Numerical Integration

Many of the integrals are obtained by numerical integration. Two grids are available: the Becke grid (default) and the old Voronoi scheme (deprecated). One can switch between the two using:

IntegrationMethod [Becke | Voronoi]

Becke Grid

This 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 key

   Quality [basic|normal|good|verygood|excellent]
(Default: Normal) For a description of the various “qualities” and the associated numerical accuracy see reference 52. The integration grid quality defined in the BECKEGRID block key overrules the NumericalQuality.

Advanced options:

     Ia1 [basic|normal|good|verygood|excellent]
     Ia2 [basic|normal|good|verygood|excellent]
  {RadialGridBoost boost}
One can define a different grid quality for each atom, with input numbers Ia1, Ia2, etc. If 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 this option.
(Default: 1.0 (or 3.0 if a numerically sensitive functional is used)) The number of radial integration points will be boosted by this factor. Some XC functionals require very accurate radial integration grids, so BAND will automatically boost the radial grid 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


  • 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):
\[\begin{split}\mathcal{P}_{i,U} = \begin{cases} 1 & \text{if $r_{i,U}<\Omega_I$} \\ 0 & \text{if $\exists j : r_{j,U}<\Omega_J$ } \\ \eta_i \frac{e^{-2 (r_{i,U}-\Omega_I) / a_0}}{(r_{i,U}-\Omega_I)^2} & \text{elsewhere} \end{cases}\end{split}\]
  • A Becke grid of normal quality is roughly equivalent (in both absolute accuracy and computation time) to INTEGRATION 4 (Voronoi scheme), and a Becke grid of good quality is roughly equivalent to INTEGRATION 6 (Voronoi scheme).
  • 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

With this keyword the radial grid part of the basis function evaluation with the subprogram Dirac can be controlled.

  NR nr
  RMin rmin
  RMax rmax
(Default: 3000) This key handles the number of radial points. With very high values like 30.000 the Dirac subprogram may not converge.
RMin, RMax
These keys define the lower (Default: 1e-6) and upper (Default: 100) bound of the logarithmic grid.

Elliptic integrals

(Expert Option) The integration of the electrostatic interaction between spherical atoms is done very precisely in an elliptic grid, which depends on NUELSTAT and NVELSTAT

(Default: 50) Electrostatic interaction integrals between spherical atomic densities are computed by numerical integration over an elliptic grid. Nuelstat is the outward (parabolic) coordinate number of integration points.
(Default: 80) Electrostatic interaction integrals between spherical atomic densities are computed by numerical integration over an elliptic grid. Nvelstat is the angular (elliptic) coordinate number of integration points.

Voronoi grid (deprecated)

INTEGRATION (block-type)
Parameter-specifications for the generation of numerical integration points and weights. Most data records must be of the form ‘parameter value’. The most important parameter is accint, which is defaulted to the value of key ACCURACY. Unless one is very familiar with the details of the numerical integration package, we strongly recommend not to use the INTEGRATION key, and to specify only ACCURACY. More information can be found in the literature.
This key expects a real value, normally between 3 and 7, and will automatically set all necessary options for the numerical integration with the Voronoi grid. A value of 3 would be basic quality and a value of 7 would be good quality