Numerical Comparison

This section collects results obtained with the three exact-decoupling methods within the same implementation and follows the discussion in Ref. [647]. The number of matrix operations necessary for the implementation of different two-component approaches has been collected in Table 14.2. The multiplication of a general matrix with a diagonal matrix requires O(m ) multiplications of floating-point numbers, where m is the dimension of the matrix identical to the number of (scalar) basis functions in this context. The multiplication of two general matrices scales formally as If m is large, the cost of the 

In addition to matrix multiplication, matrix diagonalization and inversion are required. Both of them are of order 0 nfi). As we can see from Table 14.2, the X2C and BSS approaches require the same five different types of matrix diagonalization and inversion, whereas the DKH approach needs only two of them. The commonly used two operations are matrix diagonalization of dimension m, which is used for basis orthonormalization as in Eq. (14.30), and matrix inversion of dimension m, which is the calculation of the inverse basis transformation matrix 

The ten real matrix multiplications found for all three decoupling schemes are the orthonormal basis transformation, K AK, of the five potential-energy matrices, A e V, W°, W, W.  

The number of two-component complex matrix multiplications is different for the different approaches. The BSS approach differs from the X2C approach only in this term. It requires three more than the X2C approach since the off-diagonal parts of its four-component Hamiltonian matrix are not diagonal. For the DKH approach, the number of two-component complex matrix 

If the scalar approximation (neglecting spin-orbit-coupling terms) is activated, the computational demands turn out to be rather different. The operation count for the scalar-relativistic variants is also given in Table 14.2. Most important, only real matrix operations are required since one can employ real basis functions and the Hamiltonian operators are also real. Then, spin is a good quantum number and the spin symmetry can be exploited so that dimensions of all two- and four-component matrix operators are reduced by half. Finally, since the spin-orbit components of the relativistic potential matrix, i.e., W, Wy, and W, are neglected, the number of matrix multiplications required for the orthonormal basis transformation is decreased from ten to four. 

---

They compared the PME method with equivalent simulations based on a 9 A residue-based cutoflF and found that for PME the averaged RMS deviations of the nonhydrogen atoms from the X-ray structure were considerably smaller than in the non-PME case. Also, the atomic fluctuations calculated from the PME dynamics simulation were in close agreement with those derived from the crystallographic temperature factors. In the case of DNA, which is highly charged, the application of PME electrostatics leads to more stable dynamics trajectories with geometries closer to experimental data [30]. A theoretical and numerical comparison of various particle mesh routines has been published by Desemo and Holm [31]. 

Desemo M and C Holm 1998a. How to Mesh Up Ewald Sums. I. A Theoretical and Numerical Comparison of Various Particle Mesh Routines. Journal of Chemical Physics 109 7678-7693. 

In this article, all numerical comparisons of relative sweetness are given on a molar basis. 

A quick impression of a star s metallicity can often be derived from inspection -either qualitative or quantitative - of strong spectral features such as the CN band A, 4215 in giants or comparison of hydrogen and Ca+ K-line (A 3933) intensities, which has been particularly useful in the discovery of extremely metal-deficient stars (Beers, Preston Shectman 1992). Numerical comparisons of digital spectra with low signalinoise ratio can also be carried out for this purpose (Carney et al. 1987). 

Numerical Comparison of Temperatures for Screw and Barrel Rotations... 

Three analytical expressions for the spin-echo intensity as a function of the gradient in a pulsed field gradient NMR experiment for spins diffusing in a sphere with reflecting walls are reinvestigated. It is found that none of the published formulas are completely correct. By numerical comparisons the correct formula is found. 

The simplicity of the polymerization reaction is the result of intense research carried out by several groups on the importance and the fundamentals of each parameter. In particular, Matyjaszewski et al. have spent great effort on the construction of numerous comparison charts on the activity of initiators and ligands that are used in ATRP [30-32]. These published comparison tables represent the summary of hundreds of single experiments and are now a very important and reliable source of data for the ATRP technique. 

The second approach starts from the modified Langevin equation Eq. (37) and uses the equivalence of the Kramers theory to the multi-dimensional TST. It has been established by numerical comparison that there is agreement between the two approaches. 

Unfortunately, no direct numerical comparison is available for us to assess the quality of this trade-off from the WDA to the SADA. Nonetheless, we believe... 

U. H. E. Hansmann and Y. Okamoto. Numerical comparison of three recently proposed algorithms in the protein folding problem. J. Comput. Chem, 18 920,... 

Aharoni has stated that the observed rates of crystallization in polymers are inconsistent with the times required for random-coil molecules to separate themselves from the melt, and claims this as support for the collapsed coil model (43). No numerical comparisons are given, and it is difficult therefore to judge the basis for his assertion. 

A numerical comparison of SACM and quantum corrected ICVTST has been performed in Ref. 26 for the capture of HC1, HCN, or CS by H3. Permanent-and induced-dipole terms were included. For a system with aB/nh = 10 3, the following results were obtained for kT/B values of 125, 31.25, 13.89, 3.472, 0.8681, and 0.4883, ratios icvtst/ sAcm of 1.33, 1.39, 1.41, 1.40, 1.22, and 1.18, respectively, were obtained. These results are in line with the conclusions from the previous sections, that is, that the truncation in ICVTST does not sufficiently improve the CVTST treatment. 

In general, the data accuracy was surprisingly good. For example, while Deaton and Frost (1946, p. 13) specified that their pure ethane contained 2.1% propane and 0.8% methane, effects of those impurities may have counterbalanced each other those impurities were insufficient to cause the data to fall outside the line formed by other ethane data sets. On the other hand, the simple hydrate data of Hammerschmidt (1934) for propane and isobutane all appear to be outliers on such semilogarithmic plots, because they are at temperatures much too far above the upper quadruple (Q2) point. Obvious outlying data were excluded from this work less obvious outliers may be determined by inspection of the plots and subsequent numerical comparisons. The data, followed by the semilogarithmic plots... 

Increases in rates of reaction effected by enzymes can be huge, but quantification of the rate enhancement, i.e. numerical comparison with the uncatalysed reaction rate, is seldom straightforward. Furthermore, the mechanism of catalysis of a given reaction by an enzyme may be very different from its catalysis by a non-enzyme. Of course, the main thrust of... 

The non-reversible behavior is plotted in Fig. 7.46, which corresponds to the corrected (Jcv/v) — E curves (dashed lines) and the (QDSCVC/AE) — E ones (symbols) of the system 4-PhenylazoPhenol. From these curves, it can be seen that although the DSCVC curves are perfectly superimposable, the CV ones clearly show smaller peak heights in both scans. This systematic decrease of the CV signals, which cannot be theoretically predicted, is 5-10 %, and it has been reported when the response of electro-active monolayers in CV has been compared with other voltammetric and chronopotentiometric electrochemical techniques [71, 72], Due to the quasi-reversible nature of the charge transfer reduction of the 4-PhenylazoPhenol, no simple equations for the peak parameters are available. So, a numerical comparison between theoretical and experimental curves for different sets of parameters should be made in order to obtain the kinetic and thermodynamic parameters of the system. 

Hasselt BWv, Lebens PJM, Calis HP, Kapteijn F, Sie ST, Moulijn JA, Bleek CMvd. A numerical comparison of alternative three-phase reactors with a conventional trickle-bed reactor. The advantages of countercurrent flow for hydrodesulfurization. Chem Eng Sci 1999 54 4791-4799. 

The numerical comparison is an open-ended problem, the scope of which must be decided by the instructor. 

The tenure data and the rank data are not exactly the same. The numbers differ by several hundred, sometimes several thousand. It is really hard to make clear numerical comparisons. What we should focus on is the fact that wherever you look, women are not advancing. You can look anywhere women are not advancing at the rates they should, and it cannot be explained by things like increasing numbers of women at lower positions, because you can subtract out those older men and women who are full professors and recalculate and you still see a problem. 

---