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• Schrader, Simon Elias; Kristiansen, Håkon Emil; Pedersen, Thomas Bondo & Kvaal, Simen (2025). Time-Dependent Gaussian Basis Sets for Many-Body Systems Using Rothe’s Method: A Mean-Field Study. Journal of Chemical Theory and Computation. ISSN 1549-9618. doi: 10.1021/acs.jctc.5c00970. Full text in Research Archive Show summary

  A challenge in modeling time-dependent strong-field processes such as high-harmonic generation for many-body systems is how to effectively represent the electronic continuum. We apply Rothe’s method to the time-dependent Hartree–Fock (TDHF) and density functional theory (TDDFT) equations of motion for the orbitals, which reformulate them as an optimization problem. We show that thawed, complex-valued Gaussian basis sets can be propagated efficiently for these orbital-based approaches, removing the need for grids. In particular, we illustrate that qualitatively correct results can often be obtained by using just a few fully flexible Gaussians that describe the unbound dynamics for both TDHF and TDDFT. Grid calculations can be reproduced quantitatively using 30–100 Gaussians for intensities up to 4 × 10^14 W/cm2 for the one-dimensional molecular systems considered in this work.

• Schrader, Simon Elias; Pedersen, Thomas Bondo & Kvaal, Simen (2025). Multidimensional quantum dynamics with explicitly correlated Gaussian wave packets using Rothe’s method. Journal of Chemical Physics. ISSN 0021-9606. 162(2). doi: 10.1063/5.0247732. Full text in Research Archive Show summary

  In a previous publication [S. E. Schrader et al., J. Chem. Phys. 161, 044105 (2024)], it has been shown that Rothe’s method can be used to solve the time-dependent Schrödinger equation for the hydrogen atom in a strong laser field using time-dependent Gaussian wave packets. Here, we generalize these results, showing that Rothe’s method can propagate arbitrary numbers of thawed, complex-valued, Explicitly Correlated Gaussian (ECGs) functions with dense correlation matrices for systems with varying dimensionality. We consider the multidimensional Hénon–Heiles potential and show that the dynamics can be quantitatively reproduced using only 30 Gaussians in 2D and that accurate spectra can be obtained using 20 Gaussians in 2D and 30–40 Gaussians in 3D and 4D. Therefore, the relevant multidimensional dynamics can be described at high quality using only a small number of ECGs that give a very compact representation of the wave function. This efficient representation, along with the demonstrated ability of Rothe’s method to propagate Gaussian wave packets in strong fields and ECGs in complex potentials, paves the way for accurate molecular dynamics calculations beyond the Born–Oppenheimer approximation in strong fields.

• Schrader, Simon Elias; Kristiansen, Håkon Emil; Pedersen, Thomas Bondo & Kvaal, Simen (2024). Time evolution as an optimization problem: The hydrogen atom in strong laser fields in a basis of time-dependent Gaussian wave packets. Journal of Chemical Physics. ISSN 0021-9606. 161(4). doi: 10.1063/5.0213576. Full text in Research Archive Show summary

  Recent advances in attosecond science have made it increasingly important to develop stable, reliable, and accurate algorithms and methods to model the time evolution of atoms and molecules in intense laser fields. A key process in attosecond science is high-harmonic generation, which is challenging to model with fixed Gaussian basis sets, as it produces high-energy electrons, with a resulting rapidly varying and highly oscillatory wave function that extends over dozens of ångström. Recently, Rothe’s method, where time evolution is rephrased as an optimization problem, has been applied to the one-dimensional Schrödinger equation. Here, we apply Rothe’s method to the hydrogen wave function and demonstrate that thawed, complex-valued Gaussian wave packets with time-dependent width, center, and momentum parameters are able to reproduce spectra obtained from essentially exact grid calculations for high-harmonic generation with only 50–181 Gaussians for field strengths up to 5 × 10^14 W/cm2. This paves the way for

• Schrader, Simon Elias & Kvaal, Simen (2023). Accelerated coupled cluster calculations with Procrustes orbital interpolation. Journal of Chemical Physics. ISSN 0021-9606. 158(11). doi: 10.1063/5.0141145. Full text in Research Archive Show summary

  The coupled cluster method is considered a gold standard in quantum chemistry, reliably giving energies that are exact within chemical accuracy (1.6 mhartree). However, even in the coupled cluster single-double (CCSD) approximation, where the cluster operator is truncated to include only single and double excitations, the method scales as O( N 6 ) in the number of electrons, and the cluster operator needs to be solved for iteratively, increasing the computation time. Inspired by eigenvector continuation, we present here an algorithm making use of the Gaussian processes that provides an improved initial guess for the coupled cluster amplitudes. The cluster operator is written as a linear combination of sample cluster operators that are obtained at particular sample geometries. By reusing the cluster operators from previous calculations in that way, it is possible to obtain a start guess for the amplitudes that surpasses both MP2 guesses and “previous geometry”-guesses in terms of the number of necessary iterations. As this improved guess is very close to the exact cluster operator, it can be used directly to calculate the CCSD energy to chemical accuracy, giving approximate CCSD energies scaling as O( N 5 ).

• Schrader, Simon Elias & Benth, Fred Espen (2022). A stochastic study of carbon emission reduction from electrification and interconnecting cable utilization. The Norway and Germany case. Energy Economics. ISSN 0140-9883. 114. doi: 10.1016/j.eneco.2022.106300. Full text in Research Archive Show summary

  We analyse how carbon emissions are reduced utilizing the new NordLink cable between Norway and Germany, taking policy plans of platform electrification in Norway into account. Building a stochastic model based on existing data for Norwegian and German electricity production and demand (Open Power system data, 2020) to account for uncertainty, we found that the cable can be used as an effective way to reduce German emissions by exporting German renewable surplus production on windy days to Norway and importing from Norway other days. We found that such flow of power will lead to a possible reduction of the countries’ total emissions by over 4.1 million tons CO annually. Based on a mean reduction of German CO emissions of about 710 g/kWh by electricity imported from Norway, we found that electrification of the Norwegian oil and gas extraction is inferior to an increased export to Germany in terms of total emission reduction, as electrification only gives a reduction of about 425 g/kWh according to numbers from the Norwegian Oil Directorate (Oljedirektoratet, 2020). A similar calculation assuming a realistic full electrification of the platforms only gave a reduction of 2.4 million tons CO annually. In the long term, we conclude that combining both approaches makes the best use of Norwegian surplus energy production in view of carbon emission reduction.

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• Pedersen, Thomas Bondo; Adamowicz, Ludwik; Kristiansen, Håkon Emil; Kvaal, Simen & Schrader, Simon Elias (2025). Challenges and opportunities in attosecond quantum dynamics. Full text in Research Archive
• Schrader, Simon Elias; Kristiansen, Håkon Emil; Pedersen, Thomas Bondo & Kvaal, Simen (2025). Rothe’s method: Solving the time-dependent Schrödinger equation with flexible Gaussians. Full text in Research Archive
• Pedersen, Thomas Bondo; Kvaal, Simen; Schrader, Simon Elias & Kristiansen, Håkon Emil (2025). Quantum Dynamics without the Born-Oppenheimer Approximation. Full text in Research Archive
• Schrader, Simon Elias; Kristiansen, Håkon Emil; Pedersen, Thomas Bondo & Kvaal, Simen (2024). Time evolution as an optimization problem - solving the time-dependent Schrödinger equation with Explicitly Correlated Gaussians. Full text in Research Archive Show summary

  A linear combination of Explicitly Correlated Gaussians (ECGs) is a very compact parameterization of multidimensional wave functions. Gaussian wave packets can be propagated variationally in specific situations (see e.g. the variational multi-configurational Gaussian (vMCG) method), but in general, propagating ECGs using a time-dependent variational principle is challenging due to ill-conditioning of the Gramian matrix that needs to be inverted at every time step. This problem can be evaded by using Rothe's method, where time evolution is rephrased as an optimization problem. We show that Rothe's method can be used to propagate linear combinations of ECGs, illustrated by calculating the spectrum of a particle in the multi-dimensional Henon-Heiles potential using only 10-20 Gaussians, and by the HHG spectrum of the Hydrogen atom in strong fields, using only 50-181 Gaussians. This paves the way for the inclusion of continuum contributions into real-time, time-dependent electronic-structure theory with ECGs for strong fields, and eventually accurate simulations of the time evolution of molecules without the Born-Oppenheimer (BO) approximation.

• Pedersen, Thomas Bondo; Schrader, Simon Elias; Kristiansen, Håkon Emil & Kvaal, Simen (2024). Attosecond quantum dynamics as an optimization problem. Full text in Research Archive Show summary

  The 2023 Nobel Prize for Physics was awarded to Agostini, Krausz, and L'Huillier for their seminal contributions to the development of attosecond laser pulses by means of high-harmonic generation. Attosecond laser pulses allow us to monitor electronic motion in atoms, molecules, and materials, providing insights into and potentially control the quantum dynamics that govern chemical processes. Moreover, attosecond spectroscopy gives new opportunities to study the structure and dynamics of short-lived matter-antimatter complexes, which may provide clues to fundamental research questions such as the nature of dark matter. Needless to say, simulations are crucial for understanding, interpreting, and predicting the quantum dynamics. However, accurate simulations are highly challenging since - ionization processes are almost unavoidable due to the inherently broad intensity distribution in the frequency domain - the population of numerous electronic states of neutral and potentially multiple ionized species necessitates computational models beyond the Born-Oppenheimer approximation - the ensuing coupled electronic-nuclear dynamics likely leads to bond breaking Highly adaptive computational models that do not rely on assumptions about the possible dynamics are needed to tackle these challenges. In this talk, I will present a novel approach developed at the Hylleraas Centre over the past $2$--$3$ years. It is based on recasting the time-dependent Schrödinger equation as an optimization problem with built-in wave-function error control and, thus, full adaptivity [1,2]. Using wave-function models expressed as linear combinations of Gaussians [3], it is demonstrated that high-harmonic generation processes and rovibrational dynamics can be simulated with high accuracy. [1] S. Kvaal, C. Lasser, T. B. Pedersen, and L. Adamowicz, No need for a grid: Adaptive fully-flexible gaussians for the time-dependent Schrödinger equation, arXiv:2207.00271 [2] S. E. Schrader, H. E. Kristiansen, T. B. Pedersen, and S. Kvaal, Time evolution as an optimization problem: The hydrogen atom in strong laser fields in a basis of time-dependent Gaussian wave packets, arXiv:2404.07699 \newline [3] A. P. Wozniak, L. Adamowicz, T. B. Pedersen, and S. Kvaal, Gaussians for Electronic and Rovibrational Quantum Dynamics, J. Phys. Chem. A 128, 3659-3671 (2024)

• Schrader, Simon Elias; Kristiansen, Håkon Emil; Pedersen, Thomas Bondo & Kvaal, Simen (2024). Time evolution of the Hydrogen atom in a strong laser field using Rothe's method. Full text in Research Archive
• Schrader, Simon Elias; Kristiansen, Håkon Emil; Pedersen, Thomas Bondo & Kvaal, Simen (2024). Rothe’s method for time evolution in strong laser fields with Explicitly Correlated Gaussians. Full text in Research Archive
• Schrader, Simon Elias; Kristiansen, Håkon Emil; Pedersen, Thomas Bondo & Kvaal, Simen (2024). Rothe’s method for time evolution in strong laser fields with Explicitly Correlated Gaussians. Full text in Research Archive
• Schrader, Simon Elias; Kristiansen, Håkon Emil; Pedersen, Thomas Bondo & Kvaal, Simen (2024). Rothe's method for quantum dynamics with Explicitly Correlated Gaussians. Full text in Research Archive
• Schrader, Simon Elias; Kristiansen, Håkon Emil; Pedersen, Thomas Bondo & Kvaal, Simen (2024). Time evolution as an optimization problem - solving the time-dependent Schrödinger equation with Explicitly Correlated Gaussians. Full text in Research Archive
• Schrader, Simon Elias; Kristiansen, Håkon Emil; Pedersen, Thomas Bondo & Kvaal, Simen (2023). Time evolution of the Hydrogen atom in a strong laser field using Rothe's method. Full text in Research Archive
• Schrader, Simon Elias; Kristiansen, Håkon Emil; Pedersen, Thomas Bondo & Kvaal, Simen (2023). Time evolution of the Hydrogen atom in a strong laser field using Rothe's method. Full text in Research Archive
• Schrader, Simon Elias; Kvaal, Simen; Pedersen, Thomas Bondo & Hjorth-Jensen, Morten (2026). Rothe’s method with Gaussians for quantum dynamics: Solving the time-dependent Schrödinger equation as an optimization problem. Universitetet i Oslo. Full text in Research Archive Show summary

  To understand how molecules and atoms interact with intense laser fields, one needs to solve the time-dependent Schrödinger equation on a computer. This equation describes how quantum systems evolve over time. However, solving it is a computationally demanding task, often requiring the use of supercomputers. This thesis develops and demonstrates a new computational approach that aims to reduce this computational cost. The main result of this thesis is the demonstration that Rothe’s method can be successfully applied to quantum dynamics using Gaussian functions. Rothe’s method is a mathematical approach for solving time-dependent equations, and Gaussian functions provide a very compact way of representing quantum states. This work develops the necessary computational protocols and theoretical considerations to apply Rothe’s method to both single-electron and many-electron systems, including approaches such as density functional theory. In addition, the method can be used to propagate nuclear quantum states represented using explicitly correlated Gaussians, a highly flexible type of Gaussian function.

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