   BF_FOURIER
 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 $$[a,b]$$

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

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.

Click on the labels of the actions for more information on what each action computes bf_fourier: BF_FOURIER MINIMUMcompulsory keyword
The minimum of the interval on which the basis functions are defined. =-pi MAXIMUMcompulsory keyword
The maximum of the interval on which the basis functions are defined. =+pi ORDERcompulsory keyword
The order of the basis function expansion. =10

Glossary of keywords and components
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.