Chemical Dynamics/Electrostatics/Fourier Transforms

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The Fourier transform is a useful mathematical transformation often utilized in many scientific and engineering fields. Here we extract useful concepts of Fourier transformation and logically arrange them to form a foundation for the Ewald summation and other related methods in electrostatics. Readers could check out other more mathematically formal introduction of Fourier transform

Definition[edit]

We use the following convention in which the Fourier transform is a unitary transformation on the 3-D Cartesian space R3, the Fourier transform and its inverse transform are symmetric:

 \hat{f}(\mathbf{k}) = \frac{1}{(2\pi)^{3/2}} \int f(\mathbf{r}) e^{- i\mathbf{k}\cdot \mathbf{r}}\,d^{3}\mathbf{r}
f(\mathbf{r}) = \frac{1}{(2\pi)^{3/2}} \int \hat{f}(\mathbf{k}) e^{ i\mathbf{k} \cdot \mathbf{r}}\,d^{3}\mathbf{k}

The translation theorem[edit]

Given a fixed position vector R0, if g(r) = ƒ(r − R0), then  

\hat{g}(\mathbf{k})= e^{- i \mathbf{k}\cdot \mathbf{R}_0 }\hat{f}(\mathbf{k}).

Proof
\hat{g}(\mathbf{k})= \frac{1}{(2\pi)^{3/2}} \int g(\mathbf{r}) e^{- i\mathbf{k}\cdot \mathbf{r}}\,d^{3}\mathbf{r}
 = \frac{1}{(2\pi)^{3/2}} \int f(\mathbf{r}-\mathbf{R}_0) e^{- i\mathbf{k}\cdot \mathbf{r}}\,d^{3}\mathbf{r}
 = \frac{1}{(2\pi)^{3/2}} \int f(\mathbf{r}-\mathbf{R}_0) e^{- i\mathbf{k}\cdot \mathbf{r}}\,d^{3}\mathbf{r}

Now, change r to a new variable by:  \mathbf{r'}= \mathbf{r}-\mathbf{R}_0

 \hat{g}(\mathbf{k}) = \frac{1}{(2\pi)^{3/2}} \int f(\mathbf{r'}) e^{- i\mathbf{k}\cdot (\mathbf{r'}+\mathbf{R}_0)}\,d^{3}\mathbf{r'}
  = \frac{1}{(2\pi)^{3/2}} e^{-i \mathbf{k}\cdot \mathbf{R}_0} \int f(\mathbf{r'}) e^{- i\mathbf{k}\cdot \mathbf{r'}}\,d^{3}\mathbf{r'}
  = \frac{1}{(2\pi)^{3/2}} e^{-i \mathbf{k}\cdot \mathbf{R}_0} \int f(\mathbf{r}) e^{- i\mathbf{k}\cdot \mathbf{r}}\,d^{3}\mathbf{r}
 = e^{- i \mathbf{k}\cdot \mathbf{R}_0 } \hat{f}(\mathbf{k}).

The convolution theorem[edit]

The convolution of f and g is usually denoted as fg, using an asterisk or star. It is defined as the integral of the product of the two functions after one is reversed and shifted:


(f * g )(t)
\stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty f(\tau)\, g(t - \tau)\, d\tau


The convolution theorem for the Fourier transform says:

If

 h(\mathbf{r}) = (f * g )(\mathbf{r})

then

 \hat{h}(\mathbf{k})= \hat{f}(\mathbf{k})\cdot \hat{g}(\mathbf{k}).
Proof
 \hat{h}(\mathbf{k}) = \int {e^{-i\mathbf{k} \cdot \mathbf{r}} h(\mathbf{r})} d\mathbf{r}
  = \int {e^{-i\mathbf{k} \cdot \mathbf{r}} \int f(\mathbf{r}) g(\mathbf{r'}-\mathbf{r})}
d^3\mathbf{r'}d^3\mathbf{r}