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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 [66]. Two-component treatments require the use of complex two-component orbitals

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

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

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

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)

$\displaystyle \vec{{m}} $($\displaystyle \bf r$) = $\displaystyle \sum_{i}^{}$ψi($\displaystyle \bf x$)$\displaystyle \vec{{{\pmb \sigma}}} $ψi($\displaystyle \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.


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Next: How to use Up: Two-component Hartree-Fock and DFT Previous: Two-component Hartree-Fock and DFT   Contents   Index
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