Self-Averaging Stochastic Kohn-Sham Density-Functional Theory

We formulate the Kohn-Sham density functional theory (KS-DFT) as a statistical theory in which the electron density is determined from an average of correlated stochastic densities in a trace formula. The key idea is that it is sufficient to converge the total energy per electron to within a predefined statistical error in order to obtain reliable estimates of the electronic band structure, the forces on nuclei, the density and its moments, etc. The fluctuations in the total energy per electron are guaranteed to decay to zero as the system size increases. This facilitates “self-averaging” which leads to the first ever report of sublinear scaling KS-DFT electronic structure. The approach sidesteps calculation of the density matrix and thus, is insensitive to its evasive sparseness, as demonstrated here for silicon nanocrystals. The formalism is not only appealing in terms of its promise to far push the limits of application of KS-DFT, but also represents a cognitive change in the way we think of electronic structure calculations as this stochastic theory seamlessly converges to the thermodynamic limit.

Figure 1

The statistical error per electron (for $I=1200$ stochastic orbitals) as a function of $N_e^{-1/2}$, where $N_e$ is the number of electrons. The dotted line is a fit to the relation $\sigma =a{N}_e^{-1/2}$ with$a\approx 250\mathrm{meV}$.

Figure 2

CPU time for full (triangles) and stochastic (circles) SCF DFT calculation for silicon NCs as a function of the number of electrons. Inset: Log-log plot of the number of stochastic orbitals required to converge the total silicon NCs energy per electron to 10 meV as a function of the number of electrons.

Figure 3

The exact (black full circles) and the 70% confidence interval for the $x$ component of the force on each of the Si (index 0 to 35) and H (index 36 to 70) nuclei in ${\mathrm{Si}}_{35}{\mathrm{H}}_{36}$. The left and right panel use 80 and 320 stochastic orbitals, respectively.

Figure 4

Left: the LDA density of states of ${\mathrm{Si}}_{35}{\mathrm{H}}_{36}$ (black line) and the stochastic estimate for 2 different runs (circles and triangles). $I=720$ stochastic orbitals. Right: the KS orbital energy, ${\epsilon }_n$, versus the index of the orbital, $n$, for the deterministic (black solid) and stochastic estimate (circles). Inset: the KS orbital energy error near the top of the valance (index 87) and bottom of the conduction (index 88) bands.