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).