First-order spin-orbit perturbation

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 ... 

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]. 

Some tables of first-order spin-orbit and spin-vibronic selection rules are given in the following pages (Tables 9.1—9.6). The first column of Tables 9.1, 9.2, 9.3, gives some possible spatial symmetries for the triplet state. The third column gives the perturbing singlet symmetry, which must be F((R) X in the absence of vibronic effects. The last three... 

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. 

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 effect of spin-orbit coupling on mixing of singlet and triplet states can be seen by considering first-order non-degenerate perturbation theory [8). Let us consider the perturbed triplet wavefunction for the state where represents the unperturbed triplet wavefunction for the... 

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... 

It should be noted that, due to the effect of spin-orbit interaction the correct initial and final states are not exactly the pure spin states. The admixture with higher electronic states j/ may be ignored only if there exists a direct coupling between the initial and final pure spin states. Otherwise, the wave function for the initial state is obtained to first order of perturbation theory as ... 

The expressions (4.22)-(4.23) found in chap. 4 for the isomer shift 5 in nonrelativ-istic form may be applied to lighter elements up to iron without causing too much of an error. In heavier elements, however, the wave function j/ is subject to considerable modification by relativistic effects, particularly near the nucleus (remember that the spin-orbit coupling coefficient increases with Z ). Therefore, the electron density at the nucleus l /(o)P will be modified as well and the aforementioned equations for the isomer shift require relativistic correction. This has been considered [1] in a somewhat restricted approach by using Dirac wave functions and first-order perturbation theory in this approximation the relativistic correction simply consists of a dimensionless factor S (Z), which is introduced in the above equations for S,... 

Using first-order perturbation theory, show that the spin-orbit interaction energy for a hydrogen atom is given by... 

As seen in the radiationless process, intercombinational radiative transitions can also be affected by spin-orbit interaction. As stated previously, spin-orbit coupling serves to mix singlet and triplet states. Although this mixing is of a highly complex nature, some insight can be gained by first-order perturbation theory. From first-order perturbation theory one can write a total wave function for the triplet state as... 

Hi H2 this is the so-called intermediate couphng. When the electrostatic and spin orbit interactions are of the same order of magnitude - and this is the case of the actinides - both should be included in first-order perturbation theory. 

The term maia a(1) is the first-order correction to the integral of the electric dipole moment operator in the a direction over orbitals a and i. The perturbed integral will depend on the change of the orbitals in the presence of a magnetic field or spin-orbit coupling. 

In the absence of an external magnetic field, orbitally nondegenerate levels with spin multiplicity greater than 2 split due to direct electron spin—spin coupling (in first order) and spin—orbit coupling (in second and higher orders of perturbation theory). This phenomenon is called zero-field splitting (ZFS). The SH that describes this phenomenon can be formulated in... 

Consider a dn configuration present in a crystal field that leaves the ground state nondegenerate except for spin. The ground state then consists of (25+ l)-spin states and the effect of the spin-orbit interaction plus the magnetic field can be computed using first- and second-order perturbation theory. If we take as the perturbation operator... 

---