atsim.potentials.spline

Package Contents

Classes

Spline_Point(potential_function, r)	Class for the attachment and detachment points of potential objects and region to be splined
Exp_Spline(detach_point, attach_point)	Class for represention splines of the form:
Buck4_Spline(detach_point, attach_point, r_min)	Class for representing the splined part of the four ranged Buckingham potential.
Custom_SplinePotential(spline)	Callable to allow splining of one potential to another
SplinePotential(startPotential, endPotential, detachmentX, attachmentX)	Callable to allow splining of one potential to another using an exponential spline
Buck4_SplinePotential(startPotential, endPotential, detachmentX, attachmentX, r_min)	Callable to allow splining of one potential to another using the Buck4 spline type

Functions

gradient(func, h=1e-06)

Function wrapper that returns derivative of func.

atsim.potentials.spline.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.

Parameters:

• func – Function to be wrapped
• h – Step size used when performing numerical differentiation

Returns:

Function that returns derivative of func

atsim.potentials.spline.polynomial

atsim.potentials.spline.exp_spline

class atsim.potentials.spline.Spline_Point(potential_function, r)

Bases: object

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

potential_function

Potential function

r

Value at which splining takes place

v

Value of potential_function at r

deriv

First derivative of potential_function: dv/dr(r)

deriv2

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

deriv_callable

deriv2_callable

class atsim.potentials.spline.Exp_Spline(detach_point, attach_point)

Bases: object

Class for represention splines of the form:

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

The spline coefficients \(B_{0...5}\) and C can be obtained using the spline_coefficients() property.

detach_point

Spline_Point giving start of splined region

attach_point

Spline_Point giving end of splined region

spline_coefficients

Coefficients for spline_function

__call__(self, r)

deriv(self, r)

deriv2(self, r)

class atsim.potentials.spline.Buck4_Spline(detach_point, attach_point, r_min)

Bases: 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:

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

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

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

The spline coefficients \(A_{0..5}\) and \(B_{0..3}\) 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.

detach_point

Spline_Point giving start of splined region

attach_point

Spline_Point giving end of splined region

r_min

Position of minimum

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]

spline5

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

spline3

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

__call__(self, r)

deriv(self, r)

deriv2(self, r)

class atsim.potentials.spline.Custom_SplinePotential(spline)

Bases: object

Callable to allow splining of one potential to another

startPotential

Returns:Function defining potential for separations < detachmentX

endPotential

Returns:Function defining potential for separations > attachmentX

interpolationFunction

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

detachmentX

Returns:Point at which spline should start

attachmentX

Returns:Point at which spline should end

splineCoefficients

Returns:Tuple containing the seven coefficients of the spline polynomial

__call__(self, rij)

Parameters:rij – separation at which to evaluate splined potential Returns:spline value

class atsim.potentials.spline.SplinePotential(startPotential, endPotential, detachmentX, attachmentX)

Bases: atsim.potentials.spline.Custom_SplinePotential

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

class atsim.potentials.spline.Buck4_SplinePotential(startPotential, endPotential, detachmentX, attachmentX, r_min)

Bases: atsim.potentials.spline.Custom_SplinePotential

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