Background Theory

Two-component treatments allow for self-consistent calculations including spin-orbit interactions. These may be particularly important for compounds containing heavy elements (additionally to scalar relativistic effects). Two-component treatments were implemented within the module ridft for RI-JK-Hartree-Fock and RI-DFT (local, gradient-corrected and hybrid functionals) via effective core potentials describing both scalar and spin-orbit relativistic effects. The theoretical background and the implementation is described in [61]. Two-component treatments require the use of complex two-component orbitals

$$\bf x \begin{pmatrix}\psi^{\alpha}_i({\bf r}) \\ \psi^{\beta}_i({\bf r}) \\ \end{pmatrix}$$

instead of real (non-complex) one-component orbitals needed for non-relativistic or scalar-relativistic treatments. The Hartree-Fock and Kohn-Sham equations are now spinor equations with a complex Fock operator

$$\begin{pmatrix}\hat{F}^{\alpha \alpha} & \hat{F}^{\alpha \beta} \\ \hat{F}^{\beta \alpha} & \hat{F}^{\beta \beta} \\ \end{pmatrix} \begin{pmatrix}\psi^{\alpha}_i({\bf r}) \\ \psi^{\beta}_i({\bf r}) \\ \end{pmatrix} \begin{pmatrix}\psi^{\alpha}_i({\bf r}) \\ \psi^{\beta}_i({\bf r}) \\ \end{pmatrix}$$

The wavefunction is no longer eigenfunction of the spin operator, the spin vector is no longer an observable.

In case of DFT for open-shell systems rotational invariance of the exchange-correlation energy was ensured by the non-collinear approach. In this approach the exchange-correlation energy is a functional of the particle density and the absolute value of the spin-vector density $\vec{{m}}\,$($\bf r$) ( $\vec{{{\pmb \sigma}}}\,$ are the Pauli matrices)

$$\vec{{m}}\, \bf r \sum_{i}^{} \bf x \vec{{{\pmb \sigma}}}\, \bf x$$

This quantity replaces the spin-density (difference between density of alpha and beta electrons) of non- or scalar-relativistic treatments.

For closed-shell species the Kramers-restricted scheme, a generalization of the RHF-scheme of one component treatments, is applicable.