Cluster Expansion Fitting Using PyTorch

The goal of the cluster expansion is to fit energy data to some Hamiltonian (using the Einstein summation convention):

$$\mathcal{H} = N_{\alpha\beta n}u_{\alpha\beta n}$$

where $N_{\alpha\beta n}$ is the number of $\alpha$-$\beta$ bonds in the $n$'th neighbor shell and $u_{\alpha\beta n}$ is the $\alpha$-$\beta$ interaction at the $n$'th nearest neighbor distance. Note that Greek letters index atom types.

We can count the number of bonds by occupation matrices $x_{i\alpha}$, where:

$$x_{i\alpha} = \begin{cases} 1 & \text{site $i$ occupied by $\alpha$} \\ 0 & \text{else} \end{cases}$$

Then, the Hamiltonian is:

$$\mathcal{H} = \frac{1}{2}u_{\alpha\beta n}A_{ijn} x_{i\alpha} x_{j\beta}$$

where $A$ is defined by:

$$A_{ijn} = \begin{cases} 1 & \text{sites $i$ and $j$ are $n$'th nearest neighbors} \\ 0 & \text{else} \end{cases}$$

Note that $u_{\alpha\beta n}$ is intensive, and therefore can be used on a larger lattice with a larger adjacency tensor.

This repository contains a library (cluster_expand.py) and an example mixed Lennard Jones solid calculation using the library (cluster_expansion_example.ipynb).

The library uses PyTorch and cooper to fit $u_{\alpha\beta n}$ in terms of configuration matrices and energies with an input adjacency tensor.

• PyTorch is used to create a Model class and accelerate the tensor operations and optimization
• Cooper is used to constrain the optimization such that $u_{\alpha\beta n} = u_{\beta\alpha n}$