Spin-orbit interaction derivative couplings

The expressions for the rotational energy levels (i.e., also involving the end-over-end rotations, not considered in the previous works) of linear triatomic molecules in doublet and triplet II electronic states that take into account a spin orbit interaction and a vibronic coupling were derived in two milestone studies by Hougen [72,32]. In them, the isomorfic Hamiltonian was inboduced, which has later been widely used in treating linear molecules (see, e.g., [55]). 

The microscopic rate constant is derived from the quantum mechanical transition probability by considering the system to be initially present in one of the vibronic levels on the initial potential surface. The initial level is coupled by spin-orbit interaction to the manifold of vibronic levels belonging to the final potential surface. The microscopic rate constant is then obtained, following the Fermi-Golden rule, as ... 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

It should be remembered, of course, that Lie groups have an independent existence apart from their role in the theory of f electrons in the lanthanides. Some of the bizarre properties that turn up in the f shell might well derive from isoscalar factors that receive a ready explanation in another context. An example of this is provided by the vanishing of the spin-orbit interaction when it is set between F and G states belonging to the irreducible representation (21) of G2. In the f shell, (21) is merely a 64-dimensional representation of no special interest. However, for mixed configurations of p and h electrons, it fits exactly into the spinor representations (iiiHi 2) of SO(14) with dimensions 2 (Judd 1970). These are the analogs of the spinor representations of eq. (130), and (21) describes the quasiparticle basis of the configurations (p-t-h)". The spinor representations of SO(3) and (Hiii) of SO(ll) provide the quasi-particle bases for the p and h shells respectively and their SO(3) structures, namely S 1,2 and 512+ 912+ is/2> when coupled, must yield the L structure of (21) of G2, namely D-l-F-l-G-l-H-t-K-t-L. In this context, the F and G terms of (21) are associated with the different irreducible representations Sj,2 and S9/2. It is this property,... 

Pacchioni has recently carried out calculations on the low-lying states of Sn2 and Pb2. This author gives the impression that he is the first to carry out a comparative ab initio Cl calculation on these systems. We would like to clarify this further. First, his calculation starts with the Hafner-Schwarz model potentials in comparison to our relativistic ab initio potentials derived from numerical Dirac-Fock solutions of the atoms. Pacchioni s calculations ignore spin-orbit interaction. Our calculations include spin-orbit interaction in a relativistic Cl scheme in comparison to the non-relativistic Cl of Pacchioni. Thus, he obtains a Z), approximately twice the experimental value which he corrects by a semi-empirical scheme to arrive at a value close to our calculated value with a relativistic Cl. Our calculations have clearly demonstrated the need to carry out an intermediate-coupling Cl calculation for Pbj as a result of large spin-orbit contamination. Calculations without spin-orbit, such as Pacchioni s, have little relationship to the real Pb2 molecule. 

These contributions are closely related to the hyperfine interactions observed in ESR spectroscopy (see O section ESR Hyperfine Coupling Tensors ). The above expressions have been derived assuming kT to be much larger than the separation of the different electronic spin states. We have also for simplicity ignored higher-order corrections arising from the spin-orbit interaction, which for many open-shell species can be quite significant. 

The indirect electron spin-spin coupling may also be analysed in terms of spin currents (Problem 11.17). Briefly, one spin dipole produces, via the spin-orbit interaction, small circulating currents around the various nuclei, the currents being different for up-spin and down-spin components the resulting spin polarization is picked up by a second spin and leads to the tensor interaction already derived. 

Approximate selection rules. The last three selection rules in Table 7.1 are derived using explicit forms for the atomic wavefunctions. Since these can only be.approximate solutions of Schrodinger s equation in the case of a many-electron atom, rules 4-6 in turn apply only approximately in many cases. Thus rule 4 is obeyed only if each of the states involved can be correctly described by means of a single configuration of electrons. Similarly rules 5 and 6 may be expected to apply only in Russell-Saunders coupling where the spin-orbit interaction is negligible compared with the electrostatic forces between the electrons. 

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