ZORA details

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 c²/(2c² + ε_i - V^KS) σ⋅p
T^ZORA = σ⋅p c²/(2c² - V^KS) σ⋅p
T^NR = p²/2

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 c²/(2c² - 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 charge 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⋅ c²/(2c² - V^KS) p + c²/(2c² - V^KS)² σ⋅ (∇ V^KS × p )

Links

ADF User Documentation: relativistic effects
ADF-GUI: model Hamiltonians
Examples: relativistic effects
References: ZORA
Related: ZORA, spin-orbit coupling