Source-linked AI summary
Real-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systems
Xavier Andrade, David A. Strubbe, Umberto De Giovannini, Ask Hjorth Larsen, Micael J. T. Oliveira, Joseba Alberdi-Rodriguez, Alejandro Varas, Iris Theophilou, Nicole Helbig, Matthieu Verstraete, Lorenzo Stella, Fernando Nogueira, Alán Aspuru-Guzik, Alberto Castro, Miguel A. L. Marques, Ángel Rubio
TL;DR
Electronic-structure simulations require methods that improve on existing tools while remaining practical for new applications. The paper examines real-space grids through the Octopus code as a flexible framework for developing and testing approaches across response, photoemission, plasmonics, and other problems. These developments include a plasmonic resonance for gold spheres with Ne > 100 and several methods intended for broader electronic-structure use.
Problem
Existing electronic-structure tools can predict many properties at modest cost but are not accurate enough for many applications.
Method
The paper uses the flexible Octopus real-space code as a framework for developing and testing new electronic-structure approaches.
Results
A clear plasmonic resonance appears in the absorption cross section of gold spheres containing Ne > 100 electrons, alongside multiple new Octopus-based simulation developments.
Takeaways & Limitations
Real-space grids and Octopus support developments spanning response properties, photoemission, plasmonics, geometry optimization, and related electronic-structure applications.
Abstract
from arXiv · showhide
Real-space grids are a powerful alternative for the simulation of electronic systems. One of the main advantages of the approach is the flexibility and simplicity of working directly in real space where the different fields are discretized on a grid, combined with competitive numerical performance and great potential for parallelization. These properties constitute a great advantage at the time of implementing and testing new physical models. Based on our experience with the Octopus code, in this article we discuss how the real-space approach has allowed for the recent development of new ideas for the simulation of electronic systems. Among these applications are approaches to calculate response properties, modeling of photoemission, optimal control of quantum systems, simulation of plasmonic systems, and the exact solution of the Schrödinger equation for low-dimensionality systems.
I. INTRODUCTION
Electronic-structure methods need continual development because existing tools are insufficient for many applications. The paper presents real-space grids and Octopus as flexible frameworks for implementing new theories and simulation capabilities.
- Motivation and real-space approach: Real-space grids simplify translating continuum electronic problems into computations through grid sums and finite-difference differential operators.Their discretization avoids dependence on a physically tailored basis set.
- Motivation and real-space approach: Grid spacing and physical extent provide systematic, continuous control of discretization error across finite, periodic, and reduced-dimensional systems.This flexibility supports simulations with different boundary conditions and dimensionalities.
- Octopus as a development framework: Real-space grids are presented as an ideal framework for implementing, developing, and testing new electronic-structure ideas.The paper emphasizes simplicity and flexibility as practical advantages for theory-development work.
- Applications enabled by Octopus: Octopus provides a real-space base for recent developments including generalized Sternheimer response methods and higher-order response calculations.The generalized formulation handles static and dynamic response in and out of resonance, while second-order response uses the 2n + 1 theorem.
- Applications enabled by Octopus: The response formalism yields optical properties, van der Waals coefficients, and vibrational properties for finite and periodic systems.Polarizabilities provide optical response and imaginary-frequency van der Waals information, while ionic-displacement perturbations produce dynamical matrices.
IV. MAGNETIC RESPONSE AND GAUGE INVARIANCE IN REAL-SPACE GRIDS
Octopus uses perturbative magnetic response with Sternheimer equations and gauge-invariant corrections for non-local potentials, enabling magnetic susceptibilities and related observables on real-space grids.
- Magnetic fields enter through orbital and spin couplings, with the Zeeman term representing electronic-spin interaction with the field.
- Magnetic susceptibility is obtained from first-order response functions to a uniform static field through the Sternheimer formalism.
- Real-space discretization can break gauge invariance, causing mesh-dependent convergence and origin-dependent magnetic observables; mesh refinement controls the grid error.
- Non-local pseudopotentials require gauge-invariant corrections, implemented through the ICL and GIPAW integration-path formulations.
- These corrections restore gauge invariance with pseudopotentials and allow magnetic susceptibility and optical-activity calculations.
- For boron fullerenes, most clusters are diamagnetic, whereas B80 is paramagnetic with strong cancellation between paramagnetic and diamagnetic terms.
V. LINEAR RESPONSE IN THE ELECTRON-HOLE BASIS
Octopus formulates linear response in an electron-hole basis using a hierarchy of TDDFT approximations, with matrix construction and parallelization designed to reduce computational cost.
- Linear response can be recast as excitation energies and residues, including dipole matrix elements and polarizability.
- The hierarchy includes RPA, Petersilka, Tamm-Dancoff, Casida, and CV(2), rather than solving the full TDDFT eigenvalue equation.
- RPA uses occupied-unoccupied Kohn-Sham energy differences, while Petersilka adds diagonal matrix corrections and can fail for degenerate or nearly degenerate transitions.
- The Tamm-Dancoff approximation neglects coupling blocks, reducing the problem to a Hermitian matrix of half the full size.
- Tamm-Dancoff results can outperform the full solution for molecular potential-energy surfaces or hybrid functionals affected by triplet instability.
- Matrix construction dominates runtime, especially for the non-local Coulomb term, which is evaluated through Poisson solves for each matrix column.
- Hermiticity nearly halves matrix-element work, and columns are distributed across processors while rows remain undistributed to avoid duplicated Poisson solves.
- Excited-state forces can be computed without additional empty-state summations, but the more complicated Casida force case is not implemented.
VI. FORCES AND GEOMETRY OPTIMIZATION ON REAL-SPACE GRIDS
Real-space grids produce translation-dependent egg-box errors, and Octopus addresses them with filtered potentials, smoother force formulations, and force-based FIRE optimization.
- Grid representation can make a translated atom have a different potential, producing artificial energy and property variations known as the egg-box effect.
- The egg-box effect is especially problematic for molecular dynamics and geometry optimization, where atoms move during the calculation.
- Octopus reduces grid artifacts through pseudopotential filtering that removes Fourier components unrepresentable on the grid.
- Rewriting force evaluation with orbital spatial derivatives is easier to implement and more precise than differentiating the ionic potential on a grid.
- For N2, ionic-potential derivatives produce considerable force oscillations, whereas orbital derivatives yield a considerably smoother force.
- The force formulation extends to second-order energy derivatives for vibrational properties through the dynamical matrix.
- Because forces converge faster than energy with grid spacing, FIRE is suitable for geometry optimization without requiring energy evaluations.
- FIRE modifies velocities and adaptive time steps, resetting parameters when the force-gradient direction changes.
VII. PHOTOEMISSION
The paper develops a real-time TDDFT photoemission scheme that combines real-space propagation near the system with momentum-space treatment of outgoing electrons. The method retrieves momentum-resolved probabilities and supports trajectories crossing the real–momentum-space boundary, while specialized discretizations address spurious periodic re-entry and long-time stability.
- Method: Photoemission is calculated with a mixed real- and momentum-space real-time TDDFT scheme implemented in Octopus.Kohn-Sham orbitals are propagated in real space on a restricted box and matched at its boundary to momentum-space representations.
- Observables: Momentum-resolved photoelectron probability P(k) is obtained directly from the momentum components and is described as the most resolved quantity available experimentally.The approach also supports energy-resolved probabilities through direct integration.
- Method: A mask function splits each orbital into bounded and unbounded components localized in spatial regions A and B.Region A uses a real-space grid, while region B uses momentum space, with an overlap region enabling matching.
- Propagation: The propagation scheme exactly describes electrons crossing the boundary when the Hamiltonian in region B is that of free electrons in the applied field.The Volkov propagator handles the momentum-space component, allowing transitions between the two representations.
- Numerical treatment: Fourier-sum discretization with ordinary FFTs imposes periodic boundaries that can spuriously reintroduce charge and make wavepackets wrap around the box.Zero padding and NFFT-based strategies push the re-entry boundary away or reduce the added real-space cost, with different momentum-space trade-offs.
- Numerical treatment: An outgoing-flow approximation removes the periodic boundary conditions and provides numerical stability for arbitrarily long propagations.Charge removal then occurs only in the bounded-region equation, and the mask function must be chosen carefully because it can cause energy-dependent reflections.
VIII. COMPLEX SCALING AND RESONANCES
The complex-scaling implementation in Octopus converts metastable resonances into localized eigenstates of a non-Hermitian Hamiltonian. This exposes resonance energies and widths while rotating the continuum, and it supports DFT-based ionization-rate calculations.
- Resonances: Resonant states are metastable states of finite systems with characteristic energies and lifetimes, but they are not eigenstates in the ordinary Hilbert space.Their outgoing-wave character makes direct calculation with standard DFT methods unavailable.
- Complex scaling: The transformed Siegert states become square-integrable eigenstates whose complex eigenvalues ε0 − iΓ/2 provide the resonance energy ε0 and width Γ.This makes resonance information accessible through eigenvalue calculations.
- Spectral signature: Under complex scaling, bound-state energies remain unchanged, the continuum rotates by −2θ, and resonances appear as isolated fourth-quadrant eigenvalues once uncovered.Localized states have matrix elements and energies independent of θ when numerically well represented.
- Complex scaling: Complex scaling analytically continues states and operators by rotating the real-space integration contour through a fixed angle θ.The transformation maps the Hamiltonian to the non-Hermitian operator Hθ = Rθ H R−θ.
- Implementation: Octopus supports complex-scaled independent-particle and selected DFT calculations by rotating the terms of the Kohn-Sham energy functional and solving for stationary points.The complex-valued functional cannot be minimized in the ordinary sense, and state occupation requires additional care.
- Application: For helium in weak electric fields, the ADK approximation approaches the accurate reference because the ionization rate is largely determined by the ionization potential.The supplied passage also notes that LDA ionization potentials are inaccurate because of the functional’s asymptotic behavior.
IX. QUANTUM OPTIMAL CONTROL
The paper extends quantum optimal-control theory to electronic systems modeled with Octopus and time-dependent electric fields. The implementation supports gradient-based and gradient-free optimization, including nonlinear TDDFT dynamics, and has been applied to manipulating states and tailoring laser-driven processes.
- Framework: QOCT searches for parameters u that maximize a target functional G(u) = F[ψ[u]] determined by the system’s evolution.The formulation also considers robustness, solution multiplicity, and algorithms for finding optima.
- Optimization: Gradient-based optimization uses a costate propagated backward in time to compute the gradient, while gradient-free algorithms use only evaluations of the objective.Octopus includes both gradient-less algorithms and standard conjugate-gradient or BFGS schemes.
- Framework: Octopus implements quantum optimal-control theory for systems controlled by time-dependent electric fields.The control parameters define the field shape, for example through Fourier coefficients or time-discretized amplitudes.
- TDDFT extension: The theory was extended because time-dependent Kohn-Sham equations are nonlinear, unlike the linear quantum-mechanical formulation usually assumed by QOCT.The resulting TDDFT optimal-control equations were implemented in Octopus.
- Applications: Applications include controlling current-carrying states in quantum rings, manipulating double-quantum-dot states, and tailoring femtosecond pulses for maximal atomic or molecular ionization.The pulse example uses approximately 5 fs high-intensity irradiation of H2+.
- Applications: The framework has also been developed for photochemical control of selected bond creation or breaking within TDDFT and Ehrenfest molecular dynamics.Earlier attempts did not provide a fully consistent combined optimal-control theory; such a theory was subsequently presented.
X. PLASMONICS
The plasmonics discussion uses real-space real-time methods and TDDFT@jellium to model quantum electronic response in nanostructures. These approaches extend from nanospheres to periodic nanowires, while nanoscale spillout and atomic-lattice nonuniformity define important limits of classical and coarse-grained descriptions.
- Motivation: Nanoplasmonic systems span tens of nanometers while strong field-enhancement regions can be smaller than 1 nm, creating a multiscale electronic-structure problem.Classical calculations predict field enhancements h > 100, with SERS intensity scaling as h^4.
- Motivation: Classical permittivity models confine electrons inside metal surfaces, but nanoscale spillout makes direct quantum electronic dynamics necessary.At this scale, the definition of a macroscopic permittivity becomes inappropriate.
- Method: TDDFT@jellium models plasmonic response by treating nuclei and core electrons as uniform positive charge while describing valence electrons explicitly.This provides a simplified quantum model for metal nanoparticles and related structures.
- Applications: Octopus has been applied to gold nanospheres, where real-time propagation produces a clear plasmonic resonance in the absorption cross section for sufficiently large spheres.The cited example uses a gold jellium model with Wigner-Seitz radius rs = 3.0 bohr.
- Applications: Periodic pairs of interacting sodium nanowires can be modeled in Octopus to assess classical permittivity-based methods, with stronger inductive interaction than corresponding nanospheres.The enhanced interaction is associated with the wires’ extended geometry.
- Implementation: Jellium geometries can serve as effective superatomic pseudopotentials, and Octopus can support scalable approaches incorporating d-electron screening in noble metals.The approach includes development of an external jellium pseudopotential generator.
- Limitations: Atomic-lattice nonuniformity is expected to affect the absorption cross section of small metal nanoparticles, requiring assessment of lattice contributions and symmetry.This limitation marks a boundary for TDDFT@jellium descriptions of small structures.
XI. DEVELOPMENT OF EXCHANGE AND CORRELATION FUNCTIONALS
The section presents exchange-correlation development within Octopus, emphasizing the unknown DFT functional and real-space benchmarking and testing of new approximations.
- DFT accuracy depends on the exchange-correlation energy Exc[n], whose exact form remains unknown and must be approximated.
- Exchange-correlation approximations are organized into Jacob’s ladder, progressing from density-only LDA and GGA forms to orbital-dependent functionals.
- Octopus supports the first three Jacob’s-ladder rungs and local hybrid components through the Libxc library.
- Octopus also implements fourth-rung approaches including exact exchange and Perdew–Zunger self-interaction correction through optimized effective potentials.
- The real-space framework enables straightforward testing of new functionals against reference data computed on the same grid.
- An Octopus method enforces an asymptotically correct exchange-correlation potential by treating it as an electrostatic potential from a fictitious exchange-correlation charge.
XII. REAL-SPACE REDUCED DENSITY-MATRIX FUNCTIONAL THEORY
This section develops a real-space implementation of reduced density-matrix functional theory in Octopus and examines its optimization and hydrogen-dissociation behavior.
- RDMFT describes electrons through the one-body reduced density matrix, whose eigenfunctions and eigenvalues are natural orbitals and occupation numbers.
- RDMFT approximates only the electron-interaction exchange-correlation contribution, while the interacting kinetic energy is explicit in the reduced density matrix.
- The Octopus implementation minimizes the energy under N-representability and orbital-orthonormality constraints using unconstrained occupation parametrization and Lagrange multipliers.
- Optimization alternates occupation-number and natural-orbital minimization until convergence, with orbital updates obtained through iterative diagonalization.
- The calculation requires suitable initial natural orbitals because unbound HF or DFT virtual states provide poor starting points for weakly occupied orbitals.
- For H2 dissociation, Octopus RDMFT resembles Gaussian-basis results, but a deviation from constant energy remains at large bond distances.
XIII. EXACT SOLUTION OF THE MANY-BODY SCHR¨ODINGER EQUATION FOR FEW ELECTRONS
The section uses Octopus’s arbitrary-dimensional real-space machinery to solve few-electron Schrödinger problems exactly and enforce particle-exchange symmetry.
- An N-electron problem in d dimensions can be mapped onto a single-particle problem in Nd dimensions, enabling direct grid-based solution for small N.
- Octopus supplies arbitrary-dimensional Schrödinger solvers, grid bookkeeping, external-potential representation, and parallelization for these calculations.
- Young-tableau symmetrization removes spatial wavefunctions incompatible with fermionic permutation symmetry and identifies allowed states.
- For a one-dimensional lithium atom, the first and fourth eigenstates have norms below 10^-13 and 10^-11 for all diagrams and are removed as bosonic.
- The implementation supports fermions, bosons, anyons, and multiple particle types with exchange requirements applied within each declared type.
- Exact few-electron calculations provide reference results for assessing approximate DFT and RDMFT exchange-correlation treatments and time-dependent spectra.
XIV. COMPRESSED SENSING AND ATOMISTIC SIMULATIONS
The section applies compressed sensing to reconstruct frequency spectra and sparse simulation matrices from shorter signals, reducing propagation or computational time.
- Real-time simulations require spectral representation of time-resolved signals, and shorter signals could reduce the associated propagation cost.
- Compressed sensing exploits spectral sparsity to reconstruct frequency-resolved quantities from fewer samples than standard Fourier analysis.
- The basis-pursuit problem selects the minimum-1-norm spectrum compatible with the short time series, favoring spectra with few frequencies.
- SPGL1 solves the compressed-sensing optimization in Octopus, with its cost remaining negligible relative to time propagation despite being more expensive than a Fourier transform.
- A 10 fs compressed-sensing propagation achieves similar methane optical-spectrum resolution to a 50 fs Fourier-transform propagation, reducing computational time by a factor of five.
- Compressed-sensing reconstruction of sparse matrices reduces computational time by a factor of three for Hessian and linear-response vibrational-frequency calculations.
XV. PARALLELIZATION, OPTIMIZATIONS AND GRAPHICS PROCESSING UNITS
Octopus combines multi-level parallelization with abstractions that simplify efficient implementation across CPUs and GPUs. Memory-local grid mappings, especially Hilbert curves, further improve finite-difference performance.
- Electronic-structure simulations require efficient execution on massively parallel platforms because electron dynamics and small spatial scales limit accessible system sizes.
- Octopus provides developer building blocks for integration, linear algebra, differential operators, Hamiltonians, and differential-equation solvers that are automatically parallel and GPU-capable.
- A higher-level interface groups several states as the basic object, exposing more data parallelism than treating individual states separately.
- Developers represent grid fields as linear arrays while auxiliary structures preserve grid information across different dimensions and grid shapes.
- Around 50% performance gain is obtained for the GPU finite-difference Laplacian by mapping grid points with a Hilbert curve.
- Octopus parallelizes through domain decomposition and additional decompositions over Kohn-Sham states, k-points, and spin, while Poisson solves require special treatment.
XVI. CONCLUSIONS
The article presents Octopus-based real-space developments as examples of how grid flexibility and simplicity support new electronic-structure ideas. It also identifies Coulomb-integral costs and non-local interactions as continuing challenges while highlighting future exaflop-scale potential.
- Octopus-based developments demonstrate that real-space grids support new electronic-structure ideas beyond straightforward implementations of existing theory.
- These advances span diverse applications beyond traditional electronic-structure calculation schemes and may help address current and future field challenges.
- Non-local potentials and pseudopotentials complicate magnetic-response formulations and introduce an additional, sometimes poorly controlled approximation.
- Two-body Coulomb integrals can have substantial numerical cost despite being computed in linear or quasi-linear time through Poisson problems.
- Real-space grid scalability makes these methods candidates for future exaflop supercomputers, provided high-performance implementations are developed.