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For questions about the usage and construction of electronic structure basis sets.

In linear algebra, a basis B is a set of vectors in a vector space V for which every element of V can be written as a linear combination of elements of B. This concept is used in quantum electronic structure theory to express the wavefunction as a linear combination of simpler functions.

The choice of functions to use can vary based on the application, with Gaussians basis sets being useful for studying molecular/localized interactions and plane wave basis sets seeing use for studying extended structures and materials. While the functions used can be optimized on fly for a given system, it is more often the case that predetermined functions are used that have been optimized for use with particular constituent atoms or moieties. These are available both within most electronic structure programs, as well as in online repositories (e.g. Basis Set Exchange)

In the language of materials modeling, a complete basis is the analog of the mathematical definition of a basis. For materials and molecules, a complete basis would need to be effectively infinite in size, so in practice finite sized "basis sets" are used which only span a subspace of the whole possible space of wavefunction solutions. The quality of the resulting wavefunction is dependent on how large/close to complete the basis set is.