Density error

Table 3.41. NBO/NRT descriptors for B4H10 (see Fig. 3.103(a)), showing (a) alternative styx codes (with specific bond types) and associated non-Lewis density error (b) NHO hu and percentage polarization for NBOs of the...

However, the relative accuracies of the two possible structural bond patterns can be assessed more quantitatively with NBO analysis. The NBO procedure allows one to specify alternative Lewis structure patterns of two- and three-center bonds158 and determine the non-Lewis density error pni, of each such structure. As shown in Table 3.41(a), the non-Lewis density of the 4012 structure (0.6072c) is smaller than that of the 3103 structure (0.8760c), which confirms that the 4012 structure (3.248) is indeed the superior bonding description in this case. 

As shown in Table 3.39, three standard styx codes (4120, 3211, and 2302) are possible for B5H9. Table 3.42(a) specifies representatives of each of these structures, including two distinct structures (labeled 4120 and 4120 ) corresponding to the 4120 motif. In addition, one may consider various non-standard styx motifs such as the structures labeled 4210 and 4200 in Table 3.42(a), which correspond to one or more B atoms lacking the usual tetracoordinate pattern. Table 3.42(a) summarizes the NBO pnl density errors associated with the six listed structures, showing that the 4120 structure (3.250) has the smallest pnl value and is indeed the best single NBO structure. 

Fig. 9.5 Values of pore wall thickness from Fig. 9.4 a-d as a function of doping density (error bars), together with the SCR width for a potential (Vbl + V-2kT/e) of 0.1 V (line). After [Le21].

A second source of error encountered with high area powders, is the annulus volume which exists between the powder surface and the closest approach distance of a gas molecule. Assuming that the closest approach of the helium atom to the powder surface is 0.5 A or 5 x 10" meter and that the specific area of the powder is lOOOm g", there will exist an annulus volume of 5 x 10" m g" or 5 x 10" cm g" This represents a density error of 5 % on materials with densities near unity. The use of gases with molecules larger than helium will exacerbate this error, again indicating the use of helium as the preferred gas. 

Density Error (g/cm ) Density Error (g/cm ) Density Error (g/cm ... 

The above observations have been made primarily in the study of the new measurements for parahydrogen. Because of the wide use of this equation in other data compilations, it is of interest to note the results of the comparison of the preliminary calculations for parahydrogen using this equation of state with more rigorous calculations that have been made subsequently. These comparisons are given by Roder et al. [ ] and indicate an average absolute error of 0.16% with maximum errors of 4% near the critical point. With the exception of the area near the critical point the density errors do not exceed 1 %. 

Truncation artifacts and density errors can be reduced by applying row-wise extrapolation techniques (Ohnesorge et al. 2000 Starman et al. 2005 Sour-BELLE et al. 2005 Zellerhoee et al. 2005 FIsieh et al. 2004). Since row-wise extrapolation only depends on neighboring data in a horizontal direction, this approach is adaptive, and it can be very effective, as shown in Fig. 3.3d. 

Slip ratio is very sensitive to measured density. A 10% er = ror in density could cause a 100% error in void fraction with a resulting 20% change in slip ratio. The effect of errors in measured density vary, depending upon the value of quality. At low qualities, the error in density measurement affects the slip ratio more than the same percentage density error at high qualities. Values of slip ratio less than one do not necessarily indicate that the stea,m is lagging behind the water. These values merely reflect the scatter of the data. 

Figure 3. Effects of (a) -dP/dt, (b) CO2 Gas Content, and (c) on Maximum Cell Density (Error Bars 3X Standard Deviation)...

The average error is about 30%. The relation can be used only if the reduced density is less than 2.5 and the reduced temperature of the mixture is greater than 0.80. 

The average error of this method is around 10%. The method is not applicable for reduced densities greater than 2.8. 

Density is generally measured at 15°C using a hydrometer in accordance with the NF T 60-101 method it is expressed in kg/1 with an error of 0.0002 to 0.0005 according to which category of hydrometer is utilized. However, in practice only three decimal places are usually retained. 

The data from Table 2 show that the algorithm developed in allows sizing of different cracks with complex cross-sections and unknown shapes for orientation angles not exceeding 45°. It is seen that the width 2a and the parameter c (or the surface density of charge m=4 r // e at the crack walls) are determined with 100% accuracy for all of the Case Symbols studied. The errors in the computation of the depths dj and di are less than 4% while the errors in the computation of d, dj, d, and d are less than 20% independent of the shape of the investigated crack and its orientation angle O <45°. 

The reason for these results is that the intensity of the leakage field and the RMS error used depend strongly on the parameter c and the crack width, and to a lesser extent on the depth profile of the crack. Also, the distribution of the density of the leakage field is measured over the centre of the crack and correspondingly changes more by varying of dj and dj rather than of d, dj, dj and d . 

According to the simple formula, the maximum bubble pressure is given by f max = 27/r where r is the radius of the circular cross-section tube, and P has been corrected for the hydrostatic head due to the depth of immersion of the tube. Using the appropriate table, show what maximum radius tube may be used if 7 computed by the simple formula is not to be more than 5% in error. Assume a liquid of 7 = 25 dyn/cm and density 0.98 g/cm. ... 

It is possible to identify particular spectral features in the modulated reflectivity spectra to band structure features. For example, in a direct band gap the joint density of states must resemble that of critical point. One of the first applications of the empirical pseudopotential method was to calculate reflectivity spectra for a given energy band. Differences between the calculated and measured reflectivity spectra could be assigned to errors in the energy band... 

Abstract. A smooth empirical potential is constructed for use in off-lattice protein folding studies. Our potential is a function of the amino acid labels and of the distances between the Ca atoms of a protein. The potential is a sum of smooth surface potential terms that model solvent interactions and of pair potentials that are functions of a distance, with a smooth cutoff at 12 Angstrom. Techniques include the use of a fully automatic and reliable estimator for smooth densities, of cluster analysis to group together amino acid pairs with similar distance distributions, and of quadratic progrmnming to find appropriate weights with which the various terms enter the total potential. For nine small test proteins, the new potential has local minima within 1.3-4.7A of the PDB geometry, with one exception that has an error of S.SA. 

Plot the probability density obtained from E in Problem 9 as a function of r, that is, simply square the function above with an appropriate scale factor as determined by trial and error. Comment on the relationship between your plot and the shell structure of the atom. 

The density determination may be carried out at the temperature of the laboratory. The liquid should stand for at least one hour and a thermometer placed either in the liquid (if practicable) or in its immediate vicinity. It is usually better to conduct the measurement at a temperature of 20° or 25° throughout this volume a standard temperature of 20° will be adopted. To determine the density of a liquid at 20°, a clean, corked test-tube containing about 5 ml. of toe liquid is immersed for about three-quarters of its length in a water thermostat at 20° for about 2 hours. An empty test-tube and a shallow beaker (e.g., a Baco beaker) are also supported in the thermostat so that only the rims protrude above the surface of the water the pycnometer is supported by its capillary arms on the rim of the test-tube, and the small crucible is placed in the beaker, which is covered with a clock glass. When the liquid has acquired the temperature of the thermostat, the small crucible is removed, charged with the liquid, the pycnometer rapidly filled and adjusted to the mark. With practice, the whole operation can be completed in about half a minute. The error introduced if the temperature of the laboratory differs by as much as 10° from that of the thermostat does not exceed 1 mg. if the temperature of the laboratory is adjusted so that it does not differ by more than 1-2° from 20°, the error is negligible. The weight of the empty pycnometer and also filled with distilled (preferably conductivity) water at 20° should also be determined. The density of the liquid can then be computed. 

The simplest approximation to the complete problem is one based only on the electron density, called a local density approximation (LDA). For high-spin systems, this is called the local spin density approximation (LSDA). LDA calculations have been widely used for band structure calculations. Their performance is less impressive for molecular calculations, where both qualitative and quantitative errors are encountered. For example, bonds tend to be too short and too strong. In recent years, LDA, LSDA, and VWN (the Vosko, Wilks, and Nusair functional) have become synonymous in the literature. 

Some density functional theory methods occasionally yield frequencies with a bit of erratic behavior, but with a smaller deviation from the experimental results than semiempirical methods give. Overall systematic error with the better DFT functionals is less than with HF. 

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