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

  $$\hat{f}(q)={\int }_{{\mathbb{R}}^3}{e}^{-iq\cdot x}f(x)dx$$

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

  $$\begin{array}{rl}\hat{f}(q)& ={\int }_{{\mathbb{R}}^3}R(x)Y_l^m(x/|x|)e^{-iq\cdot x}dx\\ & =\sum _{l=0}^{\mathrm{\infty }}4\pi i^l\sum _{m=-l}^l{\int }_{{\mathbb{R}}^3}R(x)j_{l^{\prime }}(|q||x|)Y_{l^{\prime }}^{m^{\prime }}(-q/|q|)Y_l^m(x/|x|)Y_{l^{\prime }}^{m^{\prime }\ast }(x/|x|)dx\\ & =4\pi Y_l^m(-q/|q|)i^l{\int }_{{\mathbb{R}}^{+}}{r}^{2}R(r)j_l(|q|r)dr\\ & =4\pi Y_l^m(q/|q|)(-i{)}^l{\int }_{{\mathbb{R}}^{+}}{r}^{2}R(r)j_l(|q|r)dr\end{array}$$

  This also holds true for real spherical harmonics.

Spherical harmonics

• Plane wave expansion formula

  $$e^{iq\cdot r}=4\pi \sum _{l=0}^{\mathrm{\infty }}\sum _{m=-l}^l{i}^l{j}_l(|q||r|)Y_l^m(q/|q|)Y_l^{m\ast }(r/|r|)$$

• Spherical harmonics orthogonality

  $${\int }_{{\mathbb{S}}^2}{Y}_l^{m\ast }(r)Y_{l^{\prime }}^{m^{\prime }}(r)dr={\delta }_{l,l^{\prime }}{\delta }_{m,m^{\prime }}$$

  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.