- Several metallic including for example elements from groups 5 (V, Nb, Ta) and 6 (Cr, Mo, W) have a body-centered cubic (BCC) ground state structure.
- The face-centered cubic (FCC) lattice is one of the most common crystal structures for metallic elements including e.g., the late transition metals from group 10 (Ni, Pd, Pt) and 11 (Cu, Ag, Au).
- The construction of force constants requires accurate reference data. Density functional theory (DFT) calculations are one of the most common source for such data.
Optimization and machine learning¶
- Automatic relevance determination regression
- Automatic relevance determination regression (ARDR) is an optimization algorithm provided by scikit-learn
- Compressive sensing
- Compressive sensing (CS), also known as compressive sampling, is an efficient method for constructing sparse solutions for linear systems [CanWak08] [NelOzoRee13].
- Cross validation
- Cross validation (CV) is employed to evaluated the transferability and accuracy of CEs.
- The least absolute shrinkage and selection operator (LASSO) is a method for performing variable selection and regularization in problems in statistics and machine learning.
- The leave one out cross validation (LOOCV) score is one of the most commonly adopted measures for quantifying the basic performance of a CE.
- Kernel ridge regression
- Kernel ridge regression (KRR) combines ridge regression with the kernel trick.
- Regularization, is commonly used in machine learning to combat overfitting and for solving underdetermined systems.
Crystal symmetry and clusters¶
- Crystal symmetry operation
- A crystal symmetry operation for a specific lattice means that the lattice is invariant under this operation. An operation comprises translational and rotational components.
- A cluster is defined as a set of points on a lattice.
- Cluster size
- The size of a cluster (commonly refered to as the cluster radius) is defined as the average distance to the geometrical center of the cluster.
- Cluster space
- The set of clusters into which a structure can be decomposed.
- Cutoffs define the longest allowed distance between two atoms in a cluster for each order.
- An orbit is defined as a set of symmetry equivalent clusters.
- Cluster expansion
- Cluster expansions provide a mapping between a configuration and a property of interest that can be many orders of magnitude faster than the underlying reference calculations from e.g., DFT.
- The parameters of a CE are usually referred to as effective cluster interactions (ECIs).
- Monte Carlo (MC) simulations are an effective method for sampling a multi-dimensional space.