A Structure Preserving Lanczos Algorithm for Computing the Optical Absorption Spectrum

Shao, Meiyue; Jornada, Felipe H. da; Lin, Lin; Yang, Chao; Deslippe, Jack; Louie, Steven G.

doi:10.1137/16m1102641

Cited by 14 publications

(10 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Ref. [3,25], we developed a structure preserving Lanczos algorithm for estimating the optical spectrum. The algorithm has been implemented in the BSEPACK [27] library.…”

Section: Estimating Optical Absorption Spectra Without Diagonalizationmentioning

confidence: 99%

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

Shao

Cepellotti

et al. 2018

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…In Ref. [3,25], we developed a structure preserving Lanczos algorithm for estimating the optical spectrum. The algorithm has been implemented in the BSEPACK [27] library.…”

Section: Estimating Optical Absorption Spectra Without Diagonalizationmentioning

confidence: 99%

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

Shao

Cepellotti

et al. 2018

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…In computational quantum chemistry and physics, the random phase approximation (RPA) or the Bethe-Salpeter (BS) equation describe the excitation states and absorption spectra for molecules or the surfaces of solids [1,2]. One important question in the RPA or BS equation is how to compute a few eigenpairs associated with several of the smallest positive eigenvalues of the following eigenvalue problem:…”

Section: Introductionmentioning

confidence: 99%

“…where K = A − B and M = A + B. The eigenvalue problem (2) was still referred to as the linear response eigenvalue problem (LREP) [3,6] and will be so in this paper, as well. The condition imposed upon A and B in (1) implies that both K and M are N × N real symmetric positive definite matrices.…”

Section: Introductionmentioning

confidence: 99%

“…Since the dimension N is usually very large for systems of practical interest, solving LREP (2) is a very challenging problem. The iterative methods are recent efforts for computing the partial spectrum of (2), such as the locally-optimal block preconditioned 4D conjugate gradient method (LOBP4DCG) [6] and its space-extended variation [16], the block Chebyshev-Davidson method [10], as well as the generalized Lanczos-type methods [8,9]. These algorithms are all based on the so-called pair of deflating subspaces, which is a generalization of the concept of the invariant subspace in the standard eigenvalue problems.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A FEAST Algorithm for the Linear Response Eigenvalue Problem

Teng

2019

Algorithms

View full text Add to dashboard Cite

In the linear response eigenvalue problem arising from quantum chemistry and physics, one needs to compute several positive eigenvalues together with the corresponding eigenvectors. For such a task, in this paper, we present a FEAST algorithm based on complex contour integration for the linear response eigenvalue problem. By simply dividing the spectrum into a collection of disjoint regions, the algorithm is able to parallelize the process of solving the linear response eigenvalue problem. The associated convergence results are established to reveal the accuracy of the approximated eigenspace. Numerical examples are presented to demonstrate the effectiveness of our proposed algorithm. From (2), we have Kx = λy and My = λx, and they together lead to:

show abstract

“…By keeping the interaction kernels in a decomposed form, the matrix-vector multiplications required in the iterative diagonalization procedures of the Hamiltonian H BSE can be performed efficiently. 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.…”

mentioning

confidence: 99%

Fast optical absorption spectra calculations for periodic solid state systems

Henneke,

Lin,

Vorwerk

et al. 2019

Preprint

View full text Add to dashboard Cite

We present a method to construct an efficient approximation to the bare exchange and screened direct interaction kernels of the Bethe-Salpeter Hamiltonian for periodic solid state systems via the interpolative separable density fitting technique. We show that the cost of constructing the approximate Bethe-Salpeter Hamiltonian scales nearly optimally as O(N k ) with respect to the number of samples in the Brillouin zone N k . In addition, we show that the cost for applying the Bethe-Salpeter Hamiltonian to a vector scales as O(N k log N k ). Therefore the optical absorption spectrum, as well as selected excitation energies can be efficiently computed via iterative methods such as the Lanczos method. This is a significant reduction from the O(N 2 k ) and O(N 3 k ) scaling associated with a brute force approach for constructing the Hamiltonian and diagonalizing the Hamiltonian respectively. We demonstrate the efficiency and accuracy of this approach with both one-dimensional model problems and three-dimensional real materials (graphene and diamond). For the diamond system with N k = 2197, it takes 6 hours to assemble the Bethe-Salpeter Hamiltonian and 4 hours to fully diagonalize the Hamiltonian using 169 cores when the brute force approach is used. The new method takes less than 3 minutes to set up the Hamiltonian and 24 minutes to compute the absorption spectrum on a single core.

show abstract

A Structure Preserving Lanczos Algorithm for Computing the Optical Absorption Spectrum

Cited by 14 publications

References 38 publications

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

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

A FEAST Algorithm for the Linear Response Eigenvalue Problem

Fast optical absorption spectra calculations for periodic solid state systems

Contact Info

Product

Resources

About