Orbital perturbation

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 ... 

Further insight into the mechanism of this reaction was obtained with the help of MO theory and quantum mechanical calculations." The following orbital diagram (Scheme 35)100>101 describes the interaction of two sulfide moieties, which results in dication formation after a two-electron oxidation (cases A, B and C correspond to progressive increase in orbital perturbation and interaction between the sulfur atoms). 

The theoretical interpretation of the results was made (334) in terms of the molecular orbital perturbation theory, in particular, of the FMO theory (CNDO-2 method), using the model of the concerted formation of both new bonds through the cyclic transition state. In this study, the authors provided an explanation for the regioselectivity of the process and obtained a series of comparative reactivities of dipolarophiles (methyl acrylate > styrene), which is in agreement with the experimental data. However, in spite of similar tendencies, the experimental series of comparative reactivities of nitronates (249) toward methyl acrylate (250a) and styrene (250b) are not consistent with the calculated series (see Chart 3.17). This is attributed to the fact that calculation methods are insufficiently correct and the... 

The use of the Hartree-Fock model allows the perturbation-theory equations (1.2)-(1.5) to be conveniently recast in terms of underlying orbitals (,), orbital energies (e,), and orbital occupancies (n,). Such orbital perturbation equations will allow us to treat the complex electronic interactions of the actual many-electron system (described by Fock operator F) in terms of a simpler non-interacting system (described by unperturbed Fock operator We shall make use of such one-electron perturbation expressions throughout this book to elucidate the origin of chemical bonding effects within the Hartree-Fock model (which can be further refined with post-HF perturbative procedures, if desired). 

These results have been interpreted in terms of HOMO-LUMO interactions. As a result of the orbital perturbation, the interaction of the HOMO of the cyclohexene double bond with the LUMO of the developing cation may become effective. At the first stage of this interaction, an overlap of the LUMO of the cyclobutyl cation with the p lobe of the double bond located close to the cation center is probably important. However, when the reaction progresses, the interaction with the p lobe on the remote carbon atom has been assumed to increase significantly. 

As was discussed earlier, the two intermolecular orbital perturbation terms,... 

For the moment, suffice it to say that the two-orbital perturbation schemes give the orbital energies and the sign of the MO coefficients in the supermolecule (A B) with a reasonable degree of precision. However, three-orbital perturbations are needed to determine the relative sizes of the coefficients. 

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. 

C, O and N atoms were treated as the frozen orbitals. Perturbation selection [29] was carried out at the LevelTwo level of thresholds. 

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... 

FIGURE 1. Frontier MOs in square-planar metal-dithiolene complexes formed by the in-phase (i.p.) and out-of phase (o.o.p.) combinations of C2S2 orbitals perturbed by the metal. The d metal orbitals stabilized relative to ligand orbitals lead to an inverted bonding scheme . Reprinted with permission from Reference 12. Copyright 2004 American Chemical Society... 

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