531 research outputs found
Recent mathematical advances in coupled cluster theory
This article presents an in-depth educational overview of the latest
mathematical developments in coupled cluster (CC) theory, beginning with
Schneider's seminal work from 2009 that introduced the first local analysis of
CC theory. We offer a tutorial review of second quantization and the CC ansatz,
laying the groundwork for understanding the mathematical basis of the theory.
This is followed by a detailed exploration of the most recent mathematical
advancements in CC theory.Our review starts with an in-depth look at the local
analysis pioneered by Schneider which has since been applied to analyze various
CC methods. We then move on to discuss the graph-based framework for CC methods
developed by Csirik and Laestadius. This framework provides a comprehensive
platform for comparing different CC methods, including multireference
approaches. Next, we delve into the latest numerical analysis results analyzing
the single reference CC method developed by Hassan, Maday, and Wang. This very
general approach is based on the invertibility of the CC function's Fr\'echet
derivative. We conclude the article with a discussion on the recent
incorporation of algebraic geometry into CC theory, highlighting how this novel
and fundamentally different mathematical perspective has furthered our
understanding and provides exciting pathways to new computational approaches
One-Dimensional Lieb-Oxford Bounds
We investigate and prove Lieb-Oxford bounds in one dimension by studying
convex potentials that approximate the ill-defined Coulomb potential. A
Lieb-Oxford inequality establishes a bound of the indirect interaction energy
for electrons in terms of the one-body particle density of a wave
function . Our results include modified soft Coulomb potential and
regularized Coulomb potential. For these potentials, we establish
Lieb-Oxford-type bounds utilizing logarithmic expressions of the particle
density. Furthermore, a previous conjectured form is discussed for different
convex potentials
Homotopy continuation methods for coupled-cluster theory in quantum chemistry
Homotopy methods have proven to be a powerful tool for understanding the
multitude of solutions provided by the coupled-cluster polynomial equations.
This endeavor has been pioneered by quantum chemists that have undertaken both
elaborate numerical as well as mathematical investigations. Recently, from the
perspective of applied mathematics, new interest in these approaches has
emerged using both topological degree theory and algebraically oriented tools.
This article provides an overview of describing the latter development
The Coupled-Cluster Formalism - a Mathematical Perspective
The Coupled-Cluster theory is one of the most successful high precision
methods used to solve the stationary Schr\"odinger equation. In this article,
we address the mathematical foundation of this theory with focus on the
advances made in the past decade. Rather than solely relying on spectral gap
assumptions (non-degeneracy of the ground state), we highlight the importance
of coercivity assumptions - G\aa rding type inequalities - for the local
uniqueness of the Coupled-Cluster solution. Based on local strong monotonicity,
different sufficient conditions for a local unique solution are suggested. One
of the criteria assumes the relative smallness of the total cluster amplitudes
(after possibly removing the single amplitudes) compared to the G\aa rding
constants. In the extended Coupled-Cluster theory the Lagrange multipliers are
wave function parameters and, by means of the bivariational principle, we here
derive a connection between the exact cluster amplitudes and the Lagrange
multipliers. This relation might prove useful when determining the quality of a
Coupled-Cluster solution. Furthermore, the use of an Aubin-Nitsche duality type
method in different Coupled-Cluster approaches is discussed and contrasted with
the bivariational principle
Coupled cluster theory: Towards an algebraic geometry formulation
Coupled cluster theory produced arguably the most widely used high-accuracy
computational quantum chemistry methods. Despite the approach's overall great
computational success, its mathematical understanding is so far limited to
results within the realm of functional analysis. The coupled cluster
amplitudes, which are the targeted objects in coupled cluster theory,
correspond to solutions to the coupled cluster equations, which is a system of
polynomial equations of at most degree four. The high dimensionality of the
electronic Schr\"odinger equation and the non-linearity of the coupled cluster
ansatz have so far stalled a formal analysis of this polynomial system. In this
article, we present algebraic investigations that shed light on the coupled
cluster equations and the root structure of this ansatz. This is of importance
for the a posteriori evaluation of coupled cluster calculations. To that end,
we investigate the root structure by means of Newton polytopes. We derive a
general v-description, which is subsequently turned into an h-description for
explicit examples. This perspective reveals an apparent connection between
Pauli's exclusion principle and the geometrical structure of the Newton
polytopes. We also propose an alternative characterization of the coupled
cluster equations projected onto singles and doubles as cubic polynomials on an
algebraic variety with certain sparsity patterns. Moreover, we provide
numerical simulations of two computationally tractable systems, namely, the two
electrons in four spin-orbitals system and the three electrons in six
spin-orbitals system. These simulations provide novel insight into the root
structure of the coupled cluster solutions when the coupled cluster ansatz is
truncated
Architecture of coatomer: Molecular characterization of delta-COP and protein interactions within the complex
Copyright © 2011 by The Rockefeller University Press.Coatomer is a cytosolic protein complex that forms the coat of COP I-coated transport vesicles. In our attempt to analyze the physical and functional interactions between its seven subunits (coat proteins, [COPs] alpha-zeta), we engaged in a program to clone and characterize the individual coatomer subunits. We have now cloned, sequenced, and overexpressed bovine alpha-COP, the 135-kD subunit of coatomer as well as delta-COP, the 57-kD subunit and have identified a yeast homolog of delta-COP by cDNA sequence comparison and by NH2-terminal peptide sequencing. delta-COP shows homologies to subunits of the clathrin adaptor complexes AP1 and AP2. We show that in Golgi-enriched membrane fractions, the protein is predominantly found in COP I-coated transport vesicles and in the budding regions of the Golgi membranes. A knock-out of the delta-COP gene in yeast is lethal. Immunoprecipitation, as well as analysis exploiting the two-hybrid system in a complete COP screen, showed physical interactions between alpha- and epsilon-COPs and between beta- and delta-COPs. Moreover, the two-hybrid system indicates interactions between gamma- and zeta-COPs as well as between alpha- and beta' COPs. We propose that these interactions reflect in vivo associations of those subunits and thus play a functional role in the assembly of coatomer and/or serve to maintain the molecular architecture of the complex.This work was supported by The Deutsche Forschungsgemeinschaft (SFB 352), the Human Frontier Science Program, and the Swiss National Science Foundation No. 31-43366.95
Analysis of The Tailored Coupled-Cluster Method in Quantum Chemistry
In quantum chemistry, one of the most important challenges is the static
correlation problem when solving the electronic Schr\"odinger equation for
molecules in the Born--Oppenheimer approximation. In this article, we analyze
the tailored coupled-cluster method (TCC), one particular and promising method
for treating molecular electronic-structure problems with static correlation.
The TCC method combines the single-reference coupled-cluster (CC) approach with
an approximate reference calculation in a subspace [complete active space
(CAS)] of the considered Hilbert space that covers the static correlation. A
one-particle spectral gap assumption is introduced, separating the CAS from the
remaining Hilbert space. This replaces the nonexisting or nearly nonexisting
gap between the highest occupied molecular orbital and the lowest unoccupied
molecular orbital usually encountered in standard single-reference quantum
chemistry. The analysis covers, in particular, CC methods tailored by
tensor-network states (TNS-TCC methods). The problem is formulated in a
nonlinear functional analysis framework, and, under certain conditions such as
the aforementioned gap, local uniqueness and existence are proved using
Zarantonello's lemma. From the Aubin--Nitsche-duality method, a quadratic error
bound valid for TNS-TCC methods is derived, e.g., for linear-tensor-network TCC
schemes using the density matrix renormalization group method
Continuous in-line monitoring of electrolyte concentrations in extracorporeal circuits for individualization of dialysis treatment
One objective of dialysis treatment is to normalize the blood
plasma electrolytes and remove waste products such as urea and creatinine
from blood. However, due to a shift in plasma osmolarity, a rapid or
excessive change of the electrolytes can lead to complications like
cardiovascular instability, overhydrating of cells, disequilibrium syndrome
and cardiac arrhythmias. Especially for critical ill patients in intensive
care unit with sepsis or multi-organ failure, any additional stress has to be
avoided. Since the exchange velocity of the electrolytes mainly depends on
the concentration gradients across the dialysis membrane between blood and
dialysate, it can be controlled by an individualized composition of dialysate
concentrations. In order to obtain a precise concentration gradient with the
individualized dialysate, it is necessary to continuously monitor the plasma
concentrations. However, with in-line sensors, the required hemocompatibility
is often difficult to achieve. In this work, we present a concept for
continuous in-line monitoring of electrolyte concentrations using
ion-selective electrodes separated from the blood flow by a dialysis
membrane, and therefore meeting the fluidic requirements for
hemocompatibility. First investigations of hemocompatibility with
reconfigured human blood show no increased hemolysis caused by the measuring
system. With this concept, it is possible to continuously measure the plasma
concentrations with a relative error of less than 0.5 %.</p
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