Spin-orbit operator relativistic corrections

The quasi-relativistic Hamiltonians obtained using the general ansatz have usually a metric that include spin-orbit contributions. This can be a undesirable situation since in a perturbation study of spin-orbit effects, the addition of the spin-orbit coupling requires reorthogonalization of the orbitals [69]. However, as seen in equation (18), the relativistic correction term (V) consists of two contributions f and B. B is several orders of magnitude less significant than f. Furthermore, the B operator can also be separated into scalar relativistic and spin-orbit contributions. 

The Amsterdam Density Functional (ADF) method [118,119] was used for calculations of some transactinide compounds. In a modem version of the method, the Hamiltonian contains relativistic corrections already in the zeroth order and is called the zero-order regular approximation (ZORA) [120]. Recently, the spin-orbit operator was included in the ZORA Fock operator [121]. The ZORA method uses analytical basis fimctions, and gives reliable geometries and bonding descriptions. For elements with a very large SO splitting, like 114, ZORA can deviate from the 4-component DFT results due to an improper description of the pi/2 spinors [117]. Another one-component quasirelativistic scheme [122] applied to the calculations of dimers of elements 111 and 114[116,117]isa modification of the ZORA method. 

The most important relativistic corrections are the one-electron spin-orbit operator, and the relativistic correction to the spin-Zeeman operator. 

The diagonal term gel arises from the spin-Zeeman term S Bext, while the anisotropic part has two contributions. The direct term arises from the P operator (eq. (10.79)) and the relativistic correction from the spin-Zeeman term (eq. (10.83)), with additional contributions coming from the combination of V2LG and the spin-orbit operators P and Pe . 

The first term is the Pauli operator regularized, which can be seen by setting V to zero in the denominator. The second term contains a spin-free and a spin-orbit relativistic correction to the operator. The spin-orbit correction is the Pauli operator regularized. There is no spin-free relativistic correction to the Pauli operator, but as before we must realize that the relativistic correction to the property is a second-order perturbation, which includes a relativistic correction to the wave function. 

Including also the next term of the expansion, Eq. (2.88), gives rise to additional operators including the mass-velocity, Darwin and one-electron spin-orbit operators, which can be used in perturbation theory calculations of relativistic corrections to the non-relativistic results of the Schrodinger equation and molecular properties. However, the expansion is based on the assumption that the scalar potential r) is small, which is not fulfilled for the inner electrons of heavy atoms, because close to the nucleus they are exposed to the strong Coulomb potential of the nucleus. For this situation the expansion is then no longer valid. Alternative expansions exist, which circumvent this... 

We now consider how to eliminate the spin-orbit interaction, but not scalar relativistic effects, from the Dirac equation (25). The straightforward elimination of spin-dependent terms, taken to be terms involving the Pauli spin matrices, certainly does not work as it eliminates all kinetic energy as well. A minimum requirement for a correct procedure for the elimination of spin-orbit interaction is that the remaining operator should go to the correct non-relativistic limit. However, this check does not guarantee that some scalar relativistic effects are eliminated as well, as pointed out by Visscher and van Lenthe [44]. Dyall [12] suggested the elimination of the spin-orbit interaction by the non-unitary transformation... 

It comprises the non-relativistic Hamiltonian of the form pf/2me + V and the relativistic correction terms, such as the mass-velocity operator —pf/8m c2, the Darwin term proportional to Pi E and the spin-orbit coupling term proportional... 

Let us emphasize that in single-configurational approach the terms of the Hamiltonian describing kinetic and potential energies of the electrons as well as one-electron relativistic corrections, contribute only to average energy and, therefore, are not contained in, which in the non-relativistic approximation consists only of the operators of electrostatic interaction e and the one-electron part of the spin-orbit interaction so, i.e. 

Relativistic corrections to the nuclear-elel L 12 J tron attraction (V are of on generated by its own movement. is the spin-other-orbit operator, describing the ler 1 /c (owing interaction ot an electronic spin with the magnetic field veneramd hv ths movement of ... 

Spin-orbit (SO) coupling corrections were calculated for the Pt atom since the relativistic effects are essential for species containing heavy elements. Other scalar relativistic corrections like the Darwin and mass-velocity terms are supposed to be implicitly included in (quasi)relativistic pseudopotentials because they mostly affect the core region of the considered heavy element. Their secondary influence can be seen in the contraction of the outer s-orbitals and the expansion of the d-orbitals. This is considered in the construction of the pseudoorbitals. The effective SO operator can be written within pseudopotential (PS) treatment in the form71 75... 

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