Effects second-order

Some transition ions have central hyperfine splittings somewhat greater than this value, for example, for copper one typically finds Az values in the range 30-200 gauss, and so in these systems the perturbation is not so small, and one has to develop so-called second-order corrections to the analytical expression in Equation 5.12 or 5.13 that is valid only for very small perturbations. The second-order perturbation result (Hagen 1982a) for central hyperfine splitting is  

What would happen if we were to lower the microwave frequency from X-band to L-band (1 GHz). The Zeeman term for g 2.2 (an average value for copper) would correspond to a held of circa 325 gauss at 1 GHz, and so the two interactions S B and S I would be of comparable magnitude. In such situations the perturbation expressions become extremely complicated and lose all practical significance. 

FIGURE 5.7 Second-order hyperfine shift in the X-band EPR of the Cu(II)-Tris complex. The thin solid line is the experimental spectrum of 1.5 mM CuS04 in 200 mM Tris-HCl buffer, pH 8.0 taken at v = 9420 MHz and T = 61 K. Tris is tris-(hydroxymethyl)aminomethane or 2-amino-2-hydroxymethyl-l,3-propanediol. The broken lines are simulations using the parameters g = 2.047, gN = 2.228, Atl = 185 gauss. In the lower trace the second-order correction has been omitted. 

A musician learns that on playing a note at a frequency of x Hz, sounds of reduced intensity at frequencies 2jc, 3x, etc. appear. In fact, any periodic excitation leads to the same type of behaviour. 

In an electrochemical system, Taylor s expansion shows that 

Until now we have used conditions for which the second and following terms are (or are supposed to be) negligible, in other words I is linear in E this corresponds to small sinusoidal perturbations. The terms in the Taylor expansion are called first harmonic or fundamental, second harmonic, third harmonic, etc. If we look at the form of a normal voltammogram (Fig. 6.2) the approximation of a linear system is valid close to Em much more than in other parts of the voltammogram where the curvature of the I-E profile is more pronounced. 

If we register the second harmonic current vs. d.c. potential, this will have the same form as the second derivative of the voltammetric curve, as Fig. 11.11 shows. One of the advantages of the use of the second harmonic is that, since the double layer capacity is essentially linear, it contributes much less to the second harmonic than to the fundamental frequency and the calculation of accurate kinetic parameters is much facilitated. 

Apart from the second harmonic there are other second-order effects, which are developed in the techniques of faradaic rectification and demodulation. Both these techniques are utilized to study systems with very fast kinetics. 

The first issues that must be addressed are molecular symmetry and the magnetic equivalence or nonequivalence of protons or other functional groups. Even if two protons, or groups, are chemically equivalent, they may or may not be magnetically equivalent. Although molecular symmetry can often simplify NMR spectra, one must be able to discern which protons or groups are equivalent ty symmetry. The two most useful symmetry properties (or symmetry operators) are the plane of symmetry and the axis of symmetry. 

A plane of symmetry is simply a mirror plane such that one half of the molecule is the mirror image of the other half, as in OTeso-pentane-2,4-diol  

The methyl groups are identical by symmetry, and one would expect this stereoisomer to show one methyl doublet in its NMR spectrum and one signal for a methyl group in its spectrum. 

Consider the other diastereomer of pentane-2,4-diol, the chiral d,l isomer. This isomer has an axis of symmetry. If the molecule is rotated 180° about an axis in the plane of the paper passing through the central carbon, the molecule can be converted into itself  

---

Second-order effects include experiments designed to clock chemical reactions, pioneered by Zewail and coworkers [25]. The experiments are shown schematically in figure Al.6.10. An initial 100-150 fs pulse moves population from the bound ground state to the dissociative first excited state in ICN. A second pulse, time delayed from the first then moves population from the first excited state to the second excited state, which is also dissociative. By noting the frequency of light absorbed from tlie second pulse, Zewail can estimate the distance between the two excited-state surfaces and thus infer the motion of the initially prepared wavepacket on the first excited state (figure Al.6.10 ). 

The interpretation of MAS experiments on nuclei with spin / > Fin non-cubic enviromnents is more complex than for / = Fiuiclei since the effect of the quadnipolar interaction is to spread the i <-> (i - 1) transition over a frequency range (2m. - 1)Vq. This usually means that for non-integer nuclei only the - transition is observed since, to first order in tire quadnipolar interaction, it is unaffected. Flowever, usually second-order effects are important and the angular dependence of the - ytransition has both P2(cos 0) andP Ccos 9) terms, only the first of which is cancelled by MAS. As a result, the line is narrowed by only a factor of 3.6, and it is necessary to spin faster than the residual linewidth Avq where... 

While all contributions to the spin Hamiltonian so far involve the electron spin and cause first-order energy shifts or splittings in the FPR spectmm, there are also tenns that involve only nuclear spms. Aside from their importance for the calculation of FNDOR spectra, these tenns may influence the FPR spectnim significantly in situations where the high-field approximation breaks down and second-order effects become important. The first of these interactions is the coupling of the nuclear spin to the external magnetic field, called the... 

Hache F, Zeboulon A, Gallot G and Gale G M 1995 Cascaded second-order effects in the femtosecond regime in p-barium borate self-compression in a visible femtosecond optical parametric oscillator Opt. Lett. 20 1556-8... 

Curved one-factor response surface showing (a) the limitation of a 2 factorial design for modeling second-order effects and (b) the application of a 3 factorial design for modeling second-order effects. 

If the actual response is that represented by the dashed curve, then the empirical model is in error. To fit an empirical model that includes curvature, a minimum of three levels must be included for each factor. The 3 factorial design shown in Figure 14.13b, for example, can be fit to an empirical model that includes second-order effects for the factor. 

When B = 0, the solution behaves ideally, at least through second-order effects. This means that deviations from ideality might be observed at still higher concentrations, but that the van t Hoff equation applies at least in dilute solutions for systems with B = 0. 

Proton chemical shift spectra over the range of 0—15 ppm ( 0.1 ppm) TFA, ttifuoroacetic acid DMSO, dimethyl sulfoxide. When complex spectra caused by second-order effects or overlapping resonances were encountered, the range was record (11,12). 

These estimates are frequently inaccurate because of second-order effects such as rotor saturation and harmonics. If the apphcation is at all critical, the motor manufac turer should be consiilted. 

Again it is seen that only when second order effects need to be considered does the relationship become more complicated. The dead volume is made up of many components, and they need not be identified and understood, particularly if the thermodynamic properties of a distribution system are to be examined. As a consequence, the subject of the column dead volume and its measurement in chromatography systems will need to be extensively investigated. Initially, however, the retention volume equation will be examined in more detail. 

Equation (34) is generally quite correct and useful. However, if highly accurate retention measurements are important, then second order effects must be taken into account and equation (33) indicates that, for accurate data, equation (34) is grossly over simplified. From equation (33), a more accurate expression for solute retention would be... 

In this chapter, the elution curve equation and the plate theory will be used to explain some specific features of a chromatogram, certain chromatographic operating procedures, and some specific column properties. Some of the subjects treated will be second-order effects and, therefore, the mathematics will be more complex and some of the physical systems more involved. Firstly, it will be necessary to express certain mathematical concepts, such as the elution curve equation, in an alternative form. For example, the Poisson equation for the elution curve will be put into the simpler Gaussian or Error function form. 

So far the plate theory has been used to examine first-order effects in chromatography. However, it can also be used in a number of other interesting ways to investigate second-order effects in both the chromatographic system itself and in ancillary apparatus such as the detector. The plate theory will now be used to examine the temperature effects that result from solute distribution between two phases. This theoretical treatment not only provides information on the thermal effects that occur in a column per se, but also gives further examples of the use of the plate theory to examine dynamic distribution systems and the different ways that it can be employed. 

Thermal changes resulting from solute interactions with the two phases are definitely second-order effects and, consequently, their theoretical treatment is more complex in nature. Thermal effects need to be considered, however, because heat changes can influence the peak shape, particularly in preparative chromatography, and the consequent temperature changes can also be explored for detection purposes. 

Thermal changes in a distribution system, although a second-order effect and, thus, more complex to deal with theoretically, can nevertheless sometimes be used to practical ends. The temperature changes that occurred in a dynamic distribution system were used, in the early days of LC, for detection purposes. Ultimately, the system proved to be ineffectual as a detector, but this could have been deduced... 

Note that there is no net change in the number of moles of gas in this equilibrium. Therefore, by Le Chatelier s principle, this reaction will be independent of external pressure (ignoring second-order effects due to gas imperfections). Under these conditions the N of the expl will... 

Figure 6. Channel spectram and related spectral phase shifts. Top compensated dispersion the phase is constant over the spectrum. Middle and bottom Second order effect with or without first order. The spectral phase variation induces channel in the spectrum.

Further substitution of alkyl groups in ethanoic acid has much less effect than this first introduction and, being now essentially a second-order effect, the influence on acid strength is not always regular, steric... 

How does a rigorously calculated electrostatic potential depend upon the computational level at which was obtained p(r) Most ab initio calculations of V(r) for reasonably sized molecules are based on self-consistent field (SCF) or near Hartree-Fock wavefunctions and therefore do not reflect electron correlation in the computation of p(r). It is true that the availability of supercomputers and high-powered work stations has made post-Hartree-Fock calculations of V(r) (which include electron correlation) a realistic possibility even for molecules with 5 to 10 first-row atoms however, there is reason to believe that such computational levels are usually not necessary and not warranted. The Mpller-Plesset theorem states that properties computed from Hartree-Fock wave functions using one-electron operators, as is T(r), are correct through first order (Mpller and Plesset 1934) any errors are no more than second-order effects. 

Nuclear hyperfine coupling results in a multi-line ESR spectrum, analogous to the spin-spin coupling multiplets of NMR spectra. ESR spectra are simpler to understand than NMR spectra in that second-order effects normally do not alter the intensities of components on the other hand, ESR multiplets can be much more complex when the electron interacts with several high-spin nuclei, and, as we will see in Chapter 3, there can also be considerable variation in line width within a spectrum. 

Second-order effects on hyperfine structure in organometallic compounds are discussed in Chapter 3. 

Our analysis thus far has assumed that solution of the spin Hamiltonian to first order in perturbation theory will suffice. This is often adequate, especially for spectra of organic radicals, but when coupling constants are large (greater than about 20 gauss) or when line widths are small (so that line positions can be very accurately measured) second-order effects become important. As we see from... 

Here, A is the nearly isotropic nuclear coupling constant, I is the nuclear spin (Iun = I), and m is the particular nuclear spin state. It may be observed that the zero field splitting term D has a second-order effect which must be considered at magnetic fields near 3,000 G (X-band). In addition to this complication nuclear transitions for which Am = 1 and 2 must also be considered. The analysis by Barry and Lay (171) of the Mn2+ spectrum in a CsX zeolite is shown in Fig. 35. From such spectra these authors have proposed that manganese is found in five different sites, depending upon the type of zeolite, the primary cation, and the extent of dehydration. 

---