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.
We develop an improved stochastic formalism for the Bethe–Salpeter equation (BSE), based on an exact separation of the effective-interaction W into two parts, W = ( W − v W) + v W, where the latter is formally any translationally invariant interaction, v W( r − r′). When optimizing the fit of the exchange kernel v W to W, using a stochastic sampling W, the difference W − v W becomes quite small. Then, in the main BSE routine, this small difference is stochastically sampled. The number of stochastic samples needed for an accurate spectrum is then largely independent of system size. While the method is formally cubic in scaling, the scaling prefactor is small due to the constant number of stochastic orbitals needed for sampling W.
The structural, electronic and mechanical properties of monoclinic Li2Si2O5 are explored using density functional theory. Different exchange–correlation functionals are considered and the results are correlated to experimental data. The calculated electronic band structure and density of states indicate that monoclinic Li2Si2O5 has an insulating character with an indirect band gap of 4.98 eV. Elastic stiffness coefficients and the bulk, shear and Young's moduli are also calculated. Our calculations predict that Li2Si2O5 is a ductile compound. We show that monoclinic Li2Si2O5 behaves as a specially orthotropic material, meaning that the structure can be masked by the orthorhombic form.
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