ASQPM for the localization of biradical structures
==================================================

Similar to the multistate metadynamics for the localization of conical intersections, the Automatic
Samling of Quantum Property Manifolds (ASQPM) method can be used for the localization of biradical
configurations of a molecular system. For this purpose, a modified version of the gap between the
highest occupied and lowest unoccupied natural orbital occupation numbers (NOONs) is used as the
primary CV (see J. O. Lindner, M. I. S. Röhr, `arXiv:1909.09005 <http://arxiv.org/abs/1909.09005>`_
(2019)).

.. math::
   
    \Delta n_{meta} = \sqrt{(n_H - n_L)^2 + 4V_{N}^2}

In order to set up the simulation, run the GUI application and check the `Do multistate
metadynamics` option in the metadynamics tab, followed by the selection of "noon gap" as a primary
CV.

.. figure:: gui_asqpm.png

The forms below that line refer to the construction of :math:`V_G`, while the `Gaussian height`
in the upper part of the window is used for the Gaussians added to :math:`V_{N}`. :math:`\epsilon` is
the threshold that is used for the :math:`\Theta`-function to switch between :math:`V_G` and
:math:`V_{N}`. Note that this CV is dimensionless and therefore the unit choices for the threshold and the
height of the Gaussians don't have any effect. `NOON 1` is the LUNO index and `NOON 2` is the HONO
index, with indexing starting at 0. The restriction of :math:`n_{H-1}` and :math:`n_{L+1}` to
values close to 2 and 0 is achieved by the automatic addition of two quadratic wall potentials as
described in the paper cited above. The `Collective Variables` tab refers to the metadynamics in :math:`V_{N}`.

Run the simulation::

    metaFALCON run

During the simulation, two files `deltan_meta.dat` and `deltan_org.dat` are written that contain the
modified and original gaps between HONO and LUNO occupations. Apart from that, NOONs are written to
the file `noons_all.dat`.

The handling of :math:`V_N` is comparable to that of :math:`V_{ge}` in the multistate metadynamics
with modified energy gap as CV.

The calculation of numerical derivatives of the modified noon gap with respect to nuclear
coordinates is implemented in a parallelized manner using the :mod:`multiprocessing` python
package. The number of separate processes to be run is controlled by the `nproc` keyword in the
CV section of the `meta-config.json` file.