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Chapter 2
File formats and conventions in
BigDFT
2.1
2.1.1
The basis set
One dimensional functions
The maximally symmetric Daubechies family of degree 16 is used to represent the
Kohn-Sham orbitals. The two fundamental functions of this family, the scaling
function φ and the wavelet ψ are shown in Fig. 2.1.1 for the one-dimensional case.
To form a basis set these functions have to be centered on the nodes of a regular
grid.
2.1.2
Wavelet basis sets in three dimensions
A 3-dim wavelet basis is made by products of 1-dim functions. 1 scaling function
and 7 wavelets can be centered on the nodes (i,j,k) of a regular 3-dim Cartesian
grid. They all are products of 1-dim scaling functions and wavelets:
φi, j,k (x, y, z) = φ(x − i)φ(y − j)φ(z − k)
ψ1i, j,k (x, y, z) = φ(x − i)φ(y − j)ψ(z − k)
ψ2i, j,k (x, y, z) = φ(x − i)ψ(y − j)φ(z − k)
ψ3i, j,k (x, y, z) = φ(x − i)ψ(y − j)ψ(z − k)
ψ4i, j,k (x, y, z) = ψ(x − i)φ(y − j)φ(z − k)
ψ5i, j,k (x, y, z) = ψ(x − i)φ(y − j)ψ(z − k)
ψ6i, j,k (x, y, z) = ψ(x − i)ψ(y − j)φ(z − k)
ψ7i, j,k (x, y, z) = ψ(x − i)ψ(y − j)ψ(z − k)
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