The Zeroth Order Regular Approximation (ZORA)

The ZORA equation is the zeroth order regular approximation to the Dirac equation, for some details see also ZORA details. The regular expansion, which leads to the ZORA Hamiltonian, remains valid even for a Coulombic potential. This is in contrast with the expansion that leads to the Pauli Hamiltonian, which is divergent for a Coulombic potential. A scalar relativistic Hamiltonian is a relativistic Hamiltonian without spin-orbit coupling.

• the ZORA Hamiltonian is bounded from below
• scaled ZORA spinor energies are exactly equal to Dirac energies for hydrogen like systems
• spin-orbit coupling can be taken into account
• all electron relativistic basis sets for all elements
• ZORA can be used for most spectroscopic properties
• scalar relativistic ZORA can be used for structure and reactivity

Above: geometric (left) and energetic (right) data for some group 11 compounds. The relativistic effects modeled by ZORA are greatest for compounds containing the heaviest elements. The effects for the gold containing compounds are the most pronounced

Spin-orbit coupling

Spin-orbit coupling results from the interaction of the electron magnetic moments with the magnetic field generated by their own orbital motion. The computation is more involved than for scalar relativistic or non-relativistic calculations because spin-orbit coupling does not have the ordinary molecular symmetry. Instead, it is invariant under the molecular point double group, which affects both spin and spatial coordinates. One of its effects is to split energies of the molecule that would otherwise be identical in energy. This interaction is responsible for many of the details of spectroscopic properties.

• ADF uses double group symmetry and provides symmetry labels
• spin-orbit coupling is part of the ZORA Hamiltonian
• spin-orbit coupling can be included for most spectroscopic properties
• all electron relativistic basis sets for all elements

Links

ADF User Documentation: relativistic effects, Schönflies symbols
ADF-GUI: model Hamiltonians, Tutorial: spin-orbit coupling
Examples: relativistic effects
References: ZORA
Related: ZORA details, symmetry

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