Optimized attenuated interaction: Enabling stochastic Bethe–Salpeter spectra for large systems

Abstract

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.

Sparse-Stochastic Fragmented Exchange for Large-Scale Hybrid Time-Dependent Density Functional Theory Calculations

“…Our recent formulation of the stochastic GW-BSE recasts the expensive screened exchange term to a translationally invariant fitted exchange interaction and a small difference that is sampled stochastically. 46 This shifts the bulk of the computational effort to calculating the convolutions, as in TDHF, for which we substantially improved the efficiency here.…”

“…For the Chebyshev coefficients, c n (ω), we use simple smoothly decaying weights. 46,47 Compared with the TDA, our inclusion here of the offdiagonal matrix B and the negative-frequency component f − (and therefore the addition of "detransitions" of negative frequency) doubles the spectral range of , leading to the doubling in the number of required Chebyshev terms, which is typically 1000 now.…”

“…The iterative Chebyshev approach to obtain the full, non-TDA, frequency-resolved dipole–dipole correlation function σ(ω) is obtained as σ ( ω ) ∝ ω ⟨ χ̃ | δ ( L − ω ) | χ ⟩ And the delta function is calculated as δ ( L − ω ) = ∑ n = 0 N normalC normalh normale normalb normaly normals normalh normale normalv c n ( ω ) T n ( L̃ ) where T n ( L̃ ) is the n ’th Chebyshev polynomial and scriptL̃ is a scaled Liouvillian with eigenvalues between −1 and 1. The optical absorption is obtained from the residues as σ ( ω ) = ∑ n = 0 N C h e b y s h e v c n ( ω ) R n with …”

Deterministic/Fragmented-Stochastic Exchange for Large-Scale Hybrid DFT Calculations

“…It will be useful both for real-time TDDFT and for frequencyresolved TDDFT and BSE. 36,37 We also expect applications within atomic orbital basis set-based DFT codes, where the wave function is eventually represented on a complete grid. Additionally, we anticipate that this method will have applications in auxiliary field quantum Monte Carlo methods, where the bulk of the computational effort also lies in evaluating exchange energy on many Slater determinants.…”

“…Our formalism will also apply to time-dependent hybrid-DFT, where, like in GKS–DFT SCF, the ⟨ϕ q ϕ s |ξ⟩ vectors would be evaluated once while the exchange matrix, eqs and , will be updated repeatedly, here once per time step. It will be useful both for real-time TDDFT and for frequency-resolved TDDFT and BSE. , We also expect applications within atomic orbital basis set-based DFT codes, where the wave function is eventually represented on a complete grid. Additionally, we anticipate that this method will have applications in auxiliary field quantum Monte Carlo methods, where the bulk of the computational effort also lies in evaluating exchange energy on many Slater determinants. − …”