J-shifting approximation

Employing the J-shifting approximation[31] the thermal rate constant could then be found fi om... 

Recently we have studied the O+H2 reaction. This reaction proceeds through a deep well and has no bottle neck geometry. It has proved impossible to apply or formulate a J shifting approximation for this case. We have therefore developed a capture model [14]. 

Here is the rotational partition function calculated at the transition state geometry and kj=o is the thermal rate calculated only for J=0. Thus, within the J-shifting approximation, the thermal rate constant can be obtained from a dynamical simulation for J=0 only. 

If transition state is linear (or approximately linear), the J-shifting approximation can not be employed in the simple form described above. For linear systems, the rotation around the linear axis occurs on the time scale of the internal motion. Thus, this rotation has to be treated as internal motion and only the rotation perpendicular to the linear axis can be assumed to be separable. Within this modified J-shifting approximation, K is a, good quantum number and the thermal rate constant is calculated by... 

Here is the partition function of the perpendicular rotational motion calculated at the transition state geometry and kj=K,K is the thermal rate constant calculated within the RT-conserving approximation for J = K, (It should be noted that by some authors the J-shifting approximation is called J and K-shifting, while the modified J-shifting approximation is called J-shifting only). 

As important as exact reactive scattering calculations may be, approximations are indispensable in developing useful tools for estimating the reactivity of complex systems. An important model is the J—shifting approximation [18-20] (JSA),... 

The theory we test is the J—shifting approximation (JSA), made popular in quantum reactive scattering theory by Bowman [18, 54] and Schatz [55]. The JSA assumes that K is conserved, and further that the centrifugal couphng is only important near the transition state geometry. Reaction probabilities for J,K > 0 are obtained by a (J, AT)—dependent energy shift from the J = 0 result via 118]... 

Figure 5.9 Comparison between exact theory (solid) and the J—shifting approximation (JSA—dash) of D+H2(u = l,i) rate constants for < j > (thick) and j = 0 (thin) as a function of temperature. With respect to the j = 0 rate constants, the JSA consistently underestimates the exact rate constant by ca. 35%. However, with respect to the < j > rate constants, the JSA is nearly exact at the lower temperatures T < 700 K, and is semi-quantitative throughout the entire temperature range. Noticeable error occurs at the highest temperatures as the assumptions inherent in the JSA break down.

The J—shifting approximation (JSA) was tested against the exact kv= uAT) and fcussi(J ) rate constants for r = 200 —1000 A". The [u = l,j = (0,1,2,3)]—selected JSA rate constants were qualitatively correct, but were in error by as much as 41%. The error systematically cancelled for the (u = 1,< j >)—selected rate constant, giving c.emi-quantitative description of this averaged quantity for T < 700 K. 

A related and particularly simple approximation is J-shifting [127. 128]. This method is a simple (and... 

The second contribution to the MCD that will be considered is the A term. A terms arise because of the change in the band-shape function f due to the apphed magnetic field. In this context, the rigid-shift approximation (11) is usually applied. By making the rigid-shift approximation it is assumed that the center of the band-shape function moves as u>j changes but that the shape of /otherwise does not change. If this is the case then... 

In the (1R, 3R) compound, and Hfe are not magnetic equivalent since they do not identically couple to Hc or to Hd Hc and Hd also are not magnetic equivalent since they do not identically couple to Ha or H6. But since the J values approximately average out by free rotation, the spin system is treated as A2X2 and the spectrum would show two triplets. In the (1S,3R) compound hd = hd and Jac — Jbc thus in this molecule, Hfl and Hb are magnetic equivalent. The question of magnetic equivalence of Hc and Hd is not relevant since they are not chemical-shift equivalent. The spin system is ABX2. 

The most simple approximation, the J-shifting approach [22], assumes that rotational and internal motion are separable. For direct reactions, this... 

Polar solvents shift the keto enol equilibrium toward the enol form (174b). Thus the NMR spectrum in DMSO of 2-phenyl-A-2-thiazoline-4-one is composed of three main signals +10.7 ppm (enolic proton). 7.7 ppm (aromatic protons), and 6.2 ppm (olefinic proton) associated with the enol form and a small signal associated with less than 10% of the keto form. In acetone, equal amounts of keto and enol forms were found (104). In general, a-methylene protons of keto forms appear at approximately 3.5 to 4.3 ppm as an AB spectra or a singlet (386, 419). A coupling constant, Jab - 15.5 Hz, has been reported for 2-[(S-carboxymethyl)thioimidyl]-A-2-thiazoline-4-one 175 (Scheme 92) (419). This high J b value could be of some help in the discussion on the structure of 178 (p. 423). 

Substituent effects (substituent increments) tabulated in more detail in the literature demonstrate that C chemical shifts of individual carbon nuclei in alkenes and aromatic as well as heteroaromatic compounds can be predicted approximately by means of mesomeric effects (resonance effects). Thus, an electron donor substituent D [D = OC//j, SC//j, N(C//j)2] attached to a C=C double bond shields the (l-C atom and the -proton (+M effect, smaller shift), whereas the a-position is deshielded (larger shift) as a result of substituent electronegativity (-/ effect). 

In the benzene series, an approximately linear relationship has been obtained between the chemical shifts of the para-hydrogen in substituted benzenes and Hammett s a-values of the substituents. Attempts have been made, especially by Taft, ° to use the chemical shifts as a quantitative characteristic of the substituent. It is more difficult to correlate the chemical shifts of thiophenes with chemical reactivity data since few quantitative chemical data are available (cf. Section VI,A). Comparing the chemical shifts of the 5-hydrogen in 2-substituted thiophenes and the parahydrogens in substituted benzenes, it is evident that although —I—M-substituents cause similar shifts, large differences are obtained for -j-M-substituents indicating that such substituents may have different effects on the reactivity of the two aromatic systems in question. Differences also... 

The framework, however, as introduced so far is of little help for our purpose since the shift from any subspace to its immediate in hierarchy would require to change entirely the set of basis functions. Although j x) are all created by the same function, they are different functions and, consequently, the approximation problem has to be solved from scratch with any change of subspace. The theory of wavelets and its relation to multiresolution analysis provides the ladder that allows the transition from one space to the other. 

To verify that steady state catalytic activity had been achieved, the catalyst was allowed to operate uninterrupted for approximately 8 hours. The catalyst was then removed from the reactor and the surface investigated by XPS. The results are shown in Figure 2c. The two major changes in the XPS spectrun were a shift in the iron 2p line to 706.9 eV and a new carbon Is line centered at 283.3 eV. This combination of iron and carbon lines indicates the formation of an iron carbide phase within the XPS sampling volume.(J) In fact after extended operation, XRD of the iron sample indicated that the bulk had been converted to FecC2 commonly referred to as the Hagg carbide.(2) It appears that the bulk and surface are fully carbided under differential reaction conditions. 

Fig. 7.78 Linear relation of the quadmpole splitting A q = ( jl)eqQ (1 + j /3)l/2 and the isomer shift b for aurous (a) and auric (b) compounds. Also included is a correlation with the relative change in electron density at the gold nucleus, Ali/r(o)P, as derived from Dirac-Fock atomic structure calculations for several electron configurations of gold. An approximate scale of the EFG (in the principal axes system) is given on the right-hand ordinate (from [341])...

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

---