arXiv subjects

Simen Kvaal

Publications and source records attributed to Simen Kvaal.

At least 19 recordsLinked to original sources

arXiv2026

Rothe's Method for Quantum Dynamics in Atoms and Molecules with Gaussian Wavepackets

Capable of capturing both bound and continuum quantum dynamics, Gaussian wavepackets are highly attractive basis functions for simulating laser-driven processes in atoms and molecules. Unfortunately, fully flexible Gaussian wavepackets are exceedingly challenging to propagate in a numerically stable manner within the framework of conventional time-dependent variational principles. In this chapter, we discuss the sources of the numerical issues and review an alternative approach, Rothe's method, that offers a route to improved numerical stability. Recent proof-of-concept simulations based on Rothe's method indicate that Gaussian wavepackets provide results on par with highly accurate grid-based methods for both electronic and rovibrational quantum dynamics, including ultrafast nonlinear processes that involve the continuum such as high-harmonic generation. Remarkably few Gaussian wavepackets are needed to achieve the high accuracy of grid-based approaches, indicating that further algorithmic developments and efficient implementations may enable efficient simulations of not only electronic and rovibrational phenomena but also fully coupled electronic-nuclear quantum dynamics with significantly reduced memory demands. We also point out remaining practical challenges, including matrix elements of the squared Hamiltonian and the treatment of Coulomb cusps.

physics.chem-ph↗

arXiv2026

Ultrafast laser-driven quantum dynamics in positronium chloride

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 while the presence of the positron delays electron ionization in PsH, a slight enhancement of electron ionization is observed 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 distuinguishable from Ps. In the tunelling regime, however, photopositron rescattering peaks may only be distuinguishable if the amount of Ps is sufficiently low.

physics.chem-ph↗

arXiv2025

Reduced Density Matrix Functional Theory And A Reduced Formulation Of Density Functional Theory

A mathematical framework for reduced density matrix functional theory (RDMFT) is proposed. The work is inspired by and generalizes the work by E.H.~Lieb [E.H. Lieb, Int. J. Quant. Chem. 24(1983), pp.243--277] on density-functional theory (DFT). We introduce a Banach space for density matrices with finite kinetic energy. The dual space is a rich class of single-particle potentials, i.e., Hermitian forms. The ground state energy of an $N$-fermion system with external forces given by any such Hermitian form is expressed as the Legendre--Fenchel transform of a convex and lower semicontinuous ``universal'' reduced density matrix functional. The formalism is employed to provide a mathematical framework for density-functional theory (DFT). The main tool here is a rigorous definition of diagonals of reduced density matrices. The result is a refinement of Lieb's results on DFT applicable to a wide variety of models.

math-ph↗

arXiv2025

Time-dependent Gaussian basis sets for many-body systems using Rothe's method: A mean-field study

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 reformulates 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 $4\times10^{14}$ W/cm$^2$ for the one-dimensional molecular systems considered in this work.

physics.chem-ph↗

arXiv2025

Rothe Time Propagation for Coupled Electronic and Rovibrational Quantum Dynamics

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 \emph{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 reproduce 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.

physics.chem-ph↗

arXiv2024

Multidimensional quantum dynamics with explicitly correlated Gaussian wave packets using Rothe's method

In a previous publication [J. Chem. Phys., 161, 044105 (2024)], it has been shown that Rothe's method can be used to solve the time-dependent Schr\"odinger equation (TDSE) 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 functions (ECGs) with dense correlation matrices for systems with varying dimensionality. We consider the multidimensional Henon-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 to 40 Gaussians in 3D and 4D. Thus, 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.

physics.chem-ph↗

arXiv2024

Configuration weights in coupled-cluster theory

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 $\hat{T}_1$-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.

physics.chem-ph↗

arXiv2024

The time-dependent bivariational principle: Theoretical foundation for real-time propagation methods of coupled-cluster type

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. The latter is new, and can in principle be used to develop novel propagation methods. Conservation laws and Poisson brackets are introduced, completing the analogy with classical mechanics. 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, and equation-of-motion coupled cluster theory.

physics.chem-ph↗

arXiv2024

Time evolution as an optimization problem: The hydrogen atom in strong laser fields in a basis of time-dependent Gaussian wave packets

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 {\aa}ngstr\"om. Recently, Rothe's method, where time evolution is rephrased as an optimization problem, has been applied to the one-dimensional Schr\"odinger equation. Here, we apply Rothe's method to the hydrogen wave function and demonstrate that 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 $5\times 10^{14}$W/cm$^2$. This paves the way for the inclusion of continuum contributions into real-time, time-dependent electronic-structure theory with Gaussian basis sets for strong fields, and eventually accurate simulations of the time evolution of molecules without the Born-Oppenheimer approximation.

physics.chem-ph↗

arXiv2024

Gaussians for Electronic and Rovibrational Quantum Dynamics

The assumptions underpinning the adiabatic Born-Oppenheimer (BO) approximation are broken for molecules interacting with attosecond laser pulses, which generate complicated coupled electronic-nuclear wavepackets 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.

physics.chem-ph↗

arXiv2023

Magnetic Optical Rotation from Real-Time Simulations in Finite Magnetic Fields

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.

physics.chem-ph↗

arXiv2023

Transient spectroscopy from time-dependent electronic-structure theory without multipole expansions

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.

physics.chem-ph↗

arXiv2023

Adiabatic extraction of nonlinear optical properties from real-time time-dependent electronic-structure theory

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.

physics.chem-ph↗

arXiv2023

The $S$-diagnostic -- an a posteriori error assessment for single-reference coupled-cluster methods

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 [CuCl$_4$]$^{2-}$, [Cu(NH$_3$)$_4$]$^{2+}$, and [Cu(H$_2$O)$_4$]$^{2+}$. Throughout the numerical investigations, the $S$-diagnostic is compared to other SRCC diagnostic procedures, that is, the $T_1$, $D_1$, and $D_2$ 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 $T_1$, $D_1$, and $D_2$ 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 [CuCl$_4$]$^{2-}$, [Cu(NH$_3$)$_4$]$^{2+}$, and [Cu(H$_2$O)$_4$]$^{2+}$, which have been known to be misdiagnosed by $T_1$ and $D_1$ diagnostics in the past. This shows that the $S$-diagnostic is a promising candidate for an a posteriori diagnostic for SRCC calculations.

physics.chem-ph↗

arXiv2022

Accelerated coupled cluster calculations with Procrustes orbital interpolation

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 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 computation time. Inspired by eigenvector continuation, we present here an algorithm making use of 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 which 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 $O(N^5)$.

physics.chem-ph↗

arXiv2022

Moreau--Yosida regularization in DFT

Moreau-Yosida regularization is introduced into the framework of exact DFT. Moreau-Yosida regularization is a lossless operation on lower semicontinuous proper convex functions over separable Hilbert spaces, and when applied to the universal functional of exact DFT (appropriately restricted to a bounded domain), gives a reformulation of the ubiquitous $v$-representability problem and a rigorous and illuminating derivation of Kohn-Sham theory. The chapter comprises a self-contained introduction to exact DFT, basic tools from convex analysis such as sub- and superdifferentiability and convex conjugation, as well as basic results on the Moreau-Yosida regularization. The regularization is then applied to exact DFT and Kohn-Sham theory, and a basic iteration scheme based in the Optimal Damping Algorithm is analyzed. In particular, its global convergence established. Some perspectives are offered near the end of the chapter.

math.NA↗

arXiv2022

No need for a grid: Adaptive fully-flexible gaussians for the time-dependent Schr\"odinger equation

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 Schr\"odinger 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 Schr\"odinger equation for atoms and molecules within and beyond the Born--Oppenheimer approximation.

quant-ph↗

arXiv2022

Laser-induced dynamic alignment of the HD molecule without the Born-Oppenheimer approximation

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.

physics.chem-ph↗