Automatic determination of Kerker's factor

If the keyword 'scf.Kerker.factor' is not given in your input file, OpenMX Ver. 3.9 automatically estimates a proper value of Kerker's factor $\alpha$ by the following equation:

$$\alpha = \frac{0.5}{\vert {\bf b}_{\rm min}\vert^2}\left(4\frac{Dq}{Aq}+1.0\right)$$

with

$$Aq = \frac{1}{3}\left(\vert {\bf b}_1 \vert^2 + \vert {\bf b}_2 \vert^2 + \vert {\bf b}_3 \vert^2\right),$$

$$Dq = \frac{1}{3}\sum_{i<j}\left\vert\vert {\bf b}_i \vert^2- \vert {\bf b}_j \vert^2\right\vert,$$

where ${\bf b}_i (i=1,2,3)$ is a reciprocal vector, and ${\bf b}_{\rm min}$ is the smallest vector among {b}. The equation takes account of the dependency of $\alpha$ on the size and anisotropy of the system. From a series of numerical calculations it is found that the estimated value works well in most cases.