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• Kvaal, Simen; Fredheim, Håkon Richard; Højlund, Mads Greisen & Pedersen, Thomas Bondo (2025). Time-dependent Bivariational Principle: Theoretical Foundation for Real-Time Propagation Methods of Coupled-Cluster Type. Journal of Physical Chemistry A. ISSN 1089-5639. 129(15), p. 3508–3521. doi: 10.1021/acs.jpca.4c07417. Full text in Research Archive Show summary

  Real-time propagation methods for chemistry and physics are invariably formulated using variational techniques. The time-dependent bivariational principle (TD-BIVP) is known to be the proper framework for coupled-cluster type methods, and is here studied from a differential geometric point of view. It is demonstrated how two distinct classical Hamilton’s equations of motion arise from considering the real and imaginary parts of the action integral. This in turn leads to two distinct bivariational principles for real bivariational approximation submanifolds. Conservation laws and Poisson brackets are introduced, completing the analogy with classical mechanics. Furthermore, the time-dependent univariational principles (the time-dependent variational principle, the McLachlan principle, and the Dirac–Frenkel principle) are reconstructed using the TD-BIVP and a bivariational submanifold on product form. An overview of established real-time propagation methods is given in the context of our formulation of the TD-BIVP, namely time-dependent traditional coupled-cluster theory, orbital-adaptive coupled-cluster theory, time-dependent orthogonal optimized coupled-cluster theory, Brueckner coupled-cluster theory, and equation-of-motion coupled cluster theory.

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• Kvaal, Simen; Bergan, Snorre; Fredheim, Håkon Richard; Larsen, Nadia S. & Neshveyev, Sergey (2025). A mathematical approach to CC theory. Full text in Research Archive Show summary

  In this talk, I will outline work being carried out in an interdisciplinary project with the Operator Algebra group at Department of Mathematics, UiO. We have done a complete reformulation of coupled-cluster methods from the point of view of the bivariational principle, which gives a more “geometric” view. We have formulated general strategies for a posteriori and a priori error estimation using a nonlinear inf-sup theorem. Several results and proofs are simplified compared to those found in the literature. We are able to place all the analyses presently in the literature in this framework, in particular the seminal works by Rohwedder—Schneider, and Hassan—Maday—Wang

• Fredheim, Håkon Richard (2024). A Mathematical Approach to Coupled Cluster Methods. Full text in Research Archive
• Fredheim, Håkon Richard (2024). A Mathematical Introduction to Quantum Chemistry. Full text in Research Archive
• Fredheim, Håkon Richard (2024). Quantum Technology: Recent developments, future challenges and risks, and what this implies for NBIM. Full text in Research Archive
• Fredheim, Håkon Richard (2024). A Mathematical Approach to Coupled Cluster Theories. Full text in Research Archive
• Fredheim, Håkon Richard (2023). Quantum Technology: Current research perspectives and recent developments. Full text in Research Archive
• Fredheim, Håkon Richard (2022). Quantum Many-Body Theory: A journey from Fock Spaces to Cuntz Algebras. Full text in Research Archive
• Fredheim, Håkon Richard (2026). A Mathematical Analysis of Coupled Cluster Theory and Density Functional Theory. Universitetet i Oslo. Full text in Research Archive Show summary

  This thesis contains three scientific papers concerning mathematical foundations for theoretical quantum chemistry. In these papers, mathematical tools such as functional analysis and the theory of C∗-algebras are employed in formulating and developing the fundamental mathematical framework of common computational methods used in quantum chemistry and fermionic many-particle physics. These methods include density functional theory (DFT) and coupled cluster theory (CC), which are explored in depth in the attached papers.

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