Population weights

The additional factor of Qi(V, T) in Eq. (21) makes the leading term in the sum unity, as suggested by the usual expression for the cluster expansion in terms of the grand canonical partition function. Note that i in the summand of Eq. (20) is not explicitly written in Eq. (21). It has been absorbed in the n , but its presense is reflected in the fact that the population is enhanced by one in the partition function numerator that appears in the summand. Equation (21) adopts precisely the form of a grand canonical average if we discover a factor of (9(n, V, T) in the summand for the population weight. Thus... 

In Fig. 1 the cumulative frequency of the measured mean values for induvidual houses is plotted on a log-normal scale. The aritmetric mean value in our measurements is 160 Bq/nP. Areas with high concentrations are overrepresented in this distribution, (as seen from the figure) and by population weighing the distribution for the municipalities, a population weighted average of 110 Bq/m in the heating season is obtained. 

Conformational equilibria still fast on the NMR timescale. From reference compounds (diastereomeric analogs, e.g., 4-/err-butyl or 3,5-cis-dimethyl substitution operating as a kind of conformational anchor), the NMR parameters of the purely equatorial and axial conformers (Pcq and Pax, respectively) can be determined. The equilibrium constant K [Eq. (3)] from these parameters and that of the population-weighted observed NMR parameter, P, can be estimated from the fast equilibrium [59CI(L)568] ... 

For a harmonic oscillator, the probability distribution averaged over all populated energy levels is a Gaussian function, centered at the equilibrium position. For the classical harmonic oscillator, this follows directly from the expression of a Boltzmann distribution in a quadratic potential. The result for the quantum-mechanical harmonic oscillator, referred to as Bloch s theorem, is less obvious, as a population-weighted average over all discrete levels must be evaluated (see, e.g., Prince 1982). 

Analogously, the fluorescence quantum yield of an extrinsic fluorescent probe contained in a peptide can be measured by comparison with an appropriate standard. If the fluorescent peptide exists in a conformational equilibrium, the fluorophore may be located in a number of different environments and may have a distinct quantum yield (ip,) in each environment. In this case the determined fluorescence quantum yield represents a population-weighted average of the individual [Pg.700]

Fig. 2. Evolution of twist angle around the P-bond (grey) and I-bond (black) after photoexcitation of the neutral form of GFP chromophore in the gas phase (left panel) and solvated by 150 water molecules (right panel). Solid lines are population-weighted averages over die trajectory basis functions. Dashed lines represent the twist angles for the individual trajectory basis functions. The sense of rotation for the two torsions is defined such that HT motion corresponds to both angles moving towards more negative values.

The translational environmental correction is crudely incorporated by estimating the volume V of each cluster [e.g., using the GEPOL algorithm E. Silla, F. Villar, O. Nilsson, J. L. Pascual-Ahuir, and O. Tapia. J. Mol Graphics 8, 168-72 (1990)] and evaluating the population-weighted excluded volume that is inaccessible to free translational motions, namely,... 

As detailed and comparable information about health impacts in the critical areas is not available for all areas, the population weighted exposure16 has been calculated on the basis of gridded air quality and population data [4, 5], The values obtained in this way give a rough estimate of potential health effects (Table 14). The highest levels have been calculated for Sofia, Krakow and Milan, where the levels for the annual mean are more than twice as high as in Berlin. 

Table 14 Population weighted exposure to annual mean and 36th highest daily mean levels in urban and regional background areas in pg/m3 (source ETC/ACC [4])...

Environ- Air quality Urban population weighted NO2 concentration. Urban... 

Two articles have addressed the variation of the optical rotatory power upon dimerization. In the first one, the different conformation accessible monomers and dimers of chlorofluoroiodoacetic acid (Scheme 3.30) have been theoretically calculated [153]. Using these results, the population weighted rotations for the... 

One also notices that the population-weighting procedure of Scheme 1.3, with/ = 1 and/2 = 1/2, gives a diminished bond multiplicity ... 

H-bonded systems may require additional diffuse or polarization functions. For example, the 6-311++G(d,p) basis set had been found to be suitable for H-bonded systems [78-81], It may be necessary to include Basis Set Superposition Errors (BSSE) [82] and Zero-Point-Energy (ZPE) corrections in evaluating the relative stabilities. Such corrections are often of the same magnitude as the energy differences among the dominant conformers. Moreover, the relative conformer energies may also differ noticeably with the basis sets used. All these factors will affect the Boltzmann factors predicted for different conformers and therefore the appearance of the population weighted VA and VCD spectra. Thus, an appropriate selection of DFT functionals and basis sets is very important for VCD simulations. A scale factor of 0.97-0.98 is usually applied to the calculated harmonic frequencies to account for the fact that the observed frequencies arise from an anharmonic force field instead of a harmonic one. A Lorentzian line shape is typically used in simulations of VA and VCD spectra. The full-width at half maximum (FWHM) used in the spectral simulation is usually based on the experimental VA line widths. 

Fig. 8 Comparison of the experimental (dashed line) VA (left) and VCD (right) spectra of 3a and 3b with the corresponding calculated population weighted spectra (solid line) of OPO axis ZlpSS and zipSS stereoisomers. The calculated VA and VCD spectra of xtpSfi are included for comparison. zlpf S is the same as ZtpSfi because the two chiral ligands are the same except their ACs. Adapted with permission from [22], Copyright (2010) Wiley...

Ambient air samples from 44 sites in 39 U.S. urban areas were collected from 6 a.m. to 9 a.m. during June through September of 1984, 1985, and 1986. Benzene was present in every sample. The median benzene site concentrations ranged from 4.8 to 35 ppb, with the overall median being 12.6 ppb (detection limit = 0.04 ppb). The data indicated that mobile sources were the major source of benzene in the vast majority of samples (EPA 1987e). In California, motor vehicle exhaust and motor vehicle evaporation accounted for 79.8% of the population exposure to ambient benzene (Allen 1987). The 1984 population-weighted average benzene concentration in California was estimated to be 3.3 ppb (Allen 1987). 

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