A-doubling parameters

We see that for a molecule in a 2,s+1 n state, there are in general three A-doubling parameters. However, for a singlet state (S = 0), only q is non-zero and for a doublet state, p and q are non-zero. To second order in perturbation theory, these parameters are given by the following expressions (in cnr1) ... 

Later on in this book, we discuss the properties of the CH radical in its X211 state in some detail. The electronic structure of this radical is rather more complicated than that of OH. Despite this, a simple pure precession calculation of the A -doubling parameters reproduces the experimental values (particularly q) reasonably well (see section 10.6.3). 

We first look at the form of the calculations for the A -doubling parameters, p and q, which are a measure of the mixing of the 2 ft state with the 2 + and 2 S states brought about by the combined effects of spin-orbit coupling and Coriolis effects. Specifically we showed in chapter 8 the results [190] first obtained by Mulliken and Christy ... 

Note of course that the mixing with the 2 A state does not contribute to the A-doubling parameters. Considering the approximations made, the agreement between experiment and theory is very reasonable. 

Field and Bergeman [12] chose to diagonalise the above matrix, thereby generating a modified n wave function with 3 + admixture. Using the modified function as a basis, they then calculated the admixture of the other nearby triplet states (in appropriate vibrational levels) using perturbation theory. The calculated values of the A-doubling parameters and effective electric dipole moments obtained are listed in table 11.1. 

The last three constants above are A-doubling parameters for a molecular A state. 

For OH in its X H state, there are two A-doubling parameters, p and q. Using the explicit expressions given in equations (7.142) and (7.143), these parameters can be calculated very accurately using good quahty ab initio wave fimctions. A simple estimate can be made using the pure precession values for the relevant matrix elements which we have derived earlier. This approximation assumes that the A-doubling effects arise wholly from the perturbations with the A E+ state. For this approximation,... 

For quadnipolar nuclei, the dependence of the pulse response on Vq/v has led to the development of quadnipolar nutation, which is a two-dimensional (2D) NMR experiment. The principle of 2D experiments is that a series of FIDs are acquired as a fimction of a second time parameter (e.g. here the pulse lengdi applied). A double Fourier transfomiation can then be carried out to give a 2D data set (FI, F2). For quadnipolar nuclei while the pulse is on the experiment is effectively being carried out at low field with the spin states detemiined by the quadnipolar interaction. In the limits Vq v the pulse response lies at v and... 

This equation is a reasonable model of electrokinetic behavior, although for theoretical studies many possible corrections must be considered. Correction must always be made for electrokinetic effects at the wall of the cell, since this wall also carries a double layer. There are corrections for the motion of solvated ions through the medium, surface and bulk conductivity of the particles, nonspherical shape of the particles, etc. The parameter zeta, determined by measuring the particle velocity and substituting in the above equation, is a measure of the potential at the so-called surface of shear, ie, the surface dividing the moving particle and its adherent layer of solution from the stationary bulk of the solution. This surface of shear ties at an indeterrninate distance from the tme particle surface. Thus, the measured zeta potential can be related only semiquantitatively to the curves of Figure 3. 

In Figure 2, a double-reciprocal plot is shown Figure 1 is a nonlinear plot of as a function of [S]. It can be seen how the least accurately measured data at low [S] make the deterrnination of the slope in the double-reciprocal plot difficult. The kinetic parameters obtained in this example by making linear regression on the double-reciprocal data ate =1.15 and = 0.25 (arbitrary units). The same kinetic parameters obtained by software using nonlinear regression are = 1.00 and = 0.20 (arbitrary units). 

In Eq. (2), the dihedral tenn includes parameters for the force constant, Ky, the periodicity or multiplicity, n and the phase, 8. The magnimde of Ky dictates the height of the barrier to rotation, such that Ky associated with a double bond would be significantly larger that that for a single bond. The periodicity, n, indicates the number of cycles per 360° rotation about the dihedral. In the case of an bond, as in ethane, n would... 

One can also gain an insight into the bifurcation for instance, assume that we introduce a parameter A that permits displacing the curve graphical interpretation for a bifurcation of the first kind. 

In the second group of models, the pc surface consists only of very small crystallites with a linear parameter y, whose sizes are comparable with the electrical double-layer parameters, i.e., with the effective Debye screening length in the bulk of the diffuse layer near the face j.262,263 In the case of such electrodes, inner layers at different monocrystalline areas are considered to be independent, but the diffuse layer is common for the entire surface of a pc electrode and depends on the average charge density <7pc = R ZjOjOj [Fig. 10(b)]. The capacitance Cj al is obtained by the equation... 

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