Classes/routines for pair potentials (analphipy.potential)

Classes:

Generic(phi_func[, dphidr_func, r_min, ...])	Class to define potential using callables.
Analytic()	Base class for defining analytic potentials.
LennardJones([sig, eps])	Lennard-Jones potential.
LennardJonesNM([n, m, sig, eps])	Generalized Lennard-Jones potential
Yukawa([z, sig, eps])	Hard core Yukawa potential
HardSphere([sig])	Hard-sphere pair potential
SquareWell([sig, eps, lam])	Square-well pair potential
CubicTable(bounds, phi_table[, phi_left, ...])	Cubic interpolation table potential.

Functions:

factory(potential_name[, rcut, lfs, cut])

Factory function to construct Phi object by name.

class analphipy.potential.Generic(phi_func, dphidr_func=None, *, r_min=None, phi_min=None, segments=None)

Bases: PhiBase

Class to define potential using callables.

Parameters:

• phi_func (Callable) – Function phi(r)

• dphidr (Callable, optional) – Optional function dphidr(r).

• r_min (float) – Location of minimum in potential energy.

• phi_min (float, optional) – Value of potential energy at minimum.

• segments (sequence of int) – Integration limits. For n = len(segments) integration will be performed over ranges (segments[0], segments[1]), (segments[1], segments[2]), ..., (segments[n-2], segments[n-]])

Attributes:

phi_func	Function \(\phi(r)\)
dphidr_func	Function \(d\phi(r)/dr\)

Methods:

phi(r)	Pair potential.
dphidr(r)	Derivative of pair potential.

phi_func

Function \(\phi(r)\)

dphidr_func

Function \(d\phi(r)/dr\)

phi(r)

Pair potential.

Parameters:

r (array-like) – pair separation

Returns:

phi (ndarray) – Evaluated pair potential.

dphidr(r)

Derivative of pair potential.

This returns the value of \(d \phi(r) / dr\)

Parameters:

r (array-like) – pair separation

Returns:

dphidr (ndarray) – Pair potential values.

class analphipy.potential.Analytic

Bases: PhiBase

Base class for defining analytic potentials.

Parameters:

• r_min (float) – Location of minimum in potential energy.

• phi_min (float, optional) – Value of potential energy at minimum.

• segments (sequence of int) – Integration limits. For n = len(segments) integration will be performed over ranges (segments[0], segments[1]), (segments[1], segments[2]), ..., (segments[n-2], segments[n-]])

Notes

Specific subclasses should set values for r_min, phi_min, and segments, as well as forms for phi and dphidr.

Attributes:

r_min	Position of minimum in \(\phi(r)\)
phi_min	Value of phi at minimum.
segments	Integration limits.

r_min

Position of minimum in \(\phi(r)\)

phi_min

Value of phi at minimum.

segments

Integration limits.

class analphipy.potential.LennardJones(sig=1.0, eps=1.0)

Bases: Analytic

Lennard-Jones potential.

\[\phi(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6\right]\]

Notes

Specific subclasses should set values for r_min, phi_min, and segments, as well as forms for phi and dphidr.

Attributes:

sig	Length parameter \(\sigma\)
eps	Energy parameter \(\epsilon\)

Methods:

phi(r)	Pair potential.
dphidr(r)	Calculate phi and dphi (=-1/r dphi/dr) at particular r.

sig

Length parameter \(\sigma\)

eps

Energy parameter \(\epsilon\)

phi(r)

Pair potential.

Parameters:

r (array-like) – pair separation

Returns:

phi (ndarray) – Evaluated pair potential.

dphidr(r)

Calculate phi and dphi (=-1/r dphi/dr) at particular r.

Parameters:

r (array-like) – pair separation

Returns:

dphidr (ndarray) – Pair potential values.

class analphipy.potential.LennardJonesNM(n=12, m=6, sig=1.0, eps=1.0)

Bases: Analytic

Generalized Lennard-Jones potential

\[\phi(r) = \epsilon \frac{n}{n-m} \left( \frac{n}{m} \right) ^{m / (n-m)} \left[ \left(\frac{\sigma}{r}\right)^n - \left(\frac{\sigma}{r}\right)^m\right]\]

Notes

with parameters n=12 and m=6, this is equivalent to LennardJones.

Attributes:

n	n parameter
m	m parameter
sig	Length parameter
eps	Energy parameter

Methods:

phi(r)	Pair potential.
dphidr(r)	Derivative of pair potential.

n

n parameter

m

m parameter

sig

Length parameter

eps

Energy parameter

phi(r)

Pair potential.

Parameters:

r (array-like) – pair separation

Returns:

phi (ndarray) – Evaluated pair potential.

dphidr(r)

Derivative of pair potential.

This returns the value of \(d \phi(r) / dr\)

Parameters:

r (array-like) – pair separation

Returns:

dphidr (ndarray) – Pair potential values.

class analphipy.potential.Yukawa(z=1.0, sig=1.0, eps=1.0)

Bases: Analytic

Hard core Yukawa potential

\[\begin{split}\phi(r) = \begin{cases} \infty & r \leq \sigma \\ -\epsilon \frac{\sigma}{r} \exp\left[-z (r/\sigma - 1) \right] & r > \sigma \end{cases}\end{split}\]

Notes

Specific subclasses should set values for r_min, phi_min, and segments, as well as forms for phi and dphidr.

Attributes:

z	Interaction parameter
sig	Length parameter
eps	Energy parameter

Methods:

phi(r)

Pair potential.

z

Interaction parameter

sig

Length parameter

eps

Energy parameter

phi(r)

Pair potential.

Parameters:

r (array-like) – pair separation

Returns:

phi (ndarray) – Evaluated pair potential.

class analphipy.potential.HardSphere(sig=1.0)

Bases: Analytic

Hard-sphere pair potential

\[\begin{split}\phi(r) = \begin{cases} \infty & r \leq \sigma \\ 0 & r > \sigma \end{cases}\end{split}\]

Notes

Specific subclasses should set values for r_min, phi_min, and segments, as well as forms for phi and dphidr.

Attributes:

sig	Length parameter

Methods:

phi(r)

Pair potential.

sig

Length parameter

phi(r)

Pair potential.

Parameters:

r (array-like) – pair separation

Returns:

phi (ndarray) – Evaluated pair potential.

class analphipy.potential.SquareWell(sig=1.0, eps=1.0, lam=1.5)

Bases: Analytic

Square-well pair potential

\[\begin{split}\phi(r) = \begin{cases} \infty & r \leq \sigma \\ \epsilon & \sigma < r \leq \lambda \sigma \\ 0 & r > \lambda \sigma \end{cases}\end{split}\]

Notes

Specific subclasses should set values for r_min, phi_min, and segments, as well as forms for phi and dphidr.

Attributes:

sig	Length parameter.
eps	Energy parameter.
lam	Well width parameter.

Methods:

phi(r)

Pair potential.

sig

Length parameter.

eps

Energy parameter.

lam

Well width parameter.

phi(r)

Pair potential.

Parameters:

r (array-like) – pair separation

Returns:

phi (ndarray) – Evaluated pair potential.

class analphipy.potential.CubicTable(bounds, phi_table, phi_left=inf, phi_right=0.0, dphi_left=inf, dphi_right=0.0, *, r_min=None, phi_min=None, segments=None)

Bases: PhiBase

Cubic interpolation table potential.

Parameters:

• r_min (float) – Location of minimum in potential energy.

• phi_min (float, optional) – Value of potential energy at minimum.

• bounds (sequence of float) – the minimum and maximum values of squared pair separation r**2 at which phi_table is evaluated.

• phi_table (array-like) – Values of potential evaluated on even grid of r**2 values.

• segments (sequence of float, optional) – Integration segments. Defaults to sqrt(bounds).

• phi_left, phi_right, dphi_left, dphi_right (float, optional) – Values to set for phi/-1/r dphidr if (left) r < bounds[0] or (right) r > bounds[1].

Attributes:

bounds	Minimum and maximum values of squared pair separation \(r^2\)
phi_table	Values of potential evaluated on even grid of \(r^2\) values.
phi_left	value of phi at left bound (r < bounds[0])
phi_right	value of phi at right bound (r > bounds[1])
dphi_left	value of dphi at left bound
dphi_right	value of dphi at right bound
size	Size of the array (less 1).
smin	Minimum value of s = r**2.
smax	Maximum value of s = r**2.
rsq_table	Value of r**2 where potential is defined.
r_table	Values of r where potential is defined.

Methods:

from_phi(phi, rmin, rmax, ds, **kws)	Create object from callable pair potential function.
phidphi(r)	Values of phi and dphi at r.
phi(r)	Pair potential.
dphidr(r)	Derivative of pair potential.

bounds

Minimum and maximum values of squared pair separation \(r^2\)

phi_table

Values of potential evaluated on even grid of \(r^2\) values.

phi_left

value of phi at left bound (r < bounds[0])

phi_right

value of phi at right bound (r > bounds[1])

dphi_left

value of dphi at left bound

dphi_right

value of dphi at right bound

classmethod from_phi(phi, rmin, rmax, ds, **kws)

Create object from callable pair potential function.

This will evaluate phi at even spacing in s = r**2 space.

Parameters:

• phi (callable()) – pair potential function

• rmin (float) – minimum pair separation r to evaluate at.

• rmax (float) – Maximum pair separation r to evaluate at.

• ds (float) – spaceing in s = r ** 2.

• **kws – Extra arguments to constructor.

Returns:

table (CubicTable)

property size

Size of the array (less 1).

property smin

Minimum value of s = r**2.

property smax

Maximum value of s = r**2.

phidphi(r)

Values of phi and dphi at r.

phi(r)

Pair potential.

Parameters:

r (array-like) – pair separation

Returns:

phi (ndarray) – Evaluated pair potential.

dphidr(r)

Derivative of pair potential.

This returns the value of \(d \phi(r) / dr\)

Parameters:

r (array-like) – pair separation

Returns:

dphidr (ndarray) – Pair potential values.

property rsq_table

Value of r**2 where potential is defined.

property r_table

Values of r where potential is defined.

analphipy.potential.factory(potential_name, rcut=None, lfs=False, cut=False, **kws)

Factory function to construct Phi object by name.

Parameters:

• potential_name ({lj, nm, sw, hs, yk}) – Name of potential.

• rcut (float, optional) – if passed, Construct either and ‘lfs’ or ‘cut’ version of the potential.

• lfs (bool, default False) – If True, construct a linear force shifted potential analphipy.base_potential.PhiLFS.

• cut (bool, default False) – If True, construct a cut potential analphipy.base_potential.PhiCut.

Returns:

phi (analphipy.base_potential.PhiBase subclass) – output potential energy class.

See also

LennardJones, LennardJonesNM, SquareWell, HardSphere, Yukawa