List of basis sets used in Quantum Chemistry and Condensed Matter Physics

The molecular orbitals or the crystal wave functions are expanded using following basis functions.

Slayter-type Orbitals (STO)
Gaussian-type orbitals (GTOs)
Numerical atomic functions
Finite elements (FEM)
Wavelets
Plane waves
Spherical waves

The GTOs and planes waves are widely used in many codes. The GTO originated from Gaussian code by John A Pople. Gaussian basis functions are the reason for the initial success in chemistry. Plane wave basis functions are widely used in condensed matter physics.