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
Here, the sum extends over all symmetry equivalent clusters
(orbit)
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
where
This problem can be approached by choosing the number of structures
Traditional algorithms [Wal09] then proceed by generating a series of CEs corresponding to different basis set choices, i.e. different values of
icet is agnostic to a particular optimization approach and can in principle be used in conjunction with any of these techniques. For many applications, the trainstation package, the documentation of which can be found here, provides a particular simple interface and many of our examples use this package.