Time-ordered graphs

Fig. 7. Time ordered graphs for static dispersion interaction. Left-hand diagram, electric-electric terms. Right-hand diagram, electric-magnetic terms...

Fig. 8. Time ordered graphs for static and fully retarded electric-electric resonance interaction...

Figure 10.4 Time-ordered graphs corresponding to the four contributions to when the perturbation matrix elements are given by Eqs. 10.5 and 10.6. These are shown in the same order as the energy level diagrams in Fig. 10.3.

The role implied by these time-ordered graphs for the virtual states called )n> should not be taken too literally. In the treatment that follows, these virtual states are in effect expanded in infinite series of true molecular eigenstates n>, and no virtual state in any of the processes will coincide with any single, particular true n>. Hence, while the energy conservation hcoi + ha>2 = —... 

At this point, we have developed our theoretical framework sufficiently to deal explicitly with TPA and Raman spectroscopy. Spontaneous tw6( -photon emission (which is depicted by the last of each set of time-ordered graphs in Figs. 

Figure 10.6 Time-ordered graphs corresponding to the energy level diagrams in Fig. 10.5.

For a given intermediate state n>, the first and second right-hand terms in Eq. 10.26 correspond to the first and second time-ordered graphs in Fig. 10.12, respectively. When a>i > a>2, the scattered radiation frequency CO2 is said to be... 

Figure 10.12 Time-ordered graphs for the Raman process with incident frequency cu, and scattered frequency...

We next evaluate the terms in corresponding to the first two time-ordered graphs. The contribution from graph (a) in the cw limit is... 

Figure 11.1 Time-ordered graphs representing the eight contributions to (0 o Eq. 11.7 when the perturbation matrix elements are given by Eq. 11.8.

Figure 11.3 The set of time-ordered graphs that can be drawn for the sum-frequency generation process in which a/, and are the incident frequencies and 0/3 is the scattered frequency. Six graphs result from permuting the three frequencies among the three interaction vertices.

As a first step in deriving expressions for CARS transition probabilities, we list in Fig. 11.6 the 12 time-ordered graphs corresponding to absorption of two photons at toj and scattering of photons at 0)3, 0 3. Using the diagrammatic techniques introduced in Section 11.1, we immediately get for the CARS contribution to the fourth-order coefficient in the time-dependent perturbation expansion (1.96)... 

Figure 11.6 The 12 time-ordered graphs that can be drawn for CARS with two incident photons at frequency w, and scattered photons at frequencies and Wg.

The decomposition data forjNO plotted as the reciprocal of concentration vs. time. This graph is linear, with a slope equal to the second-order rate constant. 

All data obtained on the rate of reaction of [Ni(NiL2)2]Cl2 with alkyl halides— i.e., methyl iodide, benzyl bromide, benzyl chloride, p-nitrobenzyl chloride, p-chlorobenzyl chloride, ethyl bromide, ethyl iodide, n-propyl bromide, and n-propyl iodide—conform closely to a pseudo-first-order rate law. Almost all experiments were carried out in the presence of an excess of alkyl halide. Since methanol solutions of the alkylated complexes have only negligible absorption at 495 m//, rates were obtained by graphs of log A0—A vs. time. The graphs are linear over the entire time interval, which corresponds to more than two half lives in most cases, passing through the origin at zero time. The rate is essentially the same whether measured by the spectrophotometric or conductivity method. 

FIGURE 13.10 We can test for a first-order reaction by plotting the natural logarithm of the reactant concentration against the time. The graph is linear if the reation is first order. The slope of the line, which is calculated by using the points A and B, is equal to the negative of the rate constant. 

The decomposition of the reaction graph will be formally ordered in a sequence of elementary reaction graphs. This sequence corresponds to the time order of consecutive elementary reactions. The formed set of N elementary reactions is bijectively mapped onto the set of integers 1,2,...yN. These integers assigned to each elementary reaction describe its position in the sequence of reactions that forms studied mechanism, henceforth called its serial number. The first of the above decompositions can be indexed in 4 different ways, we shall consider the following two possibilities... 

The DENDRAL project initiated in 1964 at Stanford was the prototypical application of artificial intelligence techniques - or what was understood at that time under this name - to chemical problems. Chemical structure generators were developed and information from mass spectra was used to prune the chemical graphs in order to derive the chemical structure associated with a certain mass spectrum. 

The oseillator frequeney is set by referring to a timing graph. In order to provide a 100 kHz operating frequency, the values for RT and CT are... 

The MC34067 data sheet eontains the neeessary equations and graphs in order to set the eritieal timer funetions of a ZCS QR eonverter. Some of the times, sueh as the one-shot off-timer, will have to adjusted in the breadboard stage. 

---