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The TDHF Code Sky3D

J. A. Maruhn, P. -G. Reinhard, P. D. Stevenson, A. S. Umar

arXiv:1310.5946v1nucl-thnucl-ex

TL;DR

Nuclear mean-field calculations require practical treatments of structure and dynamics across ground states, vibrations, and collisions. Sky3D addresses this with unrestricted three-dimensional Skyrme calculations and flexible initialization, while documenting scope limits and the need for user-performed spectral post-processing.

  • Problem

    The paper addresses practical nuclear mean-field modeling of ground states, collective vibrations, and heavy-ion collisions using effective Skyrme functionals.

  • Method

    Sky3D solves static and time-dependent equations on an unrestricted three-dimensional Cartesian grid, with flexible nuclear initialization and energy evaluation.

  • Results

    Energy remains conserved after excitation in the demonstrated evolution, while smaller time steps reduce fluctuations and fluctuation amplitudes scale approximately as dt^2.

  • Takeaways & Limitations

    Sky3D supports flexible three-dimensional nuclear-dynamics studies, but users must perform the final spectral-analysis steps themselves from printed time-dependent signals.

  • Takeaways & Limitations

    The code is limited by the mean-field approximation, missing time-odd terms, frozen BCS occupations, and static-calculation issues such as possible center-of-mass drift or rotation.

Abstract

from arXiv · show

The nuclear mean-field model based on Skyrme forces or related density functionals has found wide-spread application to the description of nuclear ground states, collective vibrational excitations, and heavy-ion collisions. The code Sky3D solves the static or dynamic equations in a three-dimensional Cartesian mesh with isolated or periodic boundary conditions and no further symmetry assumptions. Pairing can be included in the BCS approximation. The code is implemented with a view to allow easy modifications for including additional physics or special analysis of the results.

PROGRAM SUMMARY

Sky3D implements static and time-dependent Hartree-Fock calculations for nuclear structure, vibrations, and low-energy collisions using Skyrme functionals on a three-dimensional mesh. Its flexible initialization and boundary-condition options support varied applications, but the mean-field approximation and implementation impose explicit scope limits.

  • Implementation: The code uses LAPACK and FFTW3, is implemented in Fortran 90, and runs with OpenMP or MPI on computers with a Fortran compiler.Some build scripts require modification under Windows.
  • Capabilities: Sky3D solves static and time-dependent Hartree-Fock equations based on a Skyrme energy functional for nuclear ground states, vibrations, and low-energy collisions.Vibrational spectra are obtained by Fourier analysis of suitable time-dependent observables.
  • Numerical framework: Wave functions are represented on a three-dimensional Cartesian mesh without symmetry restrictions, using periodic wave-function boundaries and optionally isolated Coulomb boundaries.Spatial derivatives use the finite Fourier transform method.
  • Restrictions: The mean-field approximation limits applications to collision energies about a few MeV per nucleon above the Coulomb barrier and relatively short interaction times; missing time-odd terms may restrict use to even-even nuclei.These restrictions arise from absent hard nucleon-nucleon collisions and incomplete Skyrme-interaction terms.
  • Initialization: Any number of initial nuclei can be placed at arbitrary positions and initial velocities, enabling flexible collision initialization.The code also supports periodic boundaries for astrophysical nuclear-matter applications.
  • Computational requirements: A single-processor run can take from a few minutes to weeks, depending strongly on grid size, nucleon number, and collision duration.The code has no built-in processor limit.

1. Introduction

The paper presents Sky3D as an accessible implementation of unrestricted three-dimensional TDHF with the non-relativistic Skyrme energy functional. It addresses the need for broadly available 3D calculations while retaining the historical TDHF name even though modern functionals make the equations formally TDDFT.

  • 1. Introduction: Self-consistent mean-field models describe many-body systems through single-particle wave functions and generate corresponding one-body potentials.Nuclear applications use effective interactions or energy-density functionals because the free-space nucleon force has strong short-range repulsion.
  • 1. Introduction: TDHF extends Hartree-Fock to dynamical processes, including nuclear vibrations and collisions, but unrestricted large-scale calculations became practical only with increased computing capacity.Earlier applications were restricted largely to the linearized small-amplitude regime.
  • 1. Introduction: Sky3D uses a fully three-dimensional Cartesian representation with no symmetry restrictions and includes the full Skyrme functional, including spin-orbit and most important time-odd terms.Earlier TDHF studies commonly imposed axial or reflection symmetries, limiting applications.
  • 1. Introduction: The paper aims to give broader public access to unrestricted 3D TDHF through a code with the numerical methods needed for efficient Skyrme simulations.The authors present the necessary computational ingredients in a comprehensive, compact manner.
  • 1. Introduction: Although modern Skyrme density functionals can contain fractional density powers unrelated to two- or three-body interactions, the paper retains the TDHF name for historical continuity.Formally, the code therefore solves TDDFT equations in such cases.

2. General purpose and structure

Sky3D solves static and time-dependent Hartree-Fock equations with Skyrme-force interactions in unrestricted three-dimensional geometry. Its energy functional, pairing treatment, and Hamiltonian construction support nuclear-structure and dynamical calculations with extensibility for additional physics.

  • Purpose and scope: Sky3D solves static and time-dependent Hartree-Fock equations for Skyrme-force interactions in general three-dimensional geometry without symmetry assumptions.The code targets nuclear structure, collective excitations, and nuclear reactions within mean-field theory.
  • Mean-field and pairing: The mean-field description uses single-particle wave functions with fractional occupation amplitudes, while BCS amplitudes are varied only statically for even-even nuclei and frozen during dynamics.The active single-particle space determines the occupation amplitudes used in the mean-field state.
  • Energy functional: The Skyrme energy functional combines kinetic, interaction, spin-orbit, Coulomb, pairing, and correlation-related contributions expressed through local densities and currents.The implementation includes density-dependent pairing options, while additional correlation corrections are generally ignored in dynamical simulations.
  • Energy functional: The implemented model omits tensor spin-orbit terms and spin-spin couplings, which may matter particularly for magnetic excitations and odd nuclei.The functional retains the minimal terms needed for Galilean invariance and basic conservation laws in TDHF calculations.
  • Hamiltonian and excitations: The single-particle Hamiltonian contains local, effective-mass, spin-orbit, and time-odd contributions derived by variation of the energy functional.For dynamical studies, the code can add a time-dependent local external field with user-controlled temporal profile, spatial operator, and strength.

1. A Gaussian of the form

The code uses Gaussian and related external pulses to control excitation bandwidth, while solving static mean-field and BCS equations iteratively before time evolution. TDHF evolution uses frozen occupations, unitary predictor-corrector propagation, and supports multi-fragment initial states.

  • 1. A Gaussian of the form: Broad envelopes improve frequency resolution around the driving frequency, whereas short pulses excite a broad frequency band.The infinitely short-pulse limit becomes an instantaneous phase boost.
  • 1. A Gaussian of the form: The instantaneous boost applies a phase factor to stationary wave functions and permits an undamped driving field for simpler spectral analysis.Its effect can be interpreted as an initial probability-flow velocity derived from the phase gradient.
  • 2.4.1. The coupled mean-field and BCS equations: The BCS active space is the set of states included in the calculation, and results depend slightly on its size.The authors recommend using about the number of single-nucleon states associated with the dynamical pairing space.
  • 2.4.2. Iterative solution: Static mean-field and BCS equations are solved iteratively by damped-gradient wave-function updates, BCS occupation updates, density recomputation, and mean-field reconstruction.Kinetic-energy damping uses an energy-scale regulator typically set to E0 = 100 MeV and a step size of δ = 0.1...0.8.
  • 2.4.2. Iterative solution: Convergence is assessed through single-particle energy variance, but vanishing variance may identify only a local isomeric minimum rather than the absolute minimum.Different initial configurations should be tested when the minimum is uncertain.
  • 2.4.2. Iterative solution: Deformed harmonic-oscillator wave functions initialize static iterations, and changing oscillator deformation can steer convergence toward prolate or oblate minima.Which shape is the absolute minimum depends on the nucleus.
  • 2.5.1. The time-dependent mean-field equations: TDHF keeps static occupation amplitudes frozen; this is considered legitimate for moderate excitations but insufficient for truly low-energy dynamics of a few MeV.The stated applications include heavy-ion collisions and giant resonances, while soft vibrations and fission require a fuller treatment.
  • 2.5.2. Time development algorithm: Time evolution is implemented with a predictor-corrector strategy that estimates the midtime Hamiltonian while preserving orthonormality and, without time-dependent external fields, total energy.The predictor first performs a trial step using the Hamiltonian at the initial time.

2.6. Observables

Sky3D outputs energy, density, multipole, deformation, and radius observables, while using Fourier-space representations for spatial derivatives on its periodic Cartesian grid. These choices support flexible analysis but impose explicit numerical and coordinate-frame conventions.

  • 2.6. Observables: The code outputs energy and density observables, with total energy identified as the key energy observable.Single-particle energies provide an additional route to evaluating total energy.
  • 2.6. Observables: Multipole moments characterize density distributions through center-of-mass, dipole, and quadrupole information.Center-of-mass quantities can be defined for proton, neutron, total, isoscalar, or isovector densities.
  • 2.6. Observables: Quadrupole tensors are diagonalized in the principal-axis frame, where only Qxx, Qyy, and Qzz remain subject to the trace condition Qxx + Qyy + Qzz = 0.This frame removes the coordinate rotation ambiguity of the quadrupole tensor.
  • 2.6. Observables: Root-mean-square radii are defined for proton, neutron, or total densities, while the isovector variant is not used.The total-density radius is primarily used for isoscalar moments.
  • 2.6. Observables: Dimensionless moments remove overall scale and provide nuclear-shape parameters, while triaxiality is physically restricted to the 0...60° sector.The code supplies multiple multipole variants for flexible analysis.
  • 2.6. Observables: The total-energy implementation augments the usual single-particle recipe with rearrangement energy required by the density dependence of the Skyrme force.The code computes energy both from the direct Skyrme expression and the augmented recipe, yielding numerically close but nonidentical values.
  • 2.7.5. Wave function storage: Wave functions and fields use an even-sized three-dimensional Cartesian grid, with equal spacing recommended in all directions and a typical spacing of 0.5–1.0 fm.The origin is centered in a cubic cell, preserving exact parity properties.
  • 2.7.3. Derivatives: Fourier transforms impose periodicity, and mth derivatives are computed by multiplying Fourier components by (ik_n)^m.The coordinate-space integration uses the trapezoidal rule with equal weights.

2.8. Initialization

Sky3D provides several initialization modes for static and dynamical calculations, ranging from harmonic-oscillator states to imported fragments and collision parameters. The initialization choice controls the starting configuration and can affect convergence behavior.

  • 2.8. Initialization: Initialization is selected through the input variable nof, with three principal modes available.The modes cover harmonic-oscillator, imported single-fragment, and collision-specific initialization.
  • 2.8. Initialization: The harmonic-oscillator mode generates static initial wave functions from directional radii and is recommended with unequal radii for nonspherical nuclei.Its deformation is influenced more by oscillator-state occupations than by the radius parameters, and the authors note defects in this simple initialization.
  • 2.8. Initialization: The single-fragment mode reads in static wave functions for studying collective nuclear vibrations with TDHF.This mode is useful when a converged static state is available as the dynamical starting point.
  • 2.8. Initialization: The collision mode computes initial velocities from center-of-mass energy ecm and impact parameter b rather than accepting velocities directly.This provides collision kinematics through two physical input parameters.

3. User initialization: a user-supplied routine user

Sky3D supports user-defined initial wave functions and restart calculations, while numerical accuracy depends on grid, time-stepping, and propagation-order choices.

  • User initialization: User initialization can populate wave functions arbitrarily, provided index ranges are correct and the wave-function array is properly filled.Initial Gaussians arranged in geometric patterns are one example.
  • Restarting calculations: Restart files permit continuation after interruptions, with a larger new grid allowed when grid spacings agree.Restart data are written periodically according to the configured restart interval.
  • Grid accuracy: Grid results remain high quality through ∆x = 0.75 fm, while ∆x = 1 fm remains acceptable for many applications but finer meshes may be needed for high excitations and fast collisions.The maximum representable kinetic energy is about 200 MeV at ∆x = 1 fm.
  • Grid accuracy: Dynamical quadrupole oscillations show practically no difference between ∆x = 0.75 fm and ∆x = 1 fm at fixed box size.The text recommends ∆x = 1 fm for many dynamical applications because they are less demanding and more costly.
  • Time stepping: Energy conservation and stability depend on time-step parameters, with smaller dt reducing fluctuations and unstable Taylor orders potentially causing catastrophic failure.For the tested case, m = 4 remained stable, whereas m = 6 failed suddenly; these findings are not generic.
  • Time stepping: A proper time step and Taylor order must be checked for each test case before larger surveys.The acceptable choice depends on the force, excitation, and intended precision.

3. Code structure

Sky3D is organized as a modular Fortran code whose components separate initialization, static and dynamic evolution, physics calculations, analysis, I/O, and parallel execution.

  • Overall organization: The modular design encapsulates functionality and is intended to ease modification, with modules reading operating parameters through NAMELISTs.The programming style keeps code close to the physical equations being solved.
  • Control flow: The main program initializes the calculation and dispatches to static or dynamic Hartree-Fock routines.Initialization may use harmonic-oscillator states, static solutions, fragments, or user-defined wave functions.
  • Physics modules: Core physics modules calculate Skyrme mean fields, Hamiltonian actions, Coulomb fields, energies, external excitations, pairing, moments, and densities.These modules divide the principal physical operations among dedicated components.
  • Supporting modules: Supporting modules define parameters, forces, grids, wave functions, derivatives, Fourier transforms, input/output, and basic operations.The Forces module also constructs selected Skyrme and pairing interactions from input.
  • Application modules: Separate Static, Dynamic, Densities, Fragments, and User modules handle evolution, output, density construction, fragment setup, and customizable initialization.The User module provides a template for more complicated setups.
  • Parallel support: Parallel execution is supported through MPI and OpenMP variants, with Fourier transforms supplied through FFTW3.The parallel modules provide MPI message passing or sequential/OpenMP replacements.

4. Parallelization

Sky3D parallelizes primarily over single-particle wave functions, supporting OpenMP and MPI while requiring communication of partial densities and selected observables.

  • Parallel model: OpenMP and MPI parallelize the application of the time-development operator or gradient iteration across wave functions.Mean fields and densities are also formed from parallel sums over wave-function contributions.
  • Parallel model: Static calculations support OpenMP but not MPI because orthogonalization is difficult to distribute across memory-separated nodes.The text identifies distributed orthogonalization as future work.
  • Execution configuration: MPI and OpenMP can be combined on nodes with multiple processors, while thread counts are set externally through OMP_THREADS.MPI builds require system-specific compiler and linker commands.
  • OpenMP implementation: OpenMP density accumulation uses reduction and combines propagation with density updates in a parallel loop.Wave functions at the half step are discarded after contributing to densities, avoiding storage of the full half-step set.
  • MPI communication: MPI nodes hold different subsets of wave functions and must collect and broadcast summed densities after time-step subcycles.Partial densities are combined in the collect_densities operation.
  • MPI communication: Single-particle properties are computed locally and then collected across nodes, while output is restricted to one node.Wave functions are written as separate files per node to avoid gathering them on one node.

5. Input description

Sky3D is configured through NAMELIST input covering files, forces, initialization, grids, static iteration, dynamics, external excitations, and fragment collisions.

  • Input organization: NAMELIST input is read from standard input, and unused namelists may be omitted or left ignored.Many variables have defaults, and input/output can be redirected through shell files.
  • Files and forces: File and force settings select wave-function, convergence, moment, spin, energy, and external-field files, plus predefined or user-defined Skyrme and pairing interactions.Pairing options include NONE, VDI, and DDDI.
  • Dynamic calculations: Dynamic settings specify the number and size of time steps, exponential-expansion order, initialization mode, and optional external perturbations.An unsuitable dt and mxpact combination can make norm and energy drift or diverge.
  • Grid configuration: Grid settings define even Cartesian dimensions, spacings, coordinate placement, and periodic or isolated boundaries.If only dx is supplied, all three spacings are made equal.
  • External excitation: External-field settings define pulse shape, amplitude, isospin character, timing, frequency, radius, and width for collective excitation.Quadrupole excitation amplitude is specified through amplq0, while isoext distinguishes equal or opposite proton-neutron driving.

6. Output description

Sky3D produces binary and text outputs for monitoring calculations, restarting runs, visualization, and downstream analysis. Static and dynamic outputs report convergence, conservation, deformation, single-particle, and fragment-related quantities at configurable intervals.

  • Output files: Binary files selectively store bulky densities, currents, or wave functions for visualization, analysis, restarting, and fragment initialization.Wave functions are retained only from the latest saved interval, while MPI runs distribute them across processor-specific files with a configuration-independent header.
  • Output files: Text *.res files provide time- or iteration-indexed quantities with comment headers, enabling immediate plotting and scripting-based extraction.Output names are configurable, and these relatively small files are always written.
  • Static calculations: Static output monitors convergence through energy, energy fluctuations, radius, and deformation, with efluct values more diagnostic than total-energy changes.The total energy can remain constant while wave functions continue changing considerably.
  • Dynamic calculations: Dynamic monitoring tracks particle-number stability, total-energy conservation, kinetic and collective energies, radii, center-of-mass motion, multipole observables, and fragment separation.The dynamic output also reports full quadrupole-tensor information, deformation parameters, and the separation derivative at every time step.
  • Detailed reports: Detailed static and dynamic reports echo inputs and summarize energies, density plots, single-particle states, integrated observables, fragment initialization, and external fields.Detailed output is repeated every mprint steps, while intermediate iterations receive only a progress line.

7. Utilities

Sky3D includes utilities for inspecting binary outputs, computing derived quantities, extracting planar density cuts, comparing Slater determinants, and converting data for visualization. Several programs are intentionally provided as adaptable analysis templates.

  • Inspection and analysis: fileinfo reads *.tdd or wave-function files and prints information from their headers.It is compiled as a standalone Fortran utility.
  • Inspection and analysis: Inertia demonstrates reading density data, distinguishing total from isospin-separated storage, and calculating the nine components of the inertia tensor.The program is explicitly presented as a model for similar analyses.
  • Visualization: A density-cut utility generates xy, xz, and yz planar files by averaging neighboring planes at the origin of the third coordinate.The resulting files are formatted for two-dimensional plotting with gnuplot.
  • Wave-function analysis: The overlap utility calculates neutron and proton Slater-determinant overlaps from compatible wave-function files and reports summary information including center-of-mass separation.The calculation proceeds through matrices of single-particle overlaps and determinants.
  • Visualization: Tdhf2Silo converts sets of *.tdd files into Silo datasets suitable for visualization and movie generation with VisIt.The conversion code is flexible and can be adapted to other dataset-handling libraries.

8. Running the code

Sky3D is built through sequential, OpenMP, or MPI Makefiles and depends on external numerical libraries. The documentation supplies input namelists and three test cases covering static structure, driven dynamics, and a collision.

  • Compilation: The supplied Makefiles produce sequential, OpenMP, and MPI executables, but compiler, library, and MPI settings may require local adaptation.MPI execution in particular depends on installation-specific compilation and launch commands.
  • Performance: Compiler and linker options are not optimized automatically, so speed tests are recommended before major calculations.This is a practical performance boundary rather than a change to the physics model.
  • External libraries: The code uses LAPACK, FFTW3, and included LINPACK routines for numerical linear algebra, Fourier transforms, and determinant calculations.FFTW3 may be installed, compiled locally, or replaced, but replacement requires accounting for planning and normalization differences.
  • Input: Static input uses files, force, main, grid, and static namelists, with fragments or user input added when required.Dynamic input replaces static with dynamic and may additionally include extern for external-field excitation.
  • Testing: Three test cases exercise a static 16O ground-state calculation, an externally excited 16O giant resonance, and a deep-inelastic collision of two 16O nuclei.The test directory provides further descriptions and expected features to inspect in the results.

9. Caveats concerning the code

The code’s applicability and reliability depend on parallel-runtime settings, initialization quality, and the calculation regime. Static calculations are especially affected by omitted symmetry restrictions and initialization or orientation issues.

  • Parallel execution: OpenMP runs require thread and stack-size settings because parallel density-field storage can demand substantial memory.The documented storage estimate is nx*ny*nz*24*16 bytes for 24 three-dimensional COMPLEX(8) fields.
  • Parallel execution: MPI setup is installation-dependent and generally requires modifying the Makefile, compilation commands, and execution procedure.The documentation directs users to local MPI documentation for these settings.
  • Static calculations: The code’s omitted symmetry restrictions benefit time-dependent applications but can complicate static calculations.The static component is described as less highly developed because the code is primarily intended for time-dependent work.
  • Static calculations: In static calculations, spin alignment can mix within degenerate even-even levels, the center of mass can drift, and heavy nuclei may rotate during iteration.Spin-operator diagonalization, center-of-mass-relative observables, constraints, or another static code are suggested remedies depending on the issue.
  • Initialization: Harmonic-oscillator initialization can be deficient for heavier nuclei, motivating Nilsson alternatives or imported wave functions followed by stabilizing iterations.Interpolation from axial or symmetry-restricting codes may require additional static iterations.
  • Accuracy: Accuracy improvements may require better initial configurations and dynamic adjustment of boost energies when deformed fragments make Rutherford estimates unreliable.The recommended procedure is to inspect relative-distance changes and correct boosts to match the relative velocity.

10. Modifying the code

Sky3D supports extending its force database, density-functional terms, and analysis outputs through several modification paths. The appropriate route depends on whether a change is temporary, permanent, density-based, or requires direct access to wave functions.

  • Force parametrizations: Temporary force parametrizations can be supplied directly through NAMELIST forces, while permanent additions require updating forces.data and nforce.The direct-input route is intended for parameter variations and other changes not expected to remain in the code database.
  • New physical terms: Adding a new density-functional term generally requires changes to force parameters, the single-particle Hamiltonian, densities or currents, wave-function operations, energy evaluation, and output routines.The code’s modification sequence includes extending the Force type, calculating the term’s action on wave functions, adding its energy contribution, and writing new fields.
  • Additional observables: Additional observables depending only on densities and mean-field components can be computed from *.tdd files after modifying density output when necessary.The tensor of inertia utility is given as an example of analysis based on *.tdd files.
  • Additional observables: Observables requiring wave functions can use user routines during static or dynamic loops, or separate wave-function files when repeated analysis would make in-loop routines impractical.The second approach follows the pattern used when write_wavefunctions is called and is better suited to lengthy calculations requiring multiple analysis revisions.
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