The coupled-cluster formalism – a mathematical perspective

Laestadius, Andre; Faulstich, Fabian M.

doi:10.1080/00268976.2018.1564848

Cited by 17 publications

(25 citation statements)

References 48 publications

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“…For a further discussion on spectral gap and Gårding inequalities in CC theories we refer to [20]. Moreover, in agreement with Section 2, we suppose…”

Section: Local Uniqueness and Residualsupporting

confidence: 68%

Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry

Faulstich¹,

Laestadius²,

Legeza³

et al. 2019

SIAM J. Numer. Anal.

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In quantum chemistry, one of the most important challenges is the static correlation problem when solving the electronic Schrödinger 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-Nitscheduality method, a quadratic error bound valid for TNS-TCC methods is derived, e.g., for lineartensor-network TCC schemes using the density matrix renormalization group method.Even if most molecules are single-reference systems in their equilibrium configuration, multireference character arises even in the simplest of chemical reactions, e.g., dissociation of N 2 . Yet, the static correlation problem is a long-lasting challenge in quantum chemistry. Many different MRCC approaches have been formulated to deal with the problem of static correlation. However, aside from formal difficulties and implementational complications, none of these methods have become a widely applicable tool. A review of different MRCC approaches is beyond the scope of this article, and we refer to Lyakh et al.[26] for a detailed description of the different benefits and disadvantages.We are here concerned with an MRCC method that is based on the single-reference methodology (also called an externally corrected ansatz): The tailored CC (TCC) method extends a precomputed solution for a chosen subsystem of the full system by including further electron correlations via CC theory. We refer to the subsystem as the complete active space (CAS) and to the remaining system as the external space. Given the single-reference CC method's major drawback, this subsystem needs to contain the static correlations. Consequently, the TCC method can be seen as a special type of an embedding method. Mathematically this corresponds to a division of excitation operators in two disjoint sub-algebras [19]. Nevertheless, in comparison with other "genuine" MRCC schemes, the TCC method suffers from the drawback that it is based on a single-reference theory and therewith introduces a certain bias towards a particular reference determinant....

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“…For a further discussion on spectral gap and Gårding inequalities in CC theories we refer to [20]. Moreover, in agreement with Section 2, we suppose…”

Section: Local Uniqueness and Residualsupporting

confidence: 68%

Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry

Faulstich¹,

Laestadius²,

Legeza³

et al. 2019

SIAM J. Numer. Anal.

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“…Monotonicity is an important notion in connection with the local analysis of the CC method and its variations [23][24][25][26][27]34]. The use of monotonicity in the analysis of the standard CC method was introduced by Schneider and Rohwedder [25,27].…”

Section: Truncations and Monotonicity Analysismentioning

confidence: 99%

“…It also connects spectral gaps, e.g. HOMO-LUMO gap, to stability constants within the analysis [24]. (See also the steerable CAS-ext gap connected to the tailored CC method [35] that treats quasi-degenerate systems [36].…”

Section: Truncations and Monotonicity Analysismentioning

confidence: 99%

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Guaranteed convergence for a class of coupled-cluster methods based on Arponen's extended theory

2020

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A wide class of coupled-cluster methods is introduced, based on Arponen's extended coupledcluster theory. This class of methods is formulated in terms of a coordinate transformation of the cluster operators. The mathematical framework for the error analysis of coupled-cluster methods based on Arponen's bivariational principle is presented, in which the concept of local strong monotonicity of the flipped gradient of the energy is central. A general mathematical result is presented, describing sufficient conditions for coordinate transformations to preserve the local strong monotonicity. The result is applied to the presented class of methods, which include the standard and quadratic coupled-cluster methods, and also Arponen's canonical version of extended coupledcluster theory. Some numerical experiments are presented, and the use of canonical coordinates for diagnostics is discussed.

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“…For an analysis of standard CC without the BIVP, see the works of Rohwedder and Schneider. 30,31 A key analysis tool for studying CC theory 3,[29][30][31][32][33] was that of local strong monotonicity 4 of the flipped gradient F :Ṽ ⊕ V →Ṽ ⊕ V given by…”

Section: Local Strong Monotonicity Analysismentioning

confidence: 99%

A state-specific multireference coupled-cluster method based on the bivariational principle

Bodenstein

Kvaal

2020

The Journal of Chemical Physics

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A state-specific multireference coupled-cluster method based on Arponen's bivariational principle is presented, the bivar-MRCC method. The method is based on single-reference theory, and therefore has a relatively straightforward formulation and modest computational complexity. The main difference from established methods is the bivariational formulation, in which independent parameterizations of the wavefunction (ket) and its complex conjugate (bra) are made. Importantly, this allows manifest multiplicative separability of the state (exact in the extended bivar-MRECC version of the method, and approximate otherwise), and additive separability of the energy, while preserving polynomial scaling of the working equations. A feature of the bivariational principle is that the formal bra and ket references can be included as bivariational parameters, which eliminates much of the bias towards the formal reference. A pilot implementation is described, and extensive benchmark calculations on several standard problems are performed. The results from the bivar-MRCC method are comparable to established state-specific multireference methods. Considering the relative affordability of the bivar-MRCC method, it may become a practical tool for non-experts.

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The coupled-cluster formalism – a mathematical perspective

Cited by 17 publications

References 48 publications

Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry

Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry

Guaranteed convergence for a class of coupled-cluster methods based on Arponen's extended theory

A state-specific multireference coupled-cluster method based on the bivariational principle

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