Quasi-Spin States

Further exploration of the internal symmetries of the Kf multiplets introduces us to the concept of quasi-spin. For a given / shell one defines an operator 1 with components M+,, lz as given in Eq. 17. 

The 00 superscripts indicate that the and a operators are coupled to quantities of rank zero in spin and orbit space. The scalar products in this equation can be evaluated by means of the general expansion formula in Eq. 13. For the case of an / = 1 shell the following expressions are obtained  

These operators obey commutation rules which are identical to the commutation rules for angular momentum or spin operators, hence the name quasi-spin 

The scalar product of the resultant quasi-spin vector 2 is given by  

Focusing now our attention to our set of (t2g )3 states we observe that 2D and S are the MQ = 0 components of a quasi-spin singlet while 2P is the Mc = 0 component of a quasi-spin triplet. The full quantum structure of the (t2g)3 multiplets thus involves seven labels QMQSLTMsMry, albeit the Me label is redundant since all states share the same MQ value. 

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Fig. 4.3 Ranges of isomer shifts observed for Fe compounds relative to metallic iron at room temperature (adapted from [24] and complemented with recent data). The high values above 1.4-2 mm s were obtained from Co emission experiments with insulators like NaCl, MgO or Ti02 [25-28], which yielded complex multi-component spectra. However, the assignment of subspectra for Fe(I) to Fe(III) in different spin states has never been confirmed by applied-field measurements, or other means. More recent examples of structurally characterized molecular Fe (I)-diketiminate and tris(phosphino)borate complexes with three-coordinate iron show values around 0.45-0.57 mm s [29-31]. The usual low-spin state for Fe(IV) with 3d configuration is 5 = 1 for quasi-octahedral or tetrahedral coordination. The low-low-spin state with S = 0 is found for distorted trigonal-prismatic sites with three strong ligands [30, 32]. Occurs only in ferrates. There is only one example of a molecular iron(VI) complex it is six-coordinate and has spin S = 0 [33]...

The quasi-spin label is perhaps the most exotic quantum characteristic of these states. It is in any case a true shell characteristic since it extends over several configurations. Most importantly it gives rise to very strong selection rules as we will demonstrate in the next section. 

The first coordination sphere of acido-pentamine [Cr(NH3)5X]2+ complexes has C4v symmetry and this leads to a tetragonal resolution of the 2Eg state into 2At and 1Bl components. The classical ligand field model predicts that these components will be virtually degenerate. This is based on a combination of pseudo-spherical and quasi-spin selection rules of the shell and will be discussed later on in Sect. 5.2. At present we welcome this example of a pseudo-degeneracy as another opportunity to observe fine details of the interelectronic repulsion interaction which have the proper anisotropy to induce a splitting of the 2Eg term. 

One thus would predict that such complexes are characterized by a pronounced orthorhombic perturbation of the emitting state, which thereby loses its parent quasi-spin structure. Possibly this is the most efficient mechanism to make an emitting doublet state with a hole in the tlg shell. Interesting photophysical and even photochemical properties might result. In any case such complexes should constitute sensitive probes for the phase-coupling ligand properties. 

The eigenvalue of Qz is N — 2 for a state of gN. We can now consider that the identical components of g1 and g together form a quasi-spin tensor of rank 1/ 2, whose array of ranks we can now indicate by writing G(l - - -]. The e, operators can be broken down into parts that have well-defined quasi-spin ranks however, it turns out that e2 is a quasi-spin scalar, which can be used to explain some similar matrix elements of e2 in g 2 and g 4 [10]. 

In order to calculate the spectrum, the chemical-shielding tensors of all spins, as well as their relative orientation and distances, have to be known. The assumption of a quasi-equilibrium state simplifies Equation (4.28) considerably. For long mixing times t , all elements in exp(Wr ,) have equal intensity. This leads to the following signal function for a single-crystallite orientation ... 

All the neutral single donors without d or f electrons have spin 1/2 while the double donors and acceptors have spin 0 in the ground state, but in some excited states, they have spin 1 and optically forbidden transitions between the singlet and triplet states have been observed. The spins of the neutral acceptors in the ground state depend on the electronic degeneracy of the VB at its maximum. For silicon, the threefold degeneracy of the valence band results in a quasi spin 3/2 of the acceptor ground state. 

Very often the catalyst (unsaturated TM complex) has a low-spin ground state with a high-spin state being very close in energy [23, 36, 37]. This situation is very useful for catalysis it is not necessary to produce a spin-flip and a low barrier is achieved. The metal surface is an ideal catalyst in this sense, since any spin multiplicity can be realized at the local state of the surface, perturbed in the course of the chemisorption process. In other words, a metal cluster which simulates the surface in the course of the chemisorption process has a number of quasi-degenerate state with different spin-multiplicities. 

The independence of Hgff on the time interval of the sequence under the condition that this interval lies between T2 and Tie can be used for determining proper Heff. This approach was first used in multi-pulse nuclear magnetic resonance in paper, and later, in NQR, for the analysis of quasi-stationary state in multi-pulse spin locking in the nuclear system with the spin of 5/tF at exact resonance. We extended this approach to the arbitrary offset of the pulse carrier frequency in relation to the resonance transition. 

The quasi-stationary state, established at times T2, is characterised by two spirt temperatures. One temperature corresponds to a common thermodynamic reservoir, which appears due to the thermal mixing of the quadrupole reservoir and the reservoir of components of the dipole-dipole interactions, non-secular in relation to the effective Hamiltonian. The second temperature corresponds to the reservoir of the secular part of the dipole-dipole interactions. At conditions to be determined further, both spin temperatures can coincide. 

Density matrix p" of the quasi-stationary state (p" = p"gf when = 0), formed at times T2 (T2 being the time of the spin-spin relaxation) equals... 

Finally, a sulfur-bridged iron(III)-copper(II) system has been reported with two [Fe(Tp-ClPP)] units that sandwich a [Cu(MNT)2] unit. Each iron coordinates to a sulfur of the [Cu(MNT)2] complex with Fel-S = 2.444(2) A, Fe2-S = 2.549(2) A. Fe2 additionally forms a weak sixth bond, Fe2-S = 2.956(2) A. Fel is five-coordinate with Fe-Np = 1.976(4) A and Fe"-Ct = 0.20 A, consistent with an intermediate-spin state. Fe2 is quasi-six-coordinate with Fe-Np = 1.978(2) A and Fe-"Ct = 0.12 A. The spin state, according to the original authors, is best assigned as intermediate spin. However, the [Fe(TAP)(SH)] result noted earlier, now makes the assignment more tenuous a low-spin state is now seen as possibility based on structural parameters. 

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