# More details of the ZORA relativistic Hamiltonian

The ZORA equation is the zeroth order regular approximation to the Dirac equation. The relativistic and non-relativistic Kohn-Sham DFT equations can be written as:

[ T + V^{KS}] φ_{i} = ε_{i} φ_{i}

The potential V^{KS} is the local Kohn-Sham potential, see also the Kohn-Sham MO model. T is the kinetic energy operator, which is different for each relativistic method:

T^{Dirac} = **σ⋅p** c^{2}/(2c^{2} + ε_{i} – V^{KS}) **σ⋅p**

T^{ZORA} = **σ⋅p** c^{2}/(2c^{2} – V^{KS}) **σ⋅p**

T^{NR} = p^{2}/2

T^{Dirac} is the energy dependent Dirac kinetic energy operator, T^{ZORA} is the ZORA kinetic energy operator, and T^{NR} is the non-relativistic kinetic energy operator.

The scaled ZORA spinor energy ε_{i}^{scaled} is:

ε_{i}^{scaled} = ε_{i}^{ZORA} / < φ_{i} | **σ⋅p** c^{2}/(2c^{2} – V^{KS})^{2} **σ⋅p** | φ_{i} >

The scaled ZORA spinor energies are exactly equal to the Dirac energies for hydrogen like systems,

where V^{KS} = -Z/r.

A magnetic field can be included by minimal substitution for a negatively charged particle:

**p** → **Π** = **p** + **A**/c

The ZORA kinetic energy can be split in a scalar relativistic part T^{SR} and a spin-orbit coupling T^{SO} term:

T^{ZORA} = T^{SR} + T^{SO} = **p⋅** c^{2}/(2c^{2} – V^{KS}) **p** + c^{2}/(2c^{2} – V^{KS})^{2} **σ⋅** (**∇** V^{KS} × **p** )

## 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
- spin-orbit coupling matrix elements with perturbative spin-orbit coupling TDDFT
- singlet-triplet contributions for spin-orbit coupled excitations