Pair Potentials¶

Base Class¶

class graph_pes.models.PairPotential(cutoff)[source]¶

Bases: GraphPESModel, ABC

An abstract base class for PES models that calculate total energy as a sum over pairwise interactions:

\[E = \sum_{i, j} V(r_{ij}, Z_i, Z_j)\]

where $r_{ij}$ is the distance between atoms $i$ and $j$, and $Z_i$ and $Z_j$ are their atomic numbers. This can be recast as a sum over local energy contributions, $E = \sum_i \varepsilon_i$, according to:

\[\varepsilon_i = \frac{1}{2} \sum_j V(r_{ij}, Z_i, Z_j)\]

Subclasses should implement interaction().

abstract interaction(r, Z_i, Z_j)[source]¶

Compute the interactions between pairs of atoms, given their distances and atomic numbers.

Parameters:

r (Tensor) – The pair-wise distances between the atoms.
Z_i (Tensor) – The atomic numbers of the central atoms.
Z_j (Tensor) – The atomic numbers of the neighbours.

Returns:

V – The pair-wise interactions.

Return type:

torch.Tensor

Available Pair Potentials¶

class graph_pes.models.LennardJones(cutoff=5.0, epsilon=0.1, sigma=1.0, shift=False)[source]¶

Bases: PairPotential

A pair potential with an interaction term of the form:

\[V_{LJ}(r) = 4 \varepsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} \right]\]

where $r_{ij}$ is the distance between atoms $i$ and $j$, and $\varepsilon$ and $\sigma$ are strictly positive paramters that control the depth and width of the potential well.

To ensure a continous energy surface, the final interaction is shifted by the value of the potential at the cutoff to remove the discontinuity:

\[V(r) = V_{LJ}(r) - V_{LJ}(r_c)\]

Warning

The derivative of the potential is not continuous at the cutoff, irrespective of whether the potential is shifted or not. This leads to discontinuities in this models’s forces. Consider wrapping this potential in SmoothedPairPotential to obtain a smooth and continuous potential and force field.

Parameters:

cutoff (float) – The cutoff radius.
epsilon (float) – The maximum depth of the potential.
sigma (float) – The distance at which the potential is zero.

Example

from graph_pes.utils.analysis import dimer_curve
from graph_pes.models import LennardJones

dimer_curve(LennardJones(), system="H2", rmax=3.5)

static from_ase( sigma=1.0, epsilon=1.0, rc=None, ro=None, smooth=False, )[source]¶

Create a LennardJones potential with an interface identical to the ASE ase.calculators.lj.LennardJones calculator.

Please refer to the ASE documentation for more details.

Parameters:

sigma (float) – The distance at which the potential is zero.
epsilon (float) – The maximum depth of the potential.
rc (float | None) – The cutoff radius. If not given, the default value is 3 * sigma.
ro (float | None) – The radius at which the smooth cutoff function begins.

class graph_pes.models.ZBLCoreRepulsion(cutoff=5.0, trainable=False)[source]¶

Bases: PairPotential

The ZBL repulsive potential:

\[V(r, Z_i, Z_j) = \frac{e^2}{4 \pi \epsilon_0} \cdot \frac{Z_i Z_j}{r} \cdot \phi(r / a)\]

where $\phi(x)$ is a dimensionless function given by:

\[\phi(x) = 0.1818e^{-3.2x} + 0.5099e^{-0.9423x} + 0.2802e^{-0.4029x} + 0.02817e^{-0.2016x}\]

and $a$ is the screening length:

\[a = \frac{\lambda_p \cdot a_0}{Z_i^{\lambda_e} + Z_j^{\lambda_e}}\]

where $a_0$ is the Bohr radius, $\lambda_p = 0.8854$ and $\lambda_e = 0.23$.

Parameters:

cutoff (float, optional) – The cutoff radius for the potential. Default is DEFAULT_CUTOFF.
trainable (bool, optional) – If True, the pre-factor ($\lambda_p$) and exponent ($\lambda_e$) of the screening length are trainable parameters. Default is False.

Example

import matplotlib.pyplot as plt
from graph_pes.utils.analysis import dimer_curve
from graph_pes.models import ZBL

dimer_curve(ZBL(), system="H2", rmin=0.1, rmax=3.5)
plt.xlim(0, 3.5)
plt.ylim(0.01, 100)
plt.yscale("log")

class graph_pes.models.Morse(cutoff=5.0, D=0.1, a=5.0, r0=1.5)[source]¶

Bases: PairPotential

A pair potential of the form:

\[V(r_{ij}, Z_i, Z_j) = V(r_{ij}) = D (1 - e^{-a(r_{ij} - r_0)})^2\]

where $r_{ij}$ is the distance between atoms $i$ and $j$, and $D$, $a$ and $r_0$ are strictly positive parameters that control the depth, width and center of the potential well respectively.

Parameters:

D (float) – The maximum depth of the potential.
a (float) – A measure of the width of the potential.
r0 (float) – The distance at which the potential is at its minimum.

Example

from graph_pes.utils.analysis import dimer_curve
from graph_pes.models import Morse

dimer_curve(Morse(), system="H2", rmax=3.5)

class graph_pes.models.LennardJonesMixture(cutoff=5.0, modulate_distances=True)[source]¶

Bases: PairPotential

An extension of the simple LennardJones potential to account for multiple atomic species.

Each element is associated with a unique pair of parameters, $sigma_i$ and $varepsilon_i$, which control the width and depth of the potential well for that element.

Interactions between atoms of different elements are calculated using effective parameters $\sigma_{i\neq j}$ and $\varepsilon_{i \neq j}$, which are calculated as:

$\sigma_{i\neq j} = \nu_{ij} \cdot (\sigma_i + \sigma_j) / 2$
$\varepsilon_{i\neq j} = \zeta_{ij} \cdot \sqrt{\varepsilon_i \cdot \varepsilon_j}$

where $\nu_{ij}$ is a mixing parameter that controls the width of the potential well.

For more details, see wikipedia.

class graph_pes.models.SmoothedPairPotential(potential, smoothing_onset=None)[source]¶

Bases: PairPotential

A wrapper around a PairPotential that applies a smooth cutoff function, $f$ = SmoothOnsetEnvelope, to the potential to ensure a continous energy surface:

\[V(r) = f(r, r_o, r_c) \cdot V_{\text{wrapped}}(r)\]

where

\[\begin{split}f(r, r_o, r_c) = \begin{cases} \hfill 1 \hfill & \text{if } r < r_o \\ \frac{(r_c - r)^2 (r_c + 2r - 3r_o)}{(r_c - r_o)^3} & \text{if } r_o \leq r < r_c \\ \hfill 0 \hfill & \text{if } r \geq r_c \end{cases}\end{split}\]

and $r_o$ and $r_c$ are the onset and cutoff radii respectively.

Parameters:

potential (PairPotential) – The potential to wrap.
smoothing_onset (float | None) – The radius at which the smooth cutoff function begins. Defaults to $2r_c / 3$.