Action: BF_FOURIER

Module	ves
Description	Usage
Fourier basis functions.

Details and examples

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

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

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_FOURIERFourier basis functions. More details MINIMUMThe minimum of the interval on which the basis functions are defined=-pi MAXIMUMThe maximum of the interval on which the basis functions are defined=+pi ORDERThe order of the basis function expansion=10 LABELa label for the action so that its output can be referenced in the input to other actions=bf_fourier
The BF_FOURIER action with label bf_fourier calculates something

Full list of keywords

The following table describes the keywords and options that can be used with this action

Keyword	Type	Default	Description
ORDER	compulsory	none	The order of the basis function expansion
MINIMUM	compulsory	none	The minimum of the interval on which the basis functions are defined
MAXIMUM	compulsory	none	The maximum of the interval on which the basis functions are defined
DEBUG_INFOThis keyword do not have examples	optional	false	Print out more detailed information about the basis set
NUMERICAL_INTEGRALSThis keyword do not have examples	optional	false	Calculate basis function integral for the uniform distribution numerically