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 \([a,b]\)
\[ \int_{a}^{b} dx \, f_{n}(x)\, f_{m}(x) = \begin{cases} 0 & n \neq m \\ (b-a) & n = m = 0 \\ (b-a)/2 & n = m \neq 0 \end{cases}. \]
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: BF_FOURIERMINIMUM=-picompulsory keyword The minimum of the interval on which the basis functions are defined.MAXIMUM=+picompulsory keyword The maximum of the interval on which the basis functions are defined.ORDER=10compulsory keyword The order of the basis function expansion.
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. |
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. |