Analysis of The Tailored Coupled-Cluster Method in Quantum Chemistry

TL;DR

This paper analyzes the tailored coupled-cluster (TCC) method in quantum chemistry, focusing on static correlation problems, and provides theoretical guarantees and error bounds under certain spectral gap assumptions.

Contribution

It offers a mathematical analysis of TCC methods, including existence, uniqueness, and error bounds, especially for tensor-network state implementations.

Findings

Proves local uniqueness and existence of TCC solutions under spectral gap conditions.

Derives quadratic error bounds for TCC methods using duality techniques.

Analyzes TCC with tensor-network states like DMRG in a rigorous framework.

Abstract

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.