Numerical integration

The key INTEGRATION has been introduced in its simple form in Chapter 2.2.

INTEGRATION accint

accint is a real number. The key is used as a simple key here.

Alternatively you can use it as a block key. This is activated if you give no argument. In the data block you specify which of several integration methods you want to use, and you give values for the involved parameters. Consult the literature for detailed information about the various schemes.

INTEGRATION
  data
  data
  ...
end

The block form is used to override default relations between various parameters that are applied in the generation of the integration grid in the polyhedron method [105]. All these parameters are accessible with subkeys in the data block of Integration. Most of the subkeys are simple keys with one single value as argument; a few subkeys are block-type (sub) keys themselves and hence require the usual format of a data block closed by subend.

accint

The main precision parameter
Its value defines the number of significant digits by which an internal set of standard integrals must be evaluated. The number and distribution of integration points is tuned accordingly. For normal applications this should yield a nearly optimal (given the underlying method) generation of points and weights. The default depends on the run type.

accsph

The polyhedron method of generating integration points partitions space in atomic polyhedrons, partitioned in pyramids with their tops at the atom in the center of the polyhedron. A core like atomic sphere is constructed around the atom; this truncates the tops of the pyramids. accsph specifies the test precision for the generation of points within the spheres. By default accsph=accint.

accpyr

Similarly this subkey sets the test level for the parts of the pyramids outside the atomic sphere. Default: accpyr=accint.

accpyu, accpyv, accpyw

The truncated pyramids are mathematically transformed into unit cubes. A product Gauss integration formula is applied to the cubes, with three (test precision) parameters for the three dimensions. Accpyw controls the direction that is essentially the radial integration from the surface of the atomic sphere to the base of the pyramid. The other two control the orthogonal directions (angular). By default all three equal accpyr.

accout

The region of space further away from the atoms, outside the polyhedrons, has its own precision parameter. By default accout=accint.

nouter

This outer region is treated by a product formula: outwards times parallel. The latter involves two dimensions: the surface of the molecule say. The outward integration is performed with Gauss-Legendre quadrature, in a few separate steps. The lengths of the steps are not equal, they increase by constant factors. The total length is fixed. The number of steps is controlled with this subkey; default: 2.

outrad

The parameter that defines the number of Gauss-Legendre integration points for each outward step. The precise relation between the actual number of points and this subkey, and the default relation between outrad and accout can be found in the implementation.

outpar

Similarly the integration in the directions parallel to the surface of the atomic system is controlled by a parameter. See the implementation for details.

dishul

Sets the distance between the outermost nuclei of the molecule and the boundary planes that define the boundary between the polyhedrons and the outer region. By default dishul=2.3*R, where Ris the radius of the largest atomic sphere in the molecule.

frange

The outward range of the outer region: integration is not performed to infinity but to a distance frange from the outermost atoms, where all functions can be assumed to be essentially zero. By default frange is derived both from accint, the general precision parameter, and from the present chemical elements: heavier atoms have longer-range functions than hydrogen say. The precise relations can be found in the implementation.

linrot

This parameter is significant only for symmetries with an axis of infinite rotational symmetry: Cand D
It is the highest rotational quantum number around this axis that occurs among the integrands. This depends on the employed basis functions and fit functions. By default the program finds this out for itself.

qpnear

If you specify point charges in the input file, there are two considerations implied for the numerical integration grid.
First, since the point charges create a Coulomb singularity. The integrands (of for instance the basis function products against the Coulomb potential) can only be evaluated with high precision if the grid around the point charges has spherical symmetry and uses local spherical coordinates, exactly as is done for the atomic nuclei. Second, the point charges do not carry fit or basis functions, hence they play only a role in the more diffuse tails of the actual functions involved in integrals. Therefore, a relative low precision of the integral part close to the point charge may have little effect on the total integration accuracy.
Since additional 'spherical centers' with their own surrounding grids increase the total number of points significantly, typically a few thousands per Coulomb center, this may result in high computational effort. Therefore, the program generates spherical grids only about those point charges that are close to the other atoms. The criterion, input with the qpnear subkey, is the closest distance between the point charge at hand and any real atom. Default 4.0 Angstrom. Any input value is interpreted in the unit-of-length specified with the Units key.

Next come the subkeys that require a list of data. The subkey must be placed on one line, the data on the next. This somewhat peculiar structure suggests that the subkeys are block keys; however their data blocks have no end code (subend) as for normal block type subkeys.

The list of data for such a subkey contains one value for each atom type. The data must be in the order in which the atom types were defined under atoms, implicitly or explicitly: remember that atoms belonging to different fragment types automatically have different atom types, even if their atom type names have been specified as identical under atoms.

rspher

gives the radii of the atomic spheres, one value for each atom type. By default, the radii are derived from the chemical element (heavier atoms get larger spheres) and from the environment: the sphere must not be too large for the atomic cell (polyhedron).

linteg

The maximum angular momentum quantum number of integrands centered on an atom of that type (one value for each atom type). This depends on the basis functions and on the fit functions. By default the program checks the function sets and sets the linteg values accordingly. This subkey is applied for the generation of grid points in the atomic spheres.

Items that relate to geometric lengths (dishul, frange, rspher) must be given in bohr (=atomic units), irrespective of the unit of length defined with units.