Tempering stochastic density functional theory

Abstract

We introduce a tempering approach with stochastic density functional theory (sDFT), labeled t-sDFT, which reduces the statistical errors in the estimates of observable expectation values. This is achieved by rewriting the electronic density as a sum of a “warm” component complemented by “colder” correction(s). Since the warm component is larger in magnitude but faster to evaluate, we use many more stochastic orbitals for its evaluation than for the smaller-sized colder correction(s). This results in a significant reduction in the statistical fluctuations and systematic deviation compared to sDFT for the same computational effort. We demonstrate the method’s performance on large hydrogen-passivated silicon nanocrystals, finding a reduction in the systematic deviation in the energy by more than an order of magnitude, while the systematic deviation in the forces is also quenched. Similarly, the statistical fluctuations are reduced by factors of ≈4–5 for the total energy and ≈1.5–2 for the forces on the atoms. Since the embedding in t-sDFT is fully stochastic, it is possible to combine t-sDFT with other variants of sDFT such as energy-window sDFT and embedded-fragmented sDFT.

Reduced Scaling of Optimal Regional Orbital Localization via Sequential Exhaustion of the Single-Particle Space

“…So far, we have assumed that a basis of individual single-particle states is known (e.g., obtained by a deterministic DFT calculation). However, this procedure is trivially extended even to other cases, e.g., when stochastic DFT is employed. − For simplicity (and without loss of generality), we assume the localization is performed in the occupied subspace. Here, the sF-PMWF calculation is initialized by constructing a guess of N c random vectors |ζ⟩, which are projected onto the occupied subspace as .…”

“…The computational scaling to the system’s size is not seen improved either. Further, for truly large systems with thousands of electrons, one would employ techniques that avoid the use (or knowledge) of all single-particle states. − …”

Reduced scaling of optimal regional orbital localization via sequential exhaustion of the single-particle space

“…However, this procedure is trivially extended even to other cases, e.g., when stochastic DFT is employed. [27][28][29][30][31][32] For simplicity (and without loss of generality), we assume the localization is performed in the occupied subspace. Here, the sG-PMWF calculation is initialized by constructing a guess of N rl random vectors |ζ which are projected onto the occupied subspace as |ζ c = P o |ζ .…”

“…Here, the projector P o is a low-pass filter constructed from the Fermi operator leveraging the knowledge of the chemical potential. [27][28][29][30][31][32]40,42 Next, in each outer-loop step, one creates a block of random vectors |ζ m r which have to be mutually orthogonal as well as orthogonal to the N rl core states via, e.g., Gram-Schmidt process.…”

“…Further, for truly giant systems, one would employ techniques that avoid the use (or knowledge) of all single-particle states. [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44] Herein, we present a new and complementary top-down approach leading to a fast, efficient, and robust orbital localization algorithm via sequentially exhausting the entire orbital space. It is beneficial for obtaining regionally localized orbitals for a subsystem within the G-PMWF scheme.…”