One-Dimensional Functions

Analytic one-dimensional functions

Function available for use when, e.g., applying broadening to calculated structure factors, radial distribution functions etc.

Keyword Parameters Derivative1 FT2 DKN3 Description
None – ✓ ✓ No function - returns zero.
Gaussian FWHM ✓ ✓ Un-normalised Gaussian with no prefactor $$ f(x) = \exp\left(-\frac{x^2}{2c^2}\right), c = \frac{\textrm{FWHM}}{2 \sqrt{2 \ln(2)}} $$
ScaledGaussian A
FWHM		 ✓ ✓ Un-normalised Gaussian with prefactor $$ f(x) = A\exp\left(-\frac{x^2}{2c^2}\right), c = \frac{\textrm{FWHM}}{2 \sqrt{2 \ln(2)}} $$
OmegaDependentGaussian FWHM ✓ ✓ Un-normalised Gaussian with variable FWHM and no prefactor - note that the parameter $\omega$ is set implicitly by the context using the broadening function $$ f(x) = \exp\left(-\frac{x^2}{2(c\omega)^2}\right), c = \frac{\textrm{FWHM}}{2 \sqrt{2 \ln(2)}} $$
GaussianC2 FWHM1
FWHM2		 ✓ ✓ Un-normalised Gaussian with constant and variable FWHM components and no prefactor - note that the parameter $\omega$ is set implicitly by the context using the broadening function $$ f(x) = \exp\left(-\frac{x^2}{2(c_1 + c_2\omega)^2}\right), c_n = \frac{\textrm{FWHM}_n}{2 \sqrt{2 \ln(2)}} $$
LennardJones126 $\epsilon$
$\sigma$		 ✓ Lennard-Jones 12-6 short-range potential $$ f(x) = 4\epsilon \left( \left( \frac{\sigma}{x} \right)^{12} - \left( \frac{\sigma}{x} \right)^6 \right) $$
Buckingham A
B
C		 ✓ Buckingham short-range potential $$ f(x) = A \exp\left(-B x\right) $$
GaussianPotential A
fwhm
$x_0$		 ✓ Gaussian potential centred at $x_0$, intended to be used as a potential override. $$ f(x) = A\exp\left(-\frac{\left(x-x_0\right)^2}{2c^2}\right), c = \frac{\textrm{FWHM}}{2 \sqrt{2 \ln(2)}} $$
Harmonic k ✓ Simple harmonic well potential$$ E_{ij} = \frac{1}{2} k r^2 $$ $$ F_{ij} = -k r $$

1Whether the first derivative of the function is defined

2Whether an analytic Fourier transform is defined

3Whether discrete kernel normalisation is defined (function sums to 1.0)

Last modified June 7, 2024: refactor: External potentials now use Function1D (#1928) (470743d1f)