Relativistic effects¶

Pauli¶

Specification of the Pauli formalism means that the first order relativistic corrections (the Pauli Hamiltonian) will be used [51-60]. In a scalar relativistic run ADF employs the single point group symmetry and only the so-called scalar relativistic corrections, Darwin and Mass-Velocity. The treatment is not strictly first-order, but is quasi-relativistic, in the sense that the first-order scalar relativistic Pauli Hamiltonian is diagonalized in the space of the non-relativistic solutions, i.e. in the non-relativistic basis set.

The quasi-relativistic approach improves results considerably over a first-order treatment. There are, however, theoretical deficiencies due to the singular behavior of the Pauli Hamiltonian at the nucleus. This would become manifest in a complete basis set but results are reasonable with the normally employed basis sets. However, this aspect implies that it is not recommended to apply this approach with an all-electron basis set for the heavy atoms, and for very heavy elements even a frozen core basis set often fails to give acceptable results. The problems with the quasi relativistic approach of the Pauli Hamiltonian are discussed for example in Ref.[61].

ZORA¶

The ZORA approach gives generally better results than the Pauli formalism. For all-electron calculations, and in fact also for calculations on very heavy elements (Actinides), the Pauli method is absolutely unreliable. Therefore the ZORA method is the recommended approach for relativistic calculations with ADF.

ZORA refers to the Zero Order Regular Approximation [61-65]. This formalism requires special basis sets, primarily to include much steeper core-like functions; applying the ZORA method with other, not-adapted basis sets, gives unreliable results. The ZORA basis sets can be found in subdirectories under the $ADFHOME/atomicdata/ZORA directory.

The ZORA formalism can also be used in Geometry Optimizations. However, there is a slight mismatch between the energy expression and the potential in the ZORA approach, which has the effect that the geometry where the gradients are zero does not exactly coincide with the point of lowest energy. The differences are very small, but not completely negligible, in the order of 0.0001 Angstrom.

X2C and RA-X2C¶

X2C stands for an exact transformation of the 4-component Dirac equation to 2-components [447-448]. RA-X2C also stands for an exact transformation of the 4-component Dirac equation to 2-components, in this case using a regular approach to calculate the transformation matrix. In practice, however, often approximations are made, and this is also true for the implementation in ADF. In the X2C and RA-X2C method implemented in ADF, first the 4-component Dirac equation for a model potential (MAPA) of the molecule is calculated for the given ADF basis set, using the modified Dirac equation by Dyall [430] for X2C, or using the regular approach [445] to the modified Dirac equation for RA-X2C.

With the method used in ADF in the basis set limit X2C and RA-X2C should lead to the same results for the model potential (MAPA). However, since in practice always a finite basis set is used, even for the model potential the results for X2C and RA-X2C will differ. Next the (electronic) 4-component Dirac solutions are transformed exactly to 2-components [446]. The transformation used is also used to calculate an effective one-electron 2-component kinetic energy operator in a basis set representation. This kinetic energy matrix is then assumed to be constant in further ADF SCF calculation on the molecule. The full Hamiltonian is approximated using the electron density in the 2-component picture, the so called Foldy-Wouthuysen picture.

In a similar way a spin-free (scalar relativistic) form of X2C and RA-X2C are calculated. The spin-free form of X2C and RA-X2C will be different even in the basis set limit, see Ref. [431].

X2C and RA-X2C in ADF can be used in single point calculations only. An all electron basis set is needed. For bond energies, polarizabilities and TD-DFT excitation energies the Foldy-Wouthuysen picture will be used. For the calculation of the EFG, ESR g-value, and ESR A-tensor, the Dirac picure will be used. X2C and RA-X2C is not implemented for frozen cores, optimization of nuclear coordinates, frequencies, NMR properties.

Spin-Orbit coupling¶

The Spin-Orbit option uses double-group symmetry. The symmetry-adapted orbitals are labeled by the quantum number J rather than L and any references in input to subspecies, such as a specification of occupation numbers, must refer to the double group labels.

Create runs must not use the Spin-Orbit formalism. The SFO analysis of Molecular Orbitals for a Spin-Orbit calculation is only implemented in the case of a scalar relativistic fragment file, which is the whole molecule.

In a Spin-Orbit run each level can allocate 2 electrons (times the dimension of the irreducible representation) as in a normal restricted calculation. However, contrary to the normal case these two electrons are not directly associated with spin-\(\alpha\) and spin-\(\beta\), but rather with the more general Kramer’s symmetry. Using the unrestricted feature in order to assign different numbers of electrons to a and b spin respectively cannot be applied as such. However, one can use the unrestricted option in combination with the collinear or non-collinear approximation. In that case one should use symmetry NOSYM, and each level can allocate 1 electron.