# Numerical Integration¶

Many of the integrals needed by Band are computed via numerical integration. See also: Wikipedia page on Numerical Integration.

## Becke Grid¶

The numerical integration grid is a refined version of the fuzzy cells integration scheme developed by Becke 1. The implementation in BAND is described in Ref. 2.

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

block:

```
BeckeGrid
Quality [Auto | Basic | Normal | Good | VeryGood | Excellent]
RadialGridBoost float
QualityPerRegion
Quality [Basic | Normal | Good | VeryGood | Excellent]
Region string
End
End
```

`BeckeGrid`

- Type
Block

- Description
Options for the numerical integration grid, which is a refined version of the fuzzy cells integration scheme developed by Becke.

`Quality`

- Type
Multiple Choice

- Default value
Auto

- Options
[Auto, Basic, Normal, Good, VeryGood, Excellent]

- Description
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

- Default value
1.0

- Description
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.

`QualityPerRegion`

- Type
Block

- Recurring
True

- Description
Sets the grid quality for all atoms in a region. If specified, this overwrites the globally set quality.

`Quality`

- Type
Multiple Choice

- Options
[Basic, Normal, Good, VeryGood, Excellent]

- Description
The region’s integration grid quality.

`Region`

- Type
String

- Description
The identifier of the region for which to set the quality.

Example: Multiresolution illustrates how to use the `QualityPerRegion`

option.

**Notes:**

The space-partition function used in BAND differs from the one described in Ref. 2. 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-Ubn):

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.

## Radial grid of NAOs¶

```
RadialDefaults
NR integer
NRPerType integer_list
RMax float
RMin float
RMinPerType float_list
End
```

`RadialDefaults`

- Type
Block

- Description
Options for the logarithmic radial grid of the basis functions used in the subprogram Dirac

`NR`

- Type
Integer

- Default value
3000

- Description
Number of radial points. With very high values (like 30000) the Dirac subprogram may not converge.

`NRPerType`

- Type
Integer List

- Description
If present overrides NR. The list needs to be as long as there are atom types

`RMax`

- Type
Float

- Default value
100.0

- Unit
Bohr

- Description
Upper bound of the logarithmic radial grid

`RMin`

- Type
Float

- Default value
1e-06

- Unit
Bohr

- Description
Lower bound of the logarithmic radial grid

`RMinPerType`

- Type
Float List

- Unit
Bohr

- Description
If specified overrides RMin. The list needs to be as long as there are atom types (different elements)

## 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

- Default value
Becke

- Options
[Becke, Voronoi]

- Description
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

- Description
Options for the Voronoi numerical integration scheme. Deprecated. Use BeckeGrid instead.

`AccInt`

- Type
Float

- Default value
3.5

- Description
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.

References

- 1
A.D. Becke,

*A multicenter numerical integration scheme for polyatomic molecules*, Journal of Chemical Physics 88, 2547 (1988).- 2(1,2)
M. Franchini, P.H.T. Philipsen, L. Visscher,

*The Becke Fuzzy Cells Integration Scheme in the Amsterdam Density Functional Program Suite*, Journal of Computational Chemistry 34, 1818 (2013).