In this article, we investigate the numerical and theoretical aspects of coupled-cluster method tailored by matrix-product states. We formal properties used method, such as energy size consistency equivalence linked unlinked formulation. The existing mathematical analysis is here elaborated in a quantum chemical framework. particular, highlight use what have defined complete active space-external space gap describing basis splitting between external part generalizing concept HOMO-LUMO gap....

Coupled cluster theory produced arguably the most widely used high-accuracy computational quantum chemistry methods.Despite approach's overall great success, its mathematical understanding is so far limited to results within realm of functional analysis.The coupled amplitudes, which are targeted objects in theory, correspond solutions equations, a system polynomial equations at degree four.The high dimensionality electronic Schrödinger equation and non-linearity ansatz have stalled formal...

Twisted bilayer graphene (TBG) has garnered significant interest in condensed matter physics over the past few years. Here, authors present numerical investigations of TBG implementing state-of-the-art quantum chemistry methods. Using a gauge-invariant order parameter, they show ${C}_{2z}$\ensuremath{\mathcal{T}} phase transition at charge neutrality which persists noninteger fillings near neutrality. The work is first systematic study for

Abstract This article provides the first mathematical analysis of Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) exact ground‐state density matrix is a fixed‐point DMET map for non‐interacting systems, (ii) there exists unique physical solution in weakly‐interacting regime, and (iii) up to order coupling parameter. provide numerical simulations support our results comment on meaning assumptions which they hold true. show that violation these may...

We develop a static quantum embedding scheme that utilizes different levels of approximations to coupled cluster (CC) theory for an active fragment region and its environment. To reduce the computational cost, we solve local problem using high-level CC method address environment with lower-level Møller–Plesset (MP) perturbative method. This approach inherits many conceptual developments from hybrid second-order (MP2) works by Nooijen [J. Chem. Phys. 111, 10815 (1999)] Bochevarov Sherrill...

Abstract All-electron electronic structure methods based on the linear combination of atomic orbitals method with Gaussian basis set discretization offer a well established, compact representation that forms much foundation modern correlated quantum chemistry calculations—on both classical and computers. Despite their ability to describe essential physics relatively few functions, these representations can suffer from quartic growth number integrals. Recent results have shown that, for some...

Density matrix embedding theory (DMET) formally requires the matching of density blocks obtained from high-level and low-level theories, but this is sometimes not achievable in practical calculations. In such a case, global band gap vanishes, can require additional numerical considerations. We find that both violation exact condition vanishing are related to assumption noninteracting pure-state v-representable (NI-PS-V), which assumes constructed following Aufbau principle. To relax NI-PS-V...

This article provides the first mathematical analysis of Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) exact ground-state density matrix is a fixed-point DMET map for non-interacting systems, (ii) there exists unique physical solution in weakly-interacting regime, and (iii) at order coupling parameter. provide numerical simulations to support our results comment on meaning assumptions which they hold true. show that violation these may yield...

In quantum chemistry, one of the most important challenges is static correlation problem when solving electronic Schr\"odinger equation for molecules in Born--Oppenheimer approximation. this article, we analyze tailored coupled-cluster method (TCC), particular and promising treating molecular electronic-structure problems with correlation. The TCC combines single-reference (CC) approach an approximate reference calculation a subspace [complete active space (CAS)] considered Hilbert that...

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AbstractHomotopy methods have proven to be a powerful tool for understanding the multitude of solutions provided by coupled-cluster polynomial equations. This endeavour has been pioneered quantum chemists that undertaken both elaborate numerical as well mathematical investigations. Recently, from perspective applied mathematics, new interest in these approaches emerged using topological degree theory and algebraically oriented tools. article provides an overview describing latter...

We propose a novel posteriori error assessment for the single-reference coupled-cluster (SRCC) method called S-diagnostic. provide derivation of S-diagnostic that is rooted in mathematical analysis different SRCC variants. numerically scrutinized S-diagnostic, testing its performance (1) geometry optimizations, (2) electronic correlation simulations systems with varying numerical difficulty, and (3) square-planar copper complexes [CuCl4]2–, [Cu(NH3)4]2+, [Cu(H2O)4]2+. Throughout...

We propose to improve the convergence properties of single-reference coupled cluster (CC) method through an augmented Lagrangian formalism. The conventional CC changes a linear high-dimensional eigenvalue problem with exponential size into determining roots nonlinear system equations that has manageable size. However, current numerical procedures for solving this get lowest suffer from two practical issues: First, may not converge, and second, when converging, they converge other --...

Abstract This article presents an educational overview of the latest mathematical developments in coupled cluster (CC) theory, beginning with Schneider's seminal work from 2009 that introduced first local analysis CC theory. We provide a tutorial review second quantization and ansatz, laying groundwork for understanding basis is followed by detailed exploration most recent advancements Our starts in‐depth look at pioneered Schneider which has since been applied to various methods....

Abstract We develop algebraic geometry for coupled cluster (CC) theory of quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schrödinger equation are approximated by a hierarchy polynomial systems at various levels truncation. exponential parametrization eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Plücker embedding. explain how derive Hamiltonians, we offer detailed study varieties and CC degrees, present...

The exploration of the root structure coupled cluster (CC) equations holds both foundational and practical significance for computational quantum chemistry. This study provides insight into intricate structures these nonlinear at CCD CCSD level theory. We utilize techniques from algebraic geometry, specifically monodromy parametric homotopy continuation methods, to calculate full solution set. compare computed CC roots against various established theoretical upper bounds, shedding light on...

We derive in the Heisenberg picture a widely used phenomenological coupling element to treat feedback effects quantum optical platforms. Our derivation is based on microscopic Hamiltonian, which describes mirror-emitter dynamics dielectric, mediating fully quantized electromagnetic field and single two-level system front of dielectric. The dielectric modelled as identical two-state atoms. equation yields describing differential operator equations, we solve Weisskopf–Wigner limit. Due finite...

Density matrix embedding theory (DMET) formally requires the matching of density blocks obtained from high-level and low-level theories, but this is sometimes not achievable in practical calculations. In such a case, global band gap vanishes, can require additional numerical considerations. We find that both violation exact condition vanishing are related to assumption non-interacting pure-state $v$-representable (NI-PS-V), which assumes constructed following Aufbau principle. order relax...

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 first local analysis CC theory. We offer a tutorial review second quantization and ansatz, laying groundwork for understanding basis is followed by detailed exploration most recent advancements theory.Our starts look at pioneered Schneider which has since been applied to analyze various methods. then...

We develop a static quantum embedding scheme, utilizing projection equations to solve coupled cluster (CC) amplitudes. To reduce the computational cost (for example, of large basis set calculation), we local fragment problem using high-level method and address environment with lower-level M{\o}ller-Plesset (MP) perturbative method. This approach is consistently formulated within framework will be called MP-CC. demonstrate effectiveness our through several prototypical molecular examples by...

The exploration of the root structure coupled cluster equations holds both foundational and practical significance for computational quantum chemistry. This study provides insight into intricate structures these non-linear at CCD CCSD level theory. We utilize techniques from algebraic geometry, specifically monodromy parametric homotopy continuation methods, to calculate full solution set. compare computed CC roots against various established theoretical upper bounds, shedding light on...

We propose a novel posteriori error assessment for the single-reference coupled-cluster (SRCC) method called $S$-diagnostic. provide derivation of $S$-diagnostic that is rooted in mathematical analysis different SRCC variants. numerically scrutinized $S$-diagnostic, testing its performance (1) geometry optimizations, (2) electronic correlation simulations systems with varying numerical difficulty, and (3) square-planar copper complexes [CuCl$_4$]$^{2-}$, [Cu(NH$_3$)$_4$]$^{2+}$,...

We develop algebraic geometry for coupled cluster (CC) theory of quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schr\"odinger equation are approximated by a hierarchy polynomial systems at various levels truncation. exponential parametrization eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Pl\"ucker embedding. explain how derive Hamiltonians, we offer detailed study varieties and CC degrees, present...

Coupled cluster theory produced arguably the most widely used high-accuracy computational quantum chemistry methods. Despite approach's overall great success, its mathematical understanding is so far limited to results within realm of functional analysis. The coupled amplitudes, which are targeted objects in theory, correspond solutions equations, a system polynomial equations at fourth order. high-dimensionality electronic Schrödinger equation and non-linearity ansatz have stalled formal...