Priyanka Ghosh

Generating angular symmetry functions for neural network potentials involves creating a set of features that describe the local environment of atoms in a molecular or solid system. These features are essential for accurately capturing the geometrical and electronic properties of the system. Here’s a step-by-step guide on how to generate angular symmetry functions within a defined range:

### 1. **Understanding Angular Symmetry Functions**

Angular symmetry functions are designed to encode the angles formed by triplets of atoms. They typically depend on the distances between atoms and the angles between them. Common forms include:

- **Cosine functions** that depend on angles.

- **Gaussian functions** that depend on distances.

### 2. **Define the Parameters**

- **Atom Types**: Identify the types of atoms in your system.

- **Distance Range**: Define the range for the distances you want to consider (e.g., minimum and maximum distances).

- **Angle Range**: Define the range for the angles (e.g., from \(0^\circ\) to \(180^\circ\)).

- **Cutoff Radius**: Set a cutoff radius beyond which the interactions are considered negligible.

### 3. **Generate Pairwise Distances**

For each atom, compute the distances to its neighbors. You can use a simple nested loop over the atoms to calculate the distances.

```python

import numpy as np

def compute_distances(positions):

distances = []

for i in range(len(positions)):

for j in range(i + 1, len(positions)):

dist = np.linalg.norm(positions[i] - positions[j])

distances.append((i, j, dist))

return distances

```

### 4. **Calculate Angles**

For each triplet of atoms, calculate the angles formed. You can use the law of cosines to find the angles based on the distances.

```python

def compute_angles(distances):

angles = []

for (i, j, d_ij) in distances:

# Assume you have a third atom k to form a triplet

for (l, m, d_jk) in distances: # Replace with actual logic to find triplet

# Calculate angle using cosine rule

# Use indices and distances to calculate the angle

# Example: angle_ijk = np.arccos((d_ij^2 + d_jk^2 - d_ik^2) / (2 * d_ij * d_jk))

pass

return angles

```

### 5. **Define Angular Symmetry Functions**

Implement the angular symmetry functions based on the defined ranges. A common form is:

\[

f_{\theta}(r_{ij}, r_{ik}, \theta_{ijk}) = \exp\left(-\frac{(r_{ij} - r_{s})^2}{2\sigma^2}\right) \cdot \exp\left(-\frac{(r_{ik} - r_{s})^2}{2\sigma^2}\right) \cdot \cos^n(\theta_{ijk})

\]

Where:

- \( r_{ij} \) and \( r_{ik} \) are the distances,

- \( \theta_{ijk} \) is the angle between the atoms,

- \( r_s \) is a reference distance,

- \( \sigma \) controls the width of the Gaussian.

### 6. **Generate the Functions Over Defined Ranges**

Loop over all the distances and angles within the defined ranges to generate the symmetry functions.

```python

def generate_symmetry_functions(distances, angles, r_s, sigma, n):

symmetry_functions = []

for (i, j, d_ij) in distances:

for (k, angle_ijk) in angles:

if d_ij < cutoff_radius: # Apply cutoff

f = np.exp(-(d_ij - r_s)**2 / (2 * sigma**2)) * np.cos(angle_ijk)**n

symmetry_functions.append(f)

return symmetry_functions

```

7. **Normalization and Scaling**

Normalize the symmetry functions if necessary, depending on the neural network architecture you plan to use. This can improve the convergence of training.

### 8. **Integration with Neural Network Potentials**

Feed the generated symmetry functions into the neural network model that predicts potential energy or forces. Ensure your neural network framework can handle the input format.

Partial credit ai