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Chebyshev polynomial basis functions.

Use as basis functions Chebyshev polynomials of the first kind \(T_{n}(x)\) defined on a bounded interval. You need to provide the interval \([a,b]\) on which the basis functions are to be used, and the order of the expansion \(N\) (i.e. the highest order polynomial used). The total number of basis functions is \(N+1\) as the constant \(T_{0}(x)=1\) is also included. These basis functions should not be used for periodic CVs.

Intrinsically the Chebyshev polynomials are defined on the interval \([-1,1]\). A variable \(t\) in the interval \([a,b]\) is transformed to a variable \(x\) in the intrinsic interval \([-1,1]\) by using the transform function

\[ x(t) = \frac{t-(a+b)/2} {(b-a)/2} \]

The Chebyshev polynomials are given by the recurrence relation

\begin{align} T_{0}(x) &= 1 \\ T_{1}(x) &= x \\ T_{n+1}(x) &= 2 \, x \, T_{n}(x) - T_{n-1}(x) \end{align}

The first 6 polynomials are shown below

The Chebyshev polynomial are orthogonal over the interval \([-1,1]\) with respect to the weight \(\frac{1}{\sqrt{1-x^2}}\)

\[ \int_{-1}^{1} dx \, T_{n}(x)\, T_{m}(x) \, \frac{1}{\sqrt{1-x^2}} = \begin{cases} 0 & n \neq m \\ \pi & n = m = 0 \\ \pi/2 & n = m \neq 0 \end{cases} \]

For further mathematical properties of the Chebyshev polynomials see for example the Wikipedia page.

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.

Examples

Here we employ a Chebyshev expansion of order 20 over the interval 0.0 to 10.0. 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.

bfC: BF_CHEBYSHEV MINIMUM=0.0 MAXIMUM=10.0 ORDER=20