Action: BF_GAUSSIANS

Module	ves
Description	Usage
Gaussian basis functions.

Details and examples

Gaussian basis functions.

These basis functions do not form orthogonal bases. We recommend using wavelets (BF_WAVELETS) instead that do form orthogonal bases.

Basis functions given by Gaussian distributions with shifted centers defined on a bounded interval. See the paper cited below for full details.

You need to provide the interval $[a,b]$ on which the bias is to be expanded. The ORDER keyword of the basis set $N$ determines the number of equally sized sub-intervalls to be used. On the borders of each of these sub-intervalls the mean $\mu$ of a Gaussian basis function is placed:

$$\mu_i = a + (i-1) \frac{b-a}{N}$$

The total number of basis functions is $N+4$ as the constant $f_{0}(x)=1$, as well as two additional Gaussians at the Boundaries are also included.

The basis functions are given by

$$\begin{aligned} f_0(x) &= 1 \\ f_i(x) &= \exp\left(-\frac{{\left(x-\mu_i\right)}^2}{2\sigma^2}\right) \end{aligned}$$

When the Gaussians are used for a periodic CV (with the PERIODIC keyword), the sub-intervals are chosen in the same way, but only $N+1$ functions are required to fill it (the ones at the boundary coincide and the ones outside can be omitted).

It is possible to specify the width $\sigma$ (i.e. the standard deviation) of the Gaussians using the WIDTH keyword. By default it is set to the sub-intervall length. It was found that performance can be typically improved with a smaller value (around 75 % of the sub-interval length), although a too small overlap will prevent the basis set from working correctly at all.

The optimization procedure then adjusts the heigths of the individual Gaussians. To avoid 'blind' optimization of the basis functions outside the currently sampled area, it is often beneficial to use the OPTIMIZATION_THRESHOLD keyword of the VES_LINEAR_EXPANSION (set it to a small value, e.g. 1e-6)

As an example two adjacent basis functions (with the mentioned width choice of 75% of the sub-interval length) can be seen below. The full basis consists of shifted Gaussians in the full specified interval.

Examples

The bias is expanded with Gaussian functions in the intervall from 0.0 to 10.0 using order 20. This results in 24 basis functions.

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

bfG: BF_GAUSSIANSGaussian basis functions. More details MINIMUMThe minimum of the interval on which the basis functions are defined=0.0 MAXIMUMThe maximum of the interval on which the basis functions are defined=10.0 ORDERThe order of the basis function expansion=20
The BF_GAUSSIANS action with label bfG calculates something

Because it was not specified, the width of the Gaussians is by default set to the sub-intervall length, i.e.\ $\sigma=0.5$. To e.g. enhance the overlap between neighbouring basis functions, it can be specified explicitely:

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

bfG: BF_GAUSSIANSGaussian basis functions. More details MINIMUMThe minimum of the interval on which the basis functions are defined=0.0 MAXIMUMThe maximum of the interval on which the basis functions are defined=10.0 ORDERThe order of the basis function expansion=20 WIDTHThe width (i=0.7
The BF_GAUSSIANS action with label bfG 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
WIDTH	optional	not used	The width (i
PERIODICThis keyword do not have examples	optional	false	Use periodic version of basis set

References

More information about how this action can be used is available in the following articles:

• B. Pampel, O. Valsson, Improving the Efficiency of Variationally Enhanced Sampling with Wavelet-Based Bias Potentials. Journal of Chemical Theory and Computation. 18, 4127–4141 (2022)