We propose to improve the convergence properties of the single-reference coupled cluster (CC) method through an augmented Lagrangian formalism. The conventional CC method changes a linear high-dimensional eigenvalue problem with exponential size into a problem of determining the roots of a nonlinear system of equations that has a manageable size. However, current numerical procedures for solving this system of equations to get the lowest eigenvalue suffer from two practical issues: First, solving the CC equations may not converge, and second, when converging, they may converge to other -- potentially unphysical -- states, which are stationary points of the CC energy expression. We show that both issues can be dealt with when a suitably defined energy is minimized in addition to solving the original CC equations. We further propose an augmented Lagrangian method for coupled cluster (alm-CC) to solve the resulting constrained optimization problem. We numerically investigate the proposed augmented Lagrangian formulation showing that the convergence towards the ground state is significantly more stable and that the optimization procedure is less susceptible to local minima. Furthermore, the computational cost of alm-CC is comparable to the conventional CC method.
We propose to improve the convergence properties of the single-reference coupled cluster (CC) method through an augmented Lagrangian formalism. The conventional CC method changes a linear high-dimensional eigenvalue problem with exponential size into a problem of determining the roots of a nonlinear system of equations that has a manageable size. However, current numerical procedures for solving this system of equations to get the lowest eigenvalue suffer from two practical issues: First, solving the CC equations may not converge, and second, when converging, they may converge to other -- potentially unphysical -- states, which are stationary points of the CC energy expression. We show that both issues can be dealt with when a suitably defined energy is minimized in addition to solving the original CC equations. We further propose an augmented Lagrangian method for coupled cluster (alm-CC) to solve the resulting constrained optimization problem. We numerically investigate the proposed augmented Lagrangian formulation showing that the convergence towards the ground state is significantly more stable and that the optimization procedure is less susceptible to local minima. Furthermore, the computational cost of alm-CC is comparable to the conventional CC method.
The workhorse method of computational materials science is undeniably density functional theory in the Kohn-Sham framework of approximate exchange and correlation energy functionals. However, the need for highly accurate predictions of ground and excited state properties in materials science motivates the further development and exploration of alternative as well as complementary techniques. Among these alternative approaches, quantum chemical wavefunction based theories and in particular coupled cluster theory hold the promise to fill a gap in the toolbox of computational materials scientists. Coupled cluster (CC) theory provides a compelling framework of approximate infinite-order perturbation theory in the form of an exponential of cluster operators describing the true quantum many-body effects of the electronic wave function at a computational cost that, despite being significantly more expensive than DFT, scales polynomially with system size. The hierarchy of size-extensive approximate methods established in the framework of CC theory achieves systematic improvability for many materials properties. This is in contrast to currently available density functionals that often suffer from uncontrolled approximations that limit the accuracy in the prediction of materials properties. In this tutorial-style review we will introduce basic concepts of coupled cluster theory and recent developments that increase its computational efficiency for calculations of molecules, solids and materials in general. We will touch upon the connection between coupled cluster theory and the random-phase approximation to bridge the gap between traditional quantum chemistry and many-body Green’s function theories that are widely-used in the field of solid state physics. We will discuss various approaches to improve the computational performance without compromising accuracy. These approaches include large-scale parallel design as well as techniques that reduce the prefactor of the computational complexity. A central part of this article discusses the convergence of calculated properties to the thermodynamic limit which is of significant importance for reliable predictions of materials properties and constitutes an additional challenge compared to calculations of large molecules. Furthermore we mention technical aspects of computer code implementations of periodic coupled cluster theories in different numerical frameworks of the one-electron orbital basis; the projector-augmented-wave formalism using a plane wave basis set and the numeric atom-centered-orbital with resolution-of-identity.
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.
Many-body perturbation theories (MBPT) (both single and multireference), by design, are capable of representing electron correlation very well. However, the practical implementation of these methods for many-electron systems beyond a certain order is extremely difficult, if not impossible. For instance, there are hardly any calculations beyond third order at the multireference and fifth order at the single reference of perturbation theories. Thus, these methods are typically limited to systems where the higher-order electron correlation effect is not significant. A practical solution to this problem is the coupled cluster theory, which was first introduced by Coester [20] and Coester and Kümmel [21] in nuclear physics and was later adopted in the electron structure theory by Čížek [23, 106] and Paldus and Čížek [107]. The coupled-cluster (CC) theory is a non-perturbative all-order theory. This stateof-the-art theory has now been recognized as the basis of modern quantitative quantum chemistry and atomic physics for high-precision calculations. Its superior accuracy is rooted in the exponential ansatz for the wave function, which provides an energy expression that scales correctly with the number of electrons and under certain provisions leads to a faster convergence of the many-body expansion.
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.
Coupled Cluster (CC) theory is one of the most popular correlated wavefunction methods, yet the steep computational cost scaling of CC truncated at higher rank limits its application. Modern Selected Configuration Interaction (SCI) methods successfully exploit the sparsity of ground-state wavefunctions in the determinant space, motivating an analogous approach with the exponential CC ansatz. In this work, we introduce Selected Coupled Cluster (SCC), an arbitrary-order sparse CC method that explicitly solves the CC residual equations within a compact amplitude space guided by an SCI wavefunction. SCC optimizes the CC amplitudes in the sparse amplitude space. Using a pilot Just-in-Time (JIT) compiled implementation, we benchmark SCC against SCI, SCI+PT2, standard rank-truncated dense CC (CCSD, CCSDT), and DMRG across various chemical systems. We demonstrate that SCC exhibits fast, monotonic convergence with the SCI variational threshold, outperforming SCI and surpassing SCI+PT2 in the tight-threshold regime. SCC also provides accurate energies for high-dimensional topologies, where DMRG typically converges slowly with bond dimension. SCC thus presents an efficient drop-in enhancement for modern SCI workflows.
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--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.