Useful formulas

This section holds a collection of formulae, which are helpful when working with DFTK and plane-wave DFT in general. See also Notation and conventions for a description of the conventions used in the equations.

Fourier transforms

• The Fourier transform is

  \[\widehat{f}(q) = \int_{{\mathbb R}^{3}} e^{-i q \cdot x} f(x) dx\]

• Fourier transforms of centered functions: If $f({x}) = R(x) Y_l^m(x/|x|)$, then

  \[\begin{aligned} \hat f( q) &= \int_{{\mathbb R}^3} R(x) Y_{l}^{m}(x/|x|) e^{-i {q} \cdot {x}} d{x} \\ &= \sum_{l = 0}^\infty 4 \pi i^l \sum_{m = -l}^l \int_{{\mathbb R}^3} R(x) j_{l'}(|q| |x|)Y_{l'}^{m'}(-q/|q|) Y_{l}^{m}(x/|x|) Y_{l'}^{m'\ast}(x/|x|) d{x} \\ &= 4 \pi Y_{l}^{m}(-q/|q|) i^{l} \int_{{\mathbb R}^+} r^2 R(r) \ j_{l}(|q| r) dr\\ &= 4 \pi Y_{l}^{m}(q/|q|) (-i)^{l} \int_{{\mathbb R}^+} r^2 R(r) \ j_{l}(|q| r) dr \end{aligned}\]

  This also holds true for real spherical harmonics.

Spherical harmonics

• Plane wave expansion formula

  \[e^{i {q} \cdot {r}} = 4 \pi \sum_{l = 0}^\infty \sum_{m = -l}^l i^l j_l(|q| |r|) Y_l^m(q/|q|) Y_l^{m\ast}(r/|r|)\]

• Spherical harmonics orthogonality

  \[\int_{\mathbb{S}^2} Y_l^{m*}(r)Y_{l'}^{m'}(r) dr = \delta_{l,l'} \delta_{m,m'}\]

  This also holds true for real spherical harmonics.

• Spherical harmonics parity

  \[Y_l^m(-r) = (-1)^l Y_l^m(r)\]

  This also holds true for real spherical harmonics.