# Numerical Integration¶

## Becke Grid¶

The numerical integration grid is a refined version of the fuzzy cells integration scheme developed by Becke.[51] The implementation in BAND is described in Ref. [52].

The quality of the Becke integration grid can be changed within the BeckeGrid block:

BeckeGrid
Quality [Auto | Basic | Normal | Good | VeryGood | Excellent]
AtomDepQuality # Non-standard block. See details.
...
End
End

BeckeGrid
Type: Block Options for the numerical integration grid, which is a refined version of the fuzzy cells integration scheme developed by Becke.
Quality
Type: Multiple Choice Auto [Auto, Basic, Normal, Good, VeryGood, Excellent] Quality of the integration grid. For a description of the various qualities and the associated numerical accuracy see reference. If ‘Auto’, the quality defined in the ‘NumericalQuality’ will be used.
RadialGridBoost
Type: Float 1.0 The number of radial grid points will be boosted by this factor. Some XC functionals require very accurate radial integration grids, so BAND will automatically boost the radial grid by a factor 3 for the following numerically sensitive functionals: LibXC M05, LibXC M05-2X, LibXC M06-2X, LibXC M06-HF, LibXC M06-L, LibXC M08-HX, LibXC M08-SO, LibXC M11-L, LibXC MS0, LibXC MS1, LibXC MS2, LibXC MS2H, LibXC MVS, LibXC MVSH, LibXC N12, LibXC N12-SX, LibXC SOGGA11, LibXC SOGGA11-X, LibXC TH1, LibXC TH2, LibXC WB97, LibXC WB97X, MetaGGA M06L, MetaHybrid M06-2X, MetaHybrid M06-HF, MetaGGA MVS.
AtomDepQuality
Type: Non-standard block One can define a different grid quality for each atom (one definition per line). Line format: ‘AtomIndex Quality’, e.g. ‘3 Good’ means that a grid of Good quality will be used for the third atom in input order. If the index of an atom is not present in the AtomDepQuality section, the quality defined in the Quality key will be used

Example: Multiresolution illustrates how to use the AtomDepQuality option.

Notes:

• The space-partition function used in BAND differs from the one described in Ref. [52]. The unnormalized partition function used in the program is defined as ($$\Omega_I$$ is an element-dependent parameter: 0.1 Bohr for H, 0.3 Bohr for He-Xe and 0.6 Bohr for Cs-Uuo):
$\begin{split}\mathcal{P}_{i,U} = \begin{cases} 1 & \text{if r_{i,U}<\Omega_I} \\ 0 & \text{if \exists j : r_{j,U}<\Omega_J } \\ \eta_i \frac{e^{-2 (r_{i,U}-\Omega_I) / a_0}}{(r_{i,U}-\Omega_I)^2} & \text{elsewhere} \end{cases}\end{split}$
• The Becke grid is not very well suited to calculate Voronoi deformation density (VDD) charges. For accurate calculation of VDD charges the Voronoi integration scheme is recommended.

RadialDefaults
NR integer
RMax float
RMin float
End

RadialDefaults
Type: Block Options for the logarithmic radial grid of the basis functions used in the subprogram Dirac
NR
Type: Integer 3000 Number of radial points. With very high values (like 30000) the Dirac subprogram may not converge.
RMax
Type: Float 100.0 Bohr Upper bound of the logarithmic radial grid
RMin
Type: Float 1e-06 Bohr Lower bound of the logarithmic radial grid

## Voronoi grid (deprecated)¶

It is possible to use an alternative numerical integration scheme to the Becke Grid, namely the Voronoi Grid.

IntegrationMethod [Becke | Voronoi]

IntegrationMethod
Type: Multiple Choice Becke [Becke, Voronoi] Choose the real-space numerical integration method. Note: the Voronoi integration scheme is deprecated.

The options for the Voronoi Grid are specified in the Integration block:

Integration
AccInt float
End

Integration
Type: Block Options for the Voronoi numerical integration scheme. Deprecated. Use BeckeGrid instead.
AccInt
Type: Float 3.5 General parameter controlling the accuracy of the Voronoi integration grid. A value of 3 would be basic quality and a value of 7 would be good quality.