Fractional occupancies

FIG. 12 Segment density profile as function of the distance from the wall Z for flexible (empty symbols) and semi-rigid (full symbols) living polymer chains at T = 0.4 [28]. The fractional occupancy of lattice sites by polymer segments is shown for the layers in the left half of the box. Dashed lines are guides for the eyes. 

FIGURE 3.5 Major components of classical receptor theory. Stimulus is the product of intrinsic efficacy (s), receptor number [R], and fractional occupancy as given by the Langmuir adsorption isotherm. A stimulus-response transduction function f translates this stimulus into tissue response. The curves defining receptor occupancy and response are translocated from each other by the stimulus-response function and intrinsic efficacy. 

At least for the case of a non-degenerate ground state of a closed shell system, it is possible to delineate the standard Kohn-Sham procedure quite sharply. (The caveat is directed toward issues of degeneracy at the Fermi level, fractional occupation, continuous non-integer electron number, and the like. In many but of course not all works, these aspects of the theory seem to be... 

Figure 3.2. Partition function and fractional occupation for a two-level system.

Here i/j. ng l 2mK is the fractional occupation number of the 2ma- sp/n-orbitals of Shell K and is a suitably averaged electron interaction matrix (cf. the usual... 

SCF-CI calculations were performed at 20 different intemuclear separations, from 1.2 bohr to 4-00. The lowest separate atom states are, B( P,2p) and H( S) therefore, in order to have a homolytic dissociation and three degenerate 2p orbitals on B we have adopted the closed shell Fock hamiltonian with fractional occupation [23] one electron was placed in the 3(t orbital, correlating with H(ls) at infinite separation, and 1/3 each in the 4it and Itr orbitals correlating with B(2p). 

Wang, S. G., Schwarz, W. H. E., 1996, Simulation of Nondynamical Correlation in Density Functional Calculations by the Optimized Fractional Occupation Approach Application to the Potential Energy Surfaces of 03 and... 

If we divide both sides of Equation (A2.15) by [ ]T, we obtain an equation for the fractional occupancy of the enzyme by inhibitor ... 

Table 11.4 shows that clozapine has approximately 10 times higher affinity for the D4 and 5-HT2A-receptors than the D2-receptor and shows a greater occupancy of the 5-HT2 than the D2-like receptors. The other atypical neuroleptic risperidone has a similar affinity for the two D2-like receptors but an affinity for the 5-HT2a-receptor that is just over 3 times lower than for the D2-receptor. Receptor occupancy in vivo shows a similar profile to clozapine. In contrast, haloperidol s affinity for the D4-receptor is just under 3 times lower and over 100 times lower for the 5-HT2a-receptor, with no binding to the latter in vivo. The fractional occupancy of striatal... 

The other extreme of the theoretical treatment of the metal surfaces is the approximation of the surface by a single atom or by a simple array of two or more atoms (e.g., 62-65) one can use for the description of these atoms either the electronic structure of the individual atoms [sometimes even simplified, e.g. (63)] or fractional occupancy of their atomic orbitals... 

The question arises as to the directions of these two orbitals. From Eq. (11) it was anticipated that there would be considerable occupancy of the nitrogen 2pz orbital hence, the value of 2a/Bo = 0.361 is associated with C22, the fractional occupancy of the p, orbital. The form of the ji tensor requires that the other p orbital be in the x direction or cl = 0.078. This occupancy of the px orbital was not anticipated in Eq. (11). 

From the isotropic coupling constant one may calculate C12, the fractional occupancy of the nitrogen s orbital, which is equal to AUo/Ao. The term Ao is the Fermi contact interaction for an unpaired electron in a pure nitrogen 2s orbital. For NO2 the value of ci2 = 56.5/550 = 0.103. The fraction of the unpaired electron associated with the N nucleus is then... 

Figure 5 shows a hypothetical plot of the microscopic Kd values versus fractional occupancy of SBA binding to Tn-PSM. The horizontal dashed line is the observed Kd value of 0.2 nM obtained from the ITC experiments (Table I). However, due to... 

The order-disorder transition of a binary alloy (e.g. CuZn) provides another instructive example. The body-centred lattice of this material may be described as two interpenetrating lattices, A and B. In the disordered high-temperature phase each of the sub-lattices is equally populated by Zn and Cu atoms, in that each lattice point is equally likely to be occupied by either a Zn or a Cu atom. At zero temperature each of the sub-lattices is entirely occupied by either Zn or Cu atoms. In terms of fractional occupation numbers for A sites, an appropriate order parameter may be defined as... 

Correspondingly, the fractional occupation of binding sites is of the form... 

Papke RL, Jacobs LB, Stokes C (2007) Activation of alpha7 nAChR occurs with low fractional occupancy of the agonist binding sites. Soc Neurosci Abst 37 574.9 Pauly JR, Marks MJ, Gross SD, Collins AC (1991) An autoradiographic analysis of cholinergic receptors in mouse brain after chronic nicotine treatment. J Pharmacol Exp Ther 258 1127-1136... 

Fig. 13.8 Histogram depicting the fractional occupancy of PRCC chemistry space. A significant fraction of the PRCC chemistry space (91.2%) is unoccupied and is available for compound selection. Only a very small percent (0.59%) is densely populated (lO-h compounds per cell).

However, despite their proven explanatory and predictive capabilities, all well-known MO models for the mechanisms of pericyclic reactions, including the Woodward-Hoffmann rules [1,2], Fukui s frontier orbital theory [3] and the Dewar-Zimmerman treatment [4-6] share an inherent limitation They are based on nothing more than the simplest MO wavefunction, in the form of a single Slater determinant, often under the additional oversimplifying assumptions characteristic of the Hiickel molecular orbital (HMO) approach. It is now well established that the accurate description of the potential surface for a pericyclic reaction requires a much more complicated ab initio wavefunction, of a quality comparable to, or even better than, that of an appropriate complete-active-space self-consistent field (CASSCF) expansion. A wavefunction of this type typically involves a large number of configurations built from orthogonal orbitals, the most important of which i.e. those in the active space) have fractional occupation numbers. Its complexity renders the re-introduction of qualitative ideas similar to the Woodward-Hoffmann rules virtually impossible. 

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