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:

The translation theorem[edit]

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

Proof

Now, change r to a new variable by:

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:



The convolution theorem for the Fourier transform says:

If

then

.
Proof