Spin-orbit interaction approximation

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. 

Figure 10. Low-energy vibronic levels in the X2II state of HCCS computed in various approximations [152]. Hq zeroth-order approximation (both vibronic and spin-orbit couplings neglected). Hi. vibronic coupling taken into account, spin-orbit interaction neglected. Hi + Hs0 both vibronic and spin-orbit couplings taken into account. Solid horizontal lines K = 0 vibronic levels dashed line K — 1 dash-dotted lines K = 2 dotted lines K — 3. Values of the quantum numbers V4, N of the basis functions dominating the vibronic wave function of the level in question are indicated. Approximate correlation of vibronic states computed in various approximations is indicated by thin lines. In all cases the stretching quantum numbers are assumed to be zero.

For a nonadiabatic system with spin-orbit interaction, the validity of the semiclassical approximation (based on the spin-coherent state representation) has been discussed in detail in Ref. 147. 

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 ... 

From the above discussion it becomes clear that in order to eliminate the spin-orbit interaction in four-component relativistic calculations of magnetic properties one must delete the quaternion imaginary parts from the regular Fock matrix and not from other quantities appearing in the response function (35). It is also possible to delete all spin interactions from magnetic properties, but this requires the use of the Sternheim approximation [57,73], that is calculating the diamagnetic contribution as an expectation value. 

It follows from the Eq. (10.2) that the spin-orbit interaction may be interpreted as the interaction of the appropriate momenta. If two momenta 1 and s interact with each other, then the corresponding quantum numbers become approximate. Only their vectorial sum 1 + s = j and its projection... 

Starting with the method described above, extensive tables of the numerical values of mean energies, integrals of electrostatic and constant of spin-orbit interactions are presented in [137] for the ground and a large number of excited configurations, for atoms of boron up to nobelium and their positive ions. They are obtained by approximation of the corresponding Hartree-Fock values by polynomials (21.20) and (21.22). Such data can be directly utilized for the calculation of spectral characteristics of the above-mentioned elements or they can serve as the initial parameters for semi-empirical calculations [138]. 

However, the quantum numbers L and S are not rigorous, due to the existence of the spin-orbit interaction between the respective momenta. Therefore, the above-mentioned selection rules hold only approximately. In intermediate coupling the selection rules with respect to L and S change and allow many more transitions. For example, the isolation of the condition AS = 0 leads to the occurrence of the so-called intercombination E 2- and M 1-lines. For the configuration 3d3 in intermediate coupling, instead of (27.10) and (27.11) we obtain... 

Let us emphasize that in single-configurational approach the terms of the Hamiltonian describing kinetic and potential energies of the electrons as well as one-electron relativistic corrections, contribute only to average energy and, therefore, are not contained in, which in the non-relativistic approximation consists only of the operators of electrostatic interaction e and the one-electron part of the spin-orbit interaction so, i.e. 

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... 

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