.. index:: DFT+U, LDA+U, Hubbard
.. _hubbardu:
    
============
DFT+U theory
============

The basic idea behind DFT+U is to treat the strong on-site Coulomb
interaction of localized electrons, which is not correctly described
by LDA or GGA, with an additional Hubbard-like term. The on-site
Coulomb interactions are particularly strong for localized d and f
electrons, but can be also important for p localized orbitals.  The
strength of the on-site interactions are usually described by
parameters U (on site Coulomb) and J (on site exchange). These
parameters U and J can be extracted from ab-initio calculations, but
usually are obtained semi-empirically.

The DFT+U corrections can be introduced in ab initio calculations in
different ways.  The two main branches are the one introduced by
Liechtenstein et al. [Liechtenstein]_, in which U and J enter as
independent corrections in the calculations, and the one proposed by
Anasimov et al. [Dudarev]_, where only a single effective
`U_\text{eff} = U-J` parameter accounts for the Coulomb interaction,
neglecting thereby any higher multi-polar terms.  The latter is the
one implemented in GPAW. Thus, the DFT+U totally energy in GPAW is:

.. math::
    
    E_\text{DFT+U} = E_\text{DFT} +
    \sum_a \frac{U_\text{eff}}{2}
    \text{Tr}(\rho^a - \rho^a \rho^a),

where `\rho^a` is the atomic orbital occupation matrix. This can be
understood as adding a penalty functional to the DFT total energy
expression that forces the on site occupancy matrix in the direction
of idempotency, i.e. to either fully occupied or fully unoccupied
levels.


GPAW implementation
===================

Here are some examples about how to introduce the Hubbard correction
`U_\text{eff}` in the calculator.  For instance if one wants to apply
a `U_\text{eff}=6` eV correction to the d orbitals of manganese
atoms, one should include the next parameter to the calculator::

    setups={'Mn': ':d,6.0'}

In the case of p electrons (here nitrogen atom is used as expample),
one should use::

    setups={'N': ':p,6.0'}

Here is an example of how to apply a `U_\text{eff}=6` eV to the d
orbitals of Ni in NiO:

.. literalinclude:: nio.py


Scaling the Hubbard correction
==============================

The projection of the orbitals needed to get the atomic orbiatal
coccupation matrix is truncated at the augmentation sphere radius
(this is beacause GPAW atomic setups tipically have a bound state
projector and an unbound one). Due to this truncation the projection
of the wavefunctions onto the atomic orbitals is always <1, being at
the maximum the integral of the projected atomic orbital within the
augmentation sphere.  at this point there are two choices: either we
normalize the projection to the value of the integral of the projected
atomic orbital within the augmentation sphere or we do not do it.

By default in GPAW the Hubbard corrections is normalized. However, in
other PAW codes (e.g. VASP) this is not the standard choice and the
Hubbard correction is not normalized.  Still, GPAW allows to apply the
Hubbard correction without normalization. For instance, in the case of
the Nitrogen example before one should write::

    setups={'N': ':p,6.0,0'}

The addition of the 0 at the end of the keyword deactivates the
normalization.

The normalization does not have a big influence when the Hubbard
correction is applied on d or f orbitals (repeat the calculation for
NiO, setting the normalization to 0 and check that the band gap is
similar in both cases), since more than 90% of their wavefunctions are
within the augmentation sphere. However, this is not the case when the
Hubbard correction is applied on p orbitals. Thus, one should expect
quite different results in a normalized +U correction calculation vs a
non-normalized +U correction calculation when the +U correction is
applied to p orbitals. One extreme example of this issue can be
observed when the +U corrections are applied to the p orbitals of
Nitrogen, when one calculates the atomic nitrogen.  In this case we
have three p orbital fully occupied (spin up) and three p orbitals
empty (spin down). In a first approximation, the occupied orbitals
should decrease their energy by `U_\text{eff}/2` with respect to a
calculation with no +U corrections, whereas the empty orbitals should
increase their energy by the same amount. The next example show how
this is only true if the +U correction is normalized.

.. literalinclude:: n.py

Here are the resulting 2p-splitting:
    
.. csv-table::
   :file: gaps.csv


(see also :download:`check.py`).


References
==========

.. [Liechtenstein] A. I. Liechtenstein, V. I. Anisimov and J. Zaane,
                   Phys. Rev. B 52, R5467 (1995).
.. [Dudarev] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys
             and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).
