Despite tremendous theoretical efforts, corroborated by state-of-the-art computational techniques toward the development of a genuine multireference coupled cluster (MRCC) theory, the search seems to be far from complete. This is due to the fact that a host of non-trivial problems are still present in the process of developing such approaches, and it now appears that it is really an arduous task to retain the simplistic beauty of a typical single-reference approach while formulating a multireference (MR) theory. The chief sources of difficulty in setting up MR theories are (a) preservation of size extensivity (connectedness) of the formalism, (b) the perfectly determined nature of the problem, that is, having an equal number of parameters and equations, (c) conservation of symmetry, (d) avoidance of intruder states, and (e) correlation to a solution of the projected Schrödinger equation.
Despite tremendous theoretical efforts, corroborated by state-of-the-art computational techniques toward the development of a genuine multireference coupled cluster (MRCC) theory, the search seems to be far from complete. This is due to the fact that a host of non-trivial problems are still present in the process of developing such approaches, and it now appears that it is really an arduous task to retain the simplistic beauty of a typical single-reference approach while formulating a multireference (MR) theory. The chief sources of difficulty in setting up MR theories are (a) preservation of size extensivity (connectedness) of the formalism, (b) the perfectly determined nature of the problem, that is, having an equal number of parameters and equations, (c) conservation of symmetry, (d) avoidance of intruder states, and (e) correlation to a solution of the projected Schrödinger equation.
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.
Our overarching goal is the development of wavefunction-based quantum chemistry methods that give good results for both weakly- and strongly-correlated systems, that preserve as many fundamental properties of the exact wavefunction as possible, and that are computationally affordable and black-box for the end user. Many of our contributions in this area are based on the archetypal quantum chemistry method, coupled-cluster theory, and they have resulted from our efforts to understand and remedy the failure of symmetry-adapted coupled-cluster theory in the presence of strong correlation. In this thesis, we recount these investigations and discuss the key insights and results that they have produced. We describe in detail the novel wavefunction approaches we have devised, as well as the approximations and considerations one must make in order to arrive at useful equations. We believe that this work could be important for the continued development of affordable methods capable of accurately describing a wide variety of quantum chemical systems.
In a series of two articles, we propose a comprehensive mathematical framework for Coupled-Cluster-type methods. These methods aim at accurately solving the many-body Schrodinger equation. In this first part, we rigorously describe the discretization schemes involved in Coupled-Cluster methods using graph-based concepts. This allows us to discuss different methods in a unified and more transparent manner, including multireference methods. Moreover, we derive the single-reference and the Jeziorski-Monkhorst multireference Coupled-Cluster equations in a unified and rigorous manner.
A general coupled-cluster method valid for arbitrary multideterminantal reference states is formulated. The resulting cluster expansion for the wave function is a generalization of that introduced by Silverstone and Sinano\ifmmode \check{g}\else \v{g}\fi{}lu and applied by Sinano\ifmmode \check{g}\else \v{g}\fi{}lu and collaborators. The connected nature of the cluster operators and the effective interaction is proven in the case when the reference space is complete, i.e., is invariant under unitary transformations of partly occupied orbitals. For incomplete reference spaces the disconnected terms appearing in the effective interaction are properly generated by the coupled-cluster theory. Approximate schemes for solving coupled-cluster equations are proposed and their relation with perturbation theory is briefly discussed.
In the past decade, the quantum chemical version of the density matrix renormalization group method has established itself as the method of choice for strongly correlated molecular systems. However, despite its favorable scaling, in practice, it is not suitable for computations of dynamic correlation. Several approaches to include that in post-DMRG methods exist; in our group, we focused on the tailored coupled cluster (TCC) approach. This method works well in many situations; however, in exactly degenerate cases (with two or more determinants of equal weight), it exhibits a bias toward the reference determinant representing the Fermi vacuum. Although sometimes it is possible to use a compensation scheme to avoid this bias for energy differences, it is certainly a drawback. In order to overcome this bias of the TCC method, we have developed a Hilbert-space multireference version of tailored CC, which can treat several determinants on an equal footing. We have implemented and compared the performance of three Hilbert-space multireference coupled cluster (MRCC) variants-the state universal one and the Brillouin-Wigner and Mukherjee's state specific ones. We have assessed these approaches on the cyclobutadiene and tetramethyleneethane molecules, which are both diradicals with exactly degenerate determinants at a certain geometry. We have also investigated the sensitivity of the results on the orbital rotation of the highest occupied and lowest unoccupied molecular orbital (HOMO-LUMO) pair, as it is well known that Hilbert-space MRCC methods are not invariant to such transformations.