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....1] for further references. Low-rank matrix and tensor techniques have also been used successfully for performing large-scale electronic structure calculations in computational quantum chemistry; see =-=[5, 20, 25, 24]-=- for examples. In this work, we propose a novel low-rank tensor method that allows for the computation of several smallest eigenvalues, offers rank adaptivity, and the possibility of incorporating pre...

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...e real space numerical methods combined with FEM or plane waves approximations have become attractive in (post) Hartree-Fock and DFT calculations as the possible alternative to traditional approaches =-=[87, 47, 16, 20, 32, 17, 34, 107]-=-. Literature surveys on the most frequently used tensor formats can be found in [69, 110, 42, 39]. In addition, methods of multilinear algebra and nonlinear tensor approximation have been discussed, s...

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...cker and the Kohn-Sham LDA models. Together with Y. Maday, E. Cancès and V. Ehrlacher have analyzed the computation of eigenvalues in spectral gaps of locally perturbed periodic Schrödinger operators =-=[23]-=-. In [53], they have introduced a general theoretical framework to analyze non-consistent approximations of the discrete eigenmodes of a self-adjoint operator, focusing in particular on the discrete e...

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...o compute the spectrum of operators of the form (4.6) is the supercell method. It consists in considering a large box ΓL, containing the defect, with periodic boundary conditions. In a recent article =-=[27]-=-, Cancès, Ehrlacher and Maday prove that using the supercell method with a planewave discretization gives no spectral pollution. This follows previous results in [104, 18]. The drawbacks of this metho...

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...Section 1.6.2 presents a short introduction to this issue along with a summary of our contributions. Chapters 3 and 4 detail all our theoretical results and proofs which were gathered in two articles =-=[40]-=- and [41] currently in review. Many-body Schrödinger model In the sequel, we will work in atomic units, which implies that � = 1, e = 1, me = 1, 4πε0 = 1, tel-00719466, version 2 - 11 Feb 2013 where �...

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...challenges in the numerical treatment of periodic systems and perturbed periodic systems in quantum chemical computations for crystalline, metallic and polymertype compounds, see [12, 43, 36, 47] and =-=[46, 40, 37, 6, 38]-=-. One of them is the the lattice summation over a huge number of Newton kernels distributed on large 3D computational grid. This problem is also considered to be a demanding computational task in the ...

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...on basis is increased. The phenomenon, called spectral pollution, is well-known and well documented. For instance, it is encountered when dealing with perturbations of periodic Schrödinger operators =-=[6, 8]-=- or Sturm-Liouville operators [30, 31, 1]. It also appears in elasticity, electromagnetism and hydrodynamics [9, 3, 25, 22, 24, 7, 2, 15]. In this paper we are interested in relativistic computations ...

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...tic treatment of arising integral operators [12, 40, 25]. The mathematical analysis of spectral problems for PDEs with the periodic-type coefficients was an attractive topic in the recent decade, see =-=[6, 7, 13]-=- and the references therein. However, the systematic developments and optimization of the basic numerical algorithms in the HartreeFock calculations for large lattice structured compounds still are la...

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...allenges in the numerical treatment of many-body systems in molecular dynamics, quantum chemical computations, simulations of large solvated biological systems and other applications, see for example =-=[43, 36, 6, 37, 14, 38]-=-, [42, 21, 10], and [12]. Beginning with the widely spread Ewald summation techniques [15], the development of lattice-sum methods has led to well established algorithms for numerical evaluation of lo...