Cubic-scaling iterative solution of the Bethe-Salpeter equation for finite systems

Abstract

Under the terms of the Creative Commons Attribution License 3.0 (CC-BY). ; The Bethe-Salpeter equation (BSE) is currently the state of the art in the description of neutral electronic 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. 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(N3) with the number of atoms, while the number of iterations typically is 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(N2) operation of applying the Coulomb kernel in the atomic orbital representation. ; We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) through the SFB 1083 project, the ANR ORGAVOLT project, and the Spanish MINECO MAT2013-46593-C6-2-P project. P.K. acknowledges financial support from the Fellows Gipuzkoa program of the Gipuzkoako Foru Aldundia through the FEDER funding scheme of the European Union, Una manera de hacer Europa. F.F. acknowledges support from the EXTRA programme of the "Universita degli Studi di Milano-Bicocca" and from the Erasmus Placement programme for student mobility. ; Peer Reviewed