6.11. Relativistic Calculations

ORCA features three different approximations to cover relativistic effects:

  1. The „Exact 2 component“ (X2C) Hamiltonian

  2. The Douglas-Kroll-Hess (DKH) Hamiltonian to second order

  3. The 0th order regular approximation (ZORA) with a model potential

Earlier versions of ORCA supported a number of additional approximations, which are no longer supported.

The main relativistic Hamiltonian that will be pursued in further development is the X2C Hamiltonian. Of the three alternatives, we believe that X2C has the best feature set and we recommend to all of our users to preferentially use this method.

All three relativistic model Hamiltonians are implemented for scalar relativistic energy calculations and these are carried through consistently through the entire program. Scalar relativity shows up as an additional effective potential that is added to the one-electron matrix. Scalar relativistic corrections to the two-electron interaction are not available in ORCA. Furthermore, self-consistent field calculations (HF, DFT, CASSCF) with inclusion of spin-orbit-coupling (SOC) are also not available in ORCA but we will not exclude the possibility to add this feature in a future version of the program.

A general overview and some practical recommendations are given in the next sections. For detailed documentation and all available options see Relativistic Options.

6.11.1. Basis sets for relativistic calculations

The different scalar relativistic potentials have different shapes in the core region. Consequently, each one of them requires specialized all electron basis sets that are optimized for the Hamiltonian at hand. The most common choices are listed in the sections Relativistically recontracted Karlsruhe basis sets and SARC basis sets with all available options listed in Built-in Basis Sets. An uncontracted basis set of sufficient size will always work. Likewise, uncontracted fitting basis sets in all forms of RI calculations are always appropriate.

Hint

Use the !Decontract keyword to decontract the chosen (all-electron) basis set and make it suitable for any relativistic Hamiltonian, as well as comparisons between them.

If large, uncontracted basis sets are used in scalar relativistic calculations, there is a distinct danger of variational collapse. This behavior is related to the fact that the relativistic orbitals will diverge for a point nucleus. ORCA features the Gaussian finite nucleus model of Dyall and Visscher for DKH and X2C. We recommend to always use this feature (FiniteNuc) in relativistic calculations.

Given the fact relativistic all-electron calculations on heavy element compounds feature very steep core basis functions, numeric integration, such as in DFT and COSX, may be challenging. ORCA features automatic procedures that adapt the integration grids for the presence of steep basis functions. However, in case you experience strange results, the numeric integration is one potential source of problem. The cure is to go to larger integration grids and, in particular, increase the radial integration accuracy (IntAcc).

6.11.2. Scalar-relativistic gradients and properties

Of the three model Hamiltonians, only X2C features analytic gradients. Hence, for geometry optimizations this is also the preferred methods. For DKH and ZORA, the program automatically switches to the one-center approximation. This requires some attention by the users since final single point energies obtained with the one-center approximation are inconsistent with energies obtained without it. The one-center approximation is usually of sufficient accuracy but we have observed cases in actual applications where it leads to clearly wrong geometries. Hence, we strongly recommend to use the X2C Hamiltonian in this realm.

Caution

Geometry optimizations with DKH and ZORA (but not X2C) automatically use the one-center approximation. When computing relative energies, do not mix energies from single-point calculations without the one-center approximation with those from geometry optimizations that do make use of this feature.

If relativistic calculations are used for molecular properties there is a potential mismatch between non-relativistically calculated property integrals and the relativistic Hamiltonian. The procedure to remove these inconsistencies is referred to as „picture change“. The picture change is usually carried through to the same level of approximation as the decoupling of the relativistic Hamiltonian into two-component and eventually to one-component form. We strongly recommend to use picture change in all relativistic property calculations and consequently, this is also the default. Relativistic property calculations without picture change are wildly inaccurate, in particular if operators are involved that carry inverse powers of the electorn-nucleus distance. Picture change effects are implemented for DKH and X2C and to some extent also for ZORA. However, they are not implemented for all properties that ORCA can calculate. Please pay attention to the output of the property integral and property programs. Both programs will explicitly state which picture change effects are included in the molecular integrals.

%rel 
  FiniteNuc     true   # Invoke the Gaussian finite nucleus model.
  PictureChange 1 or 2 # First or second order picture change effects.
                       # Second order is potentially more accurate and more expensive.
end

6.11.3. Exact two-component method (X2C)

Despite the name, the X2C method is implemented in ORCA only as a scalar-relativistic, effective one-component method. The theory and implementation are discussed in Exact Two-Component Theory (X2C), together with appropriate references to cite in your work. In the simplest case, it is sufficient to add the X2C simple keyword to the input and choose an appropriate basis set:

! X2C X2C-TZVPall X2C/J

The DLU approximation,[660] discussed in DLU approximation, is the recommended way to reduce the cost of the X2C transformation, particularly for gradient/Hessian calculations, with minor loss of accuracy. It is available via the simple input keyword DLU-X2C.

6.11.4. Douglas-Kroll-Hess (DKH)

The first- or second-order DKH method be requested via the simple input keywords DKH1 or DKH2, respectively (DKH is an alias for the latter), together with appropriate basis sets:

! DKH DKH-def2-TZVP SARC/J

For most calculations, no other settings are needed. See The Douglas-Kroll-Hess Method for an overview of the underlying theory.

6.11.5. ZORA and IORA

The \(0^{\text{th}}\) order regular approximation (ZORA; pioneered by van Lenthe et al., see Ref. [864] and many follow up papers by the Amsterdam group) implementation in ORCA essentially follows van Wüllen [867] and solves the ZORA equations with a suitable model potential and a model density derived from accurate atomic ZORA calculations. See Relativistic Options for explanation of the ModelPot and ModelDens keywords used to control these models. If the relevant precautions are taken (see below), the use of the ZORA or IORA methods is as easy as in the DKH/X2C case. For example:

! ZORA ZORA-def2-TZVP SARC/J
# for more detail use
%rel
     ModelPot 1,1,1,1
     ModelDens rhoZORA
end

Attention

The ZORA method is highly dependent on numerical integration and it is very important to pay attention to the subject of radial integration accuracy! By default, from ORCA 5.0 we consider that during the grid construction and the defaults should work very well. Only for very problematic cases, consider using a higher IntAcc parameter or at least to increase the radial integration accuracy around the heavy atoms using SpecialGridAtoms and SpecialGridIntAcc.