Efficient computer implementations of the GW approximation must approximate a numerically challenging frequency integral; the integral can be performed analytically, but doing so leads to an expensive implementation whose computational cost scales as O(N6), where N is the size of the system. Here, we introduce a new formulation of the full-frequency GW approximation by exactly recasting it as an eigenvalue problem in an expanded space. This new formulation (1) avoids the use of time or frequency grids, (2) naturally obviates the need for the common “diagonal” approximation, (3) enables common iterative eigensolvers that reduce the canonical scaling to O(N5), and (4) enables a density-fitted implementation that reduces the scaling to O(N4). We numerically verify these scaling behaviors and test a variety of approximations that are motivated by this new formulation. The new formulation is found to be competitive with conventional O(N4) methods based on analytic continuation or contour deformation. In this new formulation, the relation of the GW approximation to configuration interaction, coupled-cluster theory, and the algebraic diagrammatic construction is made especially apparent, providing a new direction for improvements to the GW approximation.

Inspired by Grimmeʼs simplified Tamm−Dancoff density functional theory approach [Grimme, S. J. Chem. Phys. 2013, 138, 244104], we describe a simplified approach to excited-state calculations within the GW approximation to the self-energy and the Bethe−Salpeter equation (BSE), which we call sGW/sBSE. The primary simplification to the electron repulsion integrals yields the same structure as with tensor hypercontraction, such that our method has a storage requirement that grows quadratically with system size and computational timing that grows cubically with system size. The performance of sGW is tested on the ionization potential of the molecules in the GW100 test set, for which it differs from ab initio GW calculations by only 0.2 eV. The performance of sBSE (based on the sGW input) is tested on the excitation energies of molecules in Thielʼs set, for which it differs from ab initio GW/BSE calculations by about 0.5 eV. As examples of the systems that can be routinely studied with sGW/sBSE, we calculate the band gap and excitation energy of hydrogen-passivated silicon nanocrystals with up to 2650 electrons in 4678 spatial orbitals and the absorption spectra of two large organic dye molecules with hundreds of atoms.

We present an algorithm and implementation of integral-direct, density-fitted Hartree–Fock (HF) and second-order Møller–Plesset perturbation theory (MP2) for periodic systems. The new code eliminates the formerly prohibitive storage requirements and allows us to study systems 1 order of magnitude larger than before at the periodic MP2 level. We demonstrate the significance of the development by studying the benzene crystal in both the thermodynamic limit and the complete basis set limit, for which we predict an MP2 cohesive energy of −72.8 kJ/mol, which is about 10–15 kJ/mol larger in magnitude than all previously reported MP2 calculations. Compared to the best theoretical estimate from literature, several modified MP2 models approach chemical accuracy in the predicted cohesive energy of the benzene crystal and hence may be promising cost-effective choices for future applications on molecular crystals.

The Bethe–Salpeter equation (BSE) that results from the GW approximation to the self-energy is a frequency-dependent (nonlinear) eigenvalue problem due to the dynamically screened Coulomb interaction between electrons and holes. The computational time required for a numerically exact treatment of this frequency dependence is O(N6), where N is the system size. To avoid the common static screening approximation, we show that the full-frequency dynamical BSE can be exactly reformulated as a frequency-independent eigenvalue problem in an expanded space of single and double excitations. When combined with an iterative eigensolver and the density fitting approximation to the electron repulsion integrals, this reformulation yields a dynamical BSE algorithm whose computational time is O(N5), which we verify numerically. Furthermore, the reformulation provides direct access to excited states with dominant double excitation character, which are completely absent in the spectrum of the statically screened BSE. We study the 21Ag state of butadiene, hexatriene, and octatetraene and find that GW/BSE overestimates the excitation energy by about 1.5–2 eV and significantly underestimates the double excitation character.

Inspired by Grimme's simplified Tamm-Dancoff density functional theory approach [S. Grimme, J. Chem. Phys. 138, 244104 (2013)], we describe a simplified approach to excited state calculations within the GW approximation to the self-energy and the Bethe-Salpeter equation (BSE), which we call sGW/sBSE. The primary simplification to the electron repulsion integrals yields the same structure as with tensor hypercontraction, such that our method has a storage requirement that grows quadratically with system size and computational timing that grows cubically with system size. The performance of sGW is tested on the ionization potential of the molecules in the GW100 test set, for which it differs from ab intio GW calculations by only 0.2 eV. The performance of sBSE (based on sGW input) is tested on the excitation energies of molecules in the Thiel set, for which it differs from ab intio GW/BSE calculations by about 0.5 eV. As examples of the systems that can be routinely studied with sGW/sBSE, we calculate the band gap and excitation energy of hydrogen-passivated silicon nanocrystals with up to 2650 electrons in 4678 spatial orbitals and the absorption spectra of two large organic dye molecules with hundreds of atoms.