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):

\[\begin{split}\mathcal{P}_{i,U} = \begin{cases} 1 & \text{if $r_{i,U}<\Omega_I$} \\ 0 & \text{if $\exists j : r_{j,U}<\Omega_J$ } \\ \eta_i \frac{e^{-2 (r_{i,U}-\Omega_I) / a_0}}{(r_{i,U}-\Omega_I)^2} & \text{elsewhere} \end{cases}\end{split}\]

  • 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.

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).