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Octopus, a computational framework for exploring light-driven phenomena and quantum dynamics in extended and finite systems

Nicolas Tancogne-Dejean, Micael J. T. Oliveira, Xavier Andrade, Heiko Appel, Carlos H. Borca, Guillaume Le Breton, Florian Buchholz, Alberto Castro, Stefano Corni, Alfredo A. Correa, Umberto De Giovannini, Alain Delgado, Florian G. Eich, Johannes Flick, Gabriel Gil, Adrián Gomez, Nicole Helbig, Hannes Hübener, René Jestädt, Joaquim Jornet-Somoza, Ask H. Larsen, Irina V. Lebedeva, Martin Lüders, Miguel A. L. Marques, Sebastian T. Ohlmann, Silvio Pipolo, Markus Rampp, Carlo A. Rozzi, David A. Strubbe, Shunsuke A. Sato, Christian Schäfer, Iris Theophilou, Alicia Welden, Angel Rubio

arXiv:1912.07921v1physics.comp-ph

TL;DR

The paper addresses challenges in developing accurate electronic-structure methods, including the expense of achieving few-meV energy resolution. It presents methodologies including Kubo–Greenwood linear response, local-domain contributions, real-space symmetrization, and coupled electron–photon approaches, with results indicating accuracy can be maintained while enabling more complex applications.

  • Problem

    Developing accurate electronic-structure methods can be expensive when few-meV energy resolution is required.

  • Method

    The paper uses Kubo–Greenwood linear response, local-domain methodologies, real-space symmetrization, and approaches coupling electronic systems to photons.

  • Results

    The paper presents new theoretical methodologies and frameworks for non-equilibrium phenomena and coupled electron, phonon, and photon dynamics, while maintaining accuracy in the reported results.

  • Takeaways & Limitations

    The real-space and grid methods are described as promising for future applications in more complex systems.

Abstract

from arXiv · show

Over the last years extraordinary advances in experimental and theoretical tools have allowed us to monitor and control matter at short time and atomic scales with a high-degree of precision. An appealing and challenging route towards engineering materials with tailored properties is to find ways to design or selectively manipulate materials, especially at the quantum level. To this end, having a state-of-the-art ab initio computer simulation tool that enables a reliable and accurate simulation of light-induced changes in the physical and chemical properties of complex systems is of utmost importance. The first principles real-space-based Octopus project was born with that idea in mind, providing an unique framework allowing to describe non-equilibrium phenomena in molecular complexes, low dimensional materials, and extended systems by accounting for electronic, ionic, and photon quantum mechanical effects within a generalized time-dependent density functional theory framework. The present article aims to present the new features that have been implemented over the last few years, including technical developments related to performance and massive parallelism. We also describe the major theoretical developments to address ultrafast light-driven processes, like the new theoretical framework of quantum electrodynamics density-functional formalism (QEDFT) for the description of novel light-matter hybrid states. Those advances, and other being released soon as part of the Octopus package, will enable the scientific community to simulate and characterize spatial and time-resolved spectroscopies, ultrafast phenomena in molecules and materials, and new emergent states of matter (QED-materials).

I. INTRODUCTION

The paper addresses the need for accurate simulations of increasingly complex non-equilibrium light–matter phenomena. It presents Octopus developments spanning new physical models, spectroscopy capabilities, numerical methods, and self-consistent light–matter coupling.

  • Research challenge: Real-time TDDFT is positioned as a crucial tool for exploring highly nonlinear phenomena in periodic and semi-periodic materials.The introduction highlights strong-field dynamics in solids as an active research area.
  • New capabilities: Recent Octopus additions include self-consistent light–matter couplings, solvent effects, van der Waals methods, magnon calculations, conductivities, photoelectron spectroscopy, and magneto-optical responses.The paper focuses on features added after earlier Octopus publications.
  • Numerical development: The project also develops numerical algorithms, iterative eigensolvers, periodic-system treatments, and technical improvements for performance and algorithmic stability.These developments are presented alongside physical-theory implementations for non-equilibrium phenomena.
  • Framework scope: Octopus targets non-equilibrium phenomena involving electronic, ionic, and photonic dynamics in complex systems.The framework is intended for molecules, low-dimensional materials, extended systems, nanostructures, and materials embedded in cavities.
  • Self-consistent coupling: Forward-only coupling is accurate when induced currents are small, but it breaks down when material currents and near-field effects become significant.The introduction identifies nanoplasmonic systems, surface plasmon-polaritons, and tip-enhanced spectroscopies as relevant cases requiring back-action, screening, and retardation.
  • Self-consistent coupling: Octopus implements coupled Ehrenfest-Maxwell-Pauli-Kohn-Sham equations to enable fully self-consistent forward–backward light–matter coupling in real time and real space.The implementation adds a Maxwell solver coupled to electron and nuclear dynamics and reaches the classical limit of QEDFT.

III. STRONG ELECTRON-PHOTON INTERACTIONS IN REAL SPACE: QUANTUM-ELECTRODYNAMICAL DENSITY-FUNCTIONAL THEORY

QEDFT extends time-dependent density-functional theory to include quantum photonic degrees of freedom in first-principles light–matter simulations. In Octopus, it supports ab initio treatment of weak and strong coupling and photon-dressed electronic states.

  • Scientific scope: The framework was introduced to treat ab initio weak and strong light–matter interactions and to explore, predict, and control non-equilibrium states of matter.Its applications include chemistry and materials, with electronic states dressed by photons while retaining real-space electronic properties.
  • Formalism: The length-gauge light–matter Hamiltonian treats interacting electrons coupled to photonic modes under the long-wavelength dipole approximation.Each photon mode is characterized by its elongation, frequency, and electron–photon coupling vector, which couples to the electronic dipole.
  • Formalism: QEDFT generalizes TDDFT by expanding the internal variables to include both the electronic density and mode-resolved photon contributions.The corresponding external variables are the electronic potential and photonic current, enabling an auxiliary Kohn–Sham–Maxwell system.
  • Exchange–correlation treatment: QEDFT places the quantum character of light–matter interactions in an additional exchange–correlation potential within the Kohn–Sham formulation.The auxiliary system combines electronic Kohn–Sham equations with Maxwell equations, while the photon exchange–correlation contribution is represented through the local potential.

A. The optimized effective potential (OEP)

The Octopus implementation develops an optimized effective potential framework for electron–photon systems, including Sternheimer-based orbital shifts, self-consistent potential updates, and KLI approximations. Tests on sodium dimers show close weak-coupling OEP/KLI behavior and stable convergence with constant or Barzilai–Borwein updates.

  • Formulation: The OEP formulation includes electron–electron and electron–photon effects through orbital shifts and self-consistent potential equations.The reformulation avoids explicit use of all unoccupied states by using occupied orbitals and orbital shifts.
  • Formulation: The Sternheimer reformulation replaces the unoccupied-state OEP calculation with Np+1 equations involving occupied orbitals.This makes the formulation more favorable for larger systems and easier to extend.
  • KLI approximation: KLI provides a simplified starting approximation that is exact for a single electron in exchange-only calculations and often reduces computational effort.The approximation is described as stable and sufficiently accurate for many electronic-structure calculations, but it fails to describe polarization features accurately.
  • KLI approximation: Light–matter KLI calculations can show artificial permanent-dipole dependence and violate translational invariance; using the electronic center-of-charge can improve stability.The issue arises from unbalancing photonic-excitation and self-polarization contributions.
  • Demonstration: For two sodium dimers in weak cavity coupling, KLI is close to OEP, while both constant and Barzilai–Borwein updates exhibit convergence behavior.The implementation compares electron densities, OEP/KLI results, and convergence using the two update schemes.

IV. DRESSED REDUCED DENSITY MATRIX FUNCTIONAL THEORY FOR ULTRA-STRONGLY COUPLED LIGHT-MATTER SYSTEMS

Dressed RDMFT extends reduced-density-matrix methods to coupled electron–photon systems by mapping them onto an auxiliary higher-dimensional electronic problem. Tests on stretched H2 indicate that dressed RDMFT remains accurate across coupling regimes, especially for total energy, although photonic observables become difficult at stronger coupling.

  • Auxiliary construction: The approach maps N electrons coupled to one photon mode onto N dressed fermions in d+1 dimensions with an auxiliary Hamiltonian containing one- and two-body terms.This construction permits standard electronic-structure methods to be applied to the dressed system.
  • Method development: The auxiliary construction is used to develop dressed RDMFT and dressed Hartree–Fock methods for coupled light–matter systems.The dressed first-order reduced density matrix supplies the quantities needed to evaluate the auxiliary Hamiltonian approximately.
  • Results: Dressed RDMFT improves considerably over dressed HF for total energy, while both methods capture photon number similarly in the stretched-H2 test.The comparison uses total energy and photon number against exact calculations.
  • Limitations: The current auxiliary-wave-function treatment neglects an additional exchange symmetry, motivating improved polaritonic N-representability conditions.The authors identify this as a future direction for the dressed theory.
  • Results: At small coupling, dressed RDMFT and dressed HF describe energy and photon number well; with increasing coupling, both fail to reproduce the strongly increasing photon number.Dressed RDMFT remains close to the exact total energy as coupling increases, whereas dressed-HF deviations grow.

V. TOWARDS DYNAMICS OF STRONGLY CORRELATED SYSTEMS:

Octopus extends strongly correlated and van der Waals calculations with self-consistent time-dependent Hubbard parameters and nonlocal dispersion treatments. These developments support light-driven correlated-material simulations while exposing limitations of fixed interaction parameters, local functionals, and empirical double-counting corrections.

  • DFT+U: DFT+U adds a mean-field Hubbard interaction on a localized subspace to improve the description of local electron–electron interactions.The method removes double counting of interactions already present in the DFT energy.
  • DFT+U: Octopus implements ab initio self-consistent U and J through ACBN0 and extends the method to time-dependent simulations of strongly correlated materials.The implementation targets out-of-equilibrium systems.
  • DFT+U: TDDFT+U simulations reproduce absorption spectra of transition-metal oxides including NiO and MnO.The reported examples demonstrate the method for correlated oxide spectroscopy.
  • Light-driven dynamics: The effective Hubbard U_eff = U − J changes during light-driven NiO dynamics, showing that fixed out-of-equilibrium electronic parameters are insufficient for strongly driven correlated materials.The time-dependent U_eff is evaluated for Ni 3d orbitals alongside the driving vector potential.
  • Van der Waals interactions: Octopus implements vdW-DF, vdW-TS, and vdW-D3 treatments for van der Waals interactions in isolated and periodic systems.The vdW-DF implementation uses libvdwxc and parallel Fourier transforms for nonlocal energy and potential contributions.
  • Van der Waals interactions: Nonlocal vdW interactions cannot be described by usual local and semilocal functionals, while explicit six-dimensional evaluation scales as O(N^2).The Roman-Pérez–Soler approach reduces the evaluation to three-dimensional integrals using convolutions and Fourier transforms.

B. vdW-TS

Octopus adds van der Waals corrections and implicit-solvent models to time-dependent calculations, enabling absorption-spectrum studies of interacting molecules and solvated systems. These implementations reproduce reference behavior and capture solvent-relaxation effects on spectral shifts and intensities.

  • vdW-TS: The TS-vdW correction enables real-time TDDFT calculations that include van der Waals effects in interacting molecular systems.The hydrogen fluoride dimer serves as a proof-of-concept application for supramolecular and related systems.
  • vdW-TS: A small van der Waals-induced red shift appears in the hydrogen fluoride dimer’s optical absorption spectrum.The shift is visible near the absorption feature around 133 nm.
  • vdW-TS: Octopus implements DFT-D3 through the authors’ library and validates its results against the library’s reference data.This makes the results consistent with other codes implementing the same correction.
  • Polarizable Continuum Model: Solvation shifts nitrobenzene absorption features toward lower excitation energies, while the chosen solvation scheme changes peak positions and intensities.Cavity-field effects directly affect peak intensities and can alter relative peak amplitudes under non-equilibrium solvation dynamics.
  • Polarizable Continuum Model: For slow solvents such as water, equilibrium and inertial TD-PCM spectra are nearly identical, making inertial TD-PCM preferable computationally.For faster solvents, EOM results depart from the inertial method and approach equilibrium TD-PCM behavior.
  • Polarizable Continuum Model: The PCM provides an implicit dielectric continuum environment for time-domain excited-state calculations, with multiple TD-PCM variants for different solvent relaxation times.The implementation uses IEF-PCM, apparent surface charges, and a tessellated molecular cavity.

VIII. MAGNONS FROM REAL-TIME TDDFT

Octopus implements real-time TDDFT calculations of magnons using transverse magnetic kicks and generalized Bloch boundary conditions. The approach reproduces supercell and prior results while supporting nonlinear and out-of-equilibrium spin dynamics with favorable computational scaling.

  • Real-time magnon calculations: The method extracts transverse spin susceptibilities by applying an impulsive Zeeman perturbation and Fourier-analyzing the subsequent spin-magnetization dynamics.The perturbation can target a finite wavevector q through a transverse magnetic kick.
  • Generalized Bloch theorem: Generalized Bloch boundary conditions avoid large supercells when calculating finite-momentum magnons, reducing the computational burden of meV-scale energy resolution.The implementation applies only in the absence of spin-orbit coupling.
  • Validation: Calculations using generalized Bloch and supercell approaches produce the same results up to numerical precision for tested cubic Ni, Fe, and Co systems.The results also agree very well with previous linear-response studies.
  • Capabilities: Real-time propagation does not require small perturbations, enabling studies of nonlinear strong-field magnetic dynamics and systems kicked from excited states.It also permits direct coupling to phonons or photons without new theory or code development.
  • Computational advantages: Time propagation scales linearly with the number of states, whereas sum-over-states approaches usually scale quadratically.The method also avoids requiring an exchange-correlation kernel.

IX. ORBITAL MAGNETO-OPTICAL RESPONSE OF SOLIDS AND MOLECULES FROM A STERNHEIMER APPROACH

Octopus implements Sternheimer-based magneto-optical response calculations for periodic solids and finite systems while preserving gauge invariance. Tests on bulk silicon show qualitative agreement with experiment, improved by excitonic effects, with computational costs comparable to optical-polarizability calculations.

  • Periodic systems: The periodic formalism handles uniform electric and magnetic fields through a gauge-invariant density-matrix response under purely periodic boundary conditions.The Sternheimer approach solves the response iteratively at each frequency while accounting for local-field effects self-consistently when needed.
  • Response formalism: The method computes magnetic-field corrections to polarizability and derives the dielectric tensor without explicitly calculating the second-order density-matrix derivative.The 2n + 1 theorem reduces the required response calculations to first-order derivatives and a supplementary vector-potential perturbation.
  • Validation: Bulk-silicon transverse dielectric spectra are qualitatively similar to experimental curves at the direct absorption edge even without excitonic effects.Including excitonic effects improves agreement with experiment.
  • Validation: Calculated magneto-optical peak magnitudes are about a factor of two smaller than measurements, but reducing the linewidth can correct them.The reported calculations use a finite linewidth δ, which affects the peak magnitudes.
  • Computational cost: For solids, magneto-optical spectra take only twice as long as optical-polarizability calculations, while finite-system calculations require the same computational effort.The overall efficiency is comparable to standard linear-response optical-polarizability calculations.

X. TIME-DEPENDENT ANGULAR RESOLVED PHOTOELECTRON

Octopus implements t-SURFF for momentum-resolved photoelectron observables in periodic and non-periodic systems, including pump-probe and strong-field simulations. The framework connects photoelectron probabilities to current flux through an analyzing surface and reproduces reported experimental behavior.

  • System classes: Octopus supports photoelectron spectroscopy for both periodic and non-periodic systems, using planes parallel to material surfaces or spherical surfaces around finite systems.The same framework accommodates surface-sensitive spectra and finite-system ionization.
  • Derived observables: Momentum-resolved spectra can be transformed into energy-resolved spectra and ARPES, P(E) and P(p∥, E), through the free-electron dispersion relation.ARPES is obtained by resolving the energy spectrum with respect to momentum parallel to the surface.
  • t-SURFF formalism: t-SURFF computes momentum-resolved photoelectron probabilities from the time-integrated photo-current flux through a surface separating inner and outer regions.The inner region uses the interacting Kohn-Sham Hamiltonian, while the outer region is described with Volkov states.
  • Strong-field ionization: Strong-field rescattering produces momentum rings centered at the vector potential at rescattering, and Octopus simulations agree excellently with experiments.The rings arise from electrons rescattering at the same time within one field period.
  • Scope and limitations: The method requires explicit vacuum and, for solids, surface slabs, which disadvantages it relative to unit-cell calculations despite capturing more experimental surface-layer reality.The analyzing surface must be placed sufficiently far from the system for convergence.
  • Ultrafast spectroscopy: Pump-probe simulations reveal conduction-band population transfer in the resulting ARPES spectrum and can address steady-state driven states and Floquet physics.The implementation also supports phenomena such as RABBITT and electron-phonon signatures in ARPES.

XI. ELECTRIC AND THERMAL CONDUCTIVITIES

Octopus evaluates time-dependent electrical and thermal currents and derives frequency-dependent conductivities from real-time TDDFT. Unlike Kubo-Greenwood, the approach is intended to handle nonlinear conduction, while the illustrated heat-current treatment remains incomplete.

  • Time-dependent response: Real-time TDDFT in Octopus computes current density and heat current density at every time step after an applied electric field.The field can be represented through a time-dependent vector potential compatible with periodic boundary conditions.
  • Conductivity: Fourier transforming the induced currents yields frequency-dependent electrical and thermal conductivities, including DC conductivity at ω = 0.The conductivities are obtained from the current response rather than an equilibrium Kubo-Greenwood calculation.
  • Illustrative application: In liquid hydrogen at 1400 K and 400 GPa, both the current and kinetic heat current decay to zero, while their Fourier transforms give frequency-dependent conductivities.The example uses an initial electric field E0 of 0.1 a.u.
  • Scope and limitations: The present heat-current expression includes only the kinetic contribution; potential and electron-electron interaction contributions are not yet implemented.Their magnitude remains to be assessed for particular systems.
  • Nonlinear transport: The TDDFT conductivity approach is more general than Kubo-Greenwood because it can be applied to materials exhibiting nonlinear conduction.It has been used to illustrate nonlinear conductivity effects in liquid aluminum, which standard linear-response theory cannot capture.

XII. LOCAL DOMAIN CONTRIBUTION TO PHYSICAL OBSERVABLES

Octopus partitions electronic densities into local domains to decompose additive observables, including optical responses, into contributions from different molecular or spatial regions. Applied to a benzene-fulvene dimer, the analysis identifies fulvene as the main source of distance-dependent spectral changes.

  • Local domain partitioning: Local domain partitioning assigns non-overlapping densities to domains, allowing additive observables to be decomposed into local contributions.Examples include the time-dependent dipole, optical absorption cross section, and exchange-correlation energy in LDA.
  • Domain definitions: Octopus defines domains using geometric shapes, atom-centered regions, or Bader volumes whose boundaries follow zero-flux density-gradient surfaces.Users can also specify overlapping regions, but those overlaps require additional care during analysis.
  • Benzene-fulvene results: Local analysis attributes the major spectral change to reduced oscillator strength of the fulvene peak near 6.5 eV.The results agree with prior FDE-rt-TDDFT calculations.
  • Scalability and extensions: The local-domain method requires no prior basis-set fragmentation, selection, or localization and can therefore treat large systems such as LHC-II.It has also been combined with real-time TDDFT transition densities for exciton-coupling calculations.

XIII. NEW PROPAGATORS FOR REAL-TIME TDDFT

Octopus implements commutator-free Magnus propagators for real-time TDDFT while targeting structural properties such as norm preservation, symplecticity, and long-time energy behavior. The fourth-order CFM4 scheme outperforms the second-order exponential midpoint rule across the tested errors.

  • Numerical structure: Symplectic propagators are relevant for TDDFT because they can preserve phase-space structure and keep long-time energy errors bounded by oscillation rather than divergence.The TDKS equations in the adiabatic approximation possess symplecticity as a mathematical property.
  • CFM propagators: Octopus implements commutator-free Magnus schemes, including CFM4, after identifying them as suitable for TDDFT based on accuracy, stability, and performance.CFM expansions avoid the nested commutators required by standard Magnus expansions.
  • Nonlinear implementation: CFM4 handles the nonlinear TDKS problem explicitly by extrapolating Hamiltonian components from previous time steps, avoiding algebraic equation solves while preserving fourth-order accuracy.The extrapolated components are the Hartree, exchange, and correlation parts.
  • Benchmark: CFM4 outperforms the exponential midpoint rule for every examined error, with the advantage becoming clearer as higher accuracy is demanded.The comparison uses propagation cost versus accuracy for a perturbed benzene molecule at varying time steps.
  • Benchmark: CFM4 is fourth-order accurate and requires two exponentials, whereas the exponential midpoint rule is second-order accurate and requires one.The different convergence slopes appear as the time step approaches zero.

XIV. CONJUGATE GRADIENT IMPLEMENTATION IN RDMFT

Octopus adds a conjugate-gradient approach for optimizing RDMFT natural orbitals directly on real-space grids. Compared with basis-set Piris calculations, it converges with fewer orbitals and reaches slightly lower total energies, while final results are independent of the initial states.

  • RDMFT setup: RDMFT requires more natural orbitals than electrons, with the exact number treated as a system-dependent convergence parameter.Core electrons represented by the pseudopotential do not count, and the number of unoccupied states must cover significantly occupied natural orbitals.
  • Initial orbitals: The additional localization step was removed because it can hinder convergence with respect to the number of basis functions.It improved results only for a small number of natural orbitals, whereas independent-particle orbitals provided the fastest convergence and lowest total energies among tested bases.
  • Conjugate-gradient method: The new implementation minimizes RDMFT natural-orbital functionals with conjugate gradients directly on the real-space grid.The method adapts a DFT conjugate-gradient procedure to RDMFT and permits systematic improvement through finer grids.
  • Convergence: Initial-state quality affects only the number of iterations required for convergence, not the final converged result.The converged Müller-functional result was verified to be independent of the initial state, so random states can be used to start the conjugate-gradient calculation.
  • Convergence: The conjugate-gradient calculation requires fewer natural orbitals than Piris basis-set implementations and yields slightly lower converged total energy.The result indicates that the grid method uses orbital contributions unavailable to the tested basis sets.

XV. PERIODIC SYSTEMS AND SYMMETRIES

Octopus extends real-space calculations to periodic systems by combining primitive-cell grids, Brillouin-zone sampling, symmetry reduction, and real-space symmetrization. It also supports symmetry-aware time-dependent calculations and electron–ion–lattice dynamics through stress-tensor and lattice-motion implementations.

  • Periodic real-space grids: Periodic calculations use real-space grids generated along primitive-cell axes, with finite-difference weights adapted to potentially non-orthogonal Wigner–Seitz cells.The primitive cell is combined with Brillouin-zone sampling through a k-point grid.
  • Symmetry reduction: Crystal symmetries reduce the Brillouin zone to its irreducible portion, potentially decreasing the number of k-points drastically.The space group and symmetries are obtained using spglib, while the implementation currently restricts them to symmorphic operations.
  • Symmetry limitations: Non-symmorphic symmetries remain unsupported because fractional translations are incompatible with arbitrary real-space grids.The current restriction covers inversion, rotations, and mirror planes.
  • Time-dependent symmetry: Time-dependent calculations retain only the subgroup of crystal symmetries that leaves a perturbation direction invariant.This handles symmetry breaking from laser fields, vector potentials, momentum kicks, or strain.
  • Real-space symmetrization: Real-space symmetrization of charge, current, and other observables is important for stable numerical results when symmetries are used.The implementation applies this symmetrization alongside irreducible-Brillouin-zone reduction.
  • Lattice dynamics: Periodic electron–ion–lattice dynamics combines ionic motion in reduced coordinates with primitive-cell motion and requires stress-tensor calculations.Octopus implements the stress tensor and lattice dynamics within the TDDFT framework.

XVI. ADDITIONAL TECHNICAL CODE IMPROVEMENTS

Octopus improves reliability, maintainability, and efficiency through interactive test analysis, broader automated testing, solver updates, and real-space eigensolver preconditioners. The filter preconditioner offers the best total-time trade-off, although some highly unoccupied-state calculations may still require restarting without preconditioning.

  • Code reliability: An interactive web application visualizes testsuite results, helping diagnose failures across toolchains and update the testsuite.It has already helped identify bugs causing regressions.
  • Eigensolver improvements: The updated conjugate-gradient eigensolver orthogonalizes each band only against previously computed lower-energy bands, accelerating convergence in most cases.This differs from orthogonalizing against all bands.
  • Eigensolver improvements: The filter preconditioner is most effective for ground-state calculations because it reduces SCF iterations without excessive computational cost.Its practical effectiveness depends on the system, and increasing the number of Jacobi iterations does not improve it.
  • Eigensolver improvements: Multigrid and Poisson-based preconditioners can reduce SCF iterations but are less effective overall because their iterations are substantially more expensive.The multigrid approach applies the Laplacian several times per iteration, while the Poisson approach increases total calculation time.
  • Eigensolver limitations: Some states, especially calculations with many unoccupied states, cannot be fully converged with a preconditioner and require restarting without it.The reasons for this limitation remain under investigation.
  • Code architecture: A major code refactoring introduces generic coupling infrastructure for existing and future developments involving multiple subsystems.The framework is presented in connection with fully coupled Maxwell–TDDFT functionality.

D. Memory layout

Octopus’s packed orbital layout improves memory locality and supports efficient CPU vectorization and GPU parallelization. The GPU implementation has expanded across devices and delivers substantial speedups, although multi-node scaling remains imperfect.

  • D. Memory layout: Packing states into smaller batches makes grid operations contiguous and parallel-friendly.The innermost index is the state index, allowing identical operations across states in a batch.
  • D. Memory layout: The layout supports efficient GPU warp execution and CPU vectorization, including specialized SSE, AVX, and AVX512 finite-difference kernels.More code has been ported to the layout to increase performance.
  • E. GPUs: Octopus supports CUDA, multiple devices per host, and GPU-resident state storage that minimizes memory transfers when capacity is sufficient.Custom management of temporary GPU variables further improved scaling across GPUs and nodes.
  • E. GPUs: 4.8× faster time steps and 2.3× lower energy-to-solution were obtained on a GPU node than on one full CPU node.The comparison used approximately 950 W for a GPU node versus 450 W for a CPU node.
  • E. GPUs: The GPU examples stored all states in GPU memory and parallelized over states rather than distributing the grid.Grid distribution was considered less effective because it requires more frequent communication.
  • XVII. CONCLUSIONS: The paper presents Octopus as a flexible real-space framework spanning many methods and systems, including coupled electron, phonon, and photon dynamics.Examples include molecules, nanoparticles, solvents, solids, and monolayers, alongside approaches such as coupled Maxwell-Kohn-Sham equations, OEP, dressed RDMFT, magnons, and Sternheimer orbital magneto-optical response.
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