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arXiv:2604.06429 · The Journal of Chemical Physics

Coupled-Cluster Imaginary-Time Evolution and the Coupled-Cluster Energy Variance

Yuhang Ai, Huanchen Zhai, Garnet Kin-Lic Chan

physics.chem-phquant-phSubmitted Apr 7, 2026• Announced Apr 9, 2026• 0 citations• Revised Sep 16, 2026 (v2)
v1 · Apr 7, 2026v2 · Sep 16, 2026
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Abstract

We discuss a coupled-cluster formalism for carrying out imaginary-time evolution from an arbitrary reference, and study the properties of the resulting evolution trajectories. The evolution converges to a solution of the standard coupled-cluster amplitude equations in the long-time limit if a finite valued limit exists, but when such a limit does not exist, the trajectories still contain additional information beyond the standard solutions. We introduce the coupled-cluster energy variance which through its minima identifies physically regularized coupled-cluster amplitudes when the solutions of the amplitude equations are unreasonable. We demonstrate the value of this formalism in several exploratory examples within single- and multi-reference coupled-cluster formulations.

Journal reference
J. Chem. Phys. 165, 114104 (2026)
DOI
10.1063/5.0339174
Comments
9 pages, 7 figures

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We discuss a coupled-cluster formalism for carrying out imaginary-time evolution from an arbitrary reference, and study the properties of the resulting evolution trajectories. The evolution converges to a solution of the standard coupled-cluster amplitude equations in the long-time limit if a finite valued limit exists, but when such a limit does not exist, the trajectories still contain additional information beyond the standard solutions. We introduce the coupled-cluster energy variance which through its minima identifies physically regularized coupled-cluster amplitudes when the solutions of the amplitude equations are unreasonable. We demonstrate the value of this formalism in several exploratory examples within single- and multi-reference coupled-cluster formulations.
We present a time-dependent formulation of coupled cluster theory. This theory allows for direct computation of the free energy of quantum systems at finite temperature by imaginary time integration and is closely related to the thermal cluster cumulant theory of Mukherjee and co-workers [ Chem. Phys. Lett. 1992 , 192 , 55 -61 ; Phys. Rev. E 1993 , 48 , 3373 -3389 ; Chem. Phys. Lett. 2001 , 335 , 281 -288 ; Chem. Phys. Lett. 2002 , 352 , 63 -69 ; Int. J. Mod. Phys. B 2003 , 17 , 5367 -5377 ]. Our derivation of the finite-temperature theory highlights connections to perturbation theory and to zero-temperature coupled cluster theory. We show explicitly how the finite-temperature coupled cluster singles and doubles amplitude equations can be derived in analogy with the zero-temperature theory and how response properties can be efficiently computed using a variational Lagrangian. We discuss the implementation for realistic systems and showcase the potential utility of the method with calculations of the exchange correlation energy of the uniform electron gas under warm dense matter conditions.
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This work presents a time-dependent (TD) unitary coupled-cluster (UCC) formulation for electronic quantum dynamics, including the propagation of excited states and their superpositions, in both single- and multi-reference regimes. Standard TD coupled-cluster techniques offer size-extensivity, but they rely on non-Hermitian bivariational action functionals that break time-reversibility, giving transition matrix elements and amplitude estimators that are asymmetric (though accurate and systematically improvable). Here we use the time-evolution operator as an exponential map driven by TD anti-Hermitian cluster operators and first-order generators. Applying the Dirac-Frenkel action principle, we extract equations of motion governed by Heisenberg-picture-like commutators. This approach connects to our previous non-Hermitian formulations, where observables are expressed in terms of regular and extended cluster operators. From that connection we obtain a generator cluster operator whose unperturbed TD limit leads to the UCC eigenvalue problem. Even though this problem can be exact, the time dependence of the generator holds only at short propagation times, so we use the generator to prepare the initial state and then propagate that state with the formal TD UCC equation of motion. The theory is tested on an extended hard-core Bose-Hubbard ring with connections to neutral atom chains, and we discuss the theory's present limitations and possible extensions.
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