Localized Alternatives to Becke Weights for Molecular Quadrature

In 1988, Becke introduced a method [1] for performing molecular DFT quadrature that continues to be popular almost 40 years later. In his approach, the integral of a functional of the electron density in an n-atom molecule is split into n atomic sub-integrals, each of which is treated by radial and angular quadrature on a nuclear-centred grid.

There is a potential double-counting problem — because the atomic sub-integrals overlap — but Becke avoids this by using a “stockholder” scheme that assigns a “Becke weight” to each grid point. By doing so, Becke ensures that, if the radial and angular quadratures are exact, the molecular quadrature will also be exact.

One of the weaknesses of Becke’s scheme is that the formal cost of computing all of the Becke weights is O(n3). This can become a computational bottleneck if other parts of the DFT calculation (such as the evaluation of the density at each grid point) have been carefully optimised.

In this lecture, I will introduce three alternative schemes [2] that, like Becke’s, are formally exact. However, the cost of computing all of the weights using two of the schemes is O(n2) and using the third scheme is O(n).

[1] A.D. Becke, J. Chem. Phys. 88 (1988) 2547

[2] P.M.W. Gill and K.D. Tsoukalas, J. Chem. Phys., submitted.