In this work we present a coupled-cluster theory for the propagation of multireference electronic systems initiating at general quantum mechanical states. Our formalism is based on the infinitesimal analysis of modified cluster operators, from which we extract a set of additional operators and their equations of motion. These cluster operators then can be used to compute electronic properties of interest in a size extensive manner. The equations derived herein are also studied for the free propagation of linear superpositions. In this regime, we derive asymmetric matrix elements for observables, and a resymmetrization factor so they become consistent with their quantum mechanical counterparts. Our formalism is applied to a four-electron/four-level model with a Hubbard-like Hamiltonian, where we show that it reproduces the numerically exact results for observables, populations, and coherences.
In this work we present a coupled-cluster theory for the propagation of multireference electronic systems initiating at general quantum mechanical states. Our formalism is based on the infinitesimal analysis of modified cluster operators, from which we extract a set of additional operators and their equations of motion. These cluster operators then can be used to compute electronic properties of interest in a size extensive manner. The equations derived herein are also studied for the free propagation of linear superpositions. In this regime, we derive asymmetric matrix elements for observables, and a resymmetrization factor so they become consistent with their quantum mechanical counterparts. Our formalism is applied to a four-electron/four-level model with a Hubbard-like Hamiltonian, where we show that it reproduces the numerically exact results for observables, populations, and coherences.
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.
Coupled cluster (CC) theory is the “gold standard” of molecular electronic structure theory, yet the full arc of the subject—from its variational origins in Hartree–Fock theory, through the exponential ansatz and its diagrammatic machinery, to its modern reformulation as a system of polynomial equations on an algebraic variety, its multireference generalizations, and its emerging reincarnation as a quantum circuit—is rarely presented as a single connected narrative. This review builds that narrative completely from first principles and carries it through to today’s research frontier. Starting from the electronic Schrödinger equation, we derive Hartree–Fock theory, second quantization, and Wick’s theorem; construct the coupled cluster singles-and-doubles (CCSD) equations via the similarity-transformed Hamiltonian; and show how these same equations reappear, verbatim, as a truncated eigenvalue problem on a projective variety generalizing the Grassmannian, with a computable CC degree controlling the number of solutions. We then develop the perturbative triples correction CCSD(T), the linear-response equationof-motion (EOM-CC) framework for excited, ionized, and electron-attached states, the effective-Hamiltonian formalism underlying multireference coupled cluster theory and its notorious intruder-state problem, and the unitary coupled cluster (UCC) ansatz that restores the variational principle at the cost of an infinite operator expansion—tamed, on a quantum computer, by Trotterization and the variational quantum eigensolver. We close with a synthesis of what is rigorously known about the resulting nonlinear equations, a map of directions deliberately left out of scope, and an assessment of the open question of quantum advantage in this domain. Every claim is stated formally, with a full statement and proof or proof sketch, and cited to its origin; the computational cost of every method is compared explicitly throughout.
Effective Hamiltonians, which are commonly used for fitting experimental observables, provide a coarse-grained representation of exact many-electron states obtained in quantum chemistry calculations; however, the mapping between the two is not trivial. In this contribution, we apply Bloch's formalism to equation-of-motion coupled-cluster wave functions to rigorously derive effective Hamiltonians in Bloch's and des Cloizeaux's forms. We report the key equations and illustrate the theory by application to systems with two or three unpaired electrons, which give rise to electronic states of covalent and ionic characters. We show that Hubbard's and Heisenberg's Hamiltonians can be extracted directly from the so-obtained effective Hamiltonians. By establishing a quantitative connection between many-body states and simple models, the approach facilitates the analysis of the correlated wave functions. We propose a simple diagnostic for assessing the validity of the model space choice based on the overlaps between the target- and model-space states. Artifacts affecting the quality of electronic structure calculations such as spin contamination are also discussed.
We propose a multireference linearized coupled cluster theory using matrix product states (MPSs-LCC) which provides remarkably accurate ground-state energies, at a computational cost that has the same scaling as multireference configuration interaction singles and doubles, for a wide variety of electronic Hamiltonians. These range from first-row dimers at equilibrium and stretched geometries to highly multireference systems such as the chromium dimer and lattice models such as periodic two-dimensional 1-band and 3-band Hubbard models. The MPS-LCC theory shows a speed up of several orders of magnitude over the usual Density Matrix Renormalization Group (DMRG) algorithm while delivering energies in excellent agreement with converged DMRG calculations. Also, in all the benchmark calculations presented here, MPS-LCC outperformed the commonly used multi-reference quantum chemistry methods in some cases giving energies in excess of an order of magnitude more accurate. As a size-extensive method that can treat large active spaces, MPS-LCC opens up the use of multireference quantum chemical techniques in strongly correlated ab initio Hamiltonians, including two- and three-dimensional solids.
We propose a multireference linearized coupled cluster theory using matrix product states (MPSs-LCC) which provides remarkably accurate ground-state energies, at a computational cost that has the same scaling as multireference configuration interaction singles and doubles, for a wide variety of electronic Hamiltonians. These range from first-row dimers at equilibrium and stretched geometries to highly multireference systems such as the chromium dimer and lattice models such as periodic two-dimensional 1-band and 3-band Hubbard models. The MPS-LCC theory shows a speed up of several orders of magnitude over the usual Density Matrix Renormalization Group (DMRG) algorithm while delivering energies in excellent agreement with converged DMRG calculations. Also, in all the benchmark calculations presented here, MPS-LCC outperformed the commonly used multi-reference quantum chemistry methods in some cases giving energies in excess of an order of magnitude more accurate. As a size-extensive method that can treat large active spaces, MPS-LCC opens up the use of multireference quantum chemical techniques in strongly correlated ab initio Hamiltonians, including two- and three-dimensional solids.
We introduce a simple coupled cluster approach for multireference and open-shell systems. In the restricted active space coupled cluster framework, the cluster operator only includes excitations from doubly occupied to virtual orbitals, and this ensures the individual operators commute among each other. The ground and excited states are obtained by diagonalisation of the coupled cluster similarity transformed Hamiltonian in a restricted active space. We demonstrate the framework for the RAS-CCSD model, where single and double excitations are included in the cluster operator, and likewise for a complete active space. We consider systems with multireference character and open-shell systems in a spin-adapted manner. The approach captures significantly more dynamical correlation than without the coupled cluster similarity transformation.