Chemical Dynamics/Electrostatics/Fourier Transforms

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
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
Given a fixed position vector R0, if g(r) = ƒ(r − R0), then &thinsp; $$\hat{g}(\mathbf{k})= e^{- i \mathbf{k}\cdot \mathbf{R}_0 }\hat{f}(\mathbf{k}).$$


 * $$\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
The convolution of f and g is usually denoted as f∗g, 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


 * $$ \hat{h}(\mathbf{k})= \hat{f}(\mathbf{k})\cdot \hat{g}(\mathbf{k})$$.


 * $$ \hat{h}(\mathbf{k}) = \int {e^{-i\mathbf{k} \cdot \mathbf{r}} h(\mathbf{r})} d\mathbf{r}$$


 * $$ = \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}$$