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**1 - 6**of**6**### Self-assembled quantum dots in a nanowire system for quantum photonics

, 2013

"... quantum photonics ..."

### Density Functional Theory: Fundamentals and Applications in Condensed Matter Physics

, 2011

"... A complete quantum mechanical description of a solid requires the solution of the many-body Schrödinger equation [8]. The numerical approximation of this equation, however, is impractical: using a straightforward numerical discretization, the number of degrees of freedom grows exponentially with the ..."

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A complete quantum mechanical description of a solid requires the solution of the many-body Schrödinger equation [8]. The numerical approximation of this equation, however, is impractical: using a straightforward numerical discretization, the number of degrees of freedom grows exponentially with the number of electrons, and therefore only systems with very few atoms can be considered. In [18], Hohenberg and Kohn proved that there exists a universal functional of the electronic density F [u] such that the ground state energy associated to an external potential v can be obtained by minimizing the energy E[u] = F [u] + v(x)u(x) dx. (1) Further refinements of the theory were presented by Levy [21] and Lieb [23]. In (1), however, the shape of functional F [u] is unknown, and must be approximated. There are two main approaches for its approximation, which have given origin to what are known as Orbital-Free Density-Functional Theory (OFDFT), and Kohn-Sham Density-Functional Theory (KSDFT). In OFDFT, the functional F is replaced by an explicit functional of u. The simplest such approximation is the Thomas-Fermi functional [32, 13], and by now several such approximations have been proposed [34, 35, 37, 36, 39]. To achieve good physical accuracy, however, complicated, nonlocal kinetic energy functionals must be used. An example of such functionals is the Wang-Teter functional. Although in principle it captures linear response effects, the kinetic energy is unbounded below, rendering it useless for practical use [2]. Density dependent kernels have been developed, and although they do not seem to suffer from this difficulty, they are not well understood, their numerical implementation becomes cumbersome, and there does not seem to be a systematic way of improving their accuracy. In a different direction, Kohn and Sham introduced an approximation scheme by decomposing the energy functional F [u] as [20] E[u] = 1 2 N∑ i=1

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"... Economic impact............................................................................................................................. 4 ..."

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Economic impact............................................................................................................................. 4

### 1 SCIENTIFIC HIGHLIGHT OF THE MONTH The ELPA Library – Scalable Parallel Eigenvalue Solutions for Electronic Structure Theory and Computational Science

"... Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N3) with the size of the investigated problem, N, and thus often defines the system size limit tha ..."

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Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N3) with the size of the investigated problem, N, and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the ELPA library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigen-value problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as general-ized eigenvalue problems, relying on the well documented matrix layout of the ScaLAPACK library but replacing all actual parallel solution steps with subroutines of its own. The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding