Action: BF_COSINE

Module	ves
Description	Usage
Fourier cosine basis functions.

Details and examples

Fourier cosine basis functions.

Use as basis functions Fourier cosine 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 cosine 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 even function, i.e. $F(-x)=F(x)$.

The Fourier cosine basis functions are given by

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

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 cosine 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.

Click on the labels of the actions for more information on what each action computes

BF_COSINEFourier cosine 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=bf1
The BF_COSINE action with label bf1 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