The scale factor

Since reflection intensities are commonly measured on a relative rather than an absolute scale, a scale factor is required to bring observations and calculations on 

TABLE 4.2 Local-Symmetry-Allowed Multipole Population Parameters in S4N4 

Since the scale factor is considered an unknown in the least-squares procedure, its estimate is dependent on the adequacy of the scattering model. Other parameters that correlate with k may be similarly affected. In particular, the temperature factors are positively correlated with k, the correlation being more pronounced the smaller the sin 0//. range of the data set, as for a small range the scale factor k and the temperature factor exp ( —fl sin 02ft.2) affect the structure factors in identical ways. 

As discussed in the following chapter, difference electron density maps, representing Ap = pobs — pcak, are based on the Fourier transform of the complex difference structure factors AF, defined as 

The difference density is strongly dependent on k, especially where the electron density p is large, as it is near the nuclear positions. When k is overestimated, the difference density will be underestimated, and pronounced negative holes will occur in the nuclear regions. 

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Also use constant dielectric Tor MM+aiul OPLS ciilciilatimis. Use the (lislance-flepeiident dielecinc for AMBER and BlO+to mimic the screening effects of solvation when no explicit solvent molecules are present. The scale factor for the dielectric permittivity, n. can vary from 1 to H(l. IlyperChem sets tt to 1. .5 for MM-r. Use 1.0 for AMBER and OPLS. and 1.0-2..5 for BlO-r. 

Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a uniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as e = (H/L) 0. Coordinate scale factors h, as well as dynamic variables are represented by a power series in e. It is expected that the scale factor h-, in the direction normal to the layer, is 0(e) while hi and /12, are 0(L). It is also anticipated that the leading terms in the expansion of h, are independent of the coordinate x. Similai ly, the physical velocity components, vi and V2, ai e 0(11), whei e U is a characteristic layer wise velocity, while V3, the component perpendicular to the layer, is 0(eU). Therefore we have... 

The remainder of the input file gives the basis set. The line, 1 0, specifies the atom center 1 (the only atom in this case) and is terminated by 0. The next line contains a shell type, S for the Is orbital, tells the system that there is 1 primitive Gaussian, and gives the scale factor as 1.0 (unsealed). The next line gives Y = 0.282942 for the Gaussian function and a contiaction coefficient. This is the value of Y, the Gaussian exponential parameter that we found in Computer Project 6-1, Part B. [The precise value for y comes from the closed solution for this problem S/Oir (McWeeny, 1979).] There is only one function, so the contiaction coefficient is 1.0. The line of asterisks tells the system that the input is complete. 

While the random fluctuations apparent are a function of the scaling factor for the traces, the two show different amplitudes. The top trace has a relatively small fluc tuation, while the bottom trace shows a larger one. 

Let us assume that the scaling factor for the heat exchanger is 0.6. Hence, in 1992, the cost of the 100 m shell-and-tube heat exchanger can be estimated from Eq. (I1I.2) to be... 

Most of the scale factors in this table are from the recent paper of Wong. The HF/6-31G(d) and MP2(Full) scale factors are the traditional ones computed by Pople and coworkers and cited by Wong. Note that the MP2 scale factor used in this book is the one for MP2(Full) even though our jobs are run using the (defriultj frozen core approximation. Scott and Radom computed the MP2(FC) and HF/3-21G entries in the table, but this work came to our attention only just as this book was going to press. 

The scale factor is optional. If Included, it says to scale the frequencies before performing the thermochemicai analysis. Note that including the factor affects the thermochemistry output only (including the ZPE) the frequencies printed earlier in the output remain unsealed. This parameter is the means by which scale foctors are applied to thermal energy corrections. 

All parameters other than the scale factor must be included, even if the default valut, are used. 

When comparing energy results to experiments performed at particular temperatures, the thermal energy correction given in the output should be added to the total energy (this sum is also given in the output). In order to apply the appropriate scale factor to a thermal energy correction, you must specify a scale factoi via input to the Readlsotopes option. The quantity reported in the output cannot simply be multiplied by the scale factor itself as it is composed of several terms, only some of which should be scaled. 

The carbon dioxide zero-point energies in the table are scaled, using the scaling factors listed on page 64. ... 

Compute the isomerization energy between acetaldehyde and ethylene oxide at STP with the QCISD(T)/6-31G(d) model chemistry, and compare the performance of the various model chemistries. Use HF/6-31G(d) to compute the thermal energy corrections. Remember to specify the scaling factor via the Freq=Recxllso option. (Note that we have already optimized the stmcture of acetaldehyde.)... 

Here are the predicted frequencies for the first excited state of formaldehyde, along with the corresponding experimental values (the scale factor is the same as for Hartree-Fock frequencies 0.8929) ... 

The acronym SEC refers to the case where the reference wave function is of the MCSCF type and tire correlation energy is calculated by an MR-CISD procedure. When the reference is a single determinant (HE) the SAC nomenclature is used. In the latter case the correlation energy may be calculated for example by MP2, MP4 or CCSD, producing acronyms like MP2-SAC, MP4-SAC and CCSD-SAC. In the SEC/SAC procedure the scale factor F is assumed constant over the whole surface. If more than one dissociation channel is important, a suitable average F may be used. 

These problems can be minimized in the great majority of cases by using specially designed probes. The user will need to know the frequency range (i.e. the bandwidth) over which the probe can be used satisfactorily and the scaling factor introduced by the probe. 

It may happen that many steps are needed before this iteration process converges, and the repeated numerical solution of Eqs. III.21 and III.18 becomes then a very tedious affair. In such a case, it is usually better to try to plot the approximate eigenvalue E(rj) as a function of the scale factor rj, particularly since one can use the value of the derivative BE/Brj, too. The linear system (Eq. III. 19) may be written in matrix form HC = EC and from this and the normalization condition Ct C = 1 follows... 

In the case the calculations are based on a truncated set Wlf 2,. . . containing adjustable parameters, the A splitting is of particular importance, since it permits the investigator to use different values of these parameters for different eigenvalues Xk— the relation III.95 will anyway be valid. The scale factor rj is such a parameter, and the results in Section II.C(3) and III.D(lb) show that, by means of the A splitting, it is now possible to get the virial theorem exactly fulfilled for at least one of the eigenfunctions associated with each Xk. 

We first mean center each data point, ay, and then divide it by the scale factor. If we do not wish to mean-center the data, we finish by adding the mean value back to the scaled data point... 

Most of the tests made so far have used water and Freon-12 (CC12F2), and the scaling factors implied by the various possible sets of scaling laws may be calculated from the physical properties for these two fluids. The appropriate scaling factors based on water at 1000 psia, for which pL/pv = 20.63, are listed in Table VII. As an example of how the scaling factors are calculated, the group Ahjl in Eq. (39) will have the same value for water and Freon-12 if... 

The temperature-factor parameter B and the scale factor k were determined by a least-squares procedure/ with observational equations set up in logarithmic form and with weights obtained from those in equation (9) by multiplying by (G (obs.))2. Since a semi-logarithmic plot of G2 (obs.)/Gf (calc.) against B showed a pronounced deviation from linearity for the last five lines, these lines were omitted from the subsequent treatments. They were much broader than the others, and apparently their intensities were underestimated. The temperature-factor parameter B was found by this treatment to have the value 1-47 A2. 

At first glance, these results seem fantastic. Look at the case where S = 100. When the pressure drop across the pilot reactor is large, a mere 47% increase in length gives a 100-fold increase in inventory The pressure and the density increase by a factor of about 69. Multiply the pressure increase by the length increase and the factor of 100 in inventory has been found. The reactor volume increases by a factor of only 1.47. The inventory and the throughput scale as S. The scaling factor for volume is much lower, 1.47 instead of 100 in this example. 

To ensure that no information is lost on Fio)) as the dilation is discretized, the scale factors 2 " for m g Z must cover the whole frequency axis. This can be accomplished by requiring the wavelets to satisfy the following... 

The basic experimental setup for etNOESY is practically identical to the conventional NOESY experiment shown in Eig. 9.2(B). For suppression of residual receptor signals a relaxation filter can be introduced and the mixing time has to be corrected according to the scaling factor a. 

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