Abstract
The Bethe-Salpeter equation (BSE) is currently the state of the art in the description of neutral electron excitations in both solids and large finite systems. It is capable of accurately treating charge-transfer excitations that present difficulties for simpler approaches. We present a local basis set formulation of the BSE for molecules where the optical spectrum is computed with the iterative Haydock recursion scheme, leading to a low computational complexity and memory footprint. Using a variant of the algorithm we can go beyond the Tamm-Dancoff approximation (TDA). We rederive the recursion relations for general matrix elements of a resolvent, show how they translate into continued fractions, and study the convergence of the method with the number of recursion coefficients and the role of different terminators. Due to the locality of the basis functions the computational cost of each iteration scales asymptotically as $O(N^3)$ with the number of atoms, while the number of iterations is typically much lower than the size of the underlying electron-hole basis. In practice we see that , even for systems with thousands of orbitals, the runtime will be dominated by the $O(N^2)$ operation of applying the Coulomb kernel in the atomic orbital representation
Highlights
Ab initio simulations of optical spectra are essential tools in the study of excited state electronic properties of solids, molecules, and nanostructures
We present a local basis set formulation of the Bethe-Salpeter equation (BSE) for molecules where the optical spectrum is computed with the iterative Haydock recursion scheme, leading to a low computational complexity and memory footprint
Implemented, an iterative scheme to compute the optical response of molecular systems at the Bethe-Salpeter level, using local basis sets
Summary
Ab initio simulations of optical spectra are essential tools in the study of excited state electronic properties of solids, molecules, and nanostructures. Avoiding an explicit diagonalization of the BSE Hamiltonian can be done by using an iterative method to obtain a few low-lying transitions (e.g., the Davidson method [31,32]), or to directly aim for the spectrum, which can be done frequency by frequency using, for example, the generalized minimal residual method [31,33,34] or for the full spectrum with the Haydock recursion scheme [20,35,36] Another option is to go over to the time domain and solve the equations of motion by time propagation [37,38]. Benedict and Shirley made use of the Haydock recursion method to compute optical spectra in the Tamm-Dancoff approximation (TDA) without computing the whole BSE Hamiltonian [23] This was achieved by using, in addition to the particle-hole basis, a real space grid product basis |x, y , in which the screened direct Coulomb interaction is diagonal (the exchange term is sparse in this representation). We present proof of principle calculations of our implementation, where the runtime is seen to be dominated by the O(N 2) scaling operations for systems up to several thousand orbitals, and discuss some of the bottlenecks and possible improvements of the scheme
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