Spin-orbital perturbed

The calculation of the magnetic anisotropy of non-cubic materials requires an expansion up to 1 /c . Except in the case of fully relativistic calculations, the expansion is never carried out consistently and only the spin-orbit perturbation is calculated to second order (or to infinite order), without taking account of the other terms of the expansion. In this section, we shall follow Gesztesy et al. (1984) and Grigore et al. (1989) to calculate the terms H3 and H. Hz will be found zero and H4 will give us terms that must be added to the second order spin-orbit calculation to obtain a consistent semi-relativistic expansion. 

In addition, it can be shown that second-order vibronic perturbation will make possible some intersystem crossing to the 3B3u(n, tt ) state. However, this second-order perturbation should be much less important than the first-order spin-orbit perturbation.(19) This will produce the unequal population of the spin states shown in Figure 6.1. In the absence of sir the ratios of population densities n are given by the following equations ... 

As in all perturbational approaches, the Hamiltonian is divided into an unperturbed part and a perturbation V. The operator is a spin-free, one-component Hamiltonian and the spin-orbit coupling operator takes the role of the perturbation. There is no natural perturbation parameter X in this particular case. Instead, J4 so is assumed to represent a first-order perturbation The perturbational treatment of fine structure is an inherent two-step approach. It starts with the computation of correlated wave functions and energies for pure spin states—mostly at the Cl level. In a second step, spin-orbit perturbed energies and wavefunctions are determined. 

Accordingly, the first-order spin-orbit perturbation of a triplet wave function may be written as a linear combination of unperturbed singlet, triplet, and quintet states with expansion coefficients defined in a similar way as those in Eq. [218]. 

In the picture of spin-orbit perturbed Russell-Saunders states, the dipole transition moment of a spin-forbidden radiative transition is thus a sum of spin-allowed dipole transitions weighted by spin-orbit coupling coefficients (e.g., the expansion coefficients in Eq. [218]). The fact that the transition dipole moment of a spin-forbidden radiative transition is a weighted sum of spin-allowed dipole transition moments is exactly what experimentalists mean when they speak of intensity borrowing. The contribution of perturbing states to the oscillator strength can be positive or negative. In other words, per-turbers can not only lend intensity to a spin-forbidden transition, they can also take it away. 

The product symmetries of the excited a3Bi multiplet components are Ai, Ai, and f>2 The spin-orbit perturbed excited state wave functions are therefore given by... 

The separation between adjacent spin components is therefore -2a/5 which equates with 2A(] ) from the effective Hamiltonian. Hence A(1> = -a/5 or -83.4 cm-1, using fFe = 417 cm-1. The value obtained from experiment is -77.3 cnr1 although in practice it is difficult to model the spin-rotation levels of FeH with an effective Hamiltonian because of large spin-orbit perturbations [38]. For molecules like FeH, one would expect second- and higher-order contributions to A to be significant. 

It should be pointed out that once the zf origin of the different bands is determined (in particular the true and false origins of the spectrum), a complete description could be given for the spin-orbit perturbations that give the lowest triplet state its radiative properties. A very extensive work was conducted by Tinti and El-Sayed (39) in which different perturbations—e.g., heating, applying a magnetic field, as well as saturation of the zf transitions with microwave radiation-were used to determine the property of the individual zf levels. The effect of these perturbations not only on the phosphorescence spectrum but also on the observed decays and polarizations has been examined. Limits on the importance of the different spin-orbit interactions are obtained. This spectroscopic work represents the type of experiments that can be done and the kind of information that can be obtained from PMDR and other methods. 

K is found by ESR workers (79) to be of D2 symmetry in the triplet state. The emission is thus symmetry allowed, and the 0,0 band is allowed by internal direct spin-orbit perturbation. The determination of the zf origin of this band could identify not only the special symmetry of the state but also the spin-orbit scheme responsible for the radiation. The emitting state is... 

The trends for the calculated orbital moments presented in Figure 5.6 can be understood qualitatively if we treat the spin-orbit term in first-order perturbation theory. In this order, the spin-orbit perturbation A V can be written as follows,... 

The total parity of a given class of levels (F fine structure component for E-states, upper versus lower A-doublet component for II-states) is found to alternate with 7. The second type of label, often loosely called the e// symmetry, factors out this (—l) 7 or (—l)-7-1/2 7-dependence (Brown et al., 1975) and becomes a rotation-independent label. (Note that e/f is not the parity of the symmetrized nonrotating molecule ASE) basis function. In fact, for half-integer S, it is not possible to construct eigenfunctions of crv in the form [ A, S, E) —A, S, — E)], because, for half-integer S, vice versa.) The third type of parity label arises when crv is allowed to operate only on the spatial coordinates of all electrons, resulting in a classification of A = 0 states according to their intrinsic E+ or E- symmetry. Only A = 0) basis functions have an intrinsic parity of this last type because, unlike A > 0) functions, they cannot be put into [ A) — A)] symmetrized form. The peculiarity of this E symmetry is underlined by the fact that the selection rule for spin-orbit perturbations (see Section 3.4.1) is E+ <-> E, whereas for all types of electronic states and all... 

Heavy molecules are, in principle, more favorable for detecting spin-orbit perturbations and, in this way, locating metastable states. In the NS molecule, which is isovalent with NO, a perturbation matrix element between b4E and B2n of 8 cm-1 has allowed the 4E state to be located (Jenouvrier and Pascat, 1980). In the NSe molecule, the 4n state has been detected by its interaction... 

The subband intensity anomaly arises from AS = 0, A Cl = 0, ff / 0, E+ E-spin-orbit perturbations combined with the opposite behavior of the phase factors for E+ — IIfi=1 versus E — IIn=i s transitions (Section 6.3.2). The nominal E+ and E eigenstates are... 

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