Natural orbitals
================

Summary
-------
To produce state-average natural orbitals, run ::

    qp_run save_natorb file.ezfio

The MOs will be replaced, so the two-electron integrals and the wave function
are invalidated as well.



Extracting natural orbitals
---------------------------

Once obtained the near |FCI| wave function, one can obtain many quantities related to it. 
One of these quantities are the natural orbitals which have the properties of diagonalizing the one-body density matrix: 

   .. math::

       \rho_{ij} = \delta_{ij}

where the element of the one-body density matrix :math:`\rho_{ij}` is define as:


   .. math::

       \rho_{ij} = \langle \Psi | \left( a^{\dagger}_{j,\alpha} a_{i,\alpha} + a^{\dagger}_{j,\beta} a_{i,\beta} \right) | \Psi \rangle


These orbitals are in general known to be better than the usual |HF| |MOs| as they are obtained from a correlated wave function. To use these orbitals for future calculations, one has to replace the current |MOs| by the natural orbitals. To do so, just run: 

.. code::

    qp_run save_natorb file.ezfio


Hands on
--------

.. important::

   As the |MOs| are changed, for the sake of coherence of future calculations, the save_natorb program 
   *automatically removes the current wave function* stored in the |EZFIO| database and replace 
   it by a single Slater determinant corresponding to a |HF| occupation of the new spin orbitals. 
   Also, all the keywords to read the one- and two-electron integrals on the |MO| basis are set to "None" 
   in order to be sure not to read integrals not corresponding to the current set of |MOs|. 

.. seealso:: 

    The documentation of the :c:func:`save_natorb` program.