Dealing with the exponential wall in electronic structure calculations

An alternative to Density Functional Theory are wavefunction based electronic structure calculations for solids. In order to perform them the Exponential Wall (EW) problem has to be resolved. It is caused by an exponential increase of the number of configurations with increasing electron number N. There are different routes one may follow. One is to characterize a many-electron wavefunction by a vector in Liouville space with a cumulant metric rather than in Hilbert space. This removes the EW problem. Another is to model the solid by an {\it impurity} or {\it fragment} embedded in a {\it bath} which is treated at a much lower level than the former. This is the case in Density Matrix Embedding Theory (DMET) or Density Embedding Theory (DET). The latter are closely related to a Schmidt decomposition of a system and to the determination of the associated entanglement. We show here the connection between the two approaches. It turns out that the DMET (or DET) has an identical active space as a previously used Local Ansatz, based on a projection and partitioning approach. Yet, the EW problem is resolved differently in the two cases. By studying a $H_{10}$ ring these differences are analyzed with the help of the method of increments.Comment: 19 pages, 5 figure

Physico-chemical variables determining the invasion risk of freshwater habitats by alien mollusks and crustaceans

The aim of this study was to assess the invasion risk of freshwater habitats and determine the environmental variables that are most favorable for the establishment of alien amphipods, isopods, gastropods, and bivalves. A total of 981 sites located in streams and rivers in Germany. Therefore we analyzed presence-absence data of alien and indigenous amphipods, isopods, gastropods, and bivalves from 981 sites located in small to large rivers in Germany with regard to eight environmental variables: chloride, ammonium, nitrate, oxygen, orthophosphate, distance to the next navigable waterway, and maximum and minimum temperature. Degraded sites close to navigable waters were exposed to an increased invasion risk by all major groups of alien species. Moreover, invaded sites by all four groups of alien species were similar, whereas the sites where indigenous members of the four groups occurred were more variable. Increased temperature and chloride concentration as well as decreased oxygen concentration were identified as major factors for the invasibility of a site. Species-specific analyses showed that chloride was among the three most predictive environmental variables determining species assemblage in all four taxonomic groups. Also distance to the next navigable waterways was similarly important. Additionally, the minimum temperature was among the most important variables for amphipods, isopods, and bivalves. The bias in the occurrence patterns of alien species toward similarly degraded habitats suggests that the members of all four major groups of freshwater alien species are a non-random, more tolerant set of species. Their common tolerance to salinity, high temperature, and oxygen depletion may reflect that most alien species were spread in ballast water tanks, where strong selective pressures, particularly temperature fluctuations, oxygen depletion, and increased salinity may create a bottleneck for successful invasion. Knowledge on the major factors that influence the invasion risk of a habitat is needed to develop strategies to limit the spread of invasive species

A Hamiltonian Krylov-Schur-type method based on the symplectic Lanczos process

We discuss a Krylov-Schur like restarting technique applied within the symplectic Lanczos algorithm for the Hamiltonian eigenvalue problem. This allows to easily implement a purging and locking strategy in order to improve the convergence properties of the symplectic Lanczos algorithm. The Krylov-Schur-like restarting is based on the SR algorithm. Some ingredients of the latter need to be adapted to the structure of the symplectic Lanczos recursion. We demonstrate the efficiency of the new method for several Hamiltonian eigenproblems

Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements

We consider the e�cient solution of the Cahn-Hilliard variational inequality using an implicit time discretization, which is formulated as an optimal control problem with pointwise constraints on the control. By applying a semi-smooth Newton method combined with a Moreau-Yosida regularization technique for handling the control constraints we show superlinear convergence in function space. At the heart of this method lies the solution of large and sparse linear systems for which we propose the use of preconditioned Krylov subspace solvers using an e�ective Schur complement approximation. Numerical results illustrate the competitiveness of this approach

Ab-Initio Calculation of the Metal-Insulator Transition in Lithium rings

We study how the Mott metal-insulator transition (MIT) is affected when we have to deal with electrons with different angular momentum quantum numbers. For that purpose we apply ab-initio quantum-chemical methods to lithium rings in order to investigate the analogue of a MIT. By changing the interatomic distance we analyse the character of the many-body wavefunction and discuss the importance of the $s-p$ orbital quasi-degeneracy within the metallic regime. The charge gap (ionization potential minus electron affinity) shows a minimum and the static electric dipole polarizability has a pronounced maximum at a lattice constant where the character of the wavefunction changes from significant $p$ to essentially $s$-type. In addition, we examine rings with bond alternation in order to answer the question under which conditions a Peierls distortion occurs.Comment: 9 pages, 11 figure

Obtaining Wannier Functions of a Crystalline Insulator within a Hartree-Fock approach: Applications to LiF and LiCl

An ab initio Hartree-Fock approach aimed at directly obtaining the localized orthogonal orbitals (Wannier functions) of a crystalline insulator is described in detail. The method is used to perform all-electron calculations on the ground states of crystalline lithium fluoride and lithium chloride, without the use of any pseudo or model potentials. Quantities such as total energy, x-ray structure factors and Compton profiles obtained using the localized Hartree-Fock orbitals are shown to be in excellent agreement with the corresponding quantities calculated using the conventional Bloch-orbital based Hartree-Fock approach. Localization characteristics of these orbitals are also discussed in detail.Comment: 39 Pages, RevTex, 4 postscript figures, to appear in PRB15, January 9

Wavefunction-based correlated ab initio calculations on crystalline solids

We present a wavefunction-based approach to correlated ab initio calculations on crystalline insulators of infinite extent. It uses the representation of the occupied and the unoccupied (virtual) single-particle states of the infinite solid in terms of Wannier functions. Electron correlation effects are evaluated by considering virtual excitations from a small region in and around the reference cell, keeping the electrons of the rest of the infinite crystal frozen at the Hartree-Fock level. The method is applied to study the ground state properties of the LiH crystal, and is shown to yield rapidly convergent results.Comment: 6 pages, RevTex, to appear in Phys. Rev.

Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods

Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.Comment: 29 page