Axial system

Mj = Mjmax-this is the only case, which can, in principle, lead to a singledeterminant wave function. This situation may be encountered in some limited highly axial systems, such as Dy-0+ [37]. However, usual compounds do not have any site symmetry, which means that several projections of the /-manifold will be mixed by the low-symmetry components of the CF, leading to a multideterminantal character of the wave function. 

To simplify terminology of axial systems, gzz is defined to be g(l (the g-value observed with the symmetry axis of Cu + parallel to the applied field), and gxx (= gyy) is defined to be gA (the g-value observed with the symmetry axis perpendicular to the applied field). An elongated z-axis (depicted in Figure 11 for Cu(H20)5 +) results in gjj > gj. For axially symmetric Cu + rigidly bound in a crystal, the g-value can then vary between the minimum (gj.) and maximum (g(,), depending on orientation of the crystal within the magnetic field. However, for axial Cu + bound in a powdered clay sample, all possible orientations, and therefore all g-values between gA and gj are represented in the "powder" spectrum. Therefore, electron spin resonance occurs only for field values, H, between Hjj and H, where ... 

The ambiguity involved in assigning the absolute configuration of a chiral molecule in a chiral crystal is presented in Scheme 1. Scheme la depicts a chiral molecule of, say, configuration S, with individual atomic coordinates — x — y - z, (i = 1,. . . , n, for n atoms) in a crystal axial system a,b,c. Scheme 1 b represents the enantiomeric crystal structure containing a molecule of configuration... 

R with a set of atomic coordinates x,y,z, referred to the same axial system. Schemes 1 a and b are related by an inversion through the origin. 

We may describe the centrosymmetric structure in Scheme 10c in terms of the alternative, left-handed axial system by choosing the reverse vectors —a, -b, -c as the axial set, as shown in Scheme lOg. The whole arrangement remains unchanged, the only difference being that the molecules S and R are now described by coordinate sets x y z, and —x —y —z respectively. By symmetry the same applies to the nonisometric arrangement (Schemes IQf and h). 

The use of a reference axial system, whether right- or left-handed, is completely analogous to the Cahn, Ingold, and Prelog convention (75-77) regarding the specification of the absolute configuration of chiral molecules R and S as depicted in Scheme 11 for molecules with large (L), medium (M), and small... 

ESR spectra of the Ag11 complexes were found to be typical of a d9 axial system. In each case gn and gx were well resolved although resolution of nitrogen hyperfine structure could not be achieved. 

Fig. 2.8. Angular dependence of the pseudocontact shift for an axial system, shown as a surface of constant absolute value of S1. In the example S1 is positive along the z axis and negative in the xy plane. The three-dimensional shape of the surface is similar to the representation of a d 2 orbital.

The following method for separation of pseudocontact and contact shifts, proposed by Dobson et al. [83], is valid for axial systems assuming that (1) there is a nucleus in such a position in the molecule that the contact contribution is zero because it is far from the paramagnetic center and (2) the contact hyperfine coupling is constant ( 10%) for Nd-Tm. 

Separation of pseudocontact and contact shifts in axial systems can also be achieved, again by using a variety of lanthanides, without assuming that for a given signal the contact shift is zero, as long as the expected patterns of both pseudocontact and contact shifts (Fig. 2.21) hold. Here, the hyperfine shift can be expressed as the sum of contact and pseudocontact shifts (Eqs. (2.39) and (2.37)) ... 

The programming of the formulae needs a COMPLEX 16 arithmetic since the CF potential itself could be complex. Therefore, it is easy to implement the complex spherical transforms of the magnetic field, x and B, into the (complex) Zeeman matrix elements. With the magnetic field Eref aligned parallel to the principal rotational axis of an axial system, the Zeeman matrix stays real since then B](] = Bxef. Its counterpart for the perpendicular direction is also real, and this involves the following transforms x = - (l/V2)Bref and... 

Fig. 3. Axial system of cytochrome b (type F) and of cytochrome c (type G)...

In addition to polarized optical studies, single-crystal EPR experiments have been performed on plastocyanin44). In an axial system, each individual molecule contributes to the EPR spectrum with a g value ) given by ... 

Anisotropic magnetic properties are anticipated in the axial system under consideration, where the Zeeman effect in the direction parallel to the C4 axis may be calculated with... 

Half-integer systems with. S > 1 /2 are referred to as Kramers systems. These systems have a degeneracy in the ground state that is relieved through Zeeman interaction. The simplest example to consider is S = 3/2. To make things easier, we will assume an axial system (A/D = 0) in a zero external field (Bo)- Therefore, the total energy is... 

As with pure 5=1/2 states, the g-factors for each of these transitions will be anisotropic producing two inflections for an axial system and three for a rhombic system. In other words, for our two-level 5 = 3/2 system we have the possibility of observing two or three transitions for each level or four or six inflections for axial or rhombic, respectively. Normally, not all of these inflections are observable. Using quantum mechanics to solve the above energy equation for 5 = 3/2, = 0 and D i hv yields the g-factors gx = 4.0 and gy =2.0 for the ms = 1 /2 level and gx = 0.0 and gy = 6.0 for the ms = 3/2 level. Solving this equation for various values of E/D from 0 to 1/3 allows the determination of the possible values of the six different g-factors. A plot of these g-factors is shown in Figure 13. 

Figure 16 shows plots of the energy levels for both axial (E = 0) and rhombic (D > 3E >0) systems with D hv. In the axial system, it is obvious that a transition cannot be induced between =0 and m = 1 because of the large energy separation (D). A transition also cannot be induced between m = 1 and m = —1, since this is forbidden by selection rules (i.e., this is a Am = 2 transition while only Arris = 1 transitions are allowed). Therefore, transitions are almost never observed in purely axial integer-spin systems. 

The powder EPR signal is dominated by a hyperfine doublet due to the interaction between the trapped electron and a single proton ( H, I = 1/2). The H hyperfine couplings can be more precisely determined by ENDOR, with values of A] = 2.07G, A2 = 2.00G, A3 = 0.31 G [23]. These hyperfine parameters indicate that the local symmetry of the site is lower than axial for a purely axial system, the hyperfine parameters should take the form A] = A2 = Aj and A3 = Ay. Although the difference between A] and A2 is small, the slightly rhombic nature of the parameters is very important and extremely informative. The magnitude of these hyperfine couplings also indicates that the electron-proton interaction is weak. 

Right-handed axial system A system of axes, usually orthogonal (at right angles to each other), that are designated 2 , y, and z, in such an order that x is converted to y and y is converted to z in the same manner that a right-handed screw (moving clockwise into a piece of wood) would proceed (with a to y as the clockwise motion and Z the direction into the wood). 

FIGURE 13.17. Temperature effects in naphthalene (Ref. 81). (a) Coordinate axial system, (b) Values for T and L for naphthalene at 92 K and 270 K. Note the increeise in numbers at the higher temperature, (c) Thermal ellipsoids at (left) 92 K and (right) 270 K. Note the increase in the sizes of the ellipsoids at the higher temperature where the atomic motion is higher. 

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