This is the generalized canonical ensemble, whose statistical weight function is... 

Fig. 13.3 Statistical weighting functions in a bare slab reactor in one-velocity model.

These limitations lead to electron spin multiplicity restrictions and to differing nuclear spin statistical weights for the rotational levels. Writing the electronic wavefunction as the product of an orbital fiinction and a spin fiinction there are restrictions on how these functions can be combined. The restrictions are imposed by the fact that the complete function has to be of synnnetry... 

In summary, for a homonuclear diatomic molecule there are generally (2/ + 1) (7+1) symmetric and (27+1)7 antisymmetric nuclear spin functions. For example, from Eqs. (50) and (51), the statistical weights of the symmetric and antisymmetric nuclear spin functions of Li2 will be and respectively. This is also true when one considers Li2 Li and Li2 Li. For the former, the statistical weights of the symmetric and antisymmetiic nuclear spin functions are and, respectively for the latter, they are and in the same order. 

Different tests for estimation the accuracy of fit and prediction capability of the retention models were investigated in this work. Distribution of the residuals with taking into account their statistical weights chai acterizes the goodness of fit. For the application of statistical weights the scedastic functions of retention factor were constmcted. Was established that random errors of the retention factor k ai e distributed normally that permits to use the statistical criteria for prediction capability and goodness of fit correctly. 

We assume that exploring all possible forms for the fields corresponds to exploring the overall usual phase space. To determine the partition function Z the contributions from all the p+ r) and P- r) distributions are summed up with a statistical weight, dependent on p+ r) and p (r), put in the form analogous to the Boltzmann factor exp[—p (F)]], where the effective Hamiltonian p (F)] is a functional of the fields. The... 

Statistical weight factors are present in rroi and re eci. The multiplication involving these partition functions still works, since the total statistical weight factor is the product of the statistical weight factors for the individual units. Thus g, =g,.,rans g, g,.Mb g,. elect... 

In the RIS model for PE described by Abe et al. [143], the conformational partition function, Z, of the unperturbed chain with n bonds is formulated using a well-known statistical weight matrix for bond i, Uj, that is usually written in a 3 x 3 form. 

If there are sufficient data points available for the underlying statistical assumptions to be valid, the most accurate interpolation is given by Eq. (52) with the weight function defined by Eqs. (46), (56) and (57). 

Some fundamental definitions and properties of distribution functions are summarized briefly in this section. The most important statistical weights, averages, and moments frequently encountered in polymer analysis are introduced [7], Most quantities defined here will feature later again in the discussion of the individual analytical techniques. 

Continuous distribution functions Some experiments, such as liquid chromatography or mass spectrometry, allow for the determination of continuous or quasi-continuous distribution functions, which are readily obtained by a transition from the discrete property variable X to the continuous variable X and the replacement of the discrete statistical weights g, by the continuous probability density g(X). For simplicity, we assume g(X) as being normalized J ° g(X)dX = 1. Averages and moments of a quantity Y(X) are defined by analogy to the discrete case as... 

The minimization of the statistical error can be done by examining the variance of cxp(—/ zL4) directly. Bennett [55] has studied this problem by combining the forward and reverse FEP simulations, and the same analysis can be followed for the OS case. In Bennett s analysis, the weighting function w is placed to balance the forward and reverse FEP contributions [7, 55]... 

However, this analysis has been performed from a purely statistical perspective, leading to the minimal statistical error for the calculation. The phase space relationship, the staging scheme (conceptual intermediate M), and thus the accuracy of the calculation are not included in Bennett s picture. However, it turns out that the calculation is also optimal from the accuracy point of view. With this optimal choice of C = AA, the weight function w(Au) given by (6.64) has its peak exactly at the crossover between / and g, where AU = AA [cf. (6.15)]. In contrast, the weights for the low-Z w tail of / and high-Zll tail of g are diminished, thus resulting in small systematic error. 

Note that iron is sufficiently ionized in the solar atmosphere that the abundance of Fe I can be neglected and its partition function (or the ground-state statistical weight gi) and the electron pressure cancel out. 

At ordinary temperatures the excited energy levels of an atom or molecule are usually too high to make significant contribution to the partition function. If the lowest state is singlet state, the statistical weight ge is unity. Thus if lowest state is taken as the zero level, the other levels are sufficiently high and the partition function is approximately unity. 

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