Accelerating Optical Absorption Spectra and Exciton Energy Computation via Interpolative Separable Density Fitting

Abstract

We present an efficient way to solve the Bethe-Salpeter equation (BSE), a model for the computation of absorption spectra in molecules and solids that includes electron-hole excitations. Standard approaches to construct and diagonalize the Bethe-Salpeter Hamiltonian require at least O(N 5 e ) operations, where Ne is proportional to the number of electrons in the system, limiting its application to small systems. Our approach is based on the interpolative separable density fitting (ISDF) technique to construct low rank approximations to the bare and screened exchange operators associated with the BSE Hamiltonian. This approach reduces the complexity of the Hamiltonian construction to O(N 3 e ) with a much smaller pre-constant. Here, we implement the ISDF method for the BSE calculations within the Tamm-Dancoff approximation (TDA) in the BerkeleyGW software package. We show that ISDF-based BSE calculations in molecules and solids reproduce accurate exciton energies and optical absorption spectra with significantly reduced computational cost.

Overcoming the Memory Bottleneck in Auxiliary Field Quantum Monte Carlo Simulations with Interpolative Separable Density Fitting

“…are a subset of these grid points, called the interpolating points, ζ µ (r) are a set of interpolating vectors and N µ = c µ M is the number of interpolating points. Importantly, previous results suggest that c µ is roughly independent of the system size, and a value in the range between 10 − 15 is usually sufficient to reproduce total energies to sub mHa/atom accuracy 42,43,53 . Inserting…”

“…We could proceed and compute v abkl using the existing ISDF orbital products constructed using the full set of M orbitals. However, it has been found previously that total energies typically converge faster with respect to c µ when ISDF is performed on orbital products containing only the occupied set of orbitals, or at least with one set of occupied orbitals and one set of virtual orbitals 53 . Thus, we instead perform a second ISDF factorization on the 'half-transformed' orbital product set…”

“…49-51, that offers just this. The ISDF approach has so far been applied to speeding up develop a cubic scaling RPA algorithm 52 , speeding up hybrid functional calculations in DFT 42,43 and the computation of excited state properties using the Bethe-Salpether approach 53 . Here our aim is to adapt it to AFQMC.…”

Fast optical absorption spectra calculations for periodic solid state systems

“…We can further use these efficient matrix-vector multiplications in a structure preserving Lanczos algorithm [33] to obtain an approximate absorption spectrum without an explicit diagonalization of the approximate H BSE . This paper generalizes the work in [12] to periodic solid state systems. According to the Bloch decomposition, each single particle orbital in a periodic system can be characterized by an orbital index i, and a Brillouin zone index k. Compared to isolated systems, the total number of electrons N e is equal to the number of electrons per unit cell multiplied by the number of k points denoted by N k .…”

“…In a recent work [12], two of the authors have presented an efficient way to construct the BSH for molecular systems, and to efficiently solve the BSE eigenvalue problem using an iterative scheme. Our approach is based on the recently-developed interpolative separable density fitting (ISDF) decomposition [19,20].…”