Perturbation spin-orbit

CORRECT WAVEFUNCTIONS FOR PERTURBATIONS (SPIN-ORBIT, EXTERNAL FIELD, RELATIVISTIC, ETC.) WITHIN BORN-OPPENHEIMER APPROXIMATION... 

Singlet and Triplet Perturbations, Spin-Orbit Coupling... 

Figure 7.15 General problem for nonradiative decay of an excited Born-Oppenheimer state with energy in electronic state A, prepared by photon excitation of a level with energy Eg in the electronic ground state. The prepared state is connected by perturbations (spin-orbit coupling, nonradiative coupling, etc.) to a set of Born-Oppenheimer states 1, > with energies , in electronic state B. The states )/7,> are not accessible by El transitions from the ground state.

The perturbations in this case are between a singlet and a triplet state. The perturbation Hamiltonian, H, of the second-order perturbation theory is spin-orbital coupling, which has the effect of mixing singlet and triplet states. 

In this section, the spin-orbit interaction is treated in the Breit-Pauli [13,24—26] approximation and incoi porated into the Hamiltonian using quasidegenerate perturbation theory [27]. This approach, which is described in [8], is commonly used in nuclear dynamics and is adequate for molecules containing only atoms with atomic numbers no larger than that of Kr. 

Appendix A Perturbative Handling of the Renner—Teller Effect and Spin-Orbit Coupling in n Electronic States of Tetraatomic Molecules... 

The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

APPENDIX A PERTURBATIVE HANDLING OF THE RENNER-TELLER EFFECT AND SPIN-ORBIT COUPLING IN n ELECTRONIC STATES OF TETRAATOMIC MOLECULES... 

The present perturbative beatment is carried out in the framework of the minimal model we defined above. All effects that do not cincially influence the vibronic and fine (spin-orbit) stracture of spectra are neglected. The kinetic energy operator for infinitesimal vibrations [Eq. (49)] is employed and the bending potential curves are represented by the lowest order (quadratic) polynomial expansions in the bending coordinates. The spin-orbit operator is taken in the phenomenological form [Eq. (16)]. We employ as basis functions... 

We introduce the dimensionless bending coordinates qr = t/XrPr anti qc = tAcPc ith Xt = (kT -r) = PrOir, Xc = sJ kcPc) = Pc nc. where cor and fOc are the harmonic frequencies for pure trans- and cis-bending vibrations, respectively. After integrating over 0, we obtain the effective Hamiltonian H = Ho + H, which is employed in the perturbative handling of the R-T effect and the spin-orbit coupling. Its zeroth-order pait is of the foim... 

The zeroth-order Hamiltonian and the spin-orbit part of the perturbation are diagonal with respect to the quantum numbers K, E, P, Uj, It, Uc, and Ic-The terms of H involving the parameters aj, ac, and bo aie diagonal with respect to both the Ij and Ic quantum numbers, while the f>2 term connects with one another the basis functions with I j = Ij 2, 4- 2. The c terms... 

Let us return to the nonadiabatic chemical processes. When a PES has been built, a part of the total Hamiltonian may remain unaccounted for, and this part, acting as a perturbation, induces transitions from the initial to the final state. There are several types of such a perturbation, namely (i) an unaccounted part of the electronic interaction (ii) non-adiabaticity (iii) spin-orbit coupling. 

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. 

For comparison with the usual second-order perturbation in the spin-orbit coupling, we assume that the first order calculation has taken all first-order effects into account as in Eq.(l 1). The second-order perturbation due to the interaction operator W is given by... 

Theoretical analyses (75-77) of the matrix-induced changes in the optical spectra of isolated, noble-metal atoms have also been made. The spectra were studied in Ar, Kr, and Xe, and showed a pronounced, reversible-energy shift of the peaks with temperature. The authors discussed the matrix influence in terms of level shift-differences, as well as spin-orbit coupling and crystal-field effects. They concluded that an increase in the matrix temperature enhances the electronic perturbation of the entrapped atom, in contrast to earlier prejudices that the temperature dilation of the surrounding cage moves the properties of the atomic guest towards those of the free atom. 

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