Adaptive intermolecular reactive empirical bond order potential (AIREBO)
========================================================================

AIREBO method was realized according to article: Steven J. Stuart, Alan B. Tutein and Judith A. Harrison. "A reactive potential for hydrocarbons with intermolecular interactions" // Journ. of Chem. Phys., V 112,14 (2000), p. 6472-6486.

AIREBO potential can be represented by a sum over pairwise interactions, including covalent bonding REBO interactions, LJ terms, and torsion interactions:

.. math::
	:label: ai_potential

	E = \frac{1}{2}\sum_{i}\sum_{j\neq{i}}\left(E_{ij}^{REBO} + E_{ij}^{LJ} +\sum_{k\neq{i,j}}\sum_{l\neq{i,j,k}}E_{kijl}^{tors} \right)

Van-der-Vaals interaction is described through the using of the Lennard-Jones potential:

.. math::
	:label: ai_lj

	E_{ij}^{LJ} = S(t_r (r_{ij}))S(t_b (b_{ij}^* ))C_{ij}V_{ij}^{LJ}(r_{ij}) + [1 - S(t_r(r_{ij}))]C_{ij}V_{ij}^{LJ}(r_{ij})

:math:`V_{ij}^{LJ}` is the traditional LJ term:

.. math::
	:label: pure_LJ

	V_{ij}^{LJ} = 4\epsilon_{ij}\left( \left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12} - \left(\frac{\sigma_{ij}}{r_{ij}}\right)^6 \right)
	
It was modified by several sets of switching functions. S(t) can be represented in the next form:

.. math::
	:label:

	S(t) = \Theta(-t)+\Theta(t)\Theta(1-t)[1 - t^2(3-2t)]

Magnitude of LJ interactions depends on bonding environment. Gradual exclusion of Lennard-Jones interactions with changings of :math:`r_{ij}` is controlled by scalling function :math:`t_{b}`:

.. math::
	:label:

	t_b(b_{ij}) = \frac{b_{ij} - b_{ij}^{min}}{b_{ij}^{max} - b_{ij}^{min}}

If atoms :math:`i` and :math:`j` are not connected by two or fewer intermediate atoms, LJ interactions between them are controlled by next switching function:

.. math::
	:label:

	C_{ij} = 1 - max\{w_{ij}(r_{ij}),w_{ik}(r_{ik})w_{kj}(r_{kj}), \forall k, \\
	w_{ik}(r_{ik})w_{kl}(r_{kl})w_{lj}(r_{lj}), \forall k,l \}

where:

.. math::

	w_{ij}(r_{ij}) = S^\prime(t_c(r_{ij}))

.. math::

	S^\prime(t_c(r_{ij})) = \Theta(-t)+\Theta(t)\Theta(1-t)\frac{1}{2}[1+cos(\pi t)]

The torsional part of equation :eq:`ai_potential` for the dihedral angle determined by atoms i, j, k, l has the next form:

.. math::
	:label:

	E^{tors}_{kijl} = w_{ki}(r_{ki})w_{ij}(r_{ij})w_{jl}(r_{jl})V^{tors}(\omega_{kijl})

where :math:`V^{tors}(\omega_{kijl})` is the torsional potential:

.. math::
	:label:

	V^{tors}(\omega_{kijl}) = \frac{256}{405}\epsilon_{kijl}cos^{10}(\omega_{kijl}/2)-\frac{1}{10}\epsilon_{kijl}

.. automodule:: kvazar.core.airebo
	:members: