Elementary transformations

This elementary transformation of Equation (2.332) allows us to visualize better geometry of the force f and level surface of its potential. Assume that the surface of... 

By elementary transformations, similar to the ones leading to Eq. (344), we then get ... 

Using the scheme shown in Fig. 56b, we find from Eq. (452) after elementary transformations ... 

After an elementary transformation, we obtain (it is recommended that the reader do it for himself)... 

Fig. 6. Six different elementary transformations of synthons from the same family 9- A)...

For a fixed family 3 (A) of isomeric synthons we construct the so-called graph of reaction distances [18, 21, 16, 25] denoted by RD(A). The vertex set of this graph is formally identical with the family A) without forbidden synthons, its two distinct vertices v and v, assigned to the synthons S(d) and S (/l), are connected by an edge [u, v ] if such an elementary transformation i = a, p exists so that the synthon S(/l) is transformed into the synthon i.e. 

For instance, the sequences of elementary transformations of the mechanisms listed in the above example are... 

Fig.10. The first (second) graph corresponds to the 1st, 5th, and 6th (2nd to 4th) mechanisms. These graphs were constructed by adding edges and loops of reaction graphs of all elementary transformations of a mechanism into one graph. The corresponding (reduced) reaction graph is given in Fig. 9...

For a moment, let us come back to the organic chemistry. An organic reaction, formally treated as the transformation S(A) => S (A), where S(A) and S (A) are stable synthons, is carried out as a sequence of transformations (not necessary elementary transformations),... 

FIGURE 1. A schematic photochemical mechanism, showing some of the possible elementary transformations. For the purpose of illustration, it is assumed that the states A and A2 have the same multiplicity, and correspond to the ground and lowest excited singlet states of most organic molecules. The state A] would then represent the lowest triplet state. Thus 21 and 11 are radiative transitions, fluorescence and phosphorescence, respectively, and 23 and 13 (intersystem crossing) and 22 (internal conversion) are nonradiative. All of 8, C, D, and F are chemical species distinct from A. Only vibrationally equilibrated electronic states are included in this mechanism (see discussion in Section III.A.l). 

The absorption of light by a substance causes the formation of excited-state molecules. This excitation is followed by various elementary transformations which eventually lead to the deactivation or to the disappearance of those excited molecules. The absorption of light as well as each one of the elementary transformations of the original molecule in an excited state is a primary step. Specifically, a primary step may be (a) a transformation of the excited molecule into a different chemical species, as in steps 24, 15, and 14 of Figure 1, or (b) a radiative or nonradiative transition between different energy levels of the molecule, e.g., steps 02, 21, 22, 23, 13, 11, and 16 of Figure 1. Those corresponding to (a) are photochemical primary steps, while those of (b) are photophysical primary steps. 

In principle, in almost all photophysical primary steps, considered strictly as elementary transformations, an excited state or the ground state is formed with vibrational (and to a less important extent, rotational) energy in excess of the amount it would have at equilibrium. 

PHOTOPHYSICAL PRIMARY STEP an elementary transformation of the originally excited species without any chemical change. 

PRIMARY STEP any one of the elementary transformations of an excited-state molecule of the species which absorbs light. The absorption step itself is also a primary step. 

Apparently, thermodynamic rushes for the series of transformations under consideration and, as a result, chemical potentials of the intermediates in their stationary states must decrease progressively while passing from one intermediate to another. When the stepwise transformation may follow sev eral parallel pathways of consecutive elementary transformations, the said relationship between the stationary chemical potentials of the intermediates must be met for each of the possible pathways of the stepwise reaction. 

For elementary reactions, the term molecularity is commonly used, Vy, of the elementary transformation, which means the number of molecules that are involved directly in the elementary act Vy = X V q( k Vy = X VjQc Evi dently, this term cannot be applied directly to the stepwise processes, but the expression for vj may be associated with the term formal molecularity vj of the stepwise reaction, which equals... 

We analyze a few of the simplest catalytic systems following. In the systems under consideration, the constituent elementary transformations usually are linear in respect to the concentrations of catalytic intermediates. Generally speaking, the results of this kind of analysis are only applicable when the inter action between the active centers, whether identical or different, of the catalyst can be ignored. This is the case mainly for homogeneous and enzymatic rather than heterogeneous catalysis. However, in some cases, the conclusions can be extended to more complex catalytic systems operating in the steady stationary states even with the lateral interaction between the active centers. 

In catalytic stepwise reactions, which involve more complex elementary transformations than scheme (4.4), the rate-determining parameters can be identified through similar considerations. Several examples of simple model schemes of catalytic transformations are given following. These schemes often are used for the microkinetic analysis of particular catalytic transformations and help to reveal the influence of various factors. 

Table 4.4 Elementary transformations at the Belousov-Zhabotinsky reaction with malonic acid as the oxidized substrate...

In order to achieve the desirable overall result, however unrealistic it may appear, an organic chemist looks for an opportunity to break down the overall transformation into a sequence of chemically possible steps. Chemically possible means that a well-defined chemical reaction can be utilized to achieve the required elementary transformation. Then it will be within the chemist s control, at least in principle, to select the proper reagents and reaction conditions ( tools and procedures ) for each step that would ensure the efficiency of every required transformation of the whole sequence and hence the viability of the conceived synthetic plan. This ideal scheme is rarely realized completely, but it is generally applicable within reasonable limits. In fact, identification of a set of simple and realistic steps in conjunction with a careful selection of the optimal reagents and reaction conditions for every single step is the underlying approach elaborated for controlled total synthesis. 

Primary process" has been used in accordance with this recently suggested definition (2) "Any continuous sequence of one or more primary steps which starts with the light absorption step." In this sense a primary step is "any one of the elementary transformations of an excited state molecule of the species which absorbs light. The absorption step Itself is also a primary step" (2). Important primary processes of OTM compounds which are described here include (1) absorption, (ii) dissociative reactions, (iii) intramolecular "twisting" isomerizations, (iv) intermolecular energy transfer, (v) inter-molecular electron transfer, (vi) luminescence. Reactions involving OTM compounds as quenchers have also been included. 

As with all such palladium-catalyzed carbon-heteroatom bond-forming chemistry, the ancillary ligand(s) (i.e., L often featuring phosphine or A-heterocyclic carbene donors " employed have a direct influence over the course of the elementary transformations. Electron-rich and sterically demanding ligands promote the formation of low-coordinate compounds of type A that are predisposed to undergo Ar—X... 

On substituting these expressions into (6.6.1) and performing some elementary transformations, we arrive at the equation... 

Expanding exponent in a series and substituting it into eq. 9.4, after elementary-transformations we obtain ... 

Kn k L). After elementary transformations, which allow us to present an expression for the field through integrals of type we obtain ... 

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