	This is part of the ves module
	It is only available if you configure PLUMED with ./configure –enable-modules=ves . Furthermore, this feature is still being developed so take care when using it and report any problems on the mailing list.

Fourier basis functions.

Use as basis functions Fourier series defined on a periodic interval. You need to provide the periodic interval \([a,b]\) on which the basis functions are to be used, and the order of the expansion \(N\) (i.e. the highest Fourier mode used). The total number of basis functions is \(2N+1\) as for each Fourier mode there is both the cosine and sine term, and the constant \(f_{0}(x)=1\) is also included. These basis functions should only be used for periodic CVs.

The Fourier series basis functions are given by

\begin{align} f_{0}(x) &= 1 \\ f_{1}(x) &= cos(\frac{2\pi }{P} x) \\ f_{2}(x) &= sin(\frac{2\pi }{P} x) \\ f_{3}(x) &= cos(2 \cdot \frac{2\pi}{P} x) \\ f_{4}(x) &= sin(2 \cdot \frac{2\pi}{P} x) \\ & \vdots \\ f_{2k-1}(x) &= cos(k \cdot \frac{2\pi}{P} x) \\ f_{2k}(x) &= sin(k \cdot \frac{2\pi}{P} x) \\ & \vdots \\ f_{2N-1}(x) &= cos(N \cdot \frac{2\pi}{P} x) \\ f_{2N}(x) &= sin(N \cdot \frac{2\pi}{P} x) \\ \end{align}

where \(P=(b-a)\) is the periodicity of the interval. They are orthogonal over the interval \([a,b]\)

\[ \int_{a}^{b} dx \, f_{n}(x)\, f_{m}(x) = \begin{cases} 0 & n \neq m \\ (b-a) & n = m = 0 \\ (b-a)/2 & n = m \neq 0 \end{cases}. \]

Compulsory keywords

ORDER	The order of the basis function expansion.
MINIMUM	The minimum of the interval on which the basis functions are defined.
MAXIMUM	The maximum of the interval on which the basis functions are defined.

Options

DEBUG_INFO

( default=off ) Print out more detailed information about the basis set. Useful for debugging.

NUMERICAL_INTEGRALS

( default=off ) Calculate basis function integral for the uniform distribution numerically. Useful for debugging.

Examples

Here we employ a Fourier expansion of order 10 over the periodic interval \(-\pi\) to \(+\pi\). This results in a total number of 21 basis functions. The label used to identify the basis function action can then be referenced later on in the input file.

BF_FOURIER MINIMUM=-pi MAXIMUM=+pi ORDER=10 LABEL=bf_fourier