Tensor Product Approximation (DMRG) and Coupled Cluster method in Quantum Chemistry

We present the Copupled Cluster (CC) method and the Density matrix Renormalization Grooup (DMRG) method in a unified way, from the perspective of recent developments in tensor product approximation. We present an introduction into recently developed hierarchical tensor representations, in particular tensor trains which are matrix product states in physics language. The discrete equations of full CI approximation applied to the electronic Schr\"odinger equation is casted into a tensorial framework in form of the second quantization. A further approximation is performed afterwards by tensor approximation within a hierarchical format or equivalently a tree tensor network. We establish the (differential) geometry of low rank hierarchical tensors and apply the Driac Frenkel principle to reduce the original high-dimensional problem to low dimensions. The DMRG algorithm is established as an optimization method in this format with alternating directional search. We briefly introduce the CC method and refer to our theoretical results. We compare this approach in the present discrete formulation with the CC method and its underlying exponential parametrization.Comment: 15 pages, 3 figure

Chelators in Iron and Copper Toxicity

Purpose of Review Chelation therapy is used for diseases causing an imbalance of iron levels (for example haemochromatosis and thalassaemia) or copper levels (for example Menkes’ and Wilson’s diseases). Currently, most pharmaceutical chelators are relatively simple but often have side effects. Some have been taken off the market. This review attempts to find theory and knowledge required to design or find better chelators. Recent Findings Recent research attempting to understand the biological mechanisms of protection against iron and copper toxicity is reviewed. Understanding of molecular mechanisms behind normal iron/copper regulation may lead to the design of more sophisticated chelators. The theory of metal ion toxicity explains why some chelators, such as EDTA, which chelate metal ions in a way which exposes the ion to the surrounding environment are shown to be unsuitable except as a means of killing cancer cells. The Lewis theory of acids and bases suggests which amino acids favour the attachment of the hard/intermediate ions Fe2+, Fe3+, Cu2+ and soft ion Cu+. Non-polar amino acids will chelate the ion in a position not in contact with the surrounding cellular environment. The conclusion is that only the soft ion binding cysteine and methionine appear as suitable chelators. Clearly, nature has developed proteins which are less restricted. Recent research on naturally produced chelators such as siderophores and phytochemicals show some promise as pharmaceuticals. Summary Although an understanding of natural mechanisms of Fe/Cu regulation continues to increase, the pharmaceutical chelators for metal overload diseases remain simple non-protein molecules. Natural and synthetic alternatives have been studied but require further research before being accepted

Geometric methods on low-rank matrix and tensor manifolds

In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors