pith. sign in

arxiv: 2606.20947 · v1 · pith:NX4Z2YOHnew · submitted 2026-06-18 · ⚛️ physics.chem-ph

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

Simon Elias Schrader , H{\aa}kon Emil Kristiansen , Aleksander P. Wozniak , Ludwik Adamowicz , Simen Kvaal , Thomas Bondo Pedersen This is my paper

Pith reviewed 2026-06-26 15:00 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords Gaussian wavepacketsRothe's methodquantum dynamicshigh-harmonic generationtime-dependent variational principleelectronic dynamicsrovibrational dynamicscontinuum processes
0
0 comments X

The pith

Rothe's method stabilizes propagation of flexible Gaussian wavepackets for laser-driven quantum dynamics in atoms and molecules.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that Gaussian wavepackets can capture both bound and continuum dynamics in ultrafast laser-driven processes but face instability under standard variational propagation. Rothe's method is presented as an alternative route to numerical stability. Proof-of-concept simulations using this approach produce results for electronic and rovibrational dynamics, including high-harmonic generation, that match highly accurate grid-based benchmarks. Only a small number of wavepackets suffice for this level of agreement, which points toward simulations with substantially lower memory requirements. The work identifies remaining obstacles in evaluating squared-Hamiltonian matrix elements and treating Coulomb cusps.

Core claim

Gaussian wavepackets propagated via Rothe's method deliver results on par with highly accurate grid-based methods for both electronic and rovibrational quantum dynamics, including ultrafast nonlinear continuum processes such as high-harmonic generation, while requiring remarkably few wavepackets to reach grid-level accuracy.

What carries the argument

Rothe's method, an alternative time-propagation scheme that improves numerical stability over conventional time-dependent variational principles when applied to fully flexible Gaussian wavepackets.

If this is right

  • Electronic and rovibrational laser-driven phenomena become simulable with far fewer basis functions than grid methods require.
  • Memory demands drop enough to allow larger systems or longer propagation times.
  • The same framework can in principle extend to fully coupled electronic-nuclear quantum dynamics.
  • Continuum processes such as high-harmonic generation are accessible without explicit grid discretization of the continuum.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the remaining matrix-element and cusp challenges are solved, the method could be combined with existing Gaussian-basis electronic-structure codes for larger molecules.
  • The low basis count suggests possible hybrid schemes that mix Gaussian wavepackets with other localized bases for multi-scale dynamics.
  • Direct benchmarking against grid results on additional nonlinear observables would provide a clear next test of accuracy retention.

Load-bearing premise

Numerical stability gains from Rothe's method will survive once matrix elements of the squared Hamiltonian and Coulomb cusps are handled without introducing errors comparable to those avoided.

What would settle it

A side-by-side computation of high-harmonic generation spectra for a simple atom or molecule in which the squared-Hamiltonian matrix elements are evaluated exactly and the Gaussian-Rothe results deviate measurably from converged grid benchmarks.

Figures

Figures reproduced from arXiv: 2606.20947 by Aleksander P. Wozniak, H{\aa}kon Emil Kristiansen, Ludwik Adamowicz, Simen Kvaal, Simon Elias Schrader, Thomas Bondo Pedersen.

Figure 1
Figure 1. Figure 1: Superposition of two complex Gaussian wave packets: The component wave packets are [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of a two-dimensional manifold with a singular line of dimensionality reduction. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A few examples of Gaussians in 𝐷 = 2 dimensions, showcasing their qualitative behavior. For the real and imaginary parts, red colors correspond to large positive values, blue colors corre￾spond to large negative values, while white is zero. All color ranges are normalized to the maximum absolute value of the Gaussian. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Condition numbers of the matrix 𝐺(𝑦) is shown for a 1D model of a single particle moving in a Gaussian potential, 𝑉(𝑥) = − exp(−𝜇𝑥2 ), 𝜇 = 0.1, for the initial state 𝑦 = 𝑦(0). The initial basis is chosen as an even-tempered Gaussian basis set with exponents given by 𝜁𝑛 = 𝑢𝑣𝑛−1, 𝑛 = 1, ⋯ , 𝑁𝑔 , with 𝑢 = 2, 𝑣 = 10 13 and 𝑁𝑔 being the number of Gaussians. The initial linear coefficients 𝒄 are chosen as the lo… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the classical PDE discretization approach vs. Rothe’s method. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: An example of a Gaussian mask consisting of [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Singularity-free approximate Coulomb potentials allow for a faster convergence of the vari [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left panel: Propagation history of the reference wavefunction obtained using the Crank [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The number of GWPs required for Rothe optimization of the 1D hydrogen wavefunction. [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Left column: expectation values as functions of time for three values the Rothe minimiza [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Left panel: Number of Gaussians required to achieve a certain level of accuracy for the 2D [PITH_FULL_IMAGE:figures/full_fig_p032_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Plot of the absolute value and the real and imaginary parts of the 2D hydrogen wavefunc [PITH_FULL_IMAGE:figures/full_fig_p033_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: HHG spectra (left), per-timestep Rothe error (center), and number of GWPs [PITH_FULL_IMAGE:figures/full_fig_p034_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Dipole moment (top left), HHG spectrum (top right), and final-time electronic density [PITH_FULL_IMAGE:figures/full_fig_p035_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Dipole moment (top left), HHG spectrum (top right), and final-time electronic density [PITH_FULL_IMAGE:figures/full_fig_p036_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Left column: expectation values as functions of time for three values the Rothe minimiza [PITH_FULL_IMAGE:figures/full_fig_p037_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Plot of the absolute value and the real and imaginary parts of the 2D Morse wavefunction at [PITH_FULL_IMAGE:figures/full_fig_p038_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Spectra 𝑆(𝜔) (Eq. (86)) for the 2D Henon-Heiles potential with 𝑀init ∈ {10, 20, 30, 40} initial Gaussians (colored lines), compared to essentially exact grid calculation (black line). (Adapted from Ref. [91] with the permission of AIP Publishing.) 40 [PITH_FULL_IMAGE:figures/full_fig_p040_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Spectra 𝑆(𝜔) (Eq. (86)) for the 4D Henon-Heiles potential with 𝑀init ∈ {10, 20, 30, 40} initial Gaussians (colored lines), compared to essentially exact grid calculation (black line). (Adapted from Ref. [91] with the permission of AIP Publishing.) 41 [PITH_FULL_IMAGE:figures/full_fig_p041_19.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript reviews numerical instabilities in propagating fully flexible Gaussian wavepackets under conventional time-dependent variational principles and proposes Rothe's method as an alternative route to improved stability. It presents proof-of-concept simulations indicating that Gaussian wavepackets achieve accuracy comparable to grid-based methods for electronic and rovibrational quantum dynamics, including continuum-involved processes such as high-harmonic generation, while requiring remarkably few basis functions. The work notes remaining practical challenges with squared-Hamiltonian matrix elements and Coulomb cusps and suggests the approach may enable memory-efficient fully coupled electron-nuclear simulations.

Significance. If the reported parity with grid benchmarks holds after the flagged challenges are resolved, the method could provide a basis-set route to high-accuracy simulations of laser-driven processes with substantially lower memory requirements than grid approaches, particularly for dynamics involving both bound and continuum states.

minor comments (2)
  1. The abstract qualifies the accuracy claim with 'indicate that' and 'may enable'; the main text should ensure all simulation results are presented with explicit quantitative comparisons (e.g., error metrics or overlap values) to the cited grid benchmarks so readers can assess the 'on par' statement directly.
  2. Because the work is framed as proof-of-concept, a dedicated subsection summarizing the specific systems, number of Gaussians employed, and observed convergence behavior would improve clarity without altering the central narrative.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its potential significance, and the recommendation for minor revision. The report does not enumerate any specific major comments, so we have no individual points requiring point-by-point rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external benchmarks

full rationale

The manuscript reviews Rothe's method as an alternative to conventional variational principles for propagating Gaussian wavepackets, then reports proof-of-concept simulations whose accuracy is assessed exclusively against independent grid-based reference calculations. No derivation step reduces a claimed result to a fitted parameter or self-citation by construction; the central claims are qualified as indicative and the open issues (squared-Hamiltonian matrix elements, Coulomb cusps) are stated explicitly rather than assumed away. Comparisons remain external and falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the premise that Rothe's method overcomes the documented instabilities of conventional TDVP while the remaining technical obstacles (squared-Hamiltonian matrix elements and Coulomb cusps) can be managed without loss of accuracy. No free parameters, ad-hoc entities, or non-standard axioms are mentioned in the abstract.

axioms (1)
  • domain assumption Conventional time-dependent variational principles for Gaussian wavepacket propagation suffer from numerical instabilities that Rothe's method can mitigate.
    Invoked in the opening discussion of numerical issues.

pith-pipeline@v0.9.1-grok · 5747 in / 1449 out tokens · 44865 ms · 2026-06-26T15:00:59.378762+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

133 extracted references · 105 canonical work pages

  1. [1]

    Born–Oppenheimer andNon-Born–Oppenheimer,AtomicandMolecularCalculationswithExplicitlyCorrelated Gaussians

    S.Bubin,M.Pavanello,W.-C.Tung,K.L.Sharkey,andL.Adamowicz.“Born–Oppenheimer andNon-Born–Oppenheimer,AtomicandMolecularCalculationswithExplicitlyCorrelated Gaussians”.Chem.Rev.113(2013),36–79.doi:10.1021/cr200419d

    work page doi:10.1021/cr200419d 2013

  2. [2]

    Theory and application of explicitly correlated Gaussians

    J. Mitroy, S. Bubin, W. Horiuchi, Y. Suzuki, L. Adamowicz, W. Cencek, K. Szalewicz, J. Ko- masa, D. Blume, and K. Varga. “Theory and application of explicitly correlated Gaussians”. Rev.Mod.Phys.85(2013),693–749.doi:10.1103/RevModPhys.85.693

    work page doi:10.1103/revmodphys.85.693 2013

  3. [3]

    Pre-Born–Oppenheimermolecularstructuretheory

    E.Mátyus.“Pre-Born–Oppenheimermolecularstructuretheory”.Mol.Phys.117(2019),590– 609.doi:10.1080/00268976.2018.1530461

    work page doi:10.1080/00268976.2018.1530461 2019

  4. [4]

    Laser-induceddynamicalignmentof theHDmoleculewithouttheBorn–Oppenheimerapproximation

    L.Adamowicz,S.Kvaal,C.Lasser,andT.B.Pedersen.“Laser-induceddynamicalignmentof theHDmoleculewithouttheBorn–Oppenheimerapproximation”.J.Chem.Phys.157(2022), 144302.doi:10.1063/5.0101352

    work page doi:10.1063/5.0101352 2022

  5. [5]

    StrongElectron-Electron-Nuclei Correlations in Two-Photon Double Ionization of H2

    K.Arteaga,J.Feist,D.Jelovina,F.Martín,andA.Palacios.“StrongElectron-Electron-Nuclei Correlations in Two-Photon Double Ionization of H2”.Phys. Rev. Lett.133 (2024), 123201. doi:10.1103/PhysRevLett.133.123201

    work page doi:10.1103/physrevlett.133.123201 2024

  6. [6]

    ZurQuantentheoriederMolekeln

    M.BornandR.Oppenheimer.“ZurQuantentheoriederMolekeln”.Ann.Phys.(Leipzig)389 (1927),457–484.doi:10.1002/ANDP.19273892002

    work page doi:10.1002/andp.19273892002 1927

  7. [7]

    M.BornandK.Huang.DynamicalTheoryofCrystalLattices.Oxford:ClarendonPress,1954

    1954

  8. [8]

    H.-D.Meyer,F.Gatti,andG.A.Worth,eds.MultidimensionalQuantumDynamics:MCTDH TheoryandApplications.Weinheim:Wiley,2009.doi:10.1002/9783527627400

    work page doi:10.1002/9783527627400 2009

  9. [9]

    Ab Initio Multiple Spawning: Photochem- istryfromFirstPrinciplesQuantumMolecularDynamics

    M. Ben-Nun, J. Quenneville, and T. J. Martínez. “Ab Initio Multiple Spawning: Photochem- istryfromFirstPrinciplesQuantumMolecularDynamics”.J.Phys.Chem.A104(2000),5161– 5175.doi:10.1021/jp994174i

    work page doi:10.1021/jp994174i 2000

  10. [10]

    Theoretical methods for attosecond electron and nuclear dynamics: applications to the H2 molecule

    A. Palacios, J. L. Sanz-Vicario, and F. Martín. “Theoretical methods for attosecond electron and nuclear dynamics: applications to the H2 molecule”.J. Phys. B48 (2015), 242001.doi: 10.1088/0953-4075/48/24/242001

    work page doi:10.1088/0953-4075/48/24/242001 2015

  11. [11]

    Real-TimeTime-DependentElec- tronic Structure Theory

    X.Li,N.Govind,C.Isborn,A.E.DePrince,andK.Lopata.“Real-TimeTime-DependentElec- tronic Structure Theory”.Chem. Rev.120 (2020), 9951–9993.doi:10.1021/acs.chemrev. 0c00223

    work page doi:10.1021/acs.chemrev 2020

  12. [12]

    Ultrafast charge migration by electron correlation

    L. Cederbaum and J. Zobeley. “Ultrafast charge migration by electron correlation”.Chem. Phys.Lett.307(1999),205–210.doi:10.1016/S0009-2614(99)00508-4

    work page doi:10.1016/s0009-2614(99)00508-4 1999

  13. [13]

    AttosecondElectronDynamics inMolecules

    M.Nisoli,P.Decleva,F.Calegari,A.Palacios,andF.Martín.“AttosecondElectronDynamics inMolecules”.Chem.Rev.117(2017),10760–10825.doi:10.1021/acs.chemrev.6b00453

    work page doi:10.1021/acs.chemrev.6b00453 2017

  14. [14]

    Theory of high- harmonic generation by low-frequency laser fields

    M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum. “Theory of high- harmonic generation by low-frequency laser fields”.Phys. Rev. A49 (1994), 2117–2132.doi: 10.1103/PhysRevA.49.2117

    work page doi:10.1103/physreva.49.2117 1994

  15. [15]

    Time-dependent theory of multiphoton ionization of xenon

    K. C. Kulander. “Time-dependent theory of multiphoton ionization of xenon”.Phys. Rev. A 38(1988),778–787.doi:10.1103/PhysRevA.38.778. 43

    work page doi:10.1103/physreva.38.778 1988

  16. [16]

    Dynamics of Short-Pulse Excitation, Ion- ization and Harmonic Conversion

    K. C. Kulander, K. J. Schafer, and J. L. Krause. “Dynamics of Short-Pulse Excitation, Ion- ization and Harmonic Conversion”.Super-Intense Laser-Atom Physics. Ed. by B. Piraux, A. L’Huillier, and K. Rzążewski. NATO ASI Series. Boston: Springer, 1993, pp. 95–110.doi: 10.1007/978-1-4615-7963-2_10

    work page doi:10.1007/978-1-4615-7963-2_10 1993

  17. [17]

    ProtonicStructureofMolecules.I.AmmoniaMolecules

    I.L.Thomas.“ProtonicStructureofMolecules.I.AmmoniaMolecules”.Phys.Rev.185(1969), 90–94.doi:10.1103/PhysRev.185.90

    work page doi:10.1103/physrev.185.90 1969

  18. [18]

    Protonic Structure of Molecules. II. Methodology, Center-of- Mass Transformation, and the Structure of Methane, Ammonia, and Water

    I. L. Thomas and H. W. Joy. “Protonic Structure of Molecules. II. Methodology, Center-of- Mass Transformation, and the Structure of Methane, Ammonia, and Water”.Phys. Rev. A2 (1970),1200–1208.doi:10.1103/PhysRevA.2.1200

    work page doi:10.1103/physreva.2.1200 1970

  19. [19]

    MulticomponentQuantumChemistry:In- tegratingElectronicandNuclearQuantumEffectsviatheNuclear-ElectronicOrbitalMethod

    F.Pavošević,T.Culpitt,andS.Hammes-Schiffer.“MulticomponentQuantumChemistry:In- tegratingElectronicandNuclearQuantumEffectsviatheNuclear-ElectronicOrbitalMethod”. Chem.Rev.120(2020),4222–4253.doi:10.1021/acs.chemrev.9b00798

    work page doi:10.1021/acs.chemrev.9b00798 2020

  20. [20]

    Real-TimeTime- DependentNuclear-ElectronicOrbitalApproach:DynamicsbeyondtheBorn-Oppenheimer Approximation

    L.Zhao,Z.Tao,F.Pavošević,A.Wildman,S.Hammes-Schiffer,andX.Li.“Real-TimeTime- DependentNuclear-ElectronicOrbitalApproach:DynamicsbeyondtheBorn-Oppenheimer Approximation”.J. Phys. Chem. Lett.11 (2020), 4052–4058.doi:10 . 1021 / acs . jpclett . 0c00701

    2020

  21. [21]

    G. D. Smith.Numerical Solution of Partial Differential Equations: Finite Difference Methods. 3rded.Oxford:ClarendonPress,1985

    1985

  22. [22]

    56.16010

    A. Natan, A. Benjamini, D. Naveh, L. Kronik, M. L. Tiago, S. P. Beckman, and J. R. Che- likowsky.“Real-spacepseudopotentialmethodforfirstprinciplescalculationsofgeneralperi- odicandpartiallyperiodicsystems”.Phys.Rev.B78(2008),075109.doi:10.1103/PhysRevB. 78.075109

    work page doi:10.1103/physrevb 2008

  23. [23]

    Time-dependentmulticonfigurationmethodsfor the numerical simulation of photoionization processes of many-electron atoms

    D.Hochstuhl,C.M.Hinz,andM.Bonitz.“Time-dependentmulticonfigurationmethodsfor the numerical simulation of photoionization processes of many-electron atoms”.Eur. Phys. J.:Spec.Top.223(2014),177–336.doi:10.1140/epjst/e2014-02092-3

    work page doi:10.1140/epjst/e2014-02092-3 2014

  24. [24]

    ApplicationsofB-splinesin atomic and molecular physics

    H.Bachau,E.Cormier,P.Decleva,J.E.Hansen,andF.Martín.“ApplicationsofB-splinesin atomic and molecular physics”.Rep.Prog.Phys.64 (2001), 1815–1943.doi:10.1088/0034- 4885/64/12/205

    work page doi:10.1088/0034- 2001

  25. [25]

    Multiresolutionoftheonedimensionalfree-particle propagator. Part 1: Construction

    E.Dinvay,Y.Zabelina,andL.Frediani.“Multiresolutionoftheonedimensionalfree-particle propagator. Part 1: Construction”.Comput. Phys. Commun.308 (2025), 109436.doi:10 . 1016/j.cpc.2024.109436

    arXiv 2025

  26. [26]

    Multiresolution of the one dimensional free-particle propagator. Part 2: Imple- mentation

    E. Dinvay. “Multiresolution of the one dimensional free-particle propagator. Part 2: Imple- mentation”.Comput. Phys. Commun.308 (2025), 109438.doi:10 . 1016 / j . cpc . 2024 . 109438

    2025

  27. [27]

    Electronic wave functions - I. A general method of calculation for the stationary states of any molecular system

    S. F. Boys. “Electronic wave functions - I. A general method of calculation for the stationary states of any molecular system”.Proc. R. Soc. Lond. A200 (1950), 542–554.doi:10.1098/ rspa.1950.0036

    arXiv 1950

  28. [28]

    Theintegralformulaeforthevariationalsolutionofthemolecularmany-electron waveequationintermsofGaussianfunctionswithdirectelectroniccorrelation

    S.F.Boys.“Theintegralformulaeforthevariationalsolutionofthemolecularmany-electron waveequationintermsofGaussianfunctionswithdirectelectroniccorrelation”.Proc.R.Soc. Lond.A.Math.Phys.Sci.258(1960),402–411.doi:10.1098/rspa.1960.0195

    work page doi:10.1098/rspa.1960.0195 1960

  29. [29]

    The use of Gaussian (exponential quadratic) wave functions in molecular prob- lems - I. General formulae for the evaluation of integrals

    K. Singer. “The use of Gaussian (exponential quadratic) wave functions in molecular prob- lems - I. General formulae for the evaluation of integrals”.Proc.R.Soc.Lond.A.Math.Phys. Sci.258(1960),412–420.doi:10.1098/rspa.1960.0196. 44

    work page doi:10.1098/rspa.1960.0196 1960

  30. [30]

    Gaussians for Electronic and RovibrationalQuantumDynamics

    A. P. Woźniak, L. Adamowicz, T. B. Pedersen, and S. Kvaal. “Gaussians for Electronic and RovibrationalQuantumDynamics”.J.Phys.Chem.A128(2024),3659–3671.doi:10.1021/ acs.jpca.4c00364

    2024

  31. [31]

    D.J.Tannor.Introductiontoquantummechanics:atime-dependentperspective.Sausalito,Cal- ifornia:UniversityScienceBooks,2007

    2007

  32. [32]

    E.J.Heller.Thesemiclassicalwaytodynamicsandspectroscopy.Princeton:PrincetonUniver- sityPress,2018

    2018

  33. [33]

    Time-dependentapproachtosemiclassicaldynamics

    E.J.Heller.“Time-dependentapproachtosemiclassicaldynamics”.J.Chem.Phys.62(1975), 1544–1555.doi:10.1063/1.430620

    work page doi:10.1063/1.430620 1975

  34. [34]

    Frozen Gaussians: A very simple semiclassical approximation

    E. J. Heller. “Frozen Gaussians: A very simple semiclassical approximation”.J. Chem. Phys. 75(1981),2923–2931.doi:10.1063/1.442382

    work page doi:10.1063/1.442382 1981

  35. [35]

    AbInitioSemiclassicalEvaluationofVibrationallyResolvedElec- tronicSpectrawithThawedGaussians

    J.VanicekandT.Begusic.“AbInitioSemiclassicalEvaluationofVibrationallyResolvedElec- tronicSpectrawithThawedGaussians”.MolecularSpectroscopyandQuantumDynamics.Ed. by R. Marquardt and M. Quack. St. Louis: Elsevier, 2021, pp. 199–229.doi:10.1016/B978- 0-12-817234-6.00011-8

    work page doi:10.1016/b978- 2021

  36. [36]

    Astrategyfortimedependentquantum mechanicalcalculationsusingaGaussianwavepacketrepresentationofthewavefunction

    S.-I.Sawada,R.Heather,B.Jackson,andH.Metiu.“Astrategyfortimedependentquantum mechanicalcalculationsusingaGaussianwavepacketrepresentationofthewavefunction”. J.Chem.Phys.83(1985),3009–3027.doi:10.1063/1.449204

    work page doi:10.1063/1.449204 1985

  37. [37]

    Thematrixsingularityprobleminthetime-dependentvariationalmethod

    K.G.Kay.“Thematrixsingularityprobleminthetime-dependentvariationalmethod”.Chem. Phys.137(1989),165–175

    1989

  38. [38]

    Matching-pursuit for simulations of quantum processes

    Y. Wu and V. S. Batista. “Matching-pursuit for simulations of quantum processes”.J. Chem. Phys.118(2003),6720–6724.doi:10.1063/1.1560636

    work page doi:10.1063/1.1560636 2003

  39. [39]

    Linear dependence and energy conservation in Gaussian wavepacket basis sets

    S. Habershon. “Linear dependence and energy conservation in Gaussian wavepacket basis sets”.J.Chem.Phys.136(2012),014109.doi:10.1063/1.3671978

    work page doi:10.1063/1.3671978 2012

  40. [40]

    BasisExpansionLeaping:ANewMethodtoSolvetheTime- Dependent Schrödinger Equation for Molecular Quantum Dynamics

    W.KochandT.J.Frankcombe.“BasisExpansionLeaping:ANewMethodtoSolvetheTime- Dependent Schrödinger Equation for Molecular Quantum Dynamics”.Phys. Rev. Lett.110 (2013),263202.doi:10.1103/PhysRevLett.110.263202

    work page doi:10.1103/physrevlett.110.263202 2013

  41. [41]

    Quantum Dy- namicsSimulationsUsingGaussianWavepackets:ThevMCGMethod

    G. Richings, I. Polyak, K. Spinlove, G. Worth, I. Burghardt, and B. Lasorne. “Quantum Dy- namicsSimulationsUsingGaussianWavepackets:ThevMCGMethod”.Int.Rev.Phys.Chem. 34(2015),269–308.doi:10.1080/0144235X.2015.1051354

    work page doi:10.1080/0144235x.2015.1051354 2015

  42. [42]

    Solving Time-dependent Schrödinger EquationUsingGaussianWavePacketDynamics

    M. H. Lee, C. W. Byun, N. N. Choi, and D. S. Kim. “Solving Time-dependent Schrödinger EquationUsingGaussianWavePacketDynamics”.J.KoreanPhys.Soc.73(2018),1269–1278. doi:10.3938/jkps.73.1269

    work page doi:10.3938/jkps.73.1269 2018

  43. [43]

    Simulation of a hydrogen atom in a laser field using the time-dependent variational principle

    K. Rowan, L. Schatzki, T. Zaklama, Y. Suzuki, K. Watanabe, and K. Varga. “Simulation of a hydrogen atom in a laser field using the time-dependent variational principle”.Phys. Rev. E 101(2020),023313.doi:10.1103/PhysRevE.101.023313

    work page doi:10.1103/physreve.101.023313 2020

  44. [44]

    Variational Dynamics of Multicomponent Wave Functions Represented in a Basis Driven by a Time- DependentGaussianWavepacket

    S. Garashchuk, J. Stetzler, C. D. Jayawardana, M. A. Safo, and V. A. Rassolov. “Variational Dynamics of Multicomponent Wave Functions Represented in a Basis Driven by a Time- DependentGaussianWavepacket”.J.Chem.TheoryComput.21(2025),7249–7266

    2025

  45. [45]

    Feischl, C

    M. Feischl, C. Lasser, C. Lubich, and J. Nick.Regularized dynamical parametric approxima- tion.arXiv:2403.19234v3[math].2026.doi:10.48550/arXiv.2403.19234

    work page doi:10.48550/arxiv.2403.19234 2026

  46. [46]

    A.SzaboandN.S.Ostlund.ModernQuantumChemistry.NewYork:Dover,1989

    1989

  47. [47]

    Helgaker, P

    T. Helgaker, P. Jørgensen, and J. Olsen.Molecular Electronic-Structure Theory. Chichester: Wiley,2000. 45

    2000

  48. [48]

    The History and Evolution of Gaussian Basis Sets

    I. Shavitt. “The History and Evolution of Gaussian Basis Sets”.Isr. J. Chem.33 (1993), 357– 367.doi:10.1002/ijch.199300044

    work page doi:10.1002/ijch.199300044 1993

  49. [49]

    Atomic orbital basis sets

    F. Jensen. “Atomic orbital basis sets”.WIREs Comput. Mol. Sci.3 (2013), 273–295.doi:10. 1002/wcms.1123

    2013

  50. [50]

    BasisSetsinQuantumChemistry

    B.NagyandF.Jensen.“BasisSetsinQuantumChemistry”.ReviewsinComputationalChem- istry.Ed.byA.L.ParrillandK.B.Lipkowitz.Vol.30.Hoboken:Wiley,2017,pp.93–149.doi: 10.1002/9781119356059.ch3

    work page doi:10.1002/9781119356059.ch3 2017

  51. [51]

    Théorie quantique des courants interatomiques dans les combinaisons aroma- tiques

    F. London. “Théorie quantique des courants interatomiques dans les combinaisons aroma- tiques”.J.Phys.Radium8(1937),397–409.doi:10.1051/jphysrad:01937008010039700

    work page doi:10.1051/jphysrad:01937008010039700 1937

  52. [52]

    Nonperturbative ab initio calculations in strong magneticfieldsusingLondonorbitals

    E. I. Tellgren, A. Soncini, and T. Helgaker. “Nonperturbative ab initio calculations in strong magneticfieldsusingLondonorbitals”.J.Chem.Phys.129(2008),154114.doi:10.1063/1. 2996525

    work page doi:10.1063/1 2008

  53. [53]

    Aparamagneticbondingmech- anism for diatomics in strong magnetic fields

    K.K.Lange,E.I.Tellgren,M.R.Hoffmann,andT.Helgaker.“Aparamagneticbondingmech- anism for diatomics in strong magnetic fields.”Science337 (2012), 327–331.doi:10.1126/ science.1219703

    2012

  54. [54]

    Recent Ad- vances in Wave Function-Based Methods of Molecular-Property Calculations

    T. Helgaker, S. Coriani, P. Jørgensen, K. Kristensen, J. Olsen, and K. Ruud. “Recent Ad- vances in Wave Function-Based Methods of Molecular-Property Calculations”.Chem. Rev. 112(2012),543–631.doi:10.1021/cr2002239

    work page doi:10.1021/cr2002239 2012

  55. [55]

    Thequantumchemistryofattosecondmolecularscience

    A.PalaciosandF.Martín.“Thequantumchemistryofattosecondmolecularscience”.WIREs Comput.Mol.Sci.10(2020),e1430.doi:10.1002/wcms.1430

    work page doi:10.1002/wcms.1430 2020

  56. [56]

    Time-dependent coupled-cluster theory

    B. S. Ofstad, E. Aurbakken, Ø. S. Schøyen, H. E. Kristiansen, S. Kvaal, and T. B. Pedersen. “Time-dependent coupled-cluster theory”.WIREs Comput. Mol. Sci.13 (2023), e1666.doi: 10.1002/wcms.1666

    work page doi:10.1002/wcms.1666 2023

  57. [57]

    UniversalGaussian-basissetsforanoptimum representationofRydbergandcontinuumwavefunctions

    K.Kaufmann,W.Baumeister,andM.Jungen.“UniversalGaussian-basissetsforanoptimum representationofRydbergandcontinuumwavefunctions”.J.Phys.B22(1989),2223–2240

    1989

  58. [58]

    Laser-induced electron dynamics including photoionization: A heuristic model within time-dependent configuration interaction the- ory

    S. Klinkusch, P. Saalfrank, and T. Klamroth. “Laser-induced electron dynamics including photoionization: A heuristic model within time-dependent configuration interaction the- ory”.J.Chem.Phys.131(2009),114304.doi:10.1063/1.3218847

    work page doi:10.1063/1.3218847 2009

  59. [59]

    TD-CI Simulationof theElectronic OpticalRe- sponse of Molecules in Intense Fields: Comparison of RPA, CIS, CIS(D), and EOM-CCSD

    J. A.Sonk, M.Caricato, andH. B. Schlegel.“TD-CI Simulationof theElectronic OpticalRe- sponse of Molecules in Intense Fields: Comparison of RPA, CIS, CIS(D), and EOM-CCSD”. J.Phys.Chem.A115(2011),4678–4690.doi:10.1021/jp107384p

    work page doi:10.1021/jp107384p 2011

  60. [60]

    TD-CISimulationoftheElectronicOpticalResponseofMolecules inIntenseFieldsII:ComparisonofDFTFunctionalsandEOM-CCSD

    J.A.SonkandH.B.Schlegel.“TD-CISimulationoftheElectronicOpticalResponseofMolecules inIntenseFieldsII:ComparisonofDFTFunctionalsandEOM-CCSD”.J.Phys.Chem.A115 (2011),11832–11840.doi:10.1021/jp206437s

    work page doi:10.1021/jp206437s 2011

  61. [61]

    Computationofhigh-harmonicgenerationspectraofH 2and N2 in intense laser pulses using quantum chemistry methods and time-dependent density functionaltheory

    E.LuppiandM.Head-Gordon.“Computationofhigh-harmonicgenerationspectraofH 2and N2 in intense laser pulses using quantum chemistry methods and time-dependent density functionaltheory”.Mol.Phys.110(2012),909–923.doi:10.1080/00268976.2012.675448

    work page doi:10.1080/00268976.2012.675448 2012

  62. [62]

    The role of Rydberg and continuum levels in computing highharmonicgenerationspectraofthehydrogenatomusingtime-dependentconfiguration interaction

    E. Luppi and M. Head-Gordon. “The role of Rydberg and continuum levels in computing highharmonicgenerationspectraofthehydrogenatomusingtime-dependentconfiguration interaction”.J.Chem.Phys.139(2013),164121.doi:10.1063/1.4824482

    work page doi:10.1063/1.4824482 2013

  63. [63]

    Strong field ionization rates simulated with time- dependent configuration interaction and an absorbing potential

    P. Krause, J. A. Sonk, and H. B. Schlegel. “Strong field ionization rates simulated with time- dependent configuration interaction and an absorbing potential”.J.Chem.Phys.140 (2014), 174113.doi:10.1063/1.4874156. 46

    work page doi:10.1063/1.4874156 2014

  64. [64]

    Computationofhigh- harmonicgenerationspectraofthehydrogenmoleculeusingtime-dependentconfiguration- interaction

    A.F.White,C.J.Heide,P.Saalfrank,M.Head-Gordon,andE.Luppi.“Computationofhigh- harmonicgenerationspectraofthehydrogenmoleculeusingtime-dependentconfiguration- interaction”.Mol.Phys.114(2016),947–956.doi:10.1080/00268976.2015.1119900

    work page doi:10.1080/00268976.2015.1119900 2016

  65. [65]

    Gaussian continuum basis functions for calculating high-harmonic generation spectra

    E. Coccia, B. Mussard, M. Labeye, J. Caillat, R. Taieb, J. Toulouse, and E. Luppi. “Gaussian continuum basis functions for calculating high-harmonic generation spectra”.Int. J. Quan- tumChem.116(2016),1120–1131.doi:10.1002/qua.25146

    work page doi:10.1002/qua.25146 2016

  66. [66]

    Optimal Basis Set for Electron Dynamics in Strong Laser Fields: The case of Molecular Ion H+ 2

    M. Labeye, F. Zapata, E. Coccia, V. Véniard, J. Toulouse, J. Caillat, R. Taieb, and E. Luppi. “Optimal Basis Set for Electron Dynamics in Strong Laser Fields: The case of Molecular Ion H+ 2”.J.Chem.TheoryComput.14(2018),5846–5858.doi:10.1021/acs.jctc.8b00656

    work page doi:10.1021/acs.jctc.8b00656 2018

  67. [67]

    Role of exchange and correlation in high-harmonic generation spectra of H2, N2, and CO2: Real-time time-dependent electronic-structure ap- proaches

    C. F. Pauletti, E. Coccia, and E. Luppi. “Role of exchange and correlation in high-harmonic generation spectra of H2, N2, and CO2: Real-time time-dependent electronic-structure ap- proaches”.J.Chem.Phys.154(2021),014101.doi:10.1063/5.0033072

    work page doi:10.1063/5.0033072 2021

  68. [68]

    Asystematiccon- structionofGaussianbasissetsforthedescriptionoflaserfieldionizationandhigh-harmonic generation

    A.P.Woźniak,M.Lesiuk,M.Przybytek,D.K.Efimov,J.S.Prauzner-Bechcicki,M.Mandrysz, M.Ciappina,E.Pisanty,J.Zakrzewski,M.Lewenstein,andR.Moszyński.“Asystematiccon- structionofGaussianbasissetsforthedescriptionoflaserfieldionizationandhigh-harmonic generation”.J.Chem.Phys.154(2021),094111.doi:10.1063/5.0040879

    work page doi:10.1063/5.0040879 2021

  69. [69]

    Effectsofelectroniccorrela- tiononthehighharmonicgenerationinhelium:Atime-dependentconfigurationinteraction singles vs time-dependent full configuration interaction study

    A.P.Woźniak,M.Przybytek,M.Lewenstein,andR.Moszyński.“Effectsofelectroniccorrela- tiononthehighharmonicgenerationinhelium:Atime-dependentconfigurationinteraction singles vs time-dependent full configuration interaction study”.J. Chem. Phys.156 (2022), 174106.doi:10.1063/5.0087384

    work page doi:10.1063/5.0087384 2022

  70. [70]

    Time-dependent ab initio approaches for high-harmonic genera- tion spectroscopy

    E. Coccia and E. Luppi. “Time-dependent ab initio approaches for high-harmonic genera- tion spectroscopy”.J. Phys. Condens. Matter34 (2022), 073001.doi:10.1088/1361- 648X/ ac3608

    work page doi:10.1088/1361- 2022

  71. [71]

    Evaluation of Diffuse Basis Sets for Simulations of Strong Field Ionization Using Time-Dependent Configuration Interaction with a Complex Absorb- ingPotential

    A. S. Durden and H. B. Schlegel. “Evaluation of Diffuse Basis Sets for Simulations of Strong Field Ionization Using Time-Dependent Configuration Interaction with a Complex Absorb- ingPotential”.J.Phys.Chem.A129(2025),3353–3367.doi:10.1021/acs.jpca.5c00195

    work page doi:10.1021/acs.jpca.5c00195 2025

  72. [72]

    M.P.doCarmoandF.J.Flaherty.Riemanniangeometry.Boston:Birkhäuser,1993

    1993

  73. [73]

    J.Marsden,T.Ratiu,andR.Abraham.Manifolds,tensoranalysis,andapplications.NewYork: Springer,2001

    2001

  74. [74]

    Avariationalsolutionofthetime-dependentSchrödingerequation

    A.D.McLachlan.“Avariationalsolutionofthetime-dependentSchrödingerequation”.Mol. Phys.8(1964),39–44.doi:10.1080/00268976400100041

    work page doi:10.1080/00268976400100041 1964

  75. [75]

    Goldstein, C

    H. Goldstein, C. P. Poole, and J. L. Safko.ClassicalMechanics. 3. ed. San Francisco: Addison Wesley,2008

    2008

  76. [76]

    C. Lubich.From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis.ZurichLecturesinAdvancedMathematicsVol.12.Zürich:EuropeanMathematical Society,2008.doi:10.4171/067

    work page doi:10.4171/067 2008

  77. [77]

    Spin-Free Quantum Chemistry

    F. A. Matsen. “Spin-Free Quantum Chemistry”.Advances in Quantum Chemistry. Ed. by P.-O. Löwdin. Vol. 1. New York: Academic Press, 1964, pp. 59–114.doi:10.1016/S0065- 3276(08)60375-5

    work page doi:10.1016/s0065- 1964

  78. [78]

    M.Hamermesh.Grouptheoryanditsapplicationtophysicalproblems.NewYork:Dover,1989

    1989

  79. [79]

    Onthecompletenessofasystemofcoherentstates

    A.M.Perelomov.“Onthecompletenessofasystemofcoherentstates”.Theor.Math.Phys.6 (1971),156–164.doi:10.1007/BF01036577

    work page doi:10.1007/bf01036577 1971

  80. [80]

    The Gaussian wave packet transform via quadrature rules

    P. Bergold and C. Lasser. “The Gaussian wave packet transform via quadrature rules”.IMA J.Numer.Anal.44(2024),1785–1820.doi:10.1093/imanum/drad049. 47

    work page doi:10.1093/imanum/drad049 2024

Showing first 80 references.