Numerical integration details
All DFT implementations employ some kind of numerical integration scheme, because the mathematical expression for any relevant density functional makes an analytical integration of matrix elements of the XC potential between basis functions of whatever type impossible. ADF uses a Gaussiantype quadrature method based on the partitioning of space in atomic cells. Each cell contains a core-region sphere, inside which the integrands with cusps and singularities at the nucleus are handled conveniently in spherical coordinates:
∫cell dr f(r) = ∫sphere dr f(r) + ∫cell-without-sphere dr f(r)
∫sphere dr f(r) = ∫R dr r2 ∫4π dΩ f(r,Ω)
The Jacobian factor r2 in the last equation removes the Coulomb singularity of the integrand at the nuclear position. The remaining cell-minus-sphere region is treated, after suitable coordinate transformations, by a Gaussian product formula in Cartesian-like coordinates.
The atomic cells fill all space 'inside' the molecule. In the remaining 'outer' region, farther away from the atoms, all integrands decay exponentially with the distance fromthe molecule are smooth, and do not have cusps. An accurate evaluation by numerical sampling techniques over that part of space is not very demanding.
The ADF numerical integration module automatically tunes the grid by varying the numbers of points in the different regions (atomic spheres, cells and the outer region), while monitoring a series of test integrals such that these are evaluated with an (input-adjustable) precision. This procedure has proven to be robust and reliable, and is able to produce any desired accuracy in the actual calculations. It is user-friendly, because one only has to specify the precision level, by a single parameter. At the same time it is flexible, as one can select any value from 0.5 (so inaccurate as to be almost meaningless) to 12.0 (effectively close to machine precision). Of course, a higher precision implies more points and a greater computational effort. As an indication of the scaling of computing time: increasing the precision parameter by 1.0 will roughly double the number of grid points, and hence, the computing time. This is a crude estimate; much depends on the actual case.
ADF applies numerical integration not only to those integrals that cannot be done analytically, such as the matrix elements of the XC potential, but to most other integrals as well. This approach has two major advantages. In the first place, simplicity, and hence efficiency, as well as reduced chances of implementation errors. Numerical integrals, assuming that the grid itself is available, are easy for the developer to implement and for the computer to evaluate: straightforward do-loops and vector constructs. In the second place, it creates the freedom to select the types of basis functions that are intrinsically the most suitable for electronic structure calculations, but which might cause difficulties in analytical integration schemes, for instance Slatertype orbitals (STOs) or numerical atomic orbitals. It makes it almost trivial to implement the computation of all kinds of properties, as long as these are expressible as integrals over space of an integrand that one can evaluate without too much difficulty at an arbitrary point r.
Text is mostly taken from: Chemistry with ADF, G. te Velde, F.M. Bickelhaupt, E.J. Baerends, C. Fonseca Guerra, S.J.A. van Gisbergen, J.G. Snijders, T. Ziegler J. Comp. Chem. 22 (2001) 931.
Links
ADF User Documentation: numerical integration
ADF-GUI: accuracy and efficiency
Related: numerical integration
