Scaling factor, calculation methods

Having collected all needed intensity data under the most favorable conditions possible, the crystallographer processes the data, applying absorption corrections and, if necessary, corrections for decomposition of the crystal, and arrives at his data set, consisting of the values of F j / or F / 2, either unsealed or with a rough scale factor calculated by statistical methods. Each datum should be accompanied by a standard deviation o that represents random error (and possible random effects of systematic errors) as derived, for example, with Eq. (13). 

The second, third, and fourth corrections to [MPd/b-Jl lG(d,p)] are analogous to A (- -). The zero point energy has been discussed in detail (scale factor 0.8929 see Scott and Radom, 1996), leaving only HLC, called the higher level correction, a purely empirical correction added to make up for the practical necessity of basis set and Cl truncation. In effect, thermodynamic variables are calculated by methods described immediately below and HLC is adjusted to give the best fit to a selected group of experimental results presumed to be reliable. 

It is possible to use computational techniques to gain insight into the vibrational motion of molecules. There are a number of computational methods available that have varying degrees of accuracy. These methods can be powerful tools if the user is aware of their strengths and weaknesses. The user is advised to use ah initio or DFT calculations with an appropriate scale factor if at all possible. Anharmonic corrections should be considered only if very-high-accuracy results are necessary. Semiempirical and molecular mechanics methods should be tried cautiously when the molecular system prevents using the other methods mentioned. 

Frequencies computed with methods other than Hartree-Fock are also scaled to similarly eliminate known systematic errors in calculated frequencies. The followng table lists the recommended scale factors for frequencies and for zero-point energies and for use in computing thermal energy corrections (the latter two items are discussed later in this chapter), for several important calculation types ... 

The introduction of various empirical corrections, such as scale factors for frequencies and energy corrections based on the number of electrons and degree of spin contamination, blurs the distinction between whether they should be considered ab initio, or as belonging to the semi-empirical class of methods, such as AMI and PM3. Nevertheless, the accuracy tiiat tiiese methods are capable of delivering makes it possible to calculate absolute stabilities (heat of formation) for small and medium sized systems which rival (or surpass) experimental data, often at a substantial lower cost than for actually performing the experiments. 

Vibrational Spectra Many of the papers quoted below deal with the determination of vibrational spectra. The method of choice is B3-LYP density functional theory. In most cases, MP2 vibrational spectra are less accurate. In order to allow for a comparison between computed frequencies within the harmonic approximation and anharmonic experimental fundamentals, calculated frequencies should be scaled by an empirical factor. This procedure accounts for systematic errors and improves the results considerably. The easiest procedure is to scale all frequencies by the same factor, e.g., 0.963 for B3-LYP/6-31G computed frequencies [95JPC3093]. A more sophisticated but still pragmatic approach is the SQM method [83JA7073], in which the underlying force constants (in internal coordinates) are scaled by different scaling factors. 

This theory appears not to involve adjustable parameters (other than the nuclear radius parameters that were taken from the literature). In particular, it was criticized that the calibration approach involved a slope that is too high by about a factor of two. However, in actual calculations with the linear response approach, it was found that the slope of the correlation line between theory and experiment (dependent on the quantum chemical method) is close to 0.5. Thus, it also requires a scaling factor of about 2 in order to reach quantitative agreement with experiment. The standard deviations between the calibration and linear response approaches are comparable thus indicating that the major error in both approaches still stems from errors in the description of the bonding that is responsible for the actual valence shell electron distribution. 

During recent years DFT methods have been used to reproduce vibrational frequencies and IR intensities (dipole moment derivatives) with high accuracy (scaling factors are close to unity).29,60,61 We therefore used the B3LYP and BLYP functionals to calculate the spectra of la and its isotopomers, and indeed the calculated frequencies, isotopic shifts, and intensities are now in excellent agreement with the experimental values (Fig. 3).62 A careful reexamination... 

A preliminary structural model of a protein is arrived at using one of the methods described above. Calculated structure factors based on the model generally are in poor agreement with the observed structure factors. The agreement is represented by an R-factor defined as found in equation 3.9 where k is a scale factor ... 

Summarizing the results we can make a conclusion that all the methods of the centre calculation are almost equal in the precision. The maximum difference between the results obtained using described methods was 2 pixels, which is less than 0.003 A (the scale factor was about 750 to 900 pixels/A) for the given texture patterns. Such small difference has no significant influence on further calculations, which was proved by calculations performed on these texture patterns. 

EXAFS Data Analysis. A key aspect of the analysis outlined above is knowledge of the correct (k) and for a particular absorber-scatterer interaction. These parameters can either be calculated ab initio (6) or can be determined by measuring the EXAFS of structurally characterized model compounds (7). The ab initio method has the advantage that one need not prepare appropriate models for all possible unknowns. Unfortunately however, the ab initio parameters must be adjusted by a scaling factor and an assignment of E, (8). For this reason, one typically calibrates the calculated... 

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