# md.pair.aniso¶

Overview

 ALJ Anistropic LJ force. AnisotropicPair Base class anisotropic pair force. Dipole Screened dipole-dipole pair forces. GayBerne Gay-Berne anisotropic pair force.

Details

Anisotropic pair forces.

Anisotropic pair force classes apply a force, torque, and virial on every particle in the simulation state commensurate with the potential energy:

$U_\mathrm{pair,total} = \frac{1}{2} \sum_{i=0}^\mathrm{N_particles-1} \sum_{j \ne i, (i,j) \notin \mathrm{exclusions}} U_\mathrm{pair}(r_{ij}, \mathbf{q}_i, \mathbf{q}_j)$

AnisotropicPair applies cuttoffs, exclusions, and assigns per particle energies and virials in the same manner as hoomd.md.pair.Pair

AnisotropicPair does not support the 'xplor' shifting mode or the r_on parameter.

class hoomd.md.pair.aniso.ALJ(nlist, default_r_cut=None)

Anistropic LJ force.

Parameters:

ALJ computes the Lennard-Jones force between anisotropic particles as described in Ramasubramani, V. et al. 2020, using the formula:

$U(r, r_c) = U_0(r) + U_c(r_c)$

The first term is the central interaction $$U_0$$, the standard center-center interaction between two Lennard-Jones particles with center-center distance $$r$$. The second term is the contact interaction $$U_c$$, computed from the smallest distance between the surfaces of the two shapes $$r_c$$. The central and contact interactions are defined as follows:

\begin{align}\begin{aligned}&U_0(r) = 4 \varepsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} \right]\\&U_c(r_c) = 4 \varepsilon_c(\varepsilon) \left[ \left( \frac{\sigma_c}{r_c} \right)^{12} - \left( \frac{\sigma_c}{r_c} \right)^{6} \right]\end{aligned}\end{align}

where $$\varepsilon$$ (epsilon) affects strength of both the central and contact interactions, $$\varepsilon_c$$ is an energy coefficient set proportional to $$\varepsilon$$ to preserve the shape, $$\sigma$$ is the interaction distance of the central term computed as the average of $$\sigma_i$$ (sigma_i) and $$\sigma_j$$ (sigma_j). Lastly, ALJ uses the contact ratios $$\beta_i$$ (contact_ratio_i) and $$\beta_j$$ (contact_ratio_j) to compute the contact sigma $$\sigma_c$$ as follows:

\begin{align}\begin{aligned}\sigma_c &= \frac{1}{2} \left[\sigma_{ci} + \sigma_{cj} \right]\\\sigma_{ci} &= \beta_i \cdot \sigma_i\\\sigma_{cj} &= \beta_j \cdot \sigma_j\end{aligned}\end{align}

The total potential energy is therefore the sum of two interactions, a central Lennard-Jones potential and a radially-shifted Lennard-Jones potential where the shift is anisotropic and depends on the extent of the shape in each direction.

Each term has an independent cutoff at which the energy is set to zero. The behavior of these cutoffs is dependent on whether a user requires LJ or Weeks-Chandler-Anderson (WCA)-like (repulsive-only) behavior. This behavior is controlled using the alpha parameter, which can take on the following values:

Set alpha based on range of the center-center and contact-contact interactions.

center-center repulsive only

center-center full-range

contact-contact repulsive only

alpha = 0

alpha = 1

contact-contact full-range

alpha = 2

alpha = 3

For polytopes, computing interactions using a single contact point leads to significant instabilities in the torques because the contact point can jump from one end of a face to another in an arbitrarily small time interval. To ameliorate this, the ALJ potential performs a local averaging over all the features associated with the closest simplices on two polytopes. This averaging can be turned off by setting the average_simplices key for the type pair to False.

Specifying only rounding_radii creates an ellipsoid, while specifying only vertices creates a convex polytope (set vertices and faces to empty lists to create the ellipsoid).

Important

The repulsive part of the contact interaction $$U_c(r_c)$$ prevents two ALJ particles from approaching closely, effectively rounding the shape by a radius $$\sigma_c$$. For this reason, the shape written by type_shapes includes the rounding due to rounding_radii and that due to $$\sigma_c$$.

Choosing r_cut:

Set r_cut for each pair of particle types so that ALJ can compute interactions for all possible relative placements and orientations of the particles. The farthest apart two particles can be while still interacting depends on the value of alpha.

In the following list, the first argument to the $$\max$$ function is for the center-center interaction. The second argument is for the contact-contact interaction, where $$R_i$$ is the radius of the shape’s minimal origin-centered bounding sphere of the particle with type $$i$$.

Let $$\lambda_{min} = 2^{1/6}$$ be the position of the potential energy minimum of the Lennard-Jones potential and $$\lambda_{cut}^{attractive}$$ be a larger value, such as 2.5 (typically used in isotropic LJ systems).

• For alpha=0:

$\begin{split}r_{\mathrm{cut},ij} = \max \bigg( & \frac{\lambda_{min}}{2} (\sigma_i + \sigma_j), \\ & R_i + R_j + R_{\mathrm{rounding},i} + R_{\mathrm{rounding},j} + \frac{\lambda_{min}}{2} (\beta_i \cdot \sigma_i + \beta_j \cdot \sigma_j) \bigg)\end{split}$
• For alpha=1:

$\begin{split}r_{\mathrm{cut},ij} = \max \bigg( & \frac{\lambda_{cut}^{attractive}}{2} (\sigma_i + \sigma_j), \\ & R_i + R_j + R_{\mathrm{rounding},i} + R_{\mathrm{rounding},j}+ \frac{\lambda_{min}}{2} (\beta_i \cdot \sigma_i + \beta_j \cdot \sigma_j) \bigg)\end{split}$
• For alpha=2:

$\begin{split}r_{\mathrm{cut},ij} = \max \bigg( & \frac{\lambda_{min}}{2} (\sigma_i + \sigma_j)), \\ & R_i + R_j + R_{\mathrm{rounding},i} + R_{\mathrm{rounding},j} + \frac{\lambda_{cut}^{attractive}}{2} (\beta_i \cdot \sigma_i + \beta_j \cdot \sigma_j) \bigg)\end{split}$
• For alpha=3:

$\begin{split}r_{\mathrm{cut},ij} = \max \bigg( & \frac{\lambda_{cut}^{attractive}}{2} (\sigma_i + \sigma_j), \\ & R_i + R_j + R_{\mathrm{rounding},i} + R_{\mathrm{rounding},j} + \frac{\lambda_{cut}^{attractive}}{2} (\beta_i \cdot \sigma_i + \beta_j \cdot \sigma_j) \bigg)\end{split}$

Warning

Changing dimension in a simulation will invalidate this force and will lead to error or unrealistic behavior.

params

The ALJ potential parameters. The dictionary has the following keys:

• epsilon (float, required) - base energy scale $$\varepsilon$$ $$[energy]$$.

• sigma_i (float, required) - the insphere diameter of the first particle type, $$\sigma_i$$ $$[length]$$.

• sigma_j (float, required) - the insphere diameter of the second particle type, $$\sigma_j$$ $$[length]$$.

• alpha (int, required) - Integer 0-3 indicating whether or not to include the attractive component of the interaction (see above for details).

• contact_ratio_i (float, optional) - $$\beta_i$$, the ratio of the contact sphere diameter of the first type with sigma_i. Defaults to 0.15.

• contact_ratio_j (float, optional) - $$\beta_j$$, the ratio of the contact sphere diameter of the second type with sigma_j. Defaults to 0.15.

• average_simplices (bool, optional) - Whether to average over simplices. Defaults to True. See class documentation for more information.

Type: hoomd.data.typeparam.TypeParameter [tuple [particle_types, particle_types], dict]

Note

While the evaluation of the potential is symmetric with respect to the potential parameter labels i and j, the parameters which physically represent a specific particle type must appear in all sets of pair parameters which include that particle type.

shape

The shape of a given type. The dictionary has the following keys per type:

Type: hoomd.data.typeparam.TypeParameter [particle_types, dict]

Example:

nl = hoomd.md.nlist.Cell(buffer=0.4)
alj = hoomd.md.pair.aniso.ALJ(nl, r_cut=2.5)

cube_verts = [(-0.5, -0.5, -0.5),
(-0.5, -0.5, 0.5),
(-0.5, 0.5, -0.5),
(-0.5, 0.5, 0.5),
(0.5, -0.5, -0.5),
(0.5, -0.5, 0.5),
(0.5, 0.5, -0.5),
(0.5, 0.5, 0.5)];

cube_faces = [[0, 2, 6],
[6, 4, 0],
[5, 0, 4],
[5,1,0],
[5,4,6],
[5,6,7],
[3,2,0],
[3,0,1],
[3,6,2],
[3,7,6],
[3,1,5],
[3,5,7]]

alj.params[("A", "A")] = dict(epsilon=2.0,
sigma_i=1.0,
sigma_j=1.0,
alpha=1,
)
alj.shape["A"] = dict(vertices=cube_verts,
faces=cube_faces)


The following example shows how to easily get the faces, with vertex indices properly ordered, for a shape with known vertices by using the coxeter package:

Example:

import coxeter

nl = hoomd.md.nlist.Cell(buffer=0.4)
alj = hoomd.md.pair.aniso.ALJ(nl, r_cut=2.5)

cube_verts = [[-0.5, -0.5, -0.5],
[-0.5, -0.5, 0.5],
[-0.5, 0.5, -0.5],
[-0.5, 0.5, 0.5],
[0.5, -0.5, -0.5],
[0.5, -0.5, 0.5],
[0.5, 0.5, -0.5],
[0.5, 0.5, 0.5]]

cube = coxeter.shapes.ConvexPolyhedron(cube_verts)

alj.params[("A", "A")] = dict(epsilon=2.0,
sigma_i=1.0,
sigma_j=1.0,
alpha=1,
)
alj.shape["A"] = dict(vertices=cube.vertices,
faces=cube.faces)

property type_shapes

The shape specification for use with GSD files for visualization.

This is not meant to be used for access to shape information in Python. See the attribute shape for programatic assess. Use this property to log shape for visualization and storage through the GSD file type.

(Loggable: category=”object”)

Type:

list [dict [str, any]]

class hoomd.md.pair.aniso.AnisotropicPair(nlist, default_r_cut=None, mode='none')

Bases: Pair

Base class anisotropic pair force.

AnisotropicPair is the base class for all anisotropic pair forces.

Warning

This class should not be instantiated by users. The class can be used for isinstance or issubclass checks.

Parameters:
class hoomd.md.pair.aniso.Dipole(nlist, default_r_cut=None)

Screened dipole-dipole pair forces.

Parameters:

Dipole computes the (screened) interaction between pairs of particles with dipoles and electrostatic charges:

\begin{align}\begin{aligned}U &= U_{dd} + U_{de} + U_{ee}\\U_{dd} &= A e^{-\kappa r} \left(\frac{\vec{\mu_i}\cdot\vec{\mu_j}}{r^3} - 3\frac{(\vec{\mu_i}\cdot \vec{r_{ji}}) (\vec{\mu_j}\cdot \vec{r_{ji}})} {r^5} \right)\\U_{de} &= A e^{-\kappa r} \left(\frac{(\vec{\mu_j}\cdot \vec{r_{ji}})q_i}{r^3} - \frac{(\vec{\mu_i}\cdot \vec{r_{ji}})q_j}{r^3} \right)\\U_{ee} &= A e^{-\kappa r} \frac{q_i q_j}{r}\end{aligned}\end{align}

Note

All units are documented electronic dipole moments. However, Dipole can also be used to represent magnetic dipoles.

Example:

nl = nlist.Cell()
dipole = md.pair.ansio.Dipole(nl, default_r_cut=3.0)
dipole.params[('A', 'B')] = dict(A=1.0, kappa=4.0)
dipole.mu['A'] = (4.0, 1.0, 0.0)

params

The dipole potential parameters. The dictionary has the following keys:

• A (float, required) - $$A$$ - electrostatic energy scale (default: 1.0) $$[\mathrm{energy} \cdot \mathrm{length} \cdot \mathrm{charge}^{-2}]$$

• kappa (float, required) - $$\kappa$$ - inverse screening length $$[\mathrm{length}^{-1}]$$

Type: TypeParameter [tuple [particle_type, particle_type], dict]

mu

$$\mu$$ - the magnetic magnitude of the particle local reference frame as a tuple (i.e. $$(\mu_x, \mu_y, \mu_z)$$) $$[\mathrm{charge} \cdot \mathrm{length}]$$.

Type: TypeParameter [particle_type, tuple [float, float, float ]]

class hoomd.md.pair.aniso.GayBerne(nlist, default_r_cut=None, mode='none')

Gay-Berne anisotropic pair force.

Parameters:

GayBerne computes the Gay-Berne anisotropic pair force on every particle in the simulation state. This version of the Gay-Berne force supports identical pairs of uniaxial ellipsoids, with orientation-independent energy-well depth. The potential comes from the following paper Allen et. al. 2006.

$\begin{split}U(\vec r, \vec e_i, \vec e_j) = \begin{cases} 4 \varepsilon \left[ \zeta^{-12} - \zeta^{-6} \right] & \zeta < \zeta_{\mathrm{cut}} \\ 0 & \zeta \ge \zeta_{\mathrm{cut}} \\ \end{cases}\end{split}$

where

\begin{align}\begin{aligned}\zeta &= \left(\frac{r-\sigma+\sigma_{\mathrm{min}}} {\sigma_{\mathrm{min}}}\right),\\\sigma^{-2} &= \frac{1}{2} \hat{\vec{r}} \cdot \vec{H^{-1}} \cdot \hat{\vec{r}},\\\vec{H} &= 2 \ell_\perp^2 \vec{1} + (\ell_\parallel^2 - \ell_\perp^2) (\vec{e_i} \otimes \vec{e_i} + \vec{e_j} \otimes \vec{e_j}),\end{aligned}\end{align}

and $$\sigma_{\mathrm{min}} = 2 \min(\ell_\perp, \ell_\parallel)$$.

The cut-off parameter $$r_{\mathrm{cut}}$$ is defined for two particles oriented parallel along the long axis, i.e. $$\zeta_{\mathrm{cut}} = \left(\frac{r-\sigma_{\mathrm{max}} + \sigma_{\mathrm{min}}}{\sigma_{\mathrm{min}}}\right)$$ where $$\sigma_{\mathrm{max}} = 2 \max(\ell_\perp, \ell_\parallel)$$ .

The quantities $$\ell_\parallel$$ and $$\ell_\perp$$ denote the semi-axis lengths parallel and perpendicular to particle orientation.

Example:

nl = nlist.Cell()
gay_berne = md.pair.aniso.GayBerne(nlist=nl, default_r_cut=2.5)
gay_berne.params[('A', 'A')] = dict(epsilon=1.0, lperp=0.45, lpar=0.5)
gay_berne.r_cut[('A', 'B')] = 2 ** (1.0 / 6.0)

params

The Gay-Berne potential parameters. The dictionary has the following keys:

Type: TypeParameter [tuple [particle_type, particle_type], dict]

property type_shapes

Get all the types of shapes in the current simulation.

Example

>>> gay_berne.type_shapes
[{'type': 'Ellipsoid', 'a': 1.0, 'b': 1.0, 'c': 1.5}]

Returns:

A list of dictionaries, one for each particle type in the system.

(Loggable: category=”object”)