Anisotropic term

Here e is the depth of the well and d is the distance to zero crossing point of the isotropic potential whereas at and bt are anisotropy parameters. Substituting this potential into Eq. (5.50) we get for each anisotropic term [202]... 

Fig. 2 The values of second-order P2 and fourth-order P4 Legendre polynomials as functions of [iRL. The two polynomials do not have a common root, i.e., the anisotropic terms in the second-order quadrupolar Hamiltonian cannot both be canceled at once, regardless of the orientation of the rotor axis...

As previously, the phase evolution due to anisotropic terms is refocused if... 

If the sample is spun exactly at fjRL = 54.736° and at infinite spinning speed, the first-order term /, vanishes (33) and the second-order anisotropic term yields... 

If j Rf is exactly the magic angle and infinite spinning speed is assumed, the first-order anisotropic terms are zero for both single and DQ coherence (33). This does not hold true for finite spinning speed, but a complete averaging of the first-order effect occurs at the exact rotor cycles. Therefore, the x evolution time has to match exactly a multiple of the rotor period. The second-order anisotropy refocusing occurs for... 

The first such experiment (Fig. 21a), introduced independently by Gan [258] and Bodenhausen et al. [259], used 13C as the so-called spy nuclei in a 2D experiment similar to the HMQC scheme used in solution NMR [260], The transfer of magnetization occurred by a combination of weak ]JCN (through-bond) couplings and second-order quadrupolar-dipolar (through-space) cross-terms between 14N and 13C, which contain the isotropic and anisotropic terms l = 0, 2, and 4 and are referred to as residual dipolar splittings (RDS) [261-263], Under MAS, the 1 = 0 term results in isotropic coupling, which for NH3 groups is expected to be... 

These authors not only consider the isotropic value of such a coupling but also study its tensor character, 2hJ(N,N), where all four anisotropic terms were considered, i.e., FC/SD, PSO, SD and DSO. They studied systematically the dependence of the 2hJ(N, N) coupling with the hydrogen bond length, d(N-N), and bond angle, 9. For the latter fixed at 0=180°, they find that the isotropic part of 2hJ(N, N), 2hJiso(N,N), decays exponentially when increasing... 

In conclusion, we note that for systems like HD-X, the intermolecular interactions become more anisotropic than for H2-X systems, because for HD the center of electronic charge and the center of mass do not coincide. The two centers differ by one sixth of the bond distance. Because the molecule rotates about the center of mass, new anisotropic terms appear in both the HD-X interaction potential and induced dipole components see Chapter 4 for details. 

The Fermi contact term, is proportional to the 4s character of the wave function whereas the anisotropic term 20 is proportional to the 3d character of the wave function. Since only the absolute values of a and ox may be determined from the EPR spectrum, various sign combinations are possible which result in AiSO = 2.4 or 14.3 G. The latter absolute value agrees well with an isotropic cobalt splitting of l3.3 G which was measured directly for the monomeric oxygen adduct of N N -ethylenebis-(acetylacetoniminato) cobalt (II), abbreviated Co (acacen) (11). 

In table 18, the results of the hyperfine calculations are collected. A comparison with the experimental data for Ax = Ajgo - TZZ and Ay = Aiso + Tzz is given. The isotropic hfcc Aiao is also included, because its strong dependence on the theoretical method is the reason for the variations of A and Ay between different methods. Since the anisotropic term Tzz is nearly constant with respect to the theoretical method, it is not given. For all calculations the QCISD/6-31G optimized geometries (CCO Rcc = 1-371 A, Rco = 1173 A CNN RNn = 1-231 A, Ron = 1-237 A NCN Ron = 1-245 A) were used. All... 

Hyperfine interactions between the electron and any magnetic nuclei (7>0) present (such as a proton, for example) produces hyperfine splitting, as illustrated in a very simple example in Fig. 2.51. This hyperfine interaction may be divided into an isotropic and an anisotropic component. The isotropic part arises from unpaired electron density at the nucleus and can only be nonzero for i -type orbitals. The anisotropic term corresponds to the classical part of the magnetic dipole interaction for which the Hamiltonian is ... 

For a pure dipole interaction, the anisotropic term in the above equation should have the form -a, -a, la. This implies that the unpaired electron is not purely p based, but also there must be some occupancy of the orthogonal p orbitals. In other words, the anisotropic tensor is actually the result of two dipolar interactions with two radius vectors, each of which is along a coordinate axis of the molecule. Therefore the anisotropic tensor is the sum of two dipolar coupling tensors and can be decomposed into two traceless components (-a, -a, la) and (-h, Ih, -h) as follows ... 

At first, let us consider a free radical which has an anisotropic g-tensor (g )and tumbles rapidly. The anisotropic term can be represented by... 

We have here the well-known formulae of Born whose theory leads, in addition to the Langevin anisotropic term (170a), to a supplementary dipole term of the form (170b) more strongly dependent on temperature 03 = 1/Jfc r ). 

Since the electric polarizability deviation tensor is of similar form, one has by (191), on taking the first anisotropic term only ... 

The isotropic and anisotropic hyperfine coupling terms in a arise from interactions between electron and nuclear spins, and provide information about the nature of the orbital containing the unpaired electron and the extent to which it overlaps with orbitals on adjacent atoms. The anisotropic term can cause similar difficulties to the g tensor anisotropy in analysing spectra of polycrystalline powders extracting coupling constants from spectra of transition metal ions or radicals in zeolites can be difficult or impossible without computer simulation. 

Therefore, if one needs an anisotropic potential one often includes only specific anisotropic contributions, e.g. the molecular quadnipole-quadrupole interaction making the ad hoc assumption that all other anisotropic terms are small, or one invokes model potratials with fewer parameters which are intrinsically anisotropic. Examples of the latter are ... 

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