Occupation number fluctuations

For example, if the displacement is a Gaussian random process, andif the scattering volume characteristics are as given in Section 6.6, the occupation number fluctuations are given by... 

It is important to note that occupation-number fluctuations always give information... 

This technique involves the same kind of consideration as discussed in Section 5.5 in connection with occupation number fluctuations . 

When using localized orbitals covalent interactions can be analyzed by orbital occupation number fluetuations and the local spin, calculated with respect to one or more orbitals local-ized on a specific atom [77,78]. The orbital occupation number fluctuations are defined as... 

Figure 16.11 compares the orbital occupation number fluctuations and local spin of covalently bonded system such as Ha and Aua, a van der Waals bonded system such as Hea and formally van der Waals bonded systems that exhibit significant covalent bonding contributions such as Hga and Bea to the values obtained for the 4f-7r interaction in Ce(CgH6)2. It is obvious also from these results that the latter case corresponds to a stretched covalent bond,... 

The fluctuations of the occupation numbers can be calculated with the aid of the formula (again for y = 0)... 

The mean-field approximation requires the proper occupation numbers of the valence orbitals (w = 2 for core and 111 = 0 for virtual molecular orbitals). These should be taken from a nonrelativistic ab initio calculation. Nevertheless, all reasonable choices proved to be suitable for practical purposes because they cause only negligible fluctuations in the calculated energy of spin-orbit coupling. 

Here, the factor of 1/2 in the exponent accounts for the zero-point, quantum fluctuations of the system the Bose statistics imply that any mode can have an occupation number that ranges from 0 to oo. Performing the sum, we find... 

Therefore, the occupied charge fluctuation peak carries most of the weight and the 4f occupation number... 

In this model the commtmal entropy appears as a result of fluctuations in the occupation numbers. These fluctuations may be considered as density fluctuations on a molecular scale. 

A recent breakthrough in molecular theory of hydrophobic effects was achieved by modeling the distribution of occupancy probabilities, the pn depicted in Figure 4, rather than applying a more difficult, direct theory of po for cavity statistics for liquid water (Pohorille and Pratt, 1990). This information theory (IT) approach (Hummer et al., 1996) focuses on the set of probabilities pn of finding n water centers inside the observation volume, with po being just one of the probabilities. Accurate estimates of the pn, and po in particular, are obtained using experimentally available information as constraints on the pn. The moments of the fluctuations in the number of water centers within the observation volume provide such constraints. 

Fluctuation in the number of molecules N contained in a given volume Fractional occupancy... 

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