We present a new cubic scaling algorithm for the calculation of the RPA correlation energy. Our scheme splits up the dependence between the occupied and virtual orbitals in χ 0 by use of Cauchy's integral formula. This introduces an additional integral to be carried out, for which we provide a geometrically convergent quadrature rule. Our scheme also uses the newly developed Interpolative Separable Density Fitting algorithm to further reduce the computational cost in a way analogous to that of the Resolution of Identity method.are, respectively, the kinetic energy of the effective single-particle system, the potential energy due to the external potential V ext , and the so-called Hartree energy, which represents the classical contribution of the energy from the Coulomb interaction between electrons. The remaining term in (1.2), E xc , is known as the exchange-correlation energy. It has no known simple form Date: November 5, 2018.

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.

We consider a modification of the OMM energy functional which contains an ℓ 1 penalty term in order to find a sparse representation of the low-lying eigenspace of self-adjoint operators. We analyze the local minima of the modified functional as well as the convergence of the modified functional to the original functional. Algorithms combining soft thresholding with gradient descent are proposed for minimizing this new functional. Numerical tests validate our approach. As an added bonus, we also prove the unanticipated and remarkable property that every local minimum the OMM functional without the ℓ 1 term is also a global minimum.

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