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 $\rho_\psi$ of a wave function $\psi$. 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 $I_\mathrm{xc}(\psi)\geq - C_1 \int_{\mathbb R} \rho_\psi(x)^{2} \mathrm{d}x$ 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

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&thinsp;%.</p

The SS-diagnostic -- an a posteriori error assessment for single-reference coupled-cluster methods

We propose a novel a posteriori error assessment for the single-reference coupled-cluster (SRCC) method called the $S$-diagnostic. We provide a derivation of the $S$-diagnostic that is rooted in the mathematical analysis of different SRCC variants. We numerically scrutinized the $S$-diagnostic, testing its performance for (1) geometry optimizations, (2) electronic correlation simulations of systems with varying numerical difficulty, and (3) the square-planar copper complexes [CuCl$_4$]$^{2-}$, [Cu(NH$_3$)$_4$]$^{2+}$, and [Cu(H$_2$O)$_4$]$^{2+}$. Throughout the numerical investigations, the $S$-diagnostic is compared to other SRCC diagnostic procedures, that is, the $T_1$, $D_1$, and $D_2$ diagnostics as well as different indices of multi-determinantal and multi-reference character in coupled-cluster theory. Our numerical investigations show that the $S$-diagnostic outperforms the $T_1$, $D_1$, and $D_2$ diagnostics and is comparable to the indices of multi-determinantal and multi-reference character in coupled-cluster theory in their individual fields of applicability. The experiments investigating the performance of the $S$-diagnostic for geometry optimizations using SRCC reveal that the $S$-diagnostic correlates well with different error measures at a high level of statistical relevance. The experiments investigating the performance of the $S$-diagnostic for electronic correlation simulations show that the $S$-diagnostic correctly predicts strong multi-reference regimes. The $S$-diagnostic moreover correctly detects the successful SRCC computations for [CuCl$_4$]$^{2-}$, [Cu(NH$_3$)$_4$]$^{2+}$, and [Cu(H$_2$O)$_4$]$^{2+}$, which have been known to be misdiagnosed by $T_1$ and $D_1$ diagnostics in the past. This shows that the $S$-diagnostic is a promising candidate for an a posteriori diagnostic for SRCC calculations