# An O(N-3) implementation of Hedin's GW approximation for molecules

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Daniel Sánchez-Portal, Jul 19, 2015 Available from:-
- "labels the auxiliary basis functions. The auxiliary basis functions are constructed either explicitly (e.g., a set of Gaussian-type atom-centered basis functions) or implicitly by using singular value decomposition on the overlap matrix of the set of N 2 functions ρ ij (x) [4] [5]. "

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**ABSTRACT:**Electron repulsion integral tensor has ubiquitous applications in quantum chemistry calculations. In this work, we propose an algorithm which compresses the electron repulsion tensor into the tensor hypercontraction format with $\mathcal{O}(n N^2 \log N)$ computational cost, where $N$ is the number of orbital functions and $n$ is the number of spatial grid points that the discretization of each orbital function has. The algorithm is based on a novel strategy of density fitting using a selection of a subset of spatial grid points to approximate the pair products of orbital functions on the whole domain. - [Show abstract] [Hide abstract]

**ABSTRACT:**Graphene plasmons are emerging as an alternative solution to noble metal plasmons, adding the advantages of tunability via electrostatic doping and long lifetimes. These excitations have been so far described using classical electrodynamics, with the carbon layer represented by a local conductivity. However, the question remains, how accurately is such a classical description representing graphene? What is the minimum size for which nonlocal and quantum finite-size effects can be ignored in the plasmons of small graphene structures? Here, we provide a clear answer to these questions by performing first-principles calculations of the optical response of doped nanostructured graphene obtained from a tight-binding model for the electronic structure and the random-phase approximation for the dielectric response. The resulting plasmon energies are in good agreement with classical local electromagnetic theory down to ∼10 nm sizes, below which plasmons split into several resonances that emphasize the molecular character of the carbon structures and the quantum nature of their optical excitations. Additionally, finite-size effects produce substantial plasmon broadening compared to homogeneous graphene up to sizes well above 20 nm in nanodisks and 10 nm in nanoribbons. The atomic structure of edge terminations is shown to be critical, with zigzag edges contributing to plasmon broadening significantly more than armchair edges. This study demonstrates the ability of graphene nanostructures to host well-defined plasmons down to sizes below 10 nm, and it delineates a roadmap for understanding their main characteristics, including the role of finite size and nonlocality, thus providing a solid background for the emerging field of graphene nanoplasmonics.ACS Nano 01/2012; 6(2):1766-75. DOI:10.1021/nn204780e · 12.03 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We present a systematic study of the performance of numerical pseudo-atomic orbital basis sets in the calculation of dielectric matrices of extended systems using the self-consistent Sternheimer approach of [F. Giustino et al., Phys. Rev. B 81 (11), 115105 (2010)]. In order to cover a range of systems, from more insulating to more metallic character, we discuss results for the three semiconductors diamond, silicon, and germanium. Dielectric matrices calculated using our method fall within 1-3% of reference planewaves calculations, demonstrating that this method is promising. We find that polarization orbitals are critical for achieving good agreement with planewaves calculations, and that only a few additional \zeta 's are required for obtaining converged results, provided the split norm is properly optimized. Our present work establishes the validity of local orbital basis sets and the self-consistent Sternheimer approach for the calculation of dielectric matrices in extended systems, and prepares the ground for future studies of electronic excitations using these methods.Physics of Condensed Matter 02/2012; 85(9). DOI:10.1140/epjb/e2012-30106-3 · 1.46 Impact Factor