Compact Representations of Kohn-Sham Invariant Subspaces

We present a method to compute a hierarchical approximate representation of the solutions of the Kohn-Sham equations. The method is based on a recursive bisection algorithm and yields one-particle wave functions localized on subdomains of varying sizes. The accuracy of the representation is set a priori by specifying the highest acceptable error in 2-norm for any solution. Applications to large systems are used to illustrate the achievable reduction in data size. Implications for the development of linear-scaling methods and for the acceleration of conventional iterative methods are discussed.

Figure 1

Walsh functions $W_j(x)$ used to define the projectors $P_l^{(k)}$. The values of the Walsh functions in each subinterval are shown as a binary pattern and form a Gray code sequence (see text).

Figure 2

Compression ratio in ${\mathrm{H}}_2\mathrm{O}$ obtained using one level (open squares), two levels (open triangles) and three levels (open circles) of recursive subspace bisection, for various sample sizes ($N_{\mathrm{mol}}$). The dashed line shows the compression obtained by truncation of MLWFs in cubes centered on Wannier centers containing 99.9% of the function’s norm.