Empirical scaling factor

For a flow boiling crisis, an empirical scaling factor was suggested by Stevens and Kirby (1964). It is a graphical correlation developed from uniform-flux round-tube data. They suggested an empirical function ... 

Figure 5.30 Empirical scaling factor. (From Steven and Kirby, 1964. Reprinted with permission of UK AEA Technology, Didcot, Oxfordshire, UK.)... 

Rationalization of these discrepancies between calculated and observed molecular properties has been achieved by recognizing that the accuracy problems tend to be systematic for the computational methods [726], For the same levels of theory, the errors in molecular parameters tend to be similar [79] for molecules if a variety of types and size. Building on the ideas of similarities between similar chemical functional groups in different molecules, the SQM procedure develops empirical scaling factors to correct the overestimated harmonic force constants from the computations [77], These scale factors form a diagonal matrix C which modified the calculated force matrix Fcalc according to the relationship [727] ... 

In the X-N method, the experimental electron density is determined from the X-ray diffraction data. The free atom electron density is determined by placing theoretical free atom electron densities at the atomic nuclear positions determined by a neutron diffraction experiment on the same crystal structure at the same temperature, albeit with a crystal of larger size. In the X-N method, it is frequently found that the atomic thermal parameters determined by neutron diffraction do not agree with those determined by X-ray diffraction, even though the experiments were carried out under identical conditions. This requires the introduction of an empirical scaling factor between the two sets of data, which is effective but disconcerting. 

Local correlation functionals were developed by Vosko et al., which involve a number of terms and empirical scaling factors. The most popular versions are called VTWV and VWN5. The combination of a local exchange and a local correlation energy density is the SVWN method. 

NOTE Empirical scaling factors have been developed for each model chemistry to help correct theoretical frequencies for anharmonic effects [Scott, A. P., and L. Radom,/. Phys. Chem., 100(1996) 16502]. 

Here IQ is the incident beam intensity, co is the angular speed of rotation of the crystal during the integration process, K and K" are empirical scale factors, Qhki is the diffracting power of the crystal,... 

Concerning the scaling of the SCF frequencies, another point should be made. An implicit assumption underlying the use of empirical scaling factors is... 

More sophisticated models employ molecular shaped cavities, but there is again no consensus on the exact procedure. The cavity is often defined based on van der Waals radii of the atoms in the molecule multiplied with an empirical scale factor. Alternatively, the molecular volume may be calculated directly from the electronic wave function, for example by using a contour surface corresponding to an electron density of 0.001. 

Owing to the semiempirical nature of the MNDO method, the theoretical force constants are between 10 and 20% larger than the experimental values. Therefore, following Torok and Pulay< empirical scale factors have been calculated in the case of the MNDO force field of butadiene for which molecule enough experimental information is available. The suggestion followed in the computation of these scale factors was that the geometrical mean value of the diagonal force constants be used.< ... 

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