# Gaussian Plane Wave

The primary basis in CP2K consists of Gaussian Type Orbital (GTO) functions. To take advantage of the efficient FFT algorithm for solving the Poisson equation, the electronic density must first be transferred from the GTO representation to a regular grid. This is referred to as collocation. After solving the Poisson equation the obtained electrostatic potential has to be transferred back into the GTO basis, which is referred to as integration. This ”opportunistic” switching between a Gaussian and a plane wave representation is the core idea of the Gaussian and Plane Waves (GPW) method. Its high-level operations are illustrated in the following flowchart:

            block-beta
    columns 5
    a["el. Density\n(Gaussian Basis)"]
    space
    b["el. Density\n(Regular Grid)"]
    space
    c["el. Density\n(Plane Waves)"]

    space space space space space

    f["el. Potenial\n(Gaussian Basis)"]
    space
    e["el. Potenial\n(Regular Grid)"]
    space
    d["el. Potenial\n(Plane Waves)"]

    a -- "Collocate"        --> b
    b -- "FFT"              --> c
    c -- "Poisson Solver"   --> d
    d -- "FFT<sup>-1</sup>" --> e
    e -- "Integrate"        --> f

## See also

- https://www.cp2k.org/gpw

- https://www.cp2k.org/quickstep

- VandeVondele2005