Spin-Orbit Response Functions

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. 

Also in response theory the summation over excited states is effectively replaced by solving a system of linear equations. Spin-orbit matrix elements are obtained from linear response functions, whereas quadratic response functions can most elegantly be utilized to compute spin-forbidden radiative transition probabilities. We refrain from going into details here, because an excellent review on this subject has been published by Agren et al.118 While these authors focus on response theory and its application in the framework of Cl and multiconfiguration self-consistent field (MCSCF) procedures, an analogous scheme using coupled-cluster electronic structure methods was presented lately by Christiansen et al.124... 

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),... 

The Ta <— So transition moments to particular spin sublevels for the three lowest triplet states of the ozone molecule, 3B2,3 A2 and 3B, were calculated by the MCQR method in ref. [70] using CASSCF wave functions. Table 7 recapitulates results for electric dipole radiative activity of different S-T transitions in ozone [70]. The type of information gained form this kind of spin-orbit response calculations are viz. transition electric dipole moments and oscillator strengths for each spin sublevel T , their polarization directions (7), radiative lifetimes (r ) and excitation energies (En). The most prominent features of the Chappuis band are reproduced in calculations, which simulate the photodynamics of ozone visible absorption [78, 79]. Because the CM (M2) state cannot be responsible for the Wulf bands, the only other candidates ought to... 

These angular variations are responsible for the different g values found in the FPR spectrum (i.e. qualitatively they depend on the symmetry of the electronic wave function). However these deviations from ge actually arise from the admixture of orbital angular momentum into the spin ground state via spin orbit coupling. The extent of this admixing depends on which orbital contributes to the spin ground state (p, d or f). The real components of the g matrix are then given by ... 

The calculated quantities include charge density wave (CDW), spin density wave, antiferro-orbital ordering, and electron-pairing response functions. The corresponding operators are the density operator A , the spin density operator the antiferro-orbital operator A°, and the singlet pairing operator , all of which are defined in terms of the original orbitals of e(= and 9 = d 2, .2) as... 

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