Correlated completely

The value of the correlation coefficient ranges from r = -1 to r = +1. In those cases where r =l, the data are completely correlated, either positively or negatively (see Figure 9-4). The smaller the absolute value of r, the lower is... 

The disadvantage of ah initio methods is that they are expensive. These methods often take enormous amounts of computer CPU time, memory, and disk space. The HF method scales as N, where N is the number of basis functions. This means that a calculation twice as big takes 16 times as long (2" ) to complete. Correlated calculations often scale much worse than this. In practice, extremely accurate solutions are only obtainable when the molecule contains a dozen electrons or less. However, results with an accuracy rivaling that of many experimental techniques can be obtained for moderate-size organic molecules. The minimally correlated methods, such as MP2 and GVB, are often used when correlation is important to the description of large molecules. 

The color and constitution of cyanine dyes may be understood through detailed consideration of their component parts, ie, chromophoric systems, terminal groups, and solvent sensitivity of the dyes. Resonance theories have been developed to accommodate significant trends very successfully. For an experienced dye chemist, these are useful in the design of dyes with a specified color, band shape, or solvent sensitivity. More recendy, quantitative values for reversible oxidation—reduction potentials have allowed more complete correlation of these dye properties with organic substituent constants. 

The correlation function is a number between -1 and 1, where 1 indicates that the two quantities are completely correlated, -1 that they are (completely) anti-correlated and 0 means that they are independent (uncorrelated). 

Some of these group frequencies are listed in Table 19.1, and amore complete correlation table is provided in Appendix 11. 

Random variables, 114,117 completely correlated, 147 complex-valued, 144 discrete, 196... 

Relative analysis has shown, that the results received by both methods, completely correlate ... 

This fairly complete correlation shows that effects of fuel properties and perhaps vent radiation with the appearence of II7, as the physics are not evident, while these are missing in the correlation of Law [19] (Equation (11.45)). Yet Law s correlation includes the effect of a compartment shape. Also in Equation (12.64), the dimensionless groups are indicated compared to the dimensional presentation of Babrauskas, in units of kJ, kg, m, K, s, etc. 

A series of optically active linear tetradentate ligands, which have been stereospeeifically synthesized, have been used to prepare complexes with a trans-[CoN Cl2] core and both five- and six-membered chelate rings, in an attempt to correlate the sign of the Cotton effect with the known chiralities of compounds. How ever, a complete correlation was not possible. The synthesis, resolution, and properties of some oxalato, malonato, and diacido complexes of Co " with the stereospecific flexible tetramine ligands 5-Me-3,2,3-tet and NA -bis-(2-picoyl)-l-methyl-1,2-diaminoethane (picpn) have been reported. The stereospecificity is demonstrated by comparison of the optical rotation of the ligand prepared via an asymmetric synthesis with that of the ligand isolated from a resolved complex. The stereochemistry of the complexes has been deduced. ... 

As can be seen, although the complete correlation effect of the a/ block is small, which seems to indicate that a perturbative treatment may be suitable for its estimation [11], this small quantity results from the cancellation between two large correlation matrices, which may render difficult the convergence of a perturbative treatment. 

Relation (26) should be underlined since it describes the idempotency def-fect of the 1-RDM which is the contraction of the complete correlation term corresponding to the ap block of the 2-RDM. 

Ivency or emulsification of beeswax, aqueous solubility, and vapor pressure are completely correlated with nematocidal efficacy as determined in laboratory tests. The correlation is not complete when field efficacy is considered. Thus carbon tetrachloride, benzene, and toluene should be good nematocides, but have not proved satisfactory in field tests. Likewise carbon disulfide, while a sound nematocide under optimum conditions, requires, for satisfactory nematode control, a greater quantity than would be expected on the basis of its solubilities and solvency. The answer may lie in the lack of polarity of these molecules. 

We can now assign the correct spin multiplicity, 1 or 3, to each of these states by the same methods described in detail for the octahedral case and then correlate the states on the two sides of the diagram to obtain the complete correlation diagram shown in Figure 9.4. 

Fig. 11.13. The mean rotational state < N2 > of rotor 2 if the sibling rotor 1 is produced in rotational state Ni comparison between experimental results and classical trajectory calculations. Here, N is the total angular momentum of OH(2II) excluding the spin. With increasing temperature of the H2O2 parent molecule the correlation gradually diminishes. The straight line indicates complete correlation. Adapted from Maul, Glaser, and Gericke (1989).

In conclusion, the general point of view is put forward that molecules act as self-dedicated computers , printing out measurement data which provide complete, self-consistent and completely correlated solutions of the Schrodinger equation10,11. The follow-up Shakespearean question, whether to measure or to calculate, which arises from the rapid development of both the measurement techniques as well as of numerical quantum mechanics, is best answered with reference to how successful and stimulating the combination of the two proves to be. 

After hydrolysis, separation of LiBr, and distilling off the solvent and volatile by-products, compound 65 remains in the residue. In the nmr spectrum of 65 two clearly separated peaks are found for the me2Si groups. These are due to hindrance of free rotation around the molecular axis caused by the phenyl groups. A complete correlation of all peaks in the nmr spectrum is very difficult. Derivatives were found useful in the determination of the structures, changing the substituents produces a definite chemical shift in the nmr spectra. 

In calibration it is normal to use several concentration levels to form a model. Indeed, for information on lack-of-fit and so predictive ability, this is essential. Hence two level factorial designs are inadequate and typically four or five concentration levels are required for each compound. However, chemometric techniques are most useful for multicomponent mixtures. Consider an experiment earned out in a mixture of methanol and acetone. What happens if the concentrations of acetone and methanol in a training set are completely correlated If the concentration of acetone increases, so does that of methanol, and similarly with a decrease. Such an experimental arrangement is shown in Figure 2.25. A more satisfactory design is given in Figure 2.26, in which the two... 

The temperature at which the phase transition occurs is called the critical temperature or Tg. Most, but not all, magnetic phase transitions are continuous , sometimes called second order . From a microscopic point of view, such phase transitions follow a scenario in which, upon cooling from high temperature, finite size, spin-correlated, fractal like, clusters develop from the random, paramagnetic state at temperatures above Tg, the so-called critical regime . As T Tg from above, the clusters grow in size until at least one cluster becomes infinite (i.e. it extends, uninterrapted, throughout the sample) in size at Tg. As the temperature decreases more clusters become associated with the infinite cluster until at T = 0 K all spins are completely correlated. 

The concepts discussed regarding the symmetry restrictions and their removal can be described in a number of ways. Complete correlation diagrams can be constructed and the forbiddenness illustrated by sharp orbital crossings 20). Although definitive, this approach would not as clearly illustrate the nature of the restraints to reaction. 

It is the responsibility of experimentalists to see their task in terms of producing the most complete correlation function possible, for comparison with that obtained from dynamical calculations. It is the goal of the theoreticians to seek to combine the techniques of MO theory with dynamical calculations calculations using quantum statistical mechanics are made possible by the existence of large computers. 

Again, if there is no correlation between x and j, then 02 = 0. At the other extreme, if there is complete correlation between x and y, then there is a relationship between tt], 2> 2 that is ... 

If one s goal is merely to predict the number of spin-allowed transitions expected for a comfidex, a complete correlation diagram is not needed. It is only necessary to know the number of excited states having the same multiplicitv as the ground slate. 

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