HP -- A code for the calculation of Hubbard parameters using density-functional perturbation theory

Iurii Timrov; Nicola Marzari; Matteo Cococcioni

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HP -- A code for the calculation of Hubbard parameters using density-functional perturbation theory

Iurii Timrov, Nicola Marzari, Matteo Cococcioni

arXiv:2203.15684v2Mar 29, 2022cond-mat.mtrl-sci

TL;DR

Localized-electron self-interaction errors and non-universal, method-dependent Hubbard parameters complicate predictive DFT+U calculations. The paper introduces HP, an open-source Quantum ESPRESSO implementation that uses DFPT and monochromatic unit-cell perturbations to calculate U and V, and demonstrates accurate geometry and voltage predictions for phospho-olivine. Its results also show that parameters should be recomputed as Li concentration changes, while projector choice remains an important limitation.

Problem
Hubbard parameters are not known a priori, are often fitted semi-empirically, and depend on the Hubbard manifold and computational setup.
Method
HP uses density-functional perturbation theory with monochromatic perturbations in primitive unit cells to calculate Hubbard parameters within Quantum ESPRESSO.
Results
DFT+U+V gives the best agreement with experimental Li intercalation voltages for Li_xMn_1/2Fe_1/2PO_4, outperforming HSE06 for this material class.
Takeaways & Limitations
For phospho-olivine Li_xMn_1/2Fe_1/2PO_4, U and V should be recomputed at each Li concentration rather than treated as global x-independent parameters.
Takeaways & Limitations
Computed Hubbard parameters depend strongly on the chosen projector functions, which must match between parameter calculations and subsequent DFT+U(+V) calculations.

Abstract

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We introduce HP, an implementation of density-functional perturbation theory, designed to compute Hubbard parameters (on-site $U$ and inter-site $V$) in the framework of DFT+$U$ and DFT+$U$+$V$. The code does not require the use of computationally expensive supercells of the traditional linear-response approach; instead, unit cells are used with monochromatic perturbations that significantly reduce the computational cost of determining Hubbard parameters. HP is an open-source software distributed under the terms of the GPL as a component of Quantum ESPRESSO. As with other components, HP is optimized to run on a variety of different platforms, from laptops to massively parallel architectures, using native mathematical libraries (LAPACK and FFTW) and a hierarchy of custom parallelization layers built on top of MPI. The effectiveness of the code is showcased by computing Hubbard parameters self-consistently for the phospho-olivine Li$_x$Mn$_{1/2}$Fe$_{1/2}$PO$_4$ ($x=0, 1/2, 1$) and by highlighting the accuracy of predictions of the geometry and Li intercalation voltages.

PROGRAM SUMMARY

HP calculates Hubbard interaction parameters for DFT+U and DFT+U+V using density-functional perturbation theory and reciprocal-space response calculations. Its unit-cell formulation reconstructs localized responses from monochromatic perturbations, reducing computational cost.

PROGRAM SUMMARY: HP computes Hubbard interaction parameters for DFT+U and DFT+U+V.The parameters are obtained from inverse response matrices to localized atomic-occupation perturbations.
PROGRAM SUMMARY: Monochromatic perturbations in a primitive unit cell reconstruct localized-perturbation responses without appropriately sized supercell calculations.The response matrices are obtained through first-order density-functional perturbation theory in reciprocal space.
PROGRAM SUMMARY: A self-consistent static Sternheimer equation computes response matrices without calculating virtual states.This implementation supports the density-functional perturbation-theory evaluation of Hubbard parameters.
PROGRAM SUMMARY: HP is a Fortran 95 component of Quantum ESPRESSO distributed under the GNU General Public License V 2.0.It uses BLAS, LAPACK, FFTW, and MPI through the Quantum ESPRESSO distribution.

1. Introduction

DFT approximations can substantially overdelocalize localized d- and f-electrons, while Hubbard parameters remain difficult to determine predictively and depend on methodological choices. The paper introduces HP as an open-source DFPT implementation for calculating these parameters within Quantum ESPRESSO.

1. Introduction: LDA and GGA suffer self-interaction errors that can overdelocalize strongly localized d- and f-electrons.These errors can produce quantitative and qualitative failures in systems dominated by localized electrons.
1. Introduction: DFT+U alleviates self-interaction errors by imposing piecewise linearity in energy versus atomic occupations.The correction acts selectively on localized d- or f-type manifolds through projectors.
1. Introduction: Hubbard U and V values are not known a priori, and semi-empirical fitting limits the method’s ab initio and predictive character.The parameters also depend on how the Hubbard manifold is defined and therefore are not universal for an element or material.
1. Introduction: Comparisons among cRPA, LR-cDFT/DFPT, and ACBN0 are complicated by different definitions of U and different Hubbard projectors.The paper argues that U values should be compared only under the same computational setup.
1. Introduction: HP implements DFPT for calculating Hubbard parameters as an open-source component of Quantum ESPRESSO.The code is designed for plane-wave, pseudopotential, and periodic-boundary-condition calculations.

2. Theory

DFT+U+V extends Hubbard corrections beyond on-site interactions by including inter-site couplings, whose values depend strongly on the chosen projector functions. HP computes these parameters with DFPT using monochromatic perturbations, unit cells, and extensions for metallic systems, ultrasoft or PAW calculations, and symmetry.

Hubbard-corrected density-functional theory: DFT+U+V adds inter-site interactions between an atom and neighboring ligands to the on-site correction of DFT+U.Setting V=0 recovers standard DFT+U.
Hubbard-corrected density-functional theory: On-site U favors localization, whereas inter-site V favors hybridized states on neighboring atoms, making both parameters important for covalent systems.The extended correction can therefore represent hybridized states that on-site corrections alone may suppress.
Hubbard projectors: The work uses Löwdin-orthogonalized atomic orbitals to represent inter-site hybridization and avoid counting Hubbard corrections twice in interstitial regions.All atomic states are orthogonalized, not only states belonging to the selected Hubbard manifolds.
Hubbard projectors: Hubbard parameters are defined through localized projectors, so their values should be discussed only after fixing the projector functions.Different projector choices can change computed values substantially; for β-MnO2, U differs by about 1−2 eV and V by about 0.3 eV between nonorthogonalized and orthogonalized atomic orbitals.
Hubbard parameters from DFPT: DFPT decomposes occupation responses into independently computable wavevector-specific contributions, enabling Hubbard-parameter calculations with the primitive unit cell instead of large supercells.This reformulation improves computational-cost scaling relative to isolated perturbations requiring converged supercells.
Extensions of the DFPT formalism: The formalism supports metallic ground states, ultrasoft pseudopotentials, PAW, and explicit crystal symmetry, with symmetry reducing calculations through wavevector-dependent irreducible wedges.Finite-wavelength perturbations lower crystal symmetry, so ground-state quantities are recomputed on an extended irreducible wedge determined by the small group of q.

3. Description of software components

HP integrates density-functional perturbation theory into Quantum ESPRESSO to calculate Hubbard parameters from linear-response calculations. Its workflow uses ground-state data, atom- and q-resolved perturbations, response-matrix construction, symmetry reconstruction, and inexpensive postprocessing.

Software components: HP is a tightly integrated Quantum ESPRESSO module that uses shared PW, Modules, LR Modules, and domain-specific mathematical libraries.The code resides in a self-contained HP directory within the Quantum ESPRESSO source tree.
Workflow: A preceding ground-state DFT or DFT+U(+V) calculation supplies the unperturbed quantities used as input for HP linear-response calculations.The ground-state calculation is performed with pw.x, which writes the required quantities to disk.
Linear-response calculation: The total number of linear-response calculations is NpertNq, where Npert is the number of perturbed Hubbard atoms and Nq is the q-grid size.HP computes U for DFT+U and U and V for DFT+U+V.
Linear-response calculation: Each perturbed Hubbard atom requires Nq independent q-specific Sternheimer equations, enabling parallelization over q points because different wavelengths do not couple in the linear regime.The q-grid contains Nq = nq1 × nq2 × nq3 points.
Response matrices: Responses from perturbations are summed to construct columns of the self-consistent and bare response matrices χ and χ0.The self-consistent response is assembled after convergence, whereas the bare response is evaluated after the first self-consistent iteration.
Postprocessing: HP reconstructs missing response-matrix columns using system symmetry, then obtains Hubbard parameters through inexpensive postprocessing with Eqs. (8) and (9).The postprocessing cost is negligible compared with the linear-response calculation.

4. Software installation, parallelization, and testing

HP is distributed within Quantum ESPRESSO and supports installation, hierarchical parallelization, and automated testing across serial and high-performance computing environments. Its parallel workflow distributes work over Hubbard atoms, q points, k-point pools, and plane waves.

Software installation: HP is distributed as source code inside the Quantum ESPRESSO tree, with Git-based version control and installation procedures shared with the wider distribution.GNU autoconf is used by Quantum ESPRESSO, and the HP repository resides within its source tree.
Software installation: The build process configures the environment, compiles pw.x, and then compiles hp.x, placing executable links in the bin/ directory.CMake can alternatively be used instead of ./configure.
Parallelization: HP uses MPI and hierarchical data and computation distribution to run from laptops to massively parallel architectures.The implementation uses standard Message Passing Interface libraries and parallelizes FFT-related work among other tasks.
Parallelization: Parallelization proceeds hierarchically over perturbed Hubbard atoms, q points, k-point pools, and plane waves.The corresponding controls include perturb only atom, start q, last q, and the number of k-point pools Npool.
Testing: The test suite runs HP automatically in serial and parallel configurations and compares Hubbard parameters against reference values.Tests use multiple compilers and MPI libraries across different high-performance computing environments.
Parallelization: Figure 1 presents the parallelization hierarchy while allowing k- and q-point work to be split in different ways according to available computational resources.The figure labels Nk, Nq, Npool, NPW, and Ipert and illustrates the roles of atom, q-point, k-point, and plane-wave parallelization.

5. Benchmarking

The HP workflow computes self-consistent Hubbard parameters for phospho-olivine across lithiation states and uses them to assess structural and voltage predictions. Including inter-site V produces clearer parameter trends and improves agreement with experimental geometry and voltages.

Hubbard-parameter workflow: HP computes Fe(3d) and Mn(3d) on-site U and Fe(3d)–O(2p) and Mn(3d)–O(2p) inter-site V for x = 0, 1/2, and 1.The parameters are obtained self-consistently within DFT+U and DFT+U+V after structural optimization at each concentration.
Ground-state calculation: Magnetic-insulator calculations require a two-step ground-state procedure before HP perturbations are applied.The first calculation uses smearing and the second restarts with fixed occupations to avoid a diverging metallic term.
Hubbard-parameter trends: DFT+U+V yields clearer lithiation trends: Mn(3d) U decreases by 1.46 eV from x = 0 to x = 1/2, while Fe(3d) U remains essentially constant.From x = 1/2 to x = 1, Fe(3d) U decreases by 0.16 eV and Mn(3d) U decreases by 0.23 eV.
Hubbard-parameter trends: Inter-site V decreases on average from x = 0 to x = 1 as Li insertion increases cell volume and Mn–O bond lengths.The paper relates larger interatomic bond lengths to smaller inter-site Hubbard V interactions.
Geometry: At x = 1, DFT+U+V gives the closest agreement with experimental lattice parameters, although the effect of V on the crystal structure is small.DFT underestimates the lattice parameters and volume, whereas DFT+U and DFT+U+V slightly overestimate them.
Li intercalation voltages: DFT+U+V gives the best agreement with experimental Li intercalation voltages, whereas DFT underestimates and DFT+U overestimates them.The experimental plateaus are 4.1 V for 0 < x < 1/2 and 3.5 V for 1/2 < x < 1; the paper reports DFT+U+V as more accurate than HSE06 for these voltages.

6. Conclusions

HP implements DFPT-based Hubbard-parameter calculations that reduce computational cost while supporting efficient, automated use. Self-consistent tests on Li_xMn_1/2Fe_1/2PO_4 show that U and V depend on lithium concentration and yield geometries and intercalation voltages agreeing well with experiment.

6. Conclusions: DFPT reduces HP’s computational cost by replacing localized perturbation responses with independently calculated monochromatic perturbations in the primitive cell.The approach also avoids calculating empty electronic states.
6. Conclusions: HP supports efficient high-performance and high-throughput calculations through multiple parallelization levels and automation.It can be used with high-performance computers and workflows such as AiiDA.
6. Conclusions: Self-consistent calculations for Li_xMn_1/2Fe_1/2PO_4 at x = 0, 1/2, and 1 show that Hubbard parameters change with lithium concentration.The authors therefore state that U and V should be recomputed at each concentration rather than treated as global x-independent parameters.
6. Conclusions: The predicted crystal geometry and intercalation voltages are in very good agreement with experimental data.This agreement supports recomputing Hubbard parameters for each lithium concentration.
6. Conclusions: HP provides a documented, open-source framework that can be extended with additional projector types, functionals, spin treatments, and GPU support.The code is hosted in a community-accessible repository for testing and contributions.

Appendix A. Sample input files

Appendix A provides sample input files for DFT+U, DFT+U+V, and Hubbard-parameter calculations using pw.x and hp.x. The examples specify representative system, numerical, and HP response-calculation settings.

Appendix A. Sample input files: The first sample input file is for a DFT+U calculation using pw.x.It identifies the calculation as a self-consistent-field run for the olivine system.
Appendix A. Sample input files: The DFT+U pw.x example specifies an orthorhombic cell with 28 atoms, seven atomic types, spin polarization, and plane-wave cutoffs of 65.0 and 780.0.The input also includes smearing and starting magnetizations.
Appendix A. Sample input files: The DFT+U+V pw.x example uses the same atom count, species count, spin setting, cutoffs, smearing, and starting magnetizations as the displayed DFT+U example, with slightly different cell parameters.The listed cell parameters are celldm(1) = 19.656 and celldm(2) = 0.581.
Appendix A. Sample input files: The second sample input file is for a DFT+U+V calculation using pw.x.It again uses a self-consistent-field calculation for the olivine system.
Appendix A. Sample input files: The third sample input file configures hp.x to calculate U and V using a 1 × 2 × 3 q-point grid and specified response-convergence settings.The displayed thresholds are conv_thr_chi = 1.0d-7 and dist_thr = 5.D-3.

Appendix B. Input Variables

Appendix B documents the principal hp.x input variables for controlling output, convergence, parallelization, and response-matrix collection. The appendix notes that the complete variable list is available in the HP documentation.

Appendix B. Input Variables: Verbosity controls the amount of information printed, with higher levels adding response matrices, inverse matrices, Hubbard parameters, and non-self-consistent calculation details.The listed levels progress from minimal output to response occupation matrices.
Appendix B. Input Variables: Convergence keywords control the self-consistent response matrix and the Sternheimer-equation solution.The appendix lists thresholds for the response matrix, the initial Sternheimer iteration, and KS-eigenvalue convergence.
Appendix B. Input Variables: niter_max sets the maximum number of iterations for the self-consistent Sternheimer-equation solution, while alpha_mix and nmix control modified-Broyden mixing.The displayed defaults are niter_max = 100, alpha_mix(i) = 0.3, and nmix = 4.
Appendix B. Input Variables: Parallelization variables distribute calculations over perturbed Hubbard atoms and q points, using perturb_only_atom, start_q, and last_q.The appendix also describes collecting response-occupation-matrix pieces across q points and combining them with phase factors and 1/N_q normalization.
Appendix B. Input Variables: The appendix presents the most relevant hp.x variables required in the inputhp namelist rather than the complete variable list.The full list is available in the HP documentation within the Quantum ESPRESSO tree.