Spin-orbit coupling calculations

The introductory part of this chapter, which provides a survey of Hamiltonian operators and wave function and energy evaluation methods that are cmrently in use for the calculation of spin-orbit coupling, is fairly general. In particular, we attempt to incorporate the spin-dipole spin-dipole interaction operator as an equal partner, whose appreciation is necessary for spin-orbit coupling calculations on organic molecules. 

M. Filatov, W. Zou, D. Cremer. Spin-orbit coupling calculations with the two-component normalized elimination of the smaU component. J. Chem. Phys., 139 (2013) 014106. 

An efficient spin-orbit coupling calculation will take advantage of symmetry and various computational techniques to improve the efficiency. This paper begins with a consideration of the applicable symmetry rules. This will be followed by a discussion of the computational aspects of the spin-orbit coupling matrix elements. The symmetry rules elucidated in this work are applicable to the recently developed relativistically transformed spin-orbit coupling operators, provided that the rotational properties of the transformed operators are unchanged, which is the case for transformations explicitly dependent upon momentum p (15,16,17). 

There is a nice point as to what we mean by the experimental energy. All the calculations so far have been based on non-relativistic quantum mechanics. A measure of the importance of relativistic effects for a given atom is afforded by its spin-orbit coupling parameter. This parameter can be easily determined from spectroscopic studies, and it is certainly not zero for first-row atoms. We should strictly compare the HF limit to an experimental energy that refers to a non-relativistic molecule. This is a moot point we can neither calculate molecular energies at the HF limit, nor can we easily make measurements that allow for these relativistic effects. 

To investigate the connection between Ap/p and pn and the various relativistic effects two types of model calculations were performed i) the speed of light c and that way all relativistic effects have been varied artificially, ii) the strength of the spin-orbit coupling has been manipulated separately [8]. 

As expected, Ap vanishes if the strength of the spin-orbit coupling is reduced to 0 by reducing (co/c) or respectively. Both sets "f model calculations give nearly the same results indicating that the so-called scalar relativistic effects due to the mass-velocity and Darwin-term, are of minor importance for the absolute value of Ap. 

For comparison with the usual second-order perturbation in the spin-orbit coupling, we assume that the first order calculation has taken all first-order effects into account as in Eq.(l 1). The second-order perturbation due to the interaction operator W is given by... 

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