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

${\hat {g}}(\mathbf {k} )=e^{-i\mathbf {k} \cdot \mathbf {R} _{0}}{\hat {f}}(\mathbf {k} ).$

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

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} )

.

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}

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