Spin-orbit coupling effective Hamiltonians

One-component calculations or two-component calculations including also spin-orbit coupling effects provide a firm basis for the calculations of higher-order relativistic corrections by means of perturbation theory. Several quasi-relativistic approximations have been proposed. The most successful approaches are the Douglas-Kroll-Hess method (DKH) [1-7], the relativistic direct perturbation theory (DPT) [8-24], the zeroth-order regular approximation (ZORA) [25-48], and the normalized elimination of small components methods (NESC) [49-53]. Related quasi-relativistic schemes based on the elimination of the small components (RESC) and other similar nonsingular quasi-relativistic Hamiltonians have also been proposed [54-61]. 

In most cases, however, the relativistic effects are rather weak and may be separated into spin-orbit coupling effects and scalar effects. The latter lead to compression and/or expansion of electron shells and can rather accurately be treated by modifying the one-electron part of the non-relativistic many-electron Hamiltonian. With this scalar-relativistic Hamiltonian the (modified) energies and wave functions are computed and subsequently an effective spin-orbit part is added to the Hamiltonian. The effects of the spin-orbit term on the energies and wave functions are commonly estimated using second-order perturbation theory. More information for the interested reader can be found in excellent textbooks on relativistic quantum chemistry [2, 3]. 

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. 

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... 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

We will see that a Hamiltonian term identical in form also arises from spin-orbit coupling, but first we will pause to see the effect of this Hamiltonian on the energy levels and ESR spectrum of a triplet-state molecule. The spin triplet wave functions can be written in the notation S,ms) ... 

II electronic states, 638-640 vibronic coupling, 628-631 triatomic molecules, 594-598 Hamiltonian equations, 612-615 pragmatic models, 620-621 Kramers doublets, geometric phase theory linear Jahn-Teller effect, 20-22 spin-orbit coupling, 20-22 Kramers-Kronig reciprocity, wave function analycity, 201 -205 Kramers theorem ... 

The main effect of taking spin-orbit interaction into account will be an admixture of singlet character to triplet states and triplet character to singlet states. The spin-orbit coupling Hamiltonian can to a good approximation be described by an effective one-electron operator Hso ... 

We extend the method over all three rows of TMs. No systematic study is available for the heavier atoms, where relativistic effects are more prominent. Here, we use the Douglas-Kroll-Hess (DKH) Hamiltonian [14,15] to account for scalar relativistic effects. No systematic study of spin-orbit coupling has been performed but we show in a few examples how it will affect the results. A new basis set is used in these studies, which has been devised to be used with the DKH Hamiltonian. 

The examination takes place in two stages, one corresponding to the formal interelectronic repulsion component of the Hamiltonian HER and the second to the spin-orbit coupling term Hes. As will be pointed out, in principle, and in certain cases in practice, it is not proper to separate the two components. However, the conventional procedure is to develop HLS as a perturbation following the application of Her. That suffices for most purposes, and simplifies the procedures. Any interaction between the d- or/-electron set and any other set is ignored. It is assumed that it is negligible or can be taken up within the concept of an effective d-orbital set. 

The modeling for the case of the 2T2g term is very instructive in order to visualize the effect of the individual contributions on the Hamiltonian (the CF splitting alone, the effect of the spin-orbit coupling, the tetragonal and/or trigonal distortion, and the vibronic coupling) as follows. 

---