Simen Kvaal

• Aurbakken, Einar; Kristiansen, Håkon Emil; Kvaal, Simen; Camper, Antoine Yves Marie Charles & Pedersen, Thomas Bondo (2026). Ultrafast laser-driven quantum dynamics in positronium chloride. Journal of Chemical Physics. ISSN 0021-9606. 165. doi: 10.1063/5.0335998. Full text in Research Archive Show summary

  We present a computational study of the laser-driven quantum dynamics of positronium (Ps), PsH, and PsCl at the time-dependent Hartree–Fock level of theory. To eliminate finite-basis effects and to properly capture continuum dynamics, we use a spherical polar pseudospectral representation. The multicomponent theory and its implementation are described in detail. We find that relative to H− and Cl−, respectively, the presence of the positron delays electron ionization in PsH, while slightly enhancing electron ionization in PsCl. In both cases, the positronic response is faster than that of the electrons. We propose that the formation of PsCl may be directly observed through photopositron spectra in the multiphoton regime, where PsCl peaks are expected at roughly twice the energy of Ps peaks, making PsCl clearly distinguishable from Ps. In the tunneling regime, however, photopositron rescattering peaks may only be distinguishable if the amount of Ps is sufficiently low.

• 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.

• Kristiansen, Håkon Emil; Kvernemoen, Håkon; Kvaal, Simen & Pedersen, Thomas Bondo (2025). Configuration Weights in Coupled-Cluster Theory. Journal of Physical Chemistry A. ISSN 1089-5639. 129(10), p. 2638–2654. doi: 10.1021/acs.jpca.4c07443. Full text in Research Archive Show summary

  We introduce a simple definition of the weight of any given Slater determinant in the coupled-cluster state, namely as the expectation value of the projection operator onto that determinant. The definition can be applied to any coupled-cluster formulation, including conventional coupled-cluster theory, perturbative coupled-cluster models, nonorthogonal orbital-optimized coupled-cluster theory, and extended coupled cluster theory, allowing for wave-function analyses on par with configuration-interaction-based wave functions. Numerical experiments show that for single-reference systems the coupled cluster weights are in excellent agreement with those obtained from the full configuration-interaction wave function. Moreover, the well-known insensitivity of the total energy obtained from truncated coupled-cluster models to the choice of orbital basis is clearly exposed by weights computed in the T1 transformed determinant basis. We demonstrate that the inseparability of the conventional linear parameterization of the bra (left state) for systems composed of noninteracting subsystems may lead to ill-behaved (negative or greater than unity) weights, an issue that can only be fully remedied by switching to extended coupled-cluster theory. The latter is corroborated by results obtained with quadratic coupled-cluster theory, which is shown numerically to yield a significant improvement.

• 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.

• 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.

• Wozniak, Aleksander P.; Adamowicz, Ludwik; Pedersen, Thomas Bondo & Kvaal, Simen (2025). Rothe Time Propagation for Coupled Electronic and Rovibrational Quantum Dynamics. Journal of Physical Chemistry A. ISSN 1089-5639. 129(24), p. 5391–5404. doi: 10.1021/acs.jpca.5c01732. Full text in Research Archive Show summary

  When time-propagating a wave packet representing a molecular system interacting with strong attosecond laser pulses, one needs to use an approach that is capable of describing intricate coupled electronic-nuclear events that require departure from the conventional adiabatic Born–Oppenheimer (BO) approximation. Hence, the propagation should be carried out simultaneously for the electrons and nuclei, treating both particle types on an equal footing without invoking the BO approximation. In such calculations, in order to achieve high accuracy, the wave packet needs to be expanded in basis functions that explicitly depend on interparticle distances, such as all-particle explicitly correlated Gaussians (ECGs). In our previous work, we employed basis sets consisting of ECGs with optimizable complex exponential parameters to fit time-dependent wave functions obtained from grid-based propagations of two model systems: a nucleus in a Morse potential and an electron in a central-field Coulomb-like potential, subjected to intense laser pulses. In this work, we present a proof-of-principle study of the time propagation of linear combinations of ECGs for these two models using Rothe’s method. It is shown that the approach very closely reproduces the virtually exact results of grid-based propagation for both systems. This provides further evidence that ECGs constitute a viable alternative to purely grid-based simulations of coupled nuclear-electronic dynamics driven by intense laser pulses.

• Aurbakken, Einar; Ofstad, Benedicte; Kristiansen, Håkon Emil; Schøyen, Øyvind Sigmundson; Kvaal, Simen & Sørensen, Lasse Kragh [Show all 8 contributors for this article] (2024). Transient spectroscopy from time-dependent electronic-structure theory without multipole expansions. Physical Review A (PRA). ISSN 2469-9926. 109(1). doi: 10.1103/PhysRevA.109.013109. Full text in Research Archive Show summary

  Based on the work done by an electromagnetic field on an atomic or molecular electronic system, a general gauge invariant formulation of transient absorption spectroscopy is presented within the semi-classical approximation. Avoiding multipole expansions, a computationally viable expression for the spectral response function is derived from the minimal-coupling Hamiltonian of an electronic system interacting with one or more laser pulses described by a source-free, enveloped electromagnetic vector potential. With a fixed-basis expansion of the electronic wave function, the computational cost of simulations of laser-driven electron dynamics beyond the dipole approximation is the same as simulations adopting the dipole approximation. We illustrate the theory by time-dependent configuration interaction and coupled-cluster simulations of core-level absorption and circular dichroism spectra.

• 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

• Wozniak, Aleksander P.; Adamowicz, Ludwik; Pedersen, Thomas Bondo & Kvaal, Simen (2024). Gaussians for Electronic and Rovibrational Quantum Dynamics. Journal of Physical Chemistry A. ISSN 1089-5639. 128(18), p. 3659–3671. doi: 10.1021/acs.jpca.4c00364. Full text in Research Archive Show summary

  The assumptions underpinning the adiabatic Born-Oppenheimer (BO) approximation are broken for molecules interacting with attosecond laser pulses, which generate complicated coupled electronic-nuclear wave packets that generally will have components of electronic and dissociation continua as well as bound-state contributions. The conceptually most straightforward way to overcome this challenge is to treat the electronic and nuclear degrees of freedom on equal quantum-mechanical footing by not invoking the BO approximation at all. Explicitly correlated Gaussian (ECG) basis functions have proved successful for non-BO calculations of stationary molecular states and energies, reproducing rovibrational absorption spectra with very high accuracy. In this paper, we present a proof-of-principle study of the ability of fully flexible ECGs (FFECGs) to capture the intricate electronic and rovibrational dynamics generated by short, high-intensity laser pulses. By fitting linear combinations of FFECGs to accurate wave function histories obtained on a large real-space grid for a regularized 2D model of the hydrogen atom and for the 2D Morse potential we demonstrate that FFECGs provide a very compact description of laser-driven electronic and rovibrational dynamics.

• 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 ).

• Kvaal, Simen (2023). Three Lagrangians for the complete-active space coupled-cluster method. Journal of Chemical Physics. ISSN 0021-9606. 158(24). doi: 10.1063/5.0148988. Full text in Research Archive
• Ofstad, Benedicte; Wibowo-Teale, Meilani; Kristiansen, Håkon Emil; Aurbakken, Einar; Kitsaras, Marios-Petros & Schøyen, Øyvind Sigmundson [Show all 12 contributors for this article] (2023). Magnetic optical rotation from real-time simulations in finite magnetic fields. Journal of Chemical Physics. ISSN 0021-9606. 159(20), p. 1–14. doi: 10.1063/5.0171927. Full text in Research Archive Show summary

  We present a numerical approach to magnetic optical rotation based on real-time time-dependent electronic-structure theory. Not relying on perturbation expansions in the magnetic-field strength, the formulation allows us to test the range of validity of the linear relation between the rotation angle per unit path length and the magnetic-field strength that was established empirically by Verdet 160 years ago. Results obtained from time-dependent coupled-cluster and time-dependent current density-functional theory are presented for the closed-shell molecules H2, HF, and CO in magnetic fields up to 55 kT at standard temperature and pressure conditions. We find that Verdet's linearity remains valid up to roughly 10-20 kT, above which significant deviations from linearity are observed. Among the three current density-functional approximations tested in this work, the current-dependent Tao-Perdew-Staroverov–Scuseria hybrid functional performs the best in comparison with time-dependent coupled-cluster singles and doubles results for the magnetic optical rotation.

• Faulstich, Fabian M.; Kristiansen, Håkon Emil; Csirik, Mihaly Andras; Kvaal, Simen; Pedersen, Thomas Bondo & Laestadius, Andre (2023). The S-diagnostic - an a posteriori error assessment for single-reference coupled-cluster methods. Journal of Physical Chemistry A. ISSN 1089-5639. 127(43), p. 9106–9120. doi: 10.1021/acs.jpca.3c01575. Full text in Research Archive Show summary

  We propose a novel a posteriori error assessment for the single-reference coupled-cluster (SRCC) method called the S-diagnostic. We provide a derivation of the S-diagnostic that is rooted in the mathematical analysis of different SRCC variants. We numerically scrutinized the S-diagnostic, testing its performance for (1) geometry optimizations, (2) electronic correlation simulations of systems with varying numerical difficulty, and (3) the square-planar copper complexes [CuCl4]2-, [Cu(NH3)4]2+, and [Cu(H2O)4]2+. Throughout the numerical investigations, the S-diagnostic is compared to other SRCC diagnostic procedures, that is, the T1, D1, and D2 diagnostics as well as different indices of multi-determinantal and multi-reference character in coupled-cluster theory. Our numerical investigations show that the S-diagnostic outperforms the T1, D1, and D2 diagnostics and is comparable to the indices of multi-determinantal and multi-reference character in coupled-cluster theory in their individual fields of applicability. The experiments investigating the performance of the S-diagnostic for geometry optimizations using SRCC reveal that the S-diagnostic correlates well with different error measures at a high level of statistical relevance. The experiments investigating the performance of the S-diagnostic for electronic correlation simulations show that the S-diagnostic correctly predicts strong multi-reference regimes. The S-diagnostic moreover correctly detects the successful SRCC computations for [CuCl4]2-, [Cu(NH3)4]2+, and [Cu(H2O)4]2+, which have been known to be misdiagnosed by T1 and D1 diagnostics in the past. This shows that the S-diagnostic is a promising candidate for an a posteriori diagnostic for SRCC calculations.

• Ofstad, Benedicte; Aurbakken, Einar; Schøyen, Øyvind Sigmundson; Kristiansen, Håkon Emil; Kvaal, Simen & Pedersen, Thomas Bondo (2023). Time-dependent coupled-cluster theory. Wiley Interdisciplinary Reviews. Computational Molecular Science. ISSN 1759-0876. 13(5). doi: 10.1002/wcms.1666. Full text in Research Archive Show summary

  Recent years have witnessed an increasing interest in time-dependent coupled-cluster (TDCC) theory for simulating laser-driven electronic dynamics in atoms and molecules, and for simulating molecular vibrational dynamics. Starting from the time-dependent bivariational principle, we review different flavors of single-reference TDCC theory with either orthonormal static, orthonormal time-dependent, or biorthonormal time-dependent spin orbitals. The time-dependent extension of equation-of-motion coupled-cluster theory is also discussed, along with the applications of TDCC methods to the calculation of linear absorption spectra, linear and low-order nonlinear response functions, highly nonlinear high harmonic generation spectra and ionization dynamics. In addition, the role of TDCC theory in finite-temperature many-body quantum mechanics is briefly described along with a few other application areas.

• Ofstad, Benedicte; Kristiansen, Håkon Emil; Aurbakken, Einar; Schøyen, Øyvind Sigmundson; Kvaal, Simen & Pedersen, Thomas Bondo (2023). Adiabatic extraction of nonlinear optical properties from real-time time-dependent electronic-structure theory. Journal of Chemical Physics. ISSN 0021-9606. 158(15). doi: 10.1063/5.0145521. Full text in Research Archive Show summary

  Real-time simulations of laser-driven electron dynamics contain information about molecular optical properties through all orders in response theory. These properties can be extracted by assuming convergence of the power series expansion of induced electric and magnetic multipole moments. However, the accuracy relative to analytical results from response theory quickly deteriorates for higher-order responses due to the presence of high-frequency oscillations in the induced multipole moment in the time domain. This problem has been ascribed to missing higher-order corrections. We here demonstrate that the deviations are caused by nonadiabatic effects arising from the finite-time ramping from zero to full strength of the external laser field. Three different approaches, two using a ramped wave and one using a pulsed wave, for extracting electrical properties from real-time time-dependent electronic-structure simulations are investigated. The standard linear ramp is compared to a quadratic ramp, which is found to yield highly accurate results for polarizabilities, and first and second hyperpolarizabilities, at roughly half the computational cost. Results for the third hyperpolarizability are presented along with a simple, computable measure of reliability.

• Adamowicz, Ludwik; Kvaal, Simen; Lasser, Caroline & Pedersen, Thomas Bondo (2022). No need for a grid: Adaptive fully-flexible gaussians for the time-dependent Schrödinger equation. arXiv.org. ISSN 2331-8422. doi: 10.48550/arXiv.2207.0027. Full text in Research Archive Show summary

  Linear combinations of complex gaussian functions, where the linear and nonlinear parameters are allowed to vary, are shown to provide an extremely flexible and effective approach for solving the time-dependent Schrödinger equation in one spatial dimension. The use of flexible basis sets has been proven notoriously hard within the systematics of the Dirac--Frenkel variational principle. In this work we present an alternative time-propagation scheme that de-emphasizes optimal parameter evolution but directly targets residual minimization via the method of Rothe's method, also called the method of vertical time layers. We test the scheme using a simple model system mimicking an atom subjected to an extreme laser pulse. Such a pulse produces complicated ionization dynamics of the system. The scheme is shown to perform very well on this model and notably does not rely on a computational grid. Only a handful of gaussian functions are needed to achieve an accuracy on par with a high-resolution, grid-based solver. This paves the way for accurate and affordable solution of the time-dependent Schrödinger equation for atoms and molecules within and beyond the Born--Oppenheimer approximation.

• Adamowicz, Ludwik; Kvaal, Simen; Lasser, Caroline & Pedersen, Thomas Bondo (2022). Laser-induced dynamic alignment of the HD molecule without the Born-Oppenheimer approximation. Journal of Chemical Physics. ISSN 0021-9606. 157(14). doi: 10.1063/5.0101352. Full text in Research Archive Show summary

  Laser-induced molecular alignment is well understood within the framework of the Born-Oppenheimer (BO) approximation. Without the BO approximation, however, the concept of molecular structure is lost, making alignment hard to define precisely. In this work, we demonstrate the emergence of alignment from the first-ever non-BO quantum dynamics simulations, using the HD molecule exposed to ultrashort laser pulses as a few-body test case. We extract the degree of alignment from the non-BO wave function by means of an operator expressed in terms of pseudo-proton coordinates that mimics the BO-based definition of alignment. The only essential approximation, in addition to the semiclassical electric-dipole approximation for the matter-field interaction, is the choice of time-independent explicitly correlated Gaussian basis functions. We use a variational, electric-field-dependent basis-set construction procedure, which allows us to keep the basis-set dimension low whilst capturing the main effects of electric polarization on the nuclear and electronic degrees of freedom. The basis-set construction procedure is validated by comparing with virtually exact grid-based simulations for two one-dimensional model systems: laser-driven electron dynamics in a soft attractive Coulomb potential and nuclear rovibrational dynamics in a Morse potential.

• Teale, Andrew Michael; Helgaker, Trygve; Savin, Andreas; Adamo, Carlo; Aradi, Balint & Arbuznikov, Alexei V. [Show all 70 contributors for this article] (2022). DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science. Physical Chemistry, Chemical Physics - PCCP. ISSN 1463-9076. doi: 10.1039/d2cp02827a. Full text in Research Archive Show summary

  In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

• Kristiansen, Håkon Emil; Ofstad, Benedicte; Hauge, Eirill Strand; Aurbakken, Einar; Schøyen, Øyvind Sigmundson & Kvaal, Simen [Show all 7 contributors for this article] (2022). Linear and Nonlinear Optical Properties from TDOMP2 Theory. Journal of Chemical Theory and Computation. ISSN 1549-9618. 18(6), p. 3687–3702. doi: 10.1021/acs.jctc.1c01309. Full text in Research Archive Show summary

  We present a derivation of real-time (RT) time-dependent orbital-optimized Møller–Plesset (TDOMP2) theory and its biorthogonal companion, time-dependent non-orthogonal OMP2 theory, starting from the time-dependent bivariational principle and a parametrization based on the exponential orbital-rotation operator formulation commonly used in the time-independent molecular electronic structure theory. We apply the TDOMP2 method to extract absorption spectra and frequency-dependent polarizabilities and first hyperpolarizabilities from RT simulations, comparing the results with those obtained from conventional time-dependent coupled-cluster singles and doubles (TDCCSD) simulations and from its second-order approximation, TDCC2. We also compare our results with those from CCSD and CC2 linear and quadratic response theories. Our results indicate that while TDOMP2 absorption spectra are of the same quality as TDCC2 spectra, including core excitations where optimized orbitals might be particularly important, frequency-dependent polarizabilities and hyperpolarizabilities from TDOMP2 simulations are significantly closer to TDCCSD results than those from TDCC2 simulations.

• Kvaal, Simen; Laestadius, Andre; Tellgren, Erik Ingemar & Helgaker, Trygve (2021). Lower Semicontinuity of the Universal Functional in Paramagnetic Current-Density Functional Theory. The Journal of Physical Chemistry Letters. 12(5), p. 1421–1425. doi: 10.1021/acs.jpclett.0c03422. Full text in Research Archive Show summary

  A cornerstone of current–density functional theory (CDFT) in its paramagnetic formulation is proven. After a brief outline of the mathematical structure of CDFT, the lower semicontinuity and expectation-valuedness of the CDFT constrained-search functional is proven, meaning that there is always a minimizing density matrix in the CDFT constrained-search universal density functional. These results place the mathematical framework of CDFT on the same footing as that of standard DFT.

• Pedersen, Thomas Bondo; Kristiansen, Håkon Emil; Bodenstein, Tilmann; Kvaal, Simen & Schøyen, Øyvind Sigmundson (2021). Interpretation of Coupled-Cluster Many-Electron Dynamics in Terms of Stationary States. Journal of Chemical Theory and Computation. ISSN 1549-9618. 17(1), p. 388–404. doi: 10.1021/acs.jctc.0c00977. Full text in Research Archive Show summary

  We demonstrate theoretically and numerically that laser-driven many-electron dynamics, as described by bivariational time-dependent coupled-cluster theory, may be analyzed in terms of stationary-state populations. Projectors heuristically defined from linear response theory and equation-of-motion coupled cluster theory are proposed for the calculation of stationary-state populations during interaction with laser pulses or other external forces, and conservation laws of the populations are discussed. Numerical tests of the proposed projectors, involving both linear and nonlinear optical processes for the He and Be atoms, and for the LiH, CH+, and LiF molecules, show that the laser-driven evolution of the stationary-state populations at the coupled-cluster singles-and-doubles (CCSD) level is very close to that obtained by ful configuration-interaction theory provided all stationary states actively participating in the dynamics are sufficiently well approximated. When double-excited states are important for the dynamics, the quality of the CCSD results deteriorate. Observing that populations computed from the linear-response projector may show spurious small-amplitude, high-frequency oscillations, the equation-of-motion projector emerges as the most promising approach to stationary-state populations.

• Kristiansen, Håkon Emil; Schøyen, Øyvind Sigmundson; Kvaal, Simen & Pedersen, Thomas Bondo (2020). Numerical stability of time-dependent coupled-cluster methods for many-electron dynamics in intense laser pulses. Journal of Chemical Physics. ISSN 0021-9606. 152. doi: 10.1063/1.5142276. Full text in Research Archive Show summary

  We investigate the numerical stability of time-dependent coupled-cluster theory for many-electron dynamics in intense laser pulses, comparing two coupled-cluster formulations with full configuration interaction theory. Our numerical experiments show that orbital-adaptive time-dependent coupled-cluster doubles (OATDCCD) theory offers significantly improved stability compared with the conventional Hartree-Fock-based time-dependent coupled-cluster singles-and-doubles (TDCCSD) formulation. The improved stability stems from greatly reduced oscillations in the doubles amplitudes, which, in turn, can be traced to the dynamic biorthonormal reference determinants of OATDCCD theory. As long as these are good approximations to the Brueckner determinant, OATDCCD theory is numerically stable. We propose the reference weight as a diagnostic quantity to identify situations where the TDCCSD and OATDCCD theories become unstable.

• Bodenstein, Tilmann & Kvaal, Simen (2020). A state-specific multireference coupled-cluster method based on the bivariational principle. Journal of Chemical Physics. ISSN 0021-9606. 153(2). doi: 10.1063/5.0009429. Full text in Research Archive Show summary

  A state-specific multireference coupled-cluster (MRCC) method based on Arponen’s bivariational principle is presented, the bivar-MRCC method. The method is based on single-reference theory and therefore has a relatively straightforward formulation and modest computational complexity. The main difference from established methods is the bivariational formulation, in which independent parameterizations of the wave function (ket) and its complex conjugate (bra) are made. Importantly, this allows manifest multiplicative separability of the state (exact in the extended bivar-MRECC version of the method and approximate otherwise), and additive separability of the energy, while preserving polynomial scaling of the working equations. A feature of the bivariational principle is that the formal bra and ket references can be included as bivariational parameters, which eliminates much of the bias toward the formal reference. A pilot implementation is described, and extensive benchmark calculations on several standard problems are performed. The results from the bivar-MRCC method are comparable to established state-specific multireference methods. Considering the relative affordability of the bivar-MRCC method, it may become a practical tool for non-experts.

• Kvaal, Simen; Laestadius, Andre & Bodenstein, Tilmann (2020). Guaranteed convergence for a class of coupled-cluster methods based on Arponen's extended theory. Molecular Physics. ISSN 0026-8976. 118(19-20). doi: 10.1080/00268976.2020.1810349. Full text in Research Archive Show summary

  A wide class of coupled-cluster methods is introduced, based on Arponen's extended coupled-cluster theory. This class of methods is formulated in terms of a coordinate transformation of the cluster operators. The mathematical framework for the error analysis of coupled-cluster methods based on Arponen's bivariational principle is presented, in which the concept of local strong monotonicity of the flipped gradient of the energy is central. A general mathematical result is presented, describing sufficient conditions for coordinate transformations to preserve the local strong monotonicity. The result is applied to the presented class of methods, which include the standard and quadratic coupled-cluster methods, and also Arponen's canonical version of extended coupled-cluster theory. Some numerical experiments are presented, and the use of canonical coordinates for diagnostics is discussed.

• Laestadius, Andre; Tellgren, Erik; Penz, Markus; Ruggenthaler, Michael; Kvaal, Simen & Helgaker, Trygve (2019). Kohn–Sham Theory with Paramagnetic Currents: Compatibility and Functional Differentiability. Journal of Chemical Theory and Computation. ISSN 1549-9618. 15(7), p. 4003–4020. doi: 10.1021/acs.jctc.9b00141. Full text in Research Archive
• Pedersen, Thomas Bondo & Kvaal, Simen (2019). Symplectic integration and physical interpretation of time-dependent coupled-cluster theory. Journal of Chemical Physics. ISSN 0021-9606. 150. doi: 10.1063/1.5085390. Full text in Research Archive Show summary

  The formulation of the time-dependent Schrödinger equation in terms of coupled-cluster theory is outlined, with emphasis on the bivariational framework and its classical Hamiltonian structure. An indefinite inner product is introduced, inducing physical interpretation of coupled-cluster states in the form of transition probabilities, autocorrelation functions, and explicitly real values for observables, solving interpretation issues which are present in time-dependent coupled-cluster theory and in ground-state calculations of molecular systems under the influence of external magnetic fields. The problem of the numerical integration of the equations of motion is considered, and a critical evaluation of the standard fourth-order Runge–Kutta scheme and the symplectic Gauss integrator of variable order are given, including several illustrative numerical experiments. While the Gauss integrator is stable even for laser pulses well above the perturbation limit, our experiments indicate that a system-dependent upper limit exists for the external field strengths. Above this limit, time-dependent coupled-cluster calculations become very challenging numerically, even in the full configuration interaction limit. The source of these numerical instabilities is shown to be rapid increases of the amplitudes as ultrashort high-intensity laser pulses pump the system out of the ground state into states that are virtually orthogonal to the static Hartree-Fock reference determinant.

• Faulstich, Fabian Maximilian; Máté, Mihály; Laestadius, Andre; Csirik, Mihály András; Veis, Libor & Antalik, Andrej [Show all 11 contributors for this article] (2019). Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen Dimer. Journal of Chemical Theory and Computation. ISSN 1549-9618. 15(4), p. 2206–2220. doi: 10.1021/acs.jctc.8b00960. Full text in Research Archive Show summary

  In this article, we investigate the numerical and theoretical aspects of the coupled-cluster method tailored by matrix-product states. We investigate formal properties of the used method, such as energy size consistency and the equivalence of linked and unlinked formulation. The existing mathematical analysis is here elaborated in a quantum chemical framework. In particular, we highlight the use of what we have defined as a complete active space-external space gap describing the basis splitting between the complete active space and the external part generalizing the concept of a HOMO–LUMO gap. Furthermore, the behavior of the energy error for an optimal basis splitting, i.e., an active space choice minimizing the density matrix renormalization group-tailored coupled-cluster singles doubles error, is discussed. We show numerical investigations on the robustness with respect to the bond dimensions of the single orbital entropy and the mutual information, which are quantities that are used to choose a complete active space. Moreover, the dependence of the ground-state energy error on the complete active space has been analyzed numerically in order to find an optimal split between the complete active space and external space by minimizing the density matrix renormalization group-tailored coupled-cluster error.

• Faulstich, Fabian Maximilian; Laestadius, Andre; Legeza, Örs; Schneider, Reinhold & Kvaal, Simen (2019). Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 57(6), p. 2579–2607. doi: 10.1137/18M1171436. Full text in Research Archive Show summary

  n quantum chemistry, one of the most important challenges is the static correlation problem when solving the electronic Schrödinger equation for molecules in the Born--Oppenheimer approximation. In this article, we analyze the tailored coupled-cluster method (TCC), one particular and promising method for treating molecular electronic-structure problems with static correlation. The TCC method combines the single-reference coupled-cluster (CC) approach with an approximate reference calculation in a subspace (complete active space (CAS)) of the considered Hilbert space that covers the static correlation. A one-particle spectral gap assumption is introduced, separating the CAS from the remaining Hilbert space. This replaces the nonexisting or nearly nonexisting gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital usually encountered in standard single-reference quantum chemistry. The analysis covers, in particular, CC methods tailored by tensor-network states (TNS-TCC methods). The problem is formulated in a nonlinear functional analysis framework, and, under certain conditions such as the aforementioned gap, local uniqueness and existence are proved using Zarantonello's lemma. From the Aubin--Nitsche-duality method, a quadratic error bound valid for TNS-TCC methods is derived, e.g., for linear-tensor-network TCC schemes using the density matrix renormalization group method. © 2019, Society for Industrial and Applied Mathematics

• Laestadius, Andre & Kvaal, Simen (2018). Analysis of the Extended Coupled-Cluster Method in Quantum Chemistry. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 56(2), p. 660–683. doi: 10.1137/17M1116611. Full text in Research Archive Show summary

  The mathematical foundation of the so-called extended coupled-cluster method for the solution of the many-fermion Schrödinger equation is here developed. We prove an existence and uniqueness result, both in the full infinite-dimensional amplitude space as well as for discretized versions of it. The extended coupled-cluster method is formulated as a critical point of an energy function using a generalization of the Rayleigh--Ritz principle: the bivariational principle. This gives a quadratic bound for the energy error in the discretized case. The existence and uniqueness results are proved using a type of monotonicity property for the flipped gradient of the energy function. A comparison to the analysis of the standard coupled-cluster method is made, and it is argued that the bivariational principle is a useful tool, both for studying coupled-cluster type methods and for developing new computational schemes in general. © 2018 Society for Industrial and Applied Mathematics

• Tellgren, Erik; Laestadius, Andre; Helgaker, Trygve; Kvaal, Simen & Teale, Andrew Michael (2018). Uniform magnetic fields in density-functional theory. Journal of Chemical Physics. ISSN 0021-9606. 148(2), p. 1–19. doi: 10.1063/1.5007300. Full text in Research Archive
• Laestadius, Andre; Penz, Markus; Tellgren, Erik; Ruggenthaler, Michael; Kvaal, Simen & Helgaker, Trygve (2018). Generalized Kohn-Sham iteration on Banach spaces. Journal of Chemical Physics. ISSN 0021-9606. 149(16), p. 1–9. doi: 10.1063/1.5037790. Full text in Research Archive Show summary

  A detailed account of the Kohn–Sham (KS) algorithm from quantum chemistry, formulated rigorously in the very general setting of convex analysis on Banach spaces, is given here. Starting from a Levy–Lieb-type functional, its convex and lower semi-continuous extension is regularized to obtain differentiability. This extra layer allows us to rigorously introduce, in contrast to the common unregularized approach, a well-defined KS iteration scheme. Convergence in a weak sense is then proven. This generalized formulation is applicable to a wide range of different density-functional theories and possibly even to models outside of quantum mechanics.

• Kvaal, Simen & Helgaker, Trygve (2015). Ground-state densities from the Rayleigh-Ritz variation principle and from density-functional theory. Journal of Chemical Physics. ISSN 0021-9606. 143(18). doi: 10.1063/1.4934797. Full text in Research Archive
• Kvaal, Simen; Ekström, Ulf Egil; Teale, Andrew Michael & Helgaker, Trygve (2014). Differentiable but exact formulation of density-functional theory. Journal of Chemical Physics. ISSN 0021-9606. 140(18). doi: 10.1063/1.4867005. Full text in Research Archive
• Jarlebring, Elias; Kvaal, Simen & Michiels, Wim (2014). An inverse iteration method for eigenvalue problems with eigenvector nonlinearities. SIAM Journal on Scientific Computing. ISSN 1064-8275. 36(4), p. A1978–A2001. doi: 10.1137/130910014. Full text in Research Archive
• Tellgren, Erik; Kvaal, Simen & Helgaker, Trygve (2014). Fermion N -representability for prescribed density and paramagnetic current density. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 89(1). doi: 10.1103/PhysRevA.89.012515. Full text in Research Archive Show summary

  The N-representability problem is the problem of determining whether there exists N-particle states with some prescribed property. Here we report an affirmative solution to the fermion N-representability problem when both the density and the paramagnetic current density are prescribed. This problem arises in current-density functional theory and is a generalization of the well-studied corresponding problem (only the density prescribed) in density functional theory. Given any density and paramagnetic current density satisfying a minimal regularity condition (essentially that a von Weizäcker–like canonical kinetic energy density is locally integrable), we prove that there exists a corresponding N-particle state. We prove this by constructing an explicit one-particle reduced density matrix in the form of a position-space kernel, i.e., a function of two continuous-position variables. In order to make minimal assumptions, we also address mathematical subtleties regarding the diagonal of, and how to rigorously extract paramagnetic current densities from, one-particle reduced density matrices in kernel form. © 2014 American Physical Society

• Kvaal, Simen (2013). Variational formulations of the coupled-cluster method in quantum chemistry. Molecular Physics. ISSN 0026-8976. 111(9-11), p. 1100–1108. doi: 10.1080/00268976.2013.812254. Full text in Research Archive
• Kvaal, Simen (2012). Ab initio quantum dynamics using coupled-cluster. Journal of Chemical Physics. ISSN 0021-9606. 136(19). doi: 10.1063/1.4718427. Full text in Research Archive
• Halvorsen, Tore Gunnar & Kvaal, Simen (2012). Manifestly gauge invariant discretizations of the Schrodinger equation. Physics Letters A. ISSN 0375-9601. 376(12-13), p. 1107–1114. doi: 10.1016/j.physleta.2012.02.028. Full text in Research Archive
• Tellgren, Erik; Kvaal, Simen; Sagvolden, Espen; Ekström, Ulf Egil; Teale, Andrew Michael & Helgaker, Trygve (2012). Choice of basic variables in current-density-functional theory. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 86(6). doi: 10.1103/PhysRevA.86.062506. Full text in Research Archive Show summary

  The selection of basic variables in current-density-functional theory and formal properties of the resulting formulations are critically examined. Focus is placed on the extent to which the Hohenberg-Kohn theorem, constrained-search approach, and Lieb's formulation (in terms of convex and concave conjugation) of standard density-functional theory can be generalized to provide foundations for current-density-functional theory. For the well-known case with the gauge-dependent paramagnetic current density as a basic variable, we find that the resulting total energy functional is not concave. It is shown that a simple redefinition of the scalar potential restores concavity and enables the application of convex analysis and convex (or concave) conjugation. As a result, the solution sets arising in potential-optimization problems can be given a simple characterization. We also review attempts to establish theories with the physical current density as a basic variable. Despite the appealing physical motivation behind this choice of basic variables, we find that the mathematical foundations of the theories proposed to date are unsatisfactory. Moreover, the analogy to standard density-functional theory is substantially weaker as neither the constrained-search approach nor the convex analysis framework carry over to a theory making use of the physical current density. © 2012 American Physical Society

• Selstø, Sølve; Birkeland, Tore; Kvaal, Simen; Nepstad, Raymond & Førre, Morten (2011). A master equation approach to double ionization of helium. Journal of Physics B: Atomic, Molecular and Optical Physics. ISSN 0953-4075. 44(21). doi: 10.1088/0953-4075/44/21/215003. Full text in Research Archive
• Kvaal, Simen (2011). Multiconfigurational time-dependent Hartree method to describe particle loss due to absorbing boundary conditions. Physical Review A. Atomic, Molecular, and Optical Physics (PRA). ISSN 1050-2947. 84(2). doi: 10.1103/PhysRevA.84.022512. Full text in Research Archive
• Lohne, Magnus Pedersen; Hagen, Gaute; Hjorth-Jensen, Morten; Kvaal, Simen & Pederiva, Francesco (2011). Ab initio computation of the energies of circular quantum dots. Physical Review B. Condensed Matter and Materials Physics. ISSN 1098-0121. 84. doi: 10.1103/PhysRevB.84.115302. Full text in Research Archive Show summary

  We perform coupled-cluster and diffusion Monte Carlo calculations of the energies of circular quantum dots up to 20 electrons. The coupled-cluster calculations include triples corrections and a renormalized Coulomb interaction defined for a given number of low-lying oscillator shells. Using such a renormalized Coulomb interaction brings the coupled-cluster calculations with triples correlations in excellent agreement with the diffusion Monte Carlo calculations. This opens up perspectives for doing ab initio calculations for much larger systems of electrons.

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• Pedersen, Thomas Bondo; Aurbakken, Einar; Kristiansen, Håkon Emil; Kvaal, Simen & Camper, Antoine Yves Marie Charles (2026). Mean-field theory of ultrafast laser-driven quantum dynamics in positronium chloride. Full text in Research Archive
• Pedersen, Thomas Bondo; Kvaal, Simen & Kristiansen, Håkon Emil (2026). Making Chemical Sense of Coupled-Cluster Dynamics. Full text in Research Archive
• 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

• 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
• Kvaal, Simen (2024). On coupled-cluster theory for bosons. 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; Kvaal, Simen; Adamowicz, Ludwik; Cezar, Henrique Musseli; Lang, Lucas & Lasser, Caroline [Show all 7 contributors for this article] (2024). Molecular Structure and Quantum Dynamics Without the Born-Oppenheimer Approximation. Full text in Research Archive
• 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
• Kvaal, Simen; Bodenstein, Tilmann; Laestadius, Andre & Reinhold, Schneider (2023). Development of multireference coupled-cluster methods from single-state and multi-state bivariational principles. Full text in Research Archive
• Kristiansen, Håkon Emil; Ofstad, Benedicte Sverdrup; Hauge, Eirill Strand; Aurbakken, Einar; Schøyen, Øyvind Sigmundson & Kvaal, Simen [Show all 7 contributors for this article] (2022). Correction to “Linear and Nonlinear Optical Properties from TDOMP2 Theory”. Journal of Chemical Theory and Computation. ISSN 1549-9618. 18(9), p. 5755–5757. doi: 10.1021/acs.jctc.2c00830. Full text in Research Archive
• Kvaal, Simen; Pedersen, Thomas Bondo; Lasser, Caroline & Adamowicz, Ludwik (2022). No need for a grid: Gaussians for the time-dependent Schrödinger equation. Full text in Research Archive
• Kvaal, Simen; Pedersen, Thomas Bondo; Lasser, Caroline & Adamowicz, Ludwik (2022). Time evolution using linear combinations of gaussians. Full text in Research Archive
• Kvaal, Simen (2022). Time evolution using linear combinations of gaussians. Full text in Research Archive
• Bodenstein, Tilmann & Kvaal, Simen (2020). A multireference coupled-cluster method based on the bivariational principle. Full text in Research Archive
• Laestadius, Andre; Penz, Markus; Tellgren, Erik; Ruggenthaler, Michael; Kvaal, Simen & Helgaker, Trygve (2019). Guaranteed convergence of a regularized Kohn-Sham iteration in finite dimensions. Full text in Research Archive Show summary

  Guaranteed convergence of a regularized Kohn-Sham iteration in finite dimensions M. Penz2, A. Laestadius1, E. Tellgren1, M. Ruggenthaler2, S. Kvaal1, T. Helgaker1 1. University of Oslo, Department of Chemistry, Oslo, Germany 2 .Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany The iterative Kohn-Sham scheme [1] has to date not been rigorously shown to converge to the correct ground-state density. This talk addresses the recent result of Penz et al. [2] that demonstrates the convergence of the exact Moreau-Yosida regularized theory in a finite-dimensional setting. This builds on previous work [3], where a similar iterative scheme was proposed that proved a weak type of convergence following an idea by Wagner et al. [4,5]. To obtain the desired convergence in both densities and potentials, the Moreau-Yosida regularization is key for the convergence proof in [2]. This ensures differentiability of the universal Lieb functional [6] and was introduced in density-functional theory (DFT) by Kvaal et al. [7]. It has also recently been successfully applied to paramagnetic current DFT [8]. References [1] W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965). [2] M. Penz, A. Laestadius, E.I. Tellgren, and Michael Ruggenthaler, Phys. Rev. Lett. 123, 037401 (2019). [3] A. Laestadius, M. Penz, E.I. Tellgren, M. Ruggenthaler, S. Kvaal, and T. Helgaker, J. Chem. Phys. 149, 164103 (2018). [4] L.O. Wagner, E. M. Stoudenmire, K. Burke, and S.R. White, Phys. Rev. Lett. 111, 093003 (2013). [5] L.O. Wagner, T. E. Baker, E. M. Stoudenmire, K. Burke, and S.R. White, Physical Review B 90, 045109 (2014). [6] E.H. Lieb, Int. J. Quantum Chem. 24, 243 (1983). [7] S. Kvaal, U. Ekström, A.M. Teale, and T. Helgaker, J. Chem. Phys. 140, 18A518 (2014). [8] A. Laestadius, M. Penz, E.I. Tellgren, M. Ruggenthaler, S. Kvaal, and T. Helgaker, J. Chem. Theory Comput. 15, 4003 (2019).

• Faulstich, Fabian Maximilian; Máté, Mihály; Laestadius, Andre; Csirik, Mihály András; Veis, Libor & Antalik, Andrej [Show all 11 contributors for this article] (2019). Numerical and theoretical aspects of the DMRG-TCC method exemplified by the nitrogen dimer. Full text in Research Archive
• Faulstich, Fabian Maximilian; Máté, Mihály; Laestadius, Andre; Csirik, Mihály András; Veis, Libor & Antalik, Andrej [Show all 11 contributors for this article] (2019). Mathematical and numerical aspects of the coupled-cluster method tailored by tensor network states. Full text in Research Archive
• Laestadius, Andre; Penz, Markus; Tellgren, Erik; Ruggenthaler, Michael; Kvaal, Simen & Helgaker, Trygve (2018). Generalized Kohn-Sham iteration on Banach Spaces. Full text in Research Archive
• Tellgren, Erik; Lange, Kai Kaarvann; Sagvolden, Espen; Kvaal, Simen; Austad, Jon & Reimann, Sarah [Show all 10 contributors for this article] (2018). Quantum chemistry for strong magnetic fields. Full text in Research Archive
• Laestadius, Andre; Faulstich, Fabian Maximilian; Kvaal, Simen; Legeza, Örs & Schneider, Reinhold (2018). The Study of Coupled-Cluster Methods Using Strong Monotonicity. Full text in Research Archive
• Laestadius, Andre & Kvaal, Simen (2018). ANALYSIS OF THE EXTENDED COUPLED-CLUSTER METHOD IN QUANTUM CHEMISTRY. Full text in Research Archive
• Pedersen, Thomas Bondo & Kvaal, Simen (2018). Time-Dependent Coupled-Cluster Theory. Full text in Research Archive
• Pedersen, Thomas Bondo & Kvaal, Simen (2018). Electron Dynamics with Coupled-Cluster Theory. Full text in Research Archive
• Laestadius, Andre; Faulstich, Fabian Maximilian; Kvaal, Simen; Legeza, Örs & Schneider, Reinhold (2018). Analysis of The Coupled-Cluster Method Tailored by Tensor-Network States in Quantum Chemistry. Full text in Research Archive
• Faulstich, Fabian Maximilian; Laestadius, Andre; Legeza, Örs; Schneider, Reinhold & Kvaal, Simen (2018). Quadratic error bounds for the coupled-cluster method tailored by tensor-network states. Full text in Research Archive
• Faulstich, Fabian Maximilian; Laestadius, Andre; Legeza, Örs; Schneider, Reinhold & Kvaal, Simen (2018). Mathematical analysis of the coupled-cluster method tailored by tensor-network states. Full text in Research Archive
• Faulstich, Fabian Maximilian; Laestadius, Andre; Legeza, Örs; Schneider, Reinhold & Kvaal, Simen (2017). Mathematical aspects of the coupled-cluster method tailored by tensor-network states in quantum chemistry. Full text in Research Archive
• Kvaal, Simen & Laestadius, Andre (2016). The extended coupled-cluster method and its rigorous analysis. Full text in Research Archive
• Laestadius, Andre & Kvaal, Simen (2016). Analysis of the Extended Coupled-Cluster Method. Full text in Research Archive
• Tellgren, Erik; Teale, Andrew Michael; Ekström, Ulf Egil; Kvaal, Simen; Sagvolden, Espen & Helgaker, Trygve (2015). Current density functional theory for molecular systems in strong magnetic fields. Full text in Research Archive
• Helgaker, Trygve; Kvaal, Simen; Ekström, Ulf Egil & Teale, Andy (2014). Differentiable but Exact Formulation of Density-Functional Theory. Full text in Research Archive
• Helgaker, Trygve; Kvaal, Simen; Teale, Andrew Michael & Ekström, Ulf Egil (2014). Differentiable but Exact Formulation of Density-Functional Theory. Full text in Research Archive
• Helgaker, Trygve; Kvaal, Simen; Ekström, Ulf Egil & Teale, Andrew Michael (2014). Differentiable but Exact Formulation of Density-Functional Theory. Full text in Research Archive
• Helgaker, Trygve; Kvaal, Simen; Teale, Andrew Michael; Ekström, Ulf Egil; Jørgensen, Poul & Olsen, Jeppe (2014). Differentiable but Exact Formulation of Density-Functional Theory. Full text in Research Archive
• Helgaker, Trygve; Borgoo, Alex; Ekström, Ulf Egil; Furness, James; Hoffmann, Mark & Kvaal, Simen [Show all 12 contributors for this article] (2014). Density-Functional Theory in Magnetic Fields. Full text in Research Archive
• Kvaal, Simen (2014). Bivariational approximations & the orbital-adaptive coupled-cluster method. Full text in Research Archive
• Kvaal, Simen; Ekström, Ulf Egil; Tellgren, Erik; Borgoo, Alex; Helgaker, Trygve & Sagvolden, Espen (2013). Moreau-Yosida regularization of DFT. Full text in Research Archive
• Kvaal, Simen (2013). Ab initio dynamics using the coupled cluster method. Full text in Research Archive
• Sagvolden, Espen; Tellgren, Erik; Kvaal, Simen; Ekström, Ulf Egil; Teale, Andrew Michael & Helgaker, Trygve (2013). Building blocks of Current Density Functional Theory. Full text in Research Archive
• Kvaal, Simen (2013). Abels Tårn. [Radio]. NRK P2. Full text in Research Archive
• Sagvolden, Espen; Tellgren, Erik; Kvaal, Simen; Ekström, Ulf Egil; Teale, Andrew Michael & Helgaker, Trygve (2013). Building blocks of Current Density Functional Theory. Full text in Research Archive
• Ekström, Ulf Egil; Kvaal, Simen; Borgoo, Alex; Helgaker, Trygve; Sagvolden, Espen & Tellgren, Erik (2013). Moreau-Yosida regularization of DFT. Full text in Research Archive
• Selstø, Sølve; Kvaal, Simen; Birkeland, Tore; Nepstad, Raymond & Førre, Morten (2012). Double ionization with absorbers. EurophysicsNews. ISSN 0531-7479. 43(1), p. 15–16. 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.

• Davidov, Aleksandar; Kvaal, Simen & Balcells, David (2023). Improving semiempirical quantum chemistry with graph neural networks. Universitetet i Oslo. Full text in Research Archive
• Ofstad, Benedicte Sverdrup & Kvaal, Simen (2023). Time-domain quantum dynamics: Optical properties from time-dependent electronic-structure theory. Universitetet i Oslo. Full text in Research Archive Show summary

  Within the field of nonlinear optics, nonlinear optical properties are defined as frequency-dependent responses. This definition emerged in the frequency domain, as lasers used in these early experiments emitted monochromatic continuous waves. This led to the development of time-dependent electronic-structure theory predominantly in the frequency domain, using perturbation-theory based response theory. However, in recent years, advances in computing power and the advent of ultrashort laser pulses have sparked interest in the time-domain for both electronic-structure theory and nonlinear optics. Time-domain electronic-structure theory provides a time-resolved description of light-matter interaction, closely mimicking experiment. Additionally, they offer the advantage of highly nonlinear responses being straightforward to implement. This thesis contributes to the field of time-domain time-dependent electronic-structure theory for the description of nonlinear optical properties by: Investigating the potential of dynamic (time-dependent) orbitals for improving the description of nonlinear optical properties, extending a hierarchy of time-domain time-dependent coupled-cluster methods to accommodate strong magnetic fields, and by developing an efficient approach for extracting higher-order response properties.

• Kristiansen, Håkon Emil; Pedersen, Thomas Bondo; Kvaal, Simen & Ruud, Kenneth (2022). Coupled cluster theory for electron dynamics. Universitetet i Oslo. Full text in Research Archive Show summary

  Coupled-cluster (CC) theory, often referred to as the gold standard of quantum chemistry, defines a convergent hierarchy of increasingly accurate methods for the description of molecular properties. The same hierarchy of methods can be extended to time-dependent electronic structure theory which is then referred to as time-dependent CC theory (TDCC). TDCC theory can be formulated with static orbitals or with dynamic orbitals. In his thesis, the candidate has developed software implementing both variants. An assessment of the importance of employing dynamic orbitals is given. A drawback of TDCC theory is the difficulty of interpretation due to the non-Hermitian formulation. For example, there is no fully consistent definition of excited states in CC theory, preventing the calculation of stationary-state populations. Based on equation-of-motion CC (EOMCC) and CC linear response (CCLR) theory, we propose two sets of projection operators that yield time-dependent stationary-state populations as expectation values. Furthermore, TDCC methods are computationally expensive. To bring reduce the computational cost, approximations to specific orders of the theory can be formulated based on perturbation theory. The candidate has developed a program for a second-order approximation to TDCC theory with dynamic orbitals and applied the method to the description of optical properties in small molecules.

• Hansen, Audun Skau; Pedersen, Thomas Bondo; Helgaker, Trygve & Kvaal, Simen (2021). Local correlation methods for infinite systems. Universitetet i Oslo. Full text in Research Archive Show summary

  Quantum Chemistry constitutes an extensive framework for accurately simulating the electronic wavefunction of atoms and molecules. The same framework may in principle be applied to the domain of periodic structures such as crystals, but is in practice severely limited by the infinite nature of these structures in conjunction with the computational complexity of quantum chemical methods. In his thesis, the candidate utilizes a mathematical structure known as bi-infinite block-Toeplitz matrices in order to smoothly transition between the molecular and periodic realm. Furthermore, he extends the divide-expand-consolidate methods originally devised for molecules to the periodic case, and demonstrates that this procedure can reduce the computational scaling of the simulation while retaining systematic control over the error.

• Bore, Sigbjørn Løland; Cascella, Michele & Kvaal, Simen (2020). Advances in the Hybrid Particle-Field Approach: Towards Biological Systems. Universitetet i Oslo. Full text in Research Archive Show summary

  This dissertation aims at advancing the capability of hybrid particle-field simulations of representing various physical phenomena relevant to biological systems. While hybrid particle-field simulations are computationally efficient and well adapted for studying mesoscale systems with molecular resolution, this approach has so far predominantly been applied to simple polymers. The computational investigation of systems of higher complexity, such as DNA and proteins, requires development of new models and an extension of the hybrid particle-field methodology. To this end, six research papers are presented. The main research output of these papers consists in both new methods for representing electrostatics and constant-pressure conditions, and new models for proteins and charged lipids within the hybrid particle-field formalism. The work contained in this thesis thus provides key steps towards large-scale realistic representations of biological systems

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