Generalized fractional occupation

The generalized fractional occupation Up is related to diagonal matrix elements of the first-order reduced density matrix constructed in natural... 

Fine particles of metallic Eu and Yb have been shown to exhibit valence transition with critical particle sizes of 50 A for Eu and 35 A for Yb. At these critical particle sizes, there is a reduction in the lattice constant with no change in the critical fee structure. There appears to be a change in valence from +2 to +3. This effect may be due to a shift in the position of Fermi level, E, such that E xc approaches zero resulting in the ions at the configuration crossover. ICF state can in general result in lattice constant anomaly due to the fractional occupation of the 4/ shells. Thus once ICF is identified in a system, valence fluctuation can be studied by many techniques. 

An important assumption in this theory is that there is no correlation between successive jumps and this is generally a good assumption at low H concentrations. However, at larger concentrations, this is not strictly true because correlation effects become significant, i.e. if an atom jumps from a filled site to an empty one, the site vacated is, at this instant, empty whereas the other sites are occupied with a probability c where c is the overall fractional occupation, so that the chance of jumping back is enhanced. This effect was first considered by Ross and Wilson [37] who showed by Monte Carlo simulation that, at finite concentrations, the quasi-elastic peak deviates from the Lorentzian shape. This was the first example of the need to resort to Monte Carlo simulation of the diffusion process to obtain G (r, t) in situations where the diffusion process becomes significantly complicated. This is likely to be important in efforts to understand the diffusive process in complex hydride stores. 

The computations, presented here, are based on an approach that uses the fractional occupation number concept [57-58], The original idea to exploit fractional occupation numbers in the framework of DFT is due to Janak [59] who generalized... 

For a given adsorbent-adsorbate system the relationship between s and the fractional occupation of the micropore volume is defined by the characteristic curve , which is assumed to be independent of temperature (see Fig. 5). Such an assumption should be valid for systems dominated by dispersion-repulsion forces (which are temperature independent), but cannot be expected to hold when electrostatic forces (which are temperature dependent) are important. The characteristic curve generally has a Gaussian form leading to an isotherm of the form ... 

Here is the number of occupied states, which is equal to half the number of valence electrons in the system. The factor of two comes from spin multiplicity, if the system is non-magnetic. Equation 6.3 can be easily generalized to situations where the highest occupied states have fractional occupancy or when there is an imbalance in the number of electrons for each spin component. 

During an observation period, Hg will be true with some probability Pg and Hi will be true with probability Pi, and 1 = Pq -f Pi. In single-molecule experiments. Pi corresponds to the fraction of time that a molecule is present in the probe volume in other words. Pi is the fractional occupancy of the probe volume. In general, this probability is imknown at the start of the experiment and is often the quantity sought. 

Restricted ensemble Kohn-Sham DFT Alternatively one can perform standard KS-DFT calculations on a collection of determinants with different occupations and take a weighted average of the individual energies to obtain an estimate of the mul-tideterminantal situation. To avoid the independent calculation of several KS determinants, a generalization of this approach was proposed by Filatov and Shaik based on the coupling operator technique developed by Roothaan for restricted open-shell Hartree-Fock. This restricted open-shell Kohn-Sham (ROKS) approach was later extended to situations where fractional occupation numbers are not imposed by the... 

We may generalize this by introducing an occupation number (number of electrons), n, for each MO. For a single determinant wave function this will either be 0, 1 or 2, while it may be a fractional number for a correlated wave function (Section 9.5). 

In this book and in general, the concentration, d, is used to represent the number of mobile species, ions or vacancies, per unit volume whereas is always the fraction of the crystallographically equivalent sites that are occupied by ions. Often if > 0.5 we talk of ion migration and c refers to ion concentration, whereas if < 0.5 we talk of vacancy migration and c refers to the concentration of vacancies. However, this is simply a convenient way of thinking about ion transport in solids by focusing on the minority species. It is equally possible to describe conduction always in terms of ions or of vacancies provided account is taken of the fact that both the concentration of mobile species and sites to which they may migrate are important. The importance of c and (1 — c) is emphasised and placed within a unified framework in Chapter 3. The concentration of ions Ci is related to the occupancy by = cJC, where C is the concentration of the sites. 

Exposures in the population of interest will generally reveal that incurred dose is only a small fraction, and sometimes a very tiny fraction, of that at which toxic responses has been or can be directly measured, in either epidemiology or animal studies. Occupational populations (Table 8.1, Scenario C) may be exposed at doses close to those for which data are available, but general population exposures are usually much smaller. Thus, to estimate risk it will be necessary to incorporate some form of extrapolation from the available dose-response data to estimate toxic response (risk) in the range of doses expected to be incurred by the population that is the subject of the risk assessment. 

The ensemble search in Eq. (82) is the Kohn-Sham procedure, generalized to allow fractional orbital occupation numbers [55, 57-59]. Equation (82) can... 

---