Scalar-relativistic/spin-free

Depending on commutation properties, the relativistic corrections E [Ve and E2[Ve may be separated into scalar relativistic (SR) and spin-orbit (SO) terms [77]. When SO terms, in particular those related to ia pVxp), are dropped, the two-component treatment is reverted to an effectively one-component method with a trivial spin-dependence of the wave functions such models are sometimes referred to as scalar relativistic, spin-free, or quasi-relativistic [2]. Such calculations are in many technical aspects very similar to the correspond-... 

The terms scalar-relativistic, spin-free, spin-averaged and one-component all refer to this very same procedure of eliminating all Pauli spin matrices from the Hamiltonian by simple deletion after employment of Dirac s relation. 

Various approaches can be pursued to compute spin-orbit effects. Four-component ab initio methods automatically include scalar and magnetic relativistic corrections, but they put high demands on computer resources. (For reviews on this subject, see, e.g., Refs. 18,19,81,82.) The following discussion focuses on two-component methods treating SOC either perturbationally or variationally. Most of these procedures start off with orbitals optimized for a spin-free Hamiltonian. Spin-orbit coupling is added then at a later stage. The latter approaches can be divided again into so-called one-step or two-step procedures as explained below. 

In compounds containing heavy main group elements, electron correlation depends on the particular spin-orbit component. The jj coupled 6p j2 and 6/73/2 orbitals of thallium, for example, exhibit very different radial amplitudes (Figure 13). As a consequence, electron correlation in the p shell, which has been computed at the spin-free level, is not transferable to the spin-orbit coupled case. This feature is named spin-polarization. It is best recovered in spin-orbit Cl procedures where electron correlation and spin-orbit interaction can be treated on the same footing—in principle at least. As illustrated below, complications arise when configuration selection is necessary to reduce the size of the Cl space. The relativistic contraction of the thallium 6s orbital, on the other hand, is mainly covered by scalar relativistic effects. 

Because the convenience of the one-electron formalism is retained, DFT methods can easily take into account the scalar relativistic effects and spin-orbit effects, via either perturbation or variational methods. The retention of the one-electron picture provides a convenient means of analyzing the effects of relativity on specific orbitals of a molecule. Spin-unrestricted Hartree-Fock (UHF) calculations usually suffer from spin contamination, particularly in systems that have low-lying excited states (such as metal-containing systems). By contrast, in spin-unrestricted Kohn-Sham (UKS) DFT calculations the spin-contamination problem is generally less significant for many open-shell systems (39). For example, for transition metal methyl complexes, the deviation of the calculated UKS expectation values S (S = spin angular momentum operator) from the contamination-free theoretical values are all less than 5% (32). 

The restriction to scalar terms and hence to a one-component Hamiltonian is sometimes referred to as a spin averaged formulation. However, the names spin-free or scalar relativistic formulation seem to be more appropriate. The matrix representations of all terms occurring after these manipulations are already available within every quantum chemical program package except for the PVP expressions. Their evaluation can, however, be reduced to the representation of the external potential via the relation... 

By applying the spin-free DHF formalism by Dyall [164] the spin-orbit effects can be separated from the scalar-relativistic effects and also allow for a determination of the PCE. The SO-effect is much smaller (-0.052 a.u. for the I field gradient in HI) than the picture-change error (-0.974 a.u. according to [163] but only -0.330 a.u. according to [162]). The differences in the calculated PCE can be attributed to the different approaches used. 

If spin-orbit coupling is considered as a perturbation, the total Hamiltonian decouples into a spin-free (referred as // for a scalar relativistic Hamiltonian) and a spin-orbit part H = The electrostatic and spin-orbit inter-... 

In this article, we will concentrate on NMR chemical shifts, for which the inportance of spin-free (scalar) relativistic (SFR) and spin-orbit (SO) contributions needs to be better appreciated. Reviews of relativistic calculations of spin-spin coupling constants are available [6,7]. Articles on conten jorary quantum chemical calculations of electronic g-tensors have been published elsewhere [5,8]. Other EPR parameters like zero-field splittings and hyperfine coupling constants are also strongly affected by relativity and are covered. 

Kotzian et al. (1991) and Kotzian and Rosch (1992) applied their INDO/1 and INDO/S-CI methods to hydrated cerium(III), i.e. model complexes [Ce(H20) ] (n=8,9), in order to rationalize the electronic structure and the electronic spectrum of these species. Besides the scalar relativistic effects spin-orbit coupling was also included in the INDO/S-CI studies. The spin-orbit splitting of the 4f F and 5d states of the free Ce " ion was calculated as 2175cm" and 2320cm in excellent agreement with the experimental values of 2253 cm and 2489 cm" , respectively. The calculated energy separation between the F and states of approximately 44000cm" (estimated fi-om fig. 5 in Kotzian and Rosch 1992) is somewhat lower than the experimental value of 49943 cm" (Martin et al. 1978). 

So far no approximation has been introduced at all, and Eq. (13.4) still yields, of course, the exact Dirac eigenvalues and large components. Also the non-relativistic limit (V 2meC, e 0) of Eq. (13.4) is well defined and recovers the Schrodinger equation by setting a = 1. Since a is a scalar operator, even this simple energy-dependent elimination of the small component based on Eqs. (13.4) and (13.5) permits an exact separation of the spin-free and spin-dependent terms of the Dirac Hamiltonian by application of Dirac s relation... 

To demonstrate the effect of different orders in DKH calculations. Table 16.4 presents results obtained for SnO and CsH (different ansatze for the electronic wave function were employed). As can be seen from the table, all spectroscopic parameters converge fast with increasing DKH order. The accuracy is mostly determined by the quality of the wave-function approximation. Note that DKHn denotes the scalar-relativistic variant, which has also been called the spin-averaged (i.e., the spin-free) DKH approach. Also, the two-electron terms have not been transformed. Since the nuclear charge numbers of Sn and Cs are not very high, these elements are not ultrarelativistic cases. However, also for Au2 it was found that the spectroscopic parameters are already converged with the second-order DKH2 Hamiltonian [1127], which is typical for such valence-shell dominated properties. 

L. Cheng, J. Gauss, J. F. Stanton. Treatment of scalar-relativistic effects on nudear magnetic shieldings using a spin-free exact-two-component approach. /. Chem. Phys., 139 (2013) 054105. 

The lowest-order effect of relativity on energetics of atoms and molecules—and hence usually the largest—is the spin-free relativistic effect (also called scalar relativity), which is dominated by the one-electron relativistic effect. For light atoms, this effect is relatively easily evaluated with the mass-velocity and Darwin operators of the Pauli Hamiltonian, or by direct perturbation theory. For heavier atoms, the Douglas-Kroll-Hess method or the NESC le method provide descriptions of the spin-independent relativistic effect that are satisfactory for all but the highest accuracy. 

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