Introduction


Electronic structure of NiO with LDA+U	The electronic structure of NiO calculated with LSDA

The LDA and GGA approximations to the exchange-correlation energy has a number of short comings, where the most important are

Self interaction, an electron interacts with itself and this prevents electrons from localizing.
Poor approximation for excited states, band gaps are often too low.

The mean field Hubbard correction, popularly called LDA+U or XC+U, is a semi-empirical correction which tries to improve on these deficiencies of the LDA and GGA functionals.

In the XC+U an additional energy term,

$\displaystyle E_{U} = \frac{1}{2} \sum_\mu U_\mu (n_\mu - n_\mu^2)$

is added to the Exchange-Correlation energy [1]. In this equation $n_\mu$ is the projection onto an atomic shell and $U_\mu$ is the “Hubbard U” for that shell. The $E_{U}$ energy term is zero for a fully occupied or un-occupied shell, while positive for a fractional occupied shell.

The energy is thereby lowered if states become fully occupied. This may happen if the energy levels move away from the Fermi Level, i.e. increasing the band gap, or if the broadening of the states is decreased, i.e. the electrons are localized. Thus, the Hubbard U improves on the deficiencies of the exchange-correlation energies.

The NiO crystal has a too low band gap in LDA and is one of the standard examples of how the LDA+U approximation can be used to improve the description of the electronic structure of solids[2]. In this tutorial you will compare the LDA and LDA+U model for this system.

Further details of the Hubbard U implementation in ATK can be found in the ATK reference manual.