Perturbational treatment of relativistic effects

The next step on the way towards the nonrelativistic limit is to treat SO coupling as a perturbation, based on nonrelativistic or scalar relativistic one-component wavefunctions (SFR effects may be included for example by ECP 

Let us start with the field-free SO effects. Perturbation by SO coupling mixes some triplet character into the formally closed-shell ground-state wavefunction. Therefore, electronic spin has to be dealt with as a further degree of freedom. This leads to hyperfine interactions between electronic and nuclear spins, in a BP framework expressed as Fermi-contact (FC) and spin-dipolar (SD) terms (in other quasirelativistic frameworks, the hyperfine terms may be contained in a single operator, see e.g. [34,40,39]). Thus, in addition to the first-order and second-order ct at the nonrelativistic level (eqs. 5-7), third-order contributions to nuclear shielding (8) arise, that couple the one- and two-electron SO operators (9) and (10) to the FC and SD Hamiltonians (11) and (12), respectively. Throughout this article, we will follow the notation introduced in [58,61,62], where these spin-orbit shielding contributions were denoted 

This leads to third-order perturbation expressions like [62] 

As pointed out already by Cheremisin and Schastnev [55] and en hasized again recently by Fukui et al. [63], a gauge-invariant perturbational inclusion of SO effects requires also to take into account the field dependence of the SO operators, leading to the following second-order contributions 

In [57], a classification of the individual relativistic corrections into hctive and [jassive operators has been proposed (similar to the classification into direct or indirect effects within a scalar relativistic ZORA framework in [39]). Active relativistic operators are those that carry a dependence on the magnetic fields (e.g. the field-dependent SO operators in eq. 15 or the SZ-KE term in eq. 17), whereas passive relativistic operators are field independent and act via a change of the wavefimction of the system (as, e.g., the field-free SO operators in eqs. 9 and 10 entering the SO-I contributions), whereas the additional contributions from the fields are in their noiu elativistic form. 

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As the fully relativistic (four-component) calculations demand severe computational efforts, several quasirelativistic (two-component) approximations have been proposed in which only large components are treated explicitly. The approaches with perturbative treatment of relativistic effects [507] have also been developed in which a nonrelativistic wavefunction is used as reference. The Breit-Pauli (BP) approximation uses the perturbation theory up to the (p/mc) term and gives reasonable results in the first-order perturbation calculation. Unfortunately, this method cannot be used in... 

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