Density fitting

Zlm Fit: density fitting with radial spline functions and real spherical harmonics

In ADF2013 and previous versions, a different density-fitting scheme (pair fit) was used. Include the key STOFIT if you want to use the old fitting scheme.

The basic ideas behind the so-called Zlm Fit can be described as follows. The total electron density is split into atomic densities (in a similar way as the volume is partitioned for the Becke grid). These atomic densities are then approximated by a combination of radial spline functions and real spherical harmonics (Zlm). The implementation in ADF is described in Ref. [2]. The algorithm used in ADF is related to the procedures proposed by Becke [3] and Delley [1].

The Zlm Fit scheme, which is the default fitting scheme in ADF, offers certain advantages compared to the old pair-fit method, especially the possibility of calculating the Coulomb potential to very high precision.

ZLMFIT
 Quality {Basic|Normal|Good|VeryGood|Excellent}
 {QualityPerRegion
    Region myregion
    Quality {Basic|Normal|Good|VeryGood|Excellent}
 End}
End
Quality

The default quality of the Zlm Fit is Normal. It can be changed with the Quality subkey.

QualityPerRegion

You can override the Zlm Fit quality for atoms in a particular region. Example: Multiresolution illustrates how to use this option.

The Zlm Fit method can be used for most features of the ADF program. For the calculation of Hartree-Fock exchange integrals, ADF uses a different fitting method, see the section on Hartree-Fock exchange.

Pair fit: symmetric density fit

The non-default density fitting procedure in ADF, called the pair fit method, is carried out separately for each pair of atoms. To use it, include the STOFIT keyword.

STOFIT

The implemented approach has several advantages in efficiency. Its drawback is that it necessitates the use of all available fit functions rather than only the symmetric combinations, although the final result, of course, needs only a symmetric fit because the total density is a symmetric (A1) function. For atoms far apart, the density fitting is performed with only symmetric functions. Given the implemented algorithm, this entails an approximation that can be tuned:

A1FIT atomicseparation
atomicseparation

The threshold distance between atoms, in Angstrom. The symmetric fit approximation is applied only for atoms farther apart. The default is 10.0 Angstrom.

Pair fit: fit integrals

STOFIT

For the computation of the Coulomb potential with the pair-fit method, the program uses a large number of so-called fit integrals: overlap integrals of a fit function with a product of two basis functions, where at least two of the three involved functions are centered on the same atom. In fact, these are ordinary overlap integrals of STOs, because the fit and basis functions are all STOs, and a product of STOs on one center is itself also an STO. To use this STO fitting method, which was previously the default, use the key STOFIT in the input of ADF (also include it in the Create mode of an atom, if that mode is used explicitly). For the bond energy, a first-order fit correction term is included, which makes the bond energy accurate to first order in the fit.

Obviously, when the two involved atoms are far enough apart, such overlap integrals become negligibly small. All fit integrals that are smaller (according to a rough but reasonable estimate) than a preset threshold are ignored and not computed.

The value of this threshold can be set in the input, using the CUTOFF_FIT subkey of the LINEARSCALING block keyword.

True density in XC potential

For the computation of the exchange-correlation potential (XC potential), the program uses the fitted density by default. This is an approximation. For the XC potential, the true density can be used if one includes the keyword EXACTDENSITY:

EXACTDENSITY

Using the EXACTDENSITY keyword makes the calculation more time-consuming, but it can improve accuracy in the following cases:

  • calculations that require accurate description of virtual orbitals, such as most TDDFT calculations;

  • when studying systems where weak interactions like van der Waals forces and hydrogen bonds are important. For example, EXACTDENSITY should be switched on when performing a geometry optimization of DNA pairs.

Precision of density fitting on standard output

In the output file of the ADF calculation, you can find, at the end of the SCF procedure, concise information about the density-fit precision: the error integral for the SCF density. The error integral is the integral of the difference between the exact density and the fit density, squared. Such values have very little to do with numerical integration; rather they show whether or not the employed set of fit functions is adequate to describe the SCF density. Error integral values that significantly exceed 1e-4 times the number of atoms are suspicious and may indicate some deficiency in the fit set for the actual calculation. The fit-error integrals for the initial (sum-of-fragments) density and the orthogonalized fragments are also printed. In the bond energy analysis, a first-order fit-correction term is given. In an optimization, such as a geometry optimization, all of this information is only printed for the final geometry.

References