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
Organic electronics is a rapidly developing technology. Typically, the molecules involved in organic electronics are made up of hundreds of atoms, prohibiting a theoretical description by wavefunction-based ab-initio methods. Density-functional and G
reens function type of methods scale less steeply with the number of atoms. Therefore, they provide a suitable framework for the theory of such large systems. In this contribution, we describe an implementation, for molecules, of Hedins GW approximation. The latter is the lowest order solution of a set of coupled integral equations for electronic Greens and vertex functions that was found by Lars Hedin half a century ago. Our implementation of Hedins GW approximation has two distinctive features: i) it uses sets of localized functions to describe the spatial dependence of correlation functions, and ii) it uses spectral functions to treat their frequency dependence. Using these features, we were able to achieve a favorable computational complexity of this approximation. In our implementation, the number of operations grows as N^3 with the number of atoms N.
We describe a fast parallel iterative method for computing molecular absorption spectra within TDDFT linear response and using the LCAO method. We use a local basis of dominant products to parametrize the space of orbital products that occur in the L
CAO approach. In this basis, the dynamical polarizability is computed iteratively within an appropriate Krylov subspace. The iterative procedure uses a a matrix-free GMRES method to determine the (interacting) density response. The resulting code is about one order of magnitude faster than our previous full-matrix method. This acceleration makes the speed of our TDDFT code comparable with codes based on Casidas equation. The implementation of our method uses hybrid MPI and OpenMP parallelization in which load balancing and memory access are optimized. To validate our approach and to establish benchmarks, we compute spectra of large molecules on various types of parallel machines. The methods developed here are fairly general and we believe they will find useful applications in molecular physics/chemistry, even for problems that are beyond TDDFT, such as organic semiconductors, particularly in photovoltaics.
