Analysis of the Extended Coupled-Cluster Method in Quantum Chemistry

Laestadius, Andre; Kvaal, Simen

doi:10.1137/17m1116611

Cited by 21 publications

(75 citation statements)

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“…This suggests the use of Zarantonello's lemma [48] to characterize local uniqueness and residual bounds. This is in line with previous studies on single-reference CC methods [38,37,21]. We state without proof:…”

Section: Local Uniqueness and Residualsupporting

confidence: 91%

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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Section: Local Uniqueness and Residualsupporting

confidence: 91%

Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry

Faulstich¹,

Laestadius²,

Legeza³

et al. 2019

SIAM J. Numer. Anal.

Self Cite

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show abstract

“…For the sake of simplicity we use the same symbols for the Gårding constants ofF as for the Hamiltonian. In complete analogy withĤ, the argument in [15,31] shows that…”

Section: A Gårding Inequality For the Fock Operatormentioning

confidence: 99%

“…In the literature there are two different proofs that the infinite dimensional (continuous) CC function f is locally strongly monotone [17] (see also [31] for the extended CC function). Even though spectral-gap assumptions enter the arguments, it is the so-called Gårding constants that give a sufficient condition for the local strong monotonicity, as will be demonstrated below.…”

Section: B Local Strong Monotonicity Of Thementioning

confidence: 99%

“…Additionally, we derive an optimal constant in the monotonicity proof of the CC function for the finite dimensional case, i.e., the projected CC theory. Comparing the CC Lagrangian with the extended CC formulation [31], we propose by means of the bivariational principle an alternative to measure the quality of the Lagrange multipliers, here interpreted as wave function parameters.…”

Section: Introductionmentioning

confidence: 99%

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The coupled-cluster formalism – a mathematical perspective

Laestadius

Faulstich

2019

Molecular Physics

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The Coupled-Cluster theory is one of the most successful high precision methods used to solve the stationary Schrödinger 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 -Gårding 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 Gårding 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.

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“…These states are motivated by the analytic solution of the integrable model considered in Section IV, where the excited states with minimal gap have a single quasiparticle excitation. Although mean-field techniques have been highly studied mostly for Hamiltonian systems [48], they can be extended also to non-normal operators [50] where left and right eigenvectors form a bi-orthonormal basis. Within this variational formalism we show in Appendix G that the fourbody interaction in (28) does not alter the eigenstates, which are therefore exactly given by the bare single-particle eigenstates |Ω exc.…”

Section: B Mean-field Approachmentioning

confidence: 99%

Driven Quantum Dynamics: Will It Blend?

2017

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Randomness is an essential tool in many disciplines of modern sciences, such as cryptography, black hole physics, random matrix theory, and Monte Carlo sampling. In quantum systems, random operations can be obtained via random circuits thanks to so-called q-designs and play a central role in condensed-matter physics and in the fast scrambling conjecture for black holes. Here, we consider a more physically motivated way of generating random evolutions by exploiting the many-body dynamics of a quantum system driven with stochastic external pulses. We combine techniques from quantum control, open quantum systems, and exactly solvable models (via the Bethe ansatz) to generate Haar-uniform random operations in driven many-body systems. We show that any fully controllable system converges to a unitary q-design in the long-time limit. Moreover, we study the convergence time of a driven spin chain by mapping its random evolution into a semigroup with an integrable Liouvillian and finding its gap. Remarkably, we find via Bethe-ansatz techniques that the gap is independent of q. We use mean-field techniques to argue that this property may be typical for other controllable systems, although we explicitly construct counterexamples via symmetry-breaking arguments to show that this is not always the case. Our findings open up new physical methods to transform classical randomness into quantum randomness, via a combination of quantum many-body dynamics and random driving.publishersversionPeer reviewe

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Analysis of the Extended Coupled-Cluster Method in Quantum Chemistry

Cited by 21 publications

References 20 publications

Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry

Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry

The coupled-cluster formalism – a mathematical perspective

Driven Quantum Dynamics: Will It Blend?

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