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 sine basis functions.
Use as basis functions Fourier sine 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 sine mode used). The total number of basis functions is \(N+1\) as the constant \(f_{0}(x)=1\) is also included. These basis functions should only be used for periodic CVs. They can be useful if the periodic function being expanded is an odd function, i.e. \(F(-x)=-F(x)\).
The Fourier sine basis functions are given by
\begin{align} f_{0}(x) &= 1 \\ f_{1}(x) &= sin(\frac{2\pi }{P} x) \\ f_{2}(x) &= sin(2 \cdot \frac{2\pi}{P} x) \\ f_{3}(x) &= sin(3 \cdot \frac{2\pi}{P} x) \\ & \vdots \\ f_{n}(x) &= sin(n \cdot \frac{2\pi}{P} x) \\ & \vdots \\ f_{N}(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 sine expansion of order 10 over the periodic interval \(-\pi\) to \(+\pi\). This results in a total number of 11 basis functions. The label used to identify the basis function action can then be referenced later on in the input file.
bfS: BF_SINEMINIMUM=-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. |