We introduce a Lagrangian implementation of the full coupled-cluster reduction [Xu et al., Phys. Rev. Lett. 121, 113001 (2018)], that is, a selected coupled-cluster (CC) based on an arbitrary-order full CC expansion using direct commutator expansions. In this method, the screening for the products of cluster amplitudes plays a central role to reduce the computational cost for the nonlinear commutator operations, while the convergence of the total energy in the standard energy expression is not rapid with tightening the threshold. The new implementation using Lagrangian is robust, containing error only quadratic to those of amplitudes, allowing a much larger screening threshold. We demonstrate the performance of the new implementation by investigating the calculations of N2 and C6H6. The accuracy and applicability are also demonstrated for the potential energy curve of H2O in comparison with conventional quantum chemical methods.
We introduce a Lagrangian implementation of the full coupled-cluster reduction [Xu et al., Phys. Rev. Lett. 121, 113001 (2018)], that is, a selected coupled-cluster (CC) based on an arbitrary-order full CC expansion using direct commutator expansions. In this method, the screening for the products of cluster amplitudes plays a central role to reduce the computational cost for the nonlinear commutator operations, while the convergence of the total energy in the standard energy expression is not rapid with tightening the threshold. The new implementation using Lagrangian is robust, containing error only quadratic to those of amplitudes, allowing a much larger screening threshold. We demonstrate the performance of the new implementation by investigating the calculations of N2 and C6H6. The accuracy and applicability are also demonstrated for the potential energy curve of H2O in comparison with conventional quantum chemical methods.
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.
A full coupled-cluster expansion suitable for sparse algebraic operations is developed by expanding the commutators of the Baker-Campbell-Hausdorff series explicitly for cluster operators in binary representations. A full coupled-cluster reduction that is capable of providing very accurate solutions of the many-body Schrödinger equation is then initiated employing screenings to the projection manifold and commutator operations. The projection manifold is iteratively updated through the single commutators ⟨κ|[H[over ^],T[over ^]]|0⟩ comprised of the primary clusters T[over ^]_{λ} with a substantial contribution to the connectivity. The operation of the commutators is further reduced by introducing a correction, taking into account the so-called exclusion-principle-violating terms that provides a fast and near-variational convergence in many cases.
An improved algorithm to implement our previously proposed static quantum embedding method MPCC (J. Chem. Phys. 161, 164107 (2024)) is presented. It uses the density-fitting (DF) approximation for the electron-repulsion integrals (ERIs) across all layers of the embedding method: the low-level method, the screened interaction, and the high-level method. Additionally, an approximate factorized solution of the Sylvester equation is employed based on the Laplace transformation for the low-level method. To build the screened interaction, we have explored an approximate construction of it based on the distinguishable cluster approximation (DCA) by Kats and Manby (J. Chem. Phys. 141, 061101 (2014)), which amounts to neglecting certain higher-order exchange and ladder diagrams. Combining all these techniques, we have obtained an O(N4) algorithm for the low-level method; a non-iterative NfragO(N4) algorithm for the screened interaction; and an iterative O(Nfrag6) algorithm for the fragment solver, where Nfrag is the number of fragment orbitals and N is the total number of orbitals. The resulting low-scaling implementation of the MPCC method is applied to trans polyacetylenes, that is, C2nH2n+2 molecules for chain lengths n=1−8, and the timing and accuracy of the method are compared to the original unfactorized implementation. Moreover, the approximations are further validated by applying the method to the potential energy surface of the \ce{O3} molecule. Finally, we apply the method to the selected molecules in the S22 dataset for which the performance of the MP2 method is known to be poor. The results demonstrate that the DF-based implementation of the MPCC method provides a significant reduction in computational cost while maintaining accuracy comparable to that of the original implementation.
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.
This thesis is concerned with the development and extension of various new approaches to solving the Coupled Cluster (CC) equations through Monte Carlo (MC) integration. Firstly, exact importance sampling of the CC wavefunction to arbitrary order is defined and applied as a modification to the extant unlinked Coupled Cluster Monte Carlo (CCMC) method. This is shown to be vital for the stability of the method while reducing both computational and memory costs on test calculations. Stochastic approaches utilising the diagrammatic formalism of deterministic CC theory in its unfactorised and totally factorised forms are then developed. These are found to show reduced computational costs compared to prior approaches. It is demonstrated that sampling only the connected components of the similarity-transformed Hamiltonian obtains a solution with memory cost proportional to system size for perfectly local systems and a fixed granularity of representation. For a fixed errorbar per replica calculation costs scale as quartic and linear in the number of replicas for the unfactorised and factorised approaches, respectively. This behaviour is independent of truncation level, and shown to translate into approximately local physical systems. The presence of various systematic biases resulting from wavefunction nonlinearity within these approaches is then investigated. All metrics introduced are found to be negligible compared to the accuracy required for chemical applications in an example system, and possible reasons for this suggested. By considering developments of Full Configuration Interaction Quantum Monte Carlo (FCIQMC) extensions of the totally factorised approach are formulated based upon the semistochastic and initiator approximations, and are shown to provide reduced computational and memory costs. Finally, these new approaches are combined with probabilistic considerations to allow exact solutions to the CC equations to be obtained while only considering approximate update steps.
This thesis deals with selected aspects of a new implementation of many-body methods which can be formulated in the framework of second quantization, in particular the coupled-cluster (CC) method with arbitrary excitation level. Coupled-cluster is one of the most successful and widely used quantum chemical methods for accurate calculations on small to medium-sized molecules. Since it employs a nonlinear parametrization of the wave function, its implementation is rather difficult, in particular if higher (i.e. more than double) excitations are to be included. The latter is necessary to obtain highly accurate results and also as a prerequisite for the generalization to multi-reference cases. The implementation described here has a twofold focus. One is on generality and flexibility regarding the method to be implemented, the other is on efficiency. To achieve flexibility, it is useful to have a machinery which automatically derives working equations for a given method. We realize this by applying techniques of second quantization. This work treats in particular the last step of this procedure, namely the simplification of the resulting equations by the identification of equivalent terms. The algorithm used here is based on the interpretation of algebraic terms as graphs. The derived CC equations then have to be solved iteratively. The efficiency of the program is mainly determined by the evaluation of the occurring expressions, which has to be done in each iteration step. The evaluation is split up in a sequence of tensor contractions. Their generic implementation is complicated by the particular structure of the involved tensors. We reduce each contraction to a sequence of matrix multiplications, which requires a previous data rearranging. But since matrix multiplication is the most efficient operation on modern computers, this additional effort pays off. Preliminary tests show that our program is at least as fast as the most efficient general coupled-cluster implementation so far, and the relation is expected to improve for calculations with larger basis sets where the matrix multiplication becomes the time-determining step. Finally, we give an outlook to possible further developments. In particular, the preparation of the equations before the actual evaluation (factorization) offers much potential for optimizations which we do not exploit at the moment, in contrast to the program with which we compare, which employs at least a partial optimization at this point.