Cluster expansions

This page provides a short introduction to cluster expansions. For more extensive descriptions, please consult for example [SanDucGra84], [Wal09] or [AngMunRah19]. A simpler introduction to the subject can be found in the first part of this tutorial.

Formalism

In the following, we are concerned with configurations corresponding to a distribution of \(M\) different species over \(N\) sites that can be written as a vector \(\boldsymbol{\sigma}\), whose elements can assume \(M\) different values, e.g., from \(S=\{-m, -m+1, \ldots m-1, m\}\) where \(M=2m\) (for even \(M\)) or \(M=2m+1\) (for odd \(M\)) [SanDucGra84]. One now seeks to represent a property \(Q\) of the system, such as the total energy, as a function of \(\boldsymbol{\sigma}\), i.e. \(Q = f(\boldsymbol{\sigma})\). To this end, one can construct a complete orthonormal basis of cluster functions \(\Gamma_{\alpha}(\boldsymbol{\sigma})\) [SanDucGra84] [San10], which allows one to express \(Q\) in the form [Wal09]

\[Q = \sum_\alpha m_\alpha J_\alpha \left<\Gamma_{\alpha'}(\boldsymbol{\sigma})\right>_{\alpha}.\]

Here, the sum extends over all symmetry equivalent clusters (orbit) \(\alpha\), \(m_{\alpha}\) denotes the multiplicity, whereas the coefficients \(J_{\alpha}\) are the so-called effective cluster interactions (ECIs). The last term in the above expression represents the average over cluster functions \(\Gamma_{\alpha}(\boldsymbol{\sigma})\) belonging to symmetry equivalent clusters. The cluster functions themselves are obtained as a product over point functions \(\Theta\). In icet, the point functions from [Wal09] are used:

\[\begin{split}\Theta_{n}(\sigma_p) &= \begin{cases} 1 & \qquad \text{if $n=0$} \\ - \cos\left(\pi(n+1)\sigma_p/M\right) & \qquad \text{if $n$ is odd} \\ -\sin\left(\pi n \sigma_p/M\right) & \qquad \text{if $n$ is even}, \end{cases}\end{split}\]

These point functions form an orthogonal set over all possible occupation numbers. For further details, please consult [Wal09] or [AngMunRah19].

In icet, the formalism is handled internally by the ClusterSpace class.

Cluster expansion construction

The task of training a CE can be formally written as a linear problem

\[\mathbf{\bar{\Pi}} \boldsymbol{J} = \boldsymbol{Q}\]

where \(\boldsymbol{Q}\) is the vector of target properties, the vector \(\boldsymbol{J}\) represents the ECIs, and \(\mathbf{\bar{\Pi}}\) is a matrix that is obtained by stacking the vectors that represent each structure (also referred to in this documentation as cluster vectors) of the training set. Here, the multiplicities \(m_{\alpha}\) have been included in \(\mathbf{\bar{\Pi}}\).

This problem can be approached by choosing the number of structures \(n_{\boldsymbol{Q}}\) (and thus the dimensionality of \(\boldsymbol{Q}\)), to be (much) larger than the number of ECIs \(n_{\boldsymbol{J}}\) (and thus the dimensionality of \(\boldsymbol{J}\), (\(n_{\boldsymbol{Q}}>n_{\boldsymbol{J}}\)). The set of equations is thus overdetermined. The optimal set of ECIs for fixed training set and cluster function basis is then obtained by minimizing the \(l_2\)-norm of \(\mathbf{\bar{\Pi}} \boldsymbol{J} - \boldsymbol{Q}\)

\[\boldsymbol{J}_{\text{opt}} = \arg\min_{\boldsymbol{J}} \left\{ || \mathbf{\bar{\Pi}} \boldsymbol{J} - \boldsymbol{Q} ||_2 \right\}.\]

Common algorithms [Wal09] then proceed by generating a series of CEs corresponding to different basis set choices, i.e. different values of \(n_{\boldsymbol{J}}\). By comparing the performance of each CE by means of a cross validation (CV) score the best performing CE is selected.

An alternative approach is to choose the number of ECIs larger than the number of structures (\(n_{\boldsymbol{J}}>n_{\boldsymbol{Q}}\)). The resulting problem is then underdetermined, but can be solved by means of compressive sensing [CanWak08] [NelHarZho13] [NelOzoRee13]. In practice, one obtains a sparse solution, i.e., a solution in which many ECIs are zero.

icet provides extensive functionality for the present task, see further under Optimizers.