:mod:`atsim.potentials.spline`
==============================

.. py:module:: atsim.potentials.spline


Package Contents
----------------

Classes
~~~~~~~

.. autoapisummary::

   atsim.potentials.spline.Spline_Point
   atsim.potentials.spline.Exp_Spline
   atsim.potentials.spline.Buck4_Spline
   atsim.potentials.spline.Custom_SplinePotential
   atsim.potentials.spline.SplinePotential
   atsim.potentials.spline.Buck4_SplinePotential



Functions
~~~~~~~~~

.. autoapisummary::

   atsim.potentials.spline.gradient


.. function:: gradient(func, h=1e-06)

   Function wrapper that returns derivative of func.

   If the callable, `func` provides a `.deriv(r)` method this will be used
   to evaluate the derivative of the function, if not the returned function
   will use num_deriv() in gradient evaluation.

   If the callable additionally provides a `.deriv2(r)` method, representing
   its second derivative, the function returned by this routine will have a `deriv()`
   method which will delegate to func.deriv2() when called.

   By providing .deriv() and .deriv2() on the `func` callable analytical descriptions
   of a potential's first and second derivatives may be specified.

   :param func: Function to be wrapped
   :param h: Step size used when performing numerical differentiation
   :return: Function that returns derivative of func


.. data:: polynomial
   

   

.. data:: exp_spline
   

   

.. py:class:: Spline_Point(potential_function, r)

   Bases: :class:`object`

   Class for the attachment and detachment points of potential objects and region to be splined

   .. attribute:: potential_function
      

      Potential function


   .. attribute:: r
      

      Value at which splining takes place


   .. attribute:: v
      

      Value of `potential_function` at `r`


   .. attribute:: deriv
      

      First derivative of `potential_function`: dv/dr(r)


   .. attribute:: deriv2
      

      Second derivative of `potential_function`: d2v/dr^2(r)


   .. attribute:: deriv_callable
      

      

   .. attribute:: deriv2_callable
      

      


.. py:class:: Exp_Spline(detach_point, attach_point)

   Bases: :class:`object`

   Class for represention splines of the form: 

   .. math::

             U(r_{ij}) = \exp \left( B_0 + B_1 r_{ij} + B_2 r_{ij}^2 + B_3 r_{ij}^3 + B_4 r_{ij}^4 + B_5 r_{ij}^5 \right) + C


   The spline coefficients :math:`B_{0...5}` and `C` can be obtained using the :meth:`~atsim.potentials.Exp_Spline.spline_coefficients` property.

   .. attribute:: detach_point
      

      Spline_Point giving start of splined region


   .. attribute:: attach_point
      

      Spline_Point giving end of splined region


   .. attribute:: spline_coefficients
      

      Coefficients for spline_function


   .. method:: __call__(self, r)


   .. method:: deriv(self, r)


   .. method:: deriv2(self, r)



.. py:class:: Buck4_Spline(detach_point, attach_point, r_min)

   Bases: :class:`object`

   Class for representing the splined part of the four ranged Buckingham potential.

   Between the detachment point and `r_min` this is a 5th order polynomial:

   .. math::

     U(r_{ij}) = A_0 + A_1 r_{ij} + A_2 r_{ij}^2 + A_3 r_{ij}^3 + A_4 r_{ij}^4 + A_5 r_{ij}^5

   and between `r_min` and the re-attachment point a 3rd order spline is used:

   .. math::

     U(r_{ij}) = B0 + B_1 r_{ij} + B_2 r_{ij}^2 + B_3 r_{ij}^3

   The spline coefficients :math:`A_{0..5}` and :math:`B_{0..3}` are solved such that the 
   the spline values match with the potential functions at the detach and re-attachment points and r_min.
   They are continuous in their first and second derivatives across these points and where the two
   splines meet at `r_min`. Finally, the derivative at `r_min` is set to be 0 with the aim of creating a
   minimum.

   .. attribute:: detach_point
      

      Spline_Point giving start of splined region


   .. attribute:: attach_point
      

      Spline_Point giving end of splined region


   .. attribute:: r_min
      

      Position of minimum


   .. attribute:: spline_coefficients
      

      Spline coefficients as list of form [A_0, A_1, A_2, A_3, A_4, A_5, B_0, B_1, B_2, B_3]


   .. attribute:: spline5
      

      Callable (atsim.potentials.potentialfunctions.polynomial) object representing the fifth order section of the buck4 spline - between `detach_point` and `r_min`


   .. attribute:: spline3
      

      Callable (atsim.potentials.potentialfunctions.polynomial) object representing the fifth order section of the buck4 spline - between `detach_point` and `r_min`


   .. method:: __call__(self, r)


   .. method:: deriv(self, r)


   .. method:: deriv2(self, r)



.. py:class:: Custom_SplinePotential(spline)

   Bases: :class:`object`

   Callable to allow splining of one potential to another

   .. attribute:: startPotential
      

      :return:  Function defining potential for separations < ``detachmentX``


   .. attribute:: endPotential
      

      :return:  Function defining potential for separations > ``attachmentX``


   .. attribute:: interpolationFunction
      

      :return:  Spline object connecting startPotential and endPotential for separations ``detachmentX`` < rij < ``attachmentX``


   .. attribute:: detachmentX
      

      :return:  Point at which spline should start


   .. attribute:: attachmentX
      

      :return:  Point at which spline should end


   .. attribute:: splineCoefficients
      

      :return:  Tuple containing the seven coefficients of the spline polynomial


   .. method:: __call__(self, rij)

      :param rij: separation at which to evaluate splined potential
      :return:  spline value



.. py:class:: SplinePotential(startPotential, endPotential, detachmentX, attachmentX)

   Bases: :class:`atsim.potentials.spline.Custom_SplinePotential`

   Callable to allow splining of one potential to another using an exponential spline


.. py:class:: Buck4_SplinePotential(startPotential, endPotential, detachmentX, attachmentX, r_min)

   Bases: :class:`atsim.potentials.spline.Custom_SplinePotential`

   Callable to allow splining of one potential to another using the Buck4 spline type