Response theory spin orbitals

Spin-Orbit Coupling Constants from Coupled-Cluster Response Theory. 

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. 

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... 

For the evaluation of probabilities for spin-forbidden electric dipole transitions, the length form is appropriate. The velocity form can be made equivalent by adding spin-dependent terms to the momentum operator. A sum-over-states expansion is slowly convergent and ought to be avoided, if possible. Variational perturbation theory and the use of spin-orbit Cl expansions are conventional alternatives to elegant and more recent response theory approaches. 

RESPONSE THEORY AND CALCULATIONS OF SPIN-ORBIT COUPLING PHENOMENA IN MOLECULES... 

For many types of electron spectroscopies there are still comparatively few studies of SOC effects in molecules in contrast to atoms, see, e.g., [1, 2, 3, 4, 5, 6, 7] and references therein. This can probably be referred to complexities in the molecular analysis due to the extra vibrational and rotational degrees of freedom, increased role of many-body interaction, interference and break-down effects in the spectra, but can also be referred to the more difficult nature of the spin-orbit coupling itself in polyatomic species. Modern ab initio formulations, as, e.g., spin-orbit response theory [8] reviewed here, have made such investigations possible using the full Breit-Pauli spin-orbit operator. 

In the present work we illustrate the potential of response theory for SOC in the areas briefly referred to above and review a set of recent applications that represent different physical aspects. We first give, in the next section, those aspects of linear and quadratic response theory that directly relate to spin-orbit coupling phenomena. We refer to the original work of Olsen and Jorgensen [15] and to the recent review by Luo et al. [23] for a general account of basic aspects of the response theory employed, but not... 

The results of response theory are most conveniently cast into a formalism based on second quantization. In a second quantized representation of (66), the spin-operators will transfer to triplet excitation operators which are weighted by the integrals over the orbital parts. The -component will have the form... 

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... 

Table 3 Spin-orbit coupling matrix elements between singlet and triplet states form second order response theory calculations on H20++ (10-6 au). A 6ai, 3b2, 3bj, la2 active space B 4ai, 2b2, 2bi active space. From Ref. [55],...

In contrast to polyenes the aromatic molecules exhibit not only the Ti — So absorption, but also the longlived T — So emission, which gives rise to phosphorescence phenomena of rigid solvents and crystals. This is another important field of applications of spin-orbit quadratic response theory. Such calculations refer to lifetimes, transition moments, oscillator strengths and polarization directions for the radiative decay of molecular triplet states. These quantities may either be averaged over the triplet levels or refer to specific triplet spin sublevels depending on the conditions for the relevant experimental measurements. 

Vibrational analysis of the benzene phosphorescence bands indicates that the radiative activity is induced predominantly by e2g vibrations [155, 156]. A weak but observable activity of b2g vibrations has also been found [156, 155, 157]. By introducing spin-orbit- and vibronic coupling through second order perturbation theory Albrecht [158] showed that the vibronic interaction within the triplet manifold is responsible for the larger part of the phosphorescence intensity. This also follows from comparison of the vibrational structure in phosphorescence and fluorescence spectra [159]. The benzene phosphorescence spectrum in rigid glasses [155] reveals a dominant vibronic activity of... 

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