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Developing a coupled-cluster theory based on a multiconfigurational reference wave function still is one of the most challenging problems in quantum chemistry, both from a theoretical and implementational perspective. Hence, no clear scientific consensus has been reached yet on the aspects of such theories. The main reason for this is that many different parameterizations are possible based on several theoretical choices that can be made, e.g., whether to use a contracted or uncontracted ansatz, which residual conditions to employ, how to treat the available excitation classes, whether to use a single or sequential similarity transformations of the Hamiltonian, and more. In this thesis, we further elucidate some aspects of this broad topic, to pave a path towards a theoretically rigorous, generally accepted multireference coupled-cluster method. To this end, we focus especially on an efficient implementation, the residual conditions, perturbative approximations, and a way to reduce the dimensionality of the involved tensors, i.e., foremost, density matrices.
To implement such theories, especially the internally contracted approaches, automated tools are required since the theories contain upwards of hundreds of thousands of terms, posing a formidable challenge. Consequently, we wrote a highly performant toolchain, ORCA-AGE II, which can derive and implement even the most complicated variants of multireference coupled-cluster theory. The toolchain consists of an optimized code generation part that keeps the code generation time as short as possible, as well as sophisticated algorithms to find optimal transformations of the tensor contractions so that they can be evaluated close to peak CPU efficiency.
In the multireference theories, more specifically on the topic of residual conditions, we propose a cumulant-based expansion that connects the many-body to the projective residual conditions and clearly demonstrates the more complicated nature of the projective variant. This expansion justifies the truncations present in the many-body expansions used in multireference equation-of-motion theories. These findings are then used in multireference equation-of-motion perturbation theory, which we developed as a perturbative transform-then-diagonalize method. From benchmarking the novel method on various organic and inorganic systems, we find it has an accuracy comparable to that of NEVPT2 theory, while being significantly cheaper and more stable than its parent method, multireference equation-of-motion coupled-cluster theory. On a different aspect of internally contracted methods with projective residual conditions, we developed an automated reduction scheme for high-order density matrices that can be applied to any method. The scheme works exceptionally well on multireference coupled-cluster theory, being always faster than the unreduced implementation through a combination of asymptotic CPU cost reductions and more efficient usage of the CPU caches through lower-dimensional tensors. These savings are realized even though the maximum order of the density matrices can be reduced by at most one since the structure of the equations does not allow for higher reductions. Finally, we also report highly accurate transition energies computed at the single-reference level benchmarked on indigo dyes, showcasing the applicability yet to be reached with the more complicated multireference approaches.
