Spin orbitals response properties

In this chapter, we therefore consider whether it is possible to eliminate spin-orbit coupling from four-component relativistic calculations. This is a situation quite different from that of more approximate relativistic methods where a considerable effort is required for the inclusion of spin-orbit coupling. We have previously shown that it is indeed possible to eliminate spin-orbit coupling from the calculation of spectroscopic constants [12,13]. In this chapter, we consider the extension of the previous result to the calculation of second-order electric and magnetic properties, i.e., linear response functions. Although the central question of this article may seem somewhat technical, it will be seen that its consideration throws considerable light on the fundamental interactions in molecular systems. We will even claim that four-component relativistic theory is the optimal framework for the understanding of such interactions since they are inherently relativistic. 

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. 

A large number of spin-orbit properties can now be derived from the response functions. From the linear response function we can deduce the second-order energy correction due to SOC (see section 4.1),... 

One of the most of important applications of quadratic response theory, pertaining to spin-orbit properties, is the calculation of the spin-orbit induced dipole moment (phosphorescence, see section 7), which can be derived from the residue... 

Apart from primary structural and energetic data, which can be extracted directly from four-component calculations, molecular properties, which connect measured and calculated quantities, are sought and obtained from response theory. In a pilot study, Visscher et al. (1997) used the four-component random-phase approximation for the calculation of frequency-dependent dipole polarizabilities for water, tin tetrahydride and the mercury atom. They demonstrated that for the mercury atom the frequency-dependent polarizability (in contrast with the static polarizability) cannot be well described by methods which treat relativistic effects as a perturbation. Thus, the varia-tionally stable one-component Douglas-Kroll-Hess method (Hess 1986) works better than perturbation theory, but differences to the four-component approach appear close to spin-forbidden transitions, where spin-orbit coupling, which the four-component approach implicitly takes care of, becomes important. Obviously, the random-phase approximation suffers from the lack of higher-order electron correlation. 

In describing the magneto-optical properties one has to take account of the spin-orbit splitting, which was not included in the band-structure calculations of de Groot et al. These authors note that no orbital quenching occurs at the F point in the Brillouin zone so that here one may expect the spin-orbit interaction to be a maximum. It follows from the magneto-optical spectra that the onset of the Kerr peak at about 2 eV is located near 1 eV (see fig. 37). From these considerations it follows that the transition responsible for the large Kerr intensity involves initial electron states located rather close to the top of the valence band (at F in the lower... 

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