Solution of the Skyrme-Hartree-Fock-Bogolyubov equations in the Cartesian deformed harmonic-oscillator basis. (VII) HFODD (v2.49t): a new version of the program

Typical running time

• The program is obtainable from the CPC Program Library and requires BLAS and LAPACK routines or equivalent linear-algebra subroutines.Source: NEW VERSION PROGRAM SUMMARY · p. 2, Typical running time · pp. 3–4

NEW VERSION PROGRAM SUMMARY

• Version v2.49t supersedes the previous version and is written in FORTRAN-90; the distributed program has 104,666 lines, including comments and separators.The code is listed as vectorized and parallelized.Source: NEW VERSION PROGRAM SUMMARY · p. 2
• Neutron and proton wave functions are expanded in a Cartesian harmonic-oscillator basis and determined by iterative diagonalization of density-dependent mean-field Hamiltonians or Routhians.Constraints target configurations, deformations, or angular momenta; the method also uses Skyrme and zero-range pairing interactions.Source: NEW VERSION PROGRAM SUMMARY · pp. 2–3

Summary of revisions

• The revision adds isospin mixing and projection, finite-temperature methods, the Lipkin translational correction, and shell-correction calculations.It also corrects errors in the previous published version.Source: Summary of revisions · p. 3
• New numerical tools include the two-basis HFB method, ALM and linear constraints, an HFBTHO interface, and matrix-element mixing.Source: Summary of revisions · p. 3
• MPI, scalable input, OpenMP, and ScaLAPACK parallel diagonalization support parallel calculations, including simplex-breaking HFB cases.Source: Summary of revisions · p. 3

• HFODD v2.49t adds isospin mixing and projection, finite-temperature HF+BCS and HFB, and a hybrid MPI/OpenMP model for large-scale calculations.Source: §1 · p. 4
• The paper's information supplements earlier HFODD publications, whose guidance remains valid unless the new write-up states otherwise -.Source: §1 · p. 4
• In serial mode, the input structure remains unchanged from earlier versions, while parallel mode requires hfodd.d and hfodd_mpiio.d.The parallel input files have distinct roles: the standard keyword structure and processor-dependent data, respectively.Source: §1 · p. 4

2.1.1 Isospin Mixing and Projection

• Because mean-field approximations can spuriously break isospin, HFODD projects unrestricted Slater determinants onto good isospin and rediagonalizes the Hamiltonian, with joint angular-momentum and isospin mixing available.The retained good-isospin basis is controlled by εT; εT = 10^-10 sets T ≤ |Tz| + 5 in practice.Source: §2.1.1 · pp. 5, 8

2.1.2 Finite-temperature Formalism

• Finite-temperature HFB and HF+BCS update thermal occupations while solving for λ, including ∂fµ/∂λ in particle-number derivatives to support convergence in the unpaired regime.With zero-range pairing, stability issues can occur for low cutoffs near the phase transition.Source: §2.1.2 · pp. 9–10

2.1.3 Lipkin Translational Energy Correction

• The Lipkin method estimates the rest-frame linear-momentum projected energy without explicit projection; its faster GOA estimate of k avoids calculating two-body momentum kernels.The projected energy depends parametrically on k, without variation with respect to k or a resulting contribution to the HF potentials.Source: §2.1.3 · pp. 10–11

2.1.4 Shell Correction

• HFODD computes shell corrections from occupied Hartree-Fock single-particle energies relative to Strutinsky-smoothed energies.A second prescription subtracts the smooth independent-gas contribution to remove spurious positive-energy terms near the dripline; proton calculations include the Coulomb potential.Source: §2.1.4 · p. 12

2.2.1 Two-basis Method for HFB Calculations

• With no conserved symmetry, the two-basis method is up to 30% faster; requesting only below-cutoff eigenvectors in standard HFB recovers almost half that gain.The method cuts off in single-particle space, reducing HFB dimension and avoiding formal drawbacks of quasiparticle-space cutoffs; results can still differ by up to 200 keV in total energy and 1.5 MeV in pairing energy despite equal quasiparticle counts.Source: §2.2.1 · pp. 13–14, Figure 3 · p. 13

2.2.2 Augmented Lagrangian Method

• The Augmented Lagrangian Method addresses inaccurate quadratic-penalty constraints by combining linear and quadratic penalties and adjusting Lagrange parameters during iterations.It adds almost no computational overhead, is described as robust, and gives the requested solution precisely.Source: §2.2.2 · pp. 14–15

2.2.3 Linear Constraints Based on the RPA Matrix

• The RPA-matrix-based linear-constraint method uses estimated cranking-approximation QRPA information to update Lagrange parameters, reducing iterations and improving constraint precision.Its computational overhead can be noticeable, but the authors report it is compensated by fewer iterations and near-perfect solution precision.Source: §2.2.3 · p. 15

2.2.4 Interface with hfbtho

• The HFBTHO interface transforms axial, parity-symmetric solutions into HFODD; if those symmetries persist, HFODD can stop after the transformation.For axial- and parity-symmetric solutions, restart precision ranges from 1 eV to 1 keV; HFODD continues iterating if either symmetry is broken.Source: §2.2.4 · pp. 15–17

2.2.5 Mixing of Matrix Elements of the HFB Matrix

• Matrix-element mixing lowers typical static-HFB Broyden memory from 302 MB to 168 MB, but numerical noise calls for periodic memory resets to obtain ground-state convergence.The authors report decent results when Broyden memory is erased every four iterations.Source: §2.2.5 · pp. 18–19

2.3.1 Distributed Memory Parallelism

• MPI divides work into master groups for separate nuclear configurations and optional within-group cores; most applications need only one core per group.The processor grid is divided into the two layers at program startup.Source: §2.3.1 · p. 20

2.3.2 Scalable Input Routines

• One core reads shared input and broadcasts it through linked lists, avoiding concurrent disk access while retaining flexible keyword data and compatibility across releases.The input is a few kB at most, so the broadcast is expected to add little communication overhead.Source: §2.3.2 · pp. 21–22

2.3.3 Shared Memory Parallelism

• For 180Hg (26 HO shells, 1,140 states, α20 = −0.32) using standard BLAS, three OpenMP-affected routines account for P = 0.37 of runtime; measured acceleration aligns with the empirical Amdahl model.Source: §2.3.3 · p. 23

2.3.4 Parallelization of Diagonalization Routines

• The ScaLAPACK tests address memory pressure from large symmetry-breaking calculations, since a 20-shell problem exceeds the 2 GB available per core on the cited Cray systems.Fission calculations may require 26–31 shells.Source: §2.3.4 · pp. 24–25, Figure 11 · p. 24
• Single-core ZHEEVR can be faster than multi-core ScaLAPACK diagonalization, which is limited to PZHEEV or PZHEEVD in these tests.The comparison used three HFB iterations for a symmetry-breaking 152Dy case with 14 shells and 680 basis states.Source: Figure 12 · p. 24, §2.3.4 · p. 25

2.4.1 Coulomb energies

• For ICOUDI=1/ICOUEX=2 or ICOUDI=2/ICOUEX=1, the direct or exchange Coulomb energy, respectively, had been set to zero.These parameter combinations are identified as the affected cases.Source: §2.4.1 · p. 25

2.4.2 Skyrme parameters

• The predefined SLY4, SLY5, and SLY7 parameters now follow Ref., while the previous values remain available under lowercase acronyms.The earlier SLY5 and SLY7 values came from Ref., whereas SLY4 used unrounded values communicated by the force authors.Source: §2.4.2 · p. 25

2.4.3 Quasiparticle blocking

• Errors in time reversal and blocking could select a different blocked quasiparticle than requested, although the calculated results still converged.In some situations the two errors compensated one another.Source: §2.4.3 · pp. 25–26

2.4.4 Yukawa mean fields

• For Yukawa calculations with IFCONT=0, the first iteration had incorrectly set the mean fields to zero; later iterations computed them correctly.The error affected continuation calculations from disk, while converged results remained correct.Source: §2.4.4 · p. 26

2.4.5 The Broyden method

• In dobroy, ALPHAM was reset to 1−SLOWEV, making the input parameter ineffective.Source: §2.4.5 · p. 26

3.1 Physics

• Physics keywords control isospin projection, temperature, shell corrections, translational-mass treatment, and Skyrme functional parameterizations, with isospin projection restricted to HF calculations and specified settings.Finite-temperature HFB or HF+BCS calculations run when TEMP>0; shell-correction options distinguish the traditional correction from one removing spurious positive-energy contributions.Source: §3.1 · pp. 26–27

3.2 Numerical Methods

• Numerical-method keywords select the two-basis HFB method, quasiparticle-energy cutoff, ALM or linear constraints, RPA-based multiplier updates, and matrix-element mixing.The two-basis option requires HFB and no simplex or parity symmetry; RPA-based updates are limited to HFB with conserved simplex.Source: §3.2 · pp. 28–29

3.3.1 List of active keywords in hfodd.d

• Parallel mode supports only a subset of HFODD keywords; users needing an unsupported option can modify PREDEF before compilation.The listed active options cover iterations, physics features, symmetries, pairing, basis, and restart settings.Source: §3.3.1 · pp. 29–30

3.3.2 Structure of hfodd mpiio.d

• The parallel input file can define calculation grids over proton and neutron numbers, deformation constraints, oscillator shells, basis deformations, and oscillator frequencies.MPIBAS=1 includes shell, deformation, and frequency dimensions in the grid.Source: §3.3.2 · pp. 30–31

4 Output Files

• The isospin-mixed eigenstates output lists energies and good-isospin basis expansion characteristics, including the expansion coefficient’s norm and real and imaginary parts.The listed states are within the εT cutoff.Source: §4 · p. 32
• Finite-temperature iteration output includes energy, stability, quadrupole moment, γ angle, entropy, neutron and proton Fermi levels, and pairing energies.These values, together with temperature, allow the grand canonical potential to be deduced at each iteration.Source: §4 · p. 32
• The HFBTHO restart output identifies the two-step run and reports functional nuclear-matter properties and neutron and proton shell corrections.The shell-correction section also describes the correction type and smoothing procedure.Source: §4 · p. 32

5 Fortran Source Files

• The cited source list identifies hf249t.f as the main source, functional.f90 as defining energy functionals, shell.f as shell correction, hfbtho.f90 as HFBTHO, interface.f90 as its interface, and mpiio.f90 as parallel-mode I/O.Source: §5 · p. 32

5.1 Standard Libraries

• BLAS and LAPACK are required; the older NAGLIB interface remains available, but BLAS and LAPACK are recommended for peak performance on most current computers.The code’s source files also include the ScaLAPACK module for routine HFBSIZ.Source: §5.1 · p. 33

5.2.1 Basic MPI

• MPI is required for multi-core calculations; the version was tested with MPICH-1, MPICH-2, and Open MPI.Parallel compilation is enabled by setting USE MPI to 1 in the project Makefile.Source: §5.2.1 · p. 33

5.2.2 Hybrid OpenMP/MPI Mode

• OpenMP can run alone or with MPI; the documented hybrid example uses 12 MPI processes and 3 threads each, allowing up to 36 active cores.The example reserves two cores per socket across six processors, leaving other cores available for threading.Source: §5.2.2 · p. 33

5.2.3 Scalapack

• ScaLAPACK requires MPI and a multi-core grid, so it cannot run in serial mode; the core count is the process-grid size multiplied by the number of HFB calculations.The code’s command-line syntax is unchanged from the basic MPI or hybrid model.Source: §5.2.3 · p. 34

A Test Run Input

• The official release example demonstrates isospin mixing and projection with neutron and proton counts both set to 20.The input labels these counts as IN_FIX and IZ_FIX under NUCLIDE.Source: A Test Run Input · p. 35
• The input provides a SKYRME-SET interaction block and separate pairing and symmetry blocks.The passages identify the blocks but do not state the selected pairing or symmetry settings.Source: A Test Run Input · pp. 35–36
• The run sequence includes restart, calculation, iteration, another calculation, and termination commands.The input also includes a KERNELFILE entry and SAVEKERNEL setting.Source: A Test Run Input · pp. 37–38

B Test Run Output

• The displayed build is a single-core HFODD v2.49t run using a deformed Cartesian harmonic-oscillator basis without symmetry planes or time-reversal symmetry.Source: B Test Run Output · p. 39
• The build reports NDBASE=680, NDSTAT=680, and oscillator limits of 40 for each Cartesian direction.Source: B Test Run Output · p. 40
• Broyden mixing is on and is triggered when stability is below 10000.000 MeV.The output lists an initial slowing factor of 0.50 and a post-trigger factor of 0.70.Source: B Test Run Output · p. 40
• The run uses 8 Gauss-Legendre knots for isospin projection.Source: B Test Run Output · p. 40
• 0.529748%, 1.193817%, 1.491841%, and 0.829077% are the Coulomb mixing in states 1–4, whose dominant isospins are T=0–3.The corresponding dominant amplitude-squared values are 0.9947025220, 0.9880618341, 0.9850815887, and 0.9917092280.Source: B Test Run Output · p. 41