TURBOMOLE provides two different possibilities for the treatment of relativistic effects: Via effective core potentials (ECPs) or via allelectron approaches (X2C, DKH, BSS). Both techniques can be employed in an onecomponent (scalar relativistic) or twocomponent (including spinorbit coupling) framework. The latter is only available in the module RIDFT.
Incoporation of scalarrelativistic effects leads to additional contributions to the oneelectron integrals (either from ECP or allelectron approach). The program structure is the same as in nonrelativistic theory( all quantities are real). Twocomponent treatments allow for selfconsistent calculations including spinorbit interactions. These may be particularly important for compounds containing heavy elements (additionally to scalar relativistic effects). Twocomponent treatments require the use of complex twocomponent orbitals

instead of real (noncomplex) onecomponent orbitals needed for nonrelativistic or scalarrelativistic treatments. The HartreeFock and KohnSham equations are now spinor equations with a complex Fock operator

The wavefunction is no longer eigenfunction of the spin operator, the spin vector is no longer an observable.
In case of DFT for openshell systems rotational invariance of the exchangecorrelation energy was ensured by the noncollinear approach. In this approach the exchangecorrelation energy is a functional of the particle density and the absolute value of the spinvector density (r) ( are the Pauli matrices)

This quantity replaces the spindensity (difference between density of alpha and beta electrons) of non or scalarrelativistic treatments.
For closedshell species the Kramersrestricted scheme, a generalization of the RHFscheme of one component treatments, is applicable.
Effective core potentials The most economic way to account for relativistic effects is via effective core potentials by choosing either the one or the twocomponent ECP (and for the latter additionally setting $soghf in the control file). The theoretical background and the implementation for the twocomponent SCF procedure is described in [77]. The backgrund theory of the more fundamental approach, allelectron relativistic electronic structure theory, is given in the next paragraph.
Relativistic allelectron approaches (X2C, DKH, BSS) Relativistic calculations are based on the Dirac rather than on the Schrödinger Hamiltonian. Since the Dirac Hamiltonian introduces pathological negativeenergy states and requires extensive oneelectron basis set expansions, methods have been devised which allow one to calculate a matrix representation of that part of the Dirac Hamiltonian, which describes electronic states only. For this, a unitary transformation is employed to blockdiagonalize the Dirac Hamiltonian and thus to decouple the negativeenergy from the electronic states. In order to be efficient, this transformation is carried out only for the oneelectron part of the full Hamiltonian (as a consequence, the twoelectron interaction will then be affected by a picturechange effect). The resulting quantum chemical approach has been called an exact twocomponent (X2C) one. It was developed by many groups starting with formal work in the mid 1980s. X2C is related to the stepwise DouglasKrollHess (DKH) approach, which also achieves exact decoupling but sequentially. The number of transformation steps is called the order of DKH. X2C is also related to the BaryszSadlejSnijders (BSS) method, that first applies the freeparticle FoldyWouthuysen transformation (which is the first mandatory step in DKH), and then constructs the onestep exact decoupling transformation of X2C. These three approaches have been reviewed and directly compared in terms of formalism and results, respectively, in Ref. [78] (see also this reference for a complete bibliography on exactdecoupling methods).
Essentially, X2C methods change the oneelectron Hamiltonian in basisset representation. The Schrödinger oneelectron Hamiltonian (including the external potential of the atomic nuclei) is replaced by the transformed (upperleft block of the) Dirac Hamiltonian. Since the transformation is carried out in the fully decontracted primitive basis, all matrix operations needed for the generation of the relativistic oneelectron Hamiltonian can be cumbersome and even prohibitive if the size of the molecule is large. In order to solve this unfavorable scaling problem, a rigorous local approach, called DLU, has been devised [79] and should be activated for large molecules. However, since the local (atomic) structure is defined by the atomcentered basis functions within DLU, diffuse functions in the basis set should be handled with care and the use of tailored basis sets is recommended (relativistic calculations require refitted basis sets; if these are not available for X2C or X2CDLU, standard secondorder DKH basis sets could be used).
As in DKH theory, X2C and BSS exist in full twocomponent (spin(same)orbit coupling including) and in a onecomponent scalar relativistic form. Both have been implemented into the TURBOMOLE package and all details on the efficient implementation have been described in Ref. [80], which should be cited when the module is activated.
The keyword $soghf enforces the twocomponent calculations. Keywords for specification of the method of calculation are the same as for the onecomponent case.
The DIIS scheme for complex Fock operators can be activated by inserting $gdiis in the controlfile. For closedshell species a Kramers invariant density functional formalism (only pure density functionals) can be switched on with the keyword $kramers. These keywords have to be inserted into the controlfile manually.
As start wavefunctions Hückel, UHF or RHFwavefunctions may be used. The twocomponent formalism supports Abelian point group symmetry if $kramers is set. Otherwise start wave functions may be transformed to C1 symmetry by define or the script ’uhfuse’. For openshell molecules it is often helpful to increase the value for $scforbitalshift closedshell; a value of ca. 1.0 may serve as a rough recommendation.
Effective core potentials The twocomponent formalism may be most easily prepared and applied in the following way:
Allelectron calculations The keywords $rx2c, $rbss and $rdkh [Order of DKH] are used to activate the X2C, BSS or DKH Hamiltonian. The default order of the DKH Hamiltonian is four. It is not recommended to go beyond, but to use X2C instead. For details on the arbitraryorder DKH Hamiltonians see Ref. [82] for details on the infiniteorder DKH theory, [83] for the implementation, and [84] for a conceptual review of DKH theory. The local approach (DLU) can be optionally activated by $rlocal for all one and twocomponent allelectron Hamiltonians. For symmetric molecules, pointgroup symmetry is not exploited by default, but can be used in the onecomponent case by setting $rsym.