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Questaal: a package of electronic structure methods based on the linear muffin-tin orbital technique

Dimitar Pashov, Swagata Acharya, Walter R. L. Lambrecht, Jerome Jackson, Kirill D. Belashchenko, Athanasios Chantis, Francois Jamet, Mark van Schilfgaarde

arXiv:1907.06021v2cond-mat.mtrl-scicond-mat.str-el

TL;DR

Questaal addresses the need for a unified, detailed account of LMTO theory and its implementation across electronic-structure methods and material settings. The paper presents the suite’s full-potential and short-ranged basis frameworks, Green’s-function and wave-function approaches, and associated DFT, GW, and correlation methods. It also identifies limits of QSGW when spin fluctuations are strong, where band coherence and quantitative agreement can deteriorate.

  • Problem

    The paper addresses the need to consolidate the theory, implementation, and evolving capabilities of the Questaal electronic-structure suite.

  • Method

    It develops LMTO formulations with full-potential and short-ranged basis methods and applies them across DFT, GW, and related electronic-structure calculations.

  • Results

    Questaal supports unified calculations spanning basis-set formulations, self-consistent electronic-structure methods, and material systems including layered correlated compounds.

  • Takeaways & Limitations

    The suite provides a broad framework for first-principles calculations while allowing basis and self-consistency choices to be matched to the material problem.

  • Takeaways & Limitations

    QSGW performs worst when spin fluctuations are large, as states lose coherence and the band picture begins to break down.

Abstract

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This paper summarises the theory and functionality behind Questaal, an open-source suite of codes for calculating the electronic structure and related properties of materials from first principles. The formalism of the linearised muffin-tin orbital (LMTO) method is revisited in detail and developed further by the introduction of short-ranged tight-binding basis functions for full-potential calculations. The LMTO method is presented in both Green's function and wave function formulations for bulk and layered systems. The suite's full-potential LMTO code uses a sophisticated basis and augmentation method that allows an efficient and precise solution to the band problem at different levels of theory, most importantly density functional theory, LDA+U, quasi-particle self-consistent GW and combinations of these with dynamical mean field theory. This paper details the technical and theoretical bases of these methods, their implementation in Questaal, and provides an overview of the code's design and capabilities.

PROGRAM SUMMARY

Questaal provides highly accurate first-principles calculations of electronic structure and related properties for diverse materials. It combines an extended full-potential LMTO framework with multiple levels of electronic-structure theory.

  • Questaal calculates electronic structure and resulting physical, spectroscopic, and magnetic properties for periodic solids with diverse electronic correlations.
  • Its many-electron treatments include density functional theory, GW with varying self-consistency, quasiparticle self-consistent GW, and dynamical mean field theory.
  • The suite uses an efficient, precise full-potential extension of the linear muffin-tin orbital technique for the single-particle band problem.
  • An advanced fully relativistic, non-collinear atomic-sphere implementation supports transport and magnetic-property calculations.

1. Introduction

The introduction positions Questaal within the main choices of electronic-structure basis sets and motivates its all-electron, atom-centred, compact LMTO approach. Its development combines established augmented-wave ideas with screening, full-potential extensions, and support for several calculation modes.

  • Electronic-structure methods differ mainly in basis-set type and core-level orthogonalisation, with plane-wave and atom-centred localised bases forming the dominant alternatives.
  • All-electron APW and KKR methods augment envelope functions with numerical partial waves inside augmentation spheres, whereas pseudopotential methods use smooth envelope functions.
  • Questaal follows the all-electron Slater tradition, using atom-centred envelope functions and an augmentation scheme that can converge to an exact solution for the reference potential.
  • The LMTO method linearises the energy dependence of partial waves, replacing complicated energy-dependent augmented-wave bases with efficient linear forms.
  • Screening transformations make muffin-tin orbitals short ranged, while Questaal extends screening to flexible full-potential functions and develops compact bases for demanding calculations.
  • The package spans full-potential and atomic-sphere DFT, LDA+U, GW, and transport-oriented implementations within a unified code family.

2. The Muffin-Tin Potential and the Atomic Spheres Approximation

The muffin-tin and atomic-sphere formulations represent each site with spherical partial waves and interstitial envelope functions. LMTO linearisation and screening then convert the underlying scattering problem into efficient, energy-independent and short-ranged basis descriptions, with accuracy limited by basis choices and approximations.

  • KKR represents muffin-tin solutions with spherical Hankel envelope functions augmented by partial waves inside atomic spheres.
  • Envelope functions have singular heads and tails at other sites, whose one-centre expansions enable augmentation throughout a solid.
  • Angular-momentum cutoffs control Hamiltonian rank and convergence, with the tail-expansion cutoff often chosen near ℓa = ℓb + 1 in lmf.
  • Partial waves are separable into radial functions and spherical harmonics, and their energy dependence is characterised through logarithmic derivatives with poles and band-centre parameters.
  • Linearising partial-wave energy dependence produces an energy-independent basis and a linear algebraic eigenvalue problem solvable through the variational principle.
  • In the atomic-sphere approximation, κ=0 Hankel envelopes are augmented inside muffin-tin spheres, while full-potential methods use more general envelopes.
  • The κ=0 Hankel basis remains long ranged and is not accurate at its compromise energy choice, motivating screening transformations and more flexible short-ranged bases.

2.11. MTO’s and Second Order Green’s Function

The section develops Green’s-function and energy-independent LMTO formulations, showing how screening representations, potential-function parameterizations, and basis choices control the resulting calculations. It also connects these constructions to layered-system scaling and benchmarks against wave-function calculations.

  • Screening representations: The MTO basis is complete for any screening representation α, which can be selected to produce short-ranged Hamiltonians.Changing α rotates the Hilbert space without changing the physical Green’s function.
  • Green’s-function construction: Questaal parameterizes Pα(ε) to second or third order, enabling lmgf and lmpg Green’s-function implementations.The Green’s function itself is representation-independent and can therefore test implementation correctness.
  • Energy-independent basis: The χγ basis yields eigenvalues correct to one higher order in εi−εν than alternative screened bases.This follows because φ and ˙φ combine in the exact proportion at each eigenvalue, while the χγ basis is otherwise orthogonal in the ASA.
  • Layered systems: lmpg uses either difference-equation or sparse-matrix techniques, both scaling linearly with layer number in memory and time.The difference-equation method is significantly faster for large systems, although both methods perform similarly for small systems.
  • Wave-function formulation: The wave-function formulation provides eigenfunctions and normal-mode Green’s functions but creates a difficult eigenvalue problem with near-singular matrices and widely varying decay scales.The eigenvalue formulation can double the matrix rank, while propagating and rapidly decaying modes may differ by many orders of magnitude.
  • Benchmarks: LM and lmgf show very good agreement for FePt and essentially perfect agreement for (Fe1−xCox)2B in the virtual crystal approximation.The comparison covers both second-order/two-centre and third-order/full three-centre formulations.

3. Full Potential Implementation

Questaal’s full-potential implementation combines advanced augmentation with compact, adaptable basis functions to improve accuracy, convergence, and efficiency across electronic-structure calculations.

  • Basis and augmentation: The implementation introduces three-component augmentation and a more general basis set tailored to full-potential calculations.Its basis development includes smooth Hankel functions, JPOs, and local orbitals for extending the valid energy window.
  • Basis and augmentation: JPO basis functions use augmentation information to construct envelope shapes that are compact and suitable for many-body treatments.The paper describes JPOs as a unitary transformation of the original basis in the present implementation.
  • Basis and augmentation: Envelope-function expansions provide flexibility to combine smooth Hankels and plane waves and tailor basis functions to the potential while remaining minimal.This potential adaptation is the basis of the JPO construction.
  • Basis and augmentation: The three-component augmentation converges to the exact result as ℓmax increases and converges more rapidly than conventional augmentation.The method also accurately represents envelope-function projections beginning at relatively low ℓmax + 1.
  • Basis and augmentation: Local orbitals extend the energy range of valid band structures, while additional high- and low-energy states can materially affect GW results.Low-lying core states can shift valence quasiparticle levels by ∼0.1 eV, and high-lying states can shift the ZnO gap by several tenths of an eV.
  • Numerical precision and limitations: The implementation addresses basis completeness and convergence through plane-wave augmentation, compact orbitals, and convergence diagnostics for energies and forces.The standard atom-centred basis misses about 5 mRy/atom in total energy, while adding smooth Hankels improves convergence; EHKS−EHF measures charge-density basis convergence.
  • Numerical precision and limitations: The approach retains known scope boundaries: LAPW functions are extended and slow to converge, while empirical LDA+U gap shifts are less accurate than QSGW.Relativistic treatment also requires Dirac p1/2 local orbitals to reduce errors in heavy semiconductors such as CdTe and PbTe.

4. GW and QSGW

Questaal’s QSGW framework addresses starting-point dependence in GW through quasiparticle self-consistency, improving electronic-structure predictions across diverse materials while retaining important scope limitations. Its successes include better band descriptions and broad property coverage, but performance can degrade when spin fluctuations or strong correlation undermine a single-determinant picture.

  • Need for self-consistency: Diagonal-only GW can produce unphysical band crossings when the reference Hamiltonian orders levels incorrectly.This issue is reported for Ge, InN, CuInSe2, and PbTe.
  • Need for self-consistency: QSGW modifies eigenfunctions through off-diagonal self-energy terms, affecting bandgaps, orbital character, charge density, and metal-insulator transitions.These effects are highlighted for TiSe2, CeO2, La2CuO4, and DMFT-combined calculations.
  • Need for self-consistency: QSGW determines the reference H0 self-consistently within GW, avoiding direct use of the energy-dependent, non-Hermitian self-energy.Quasiparticle self-consistency provides a reference Hamiltonian compatible with an independent-particle description.
  • Material applications: For TiSe2, QSGW predicts the ideal P¯3m1 phase to be metallic, while lattice displacements produce the insulating P¯3c1 phase.This contrasts with the small positive gap obtained at the GLDAW LDA level for P¯3m1.
  • Material applications: GLDAW LDA can yield worse electronic structures than LDA in MnAs and FeTe, including an unphysical dispersionless band and nonsensical Fermi surface.The comparison illustrates the practical importance of self-consistency and starting-point treatment in metals.
  • Limitations: QSGW’s self-consistency largely reduces starting-point dependence but does not eliminate metastable solutions or ambiguities in strongly correlated systems.Examples include competing low-spin and high-spin states in FeSe and multiple stationary points in CuO.
  • Successes of QSGW: QSGW generally improves on DFT and LDA-based GW across quasiparticle levels, gaps, magnetic properties, response functions, and other material properties.The paper reports systematic reliability across a wide range of materials, with fundamental gaps usually in good agreement but systematically overestimated.

5. DFT+DMFT and QSGW +DMFT

Questaal combines band-structure methods with DMFT to treat local electronic correlations through a self-consistent impurity embedded in a QSGW or DFT bath. Its QSGW+DMFT applications describe insulating and metallic cuprates, while single-site DMFT remains limited for nonlocal fluctuation phenomena.

  • DMFT framework: GW extends independent-electron Bloch descriptions with correlation effects, while DMFT interpolates between Bloch-wave and atomic limits through a self-consistent local impurity and bath.The bath preserves both atomic-like and Bloch-like excitations through impurity–bath feedback.
  • DMFT framework: Questaal identifies correlated d- or f-state subspaces using atom-centred partial-wave projectors, then couples them to a bath through local Green’s functions and a self-energy.The correlated subspace is chosen in augmentation spheres because the basis is localized and has definite angular momentum.
  • Interaction construction: Fully ab initio QSGW+DMFT requires a local Green’s function, partially screened U, and double-counting correction, but their consistent construction remains unsettled.Questaal estimates U and J using constrained RPA; practical calculations use static interactions and fully localized-limit double counting.
  • Applications: For La2CuO4, paramagnetic DMFT splits the Cu-3d_x2−y2 state to produce an insulating gap, complementing QSGW’s antiferromagnetic insulating solution.The calculation uses five Cu-3d partial waves and a single-site CT-QMC impurity solver embedded in the QSGW bath.
  • Applications: QSGW+DMFT recovers a metallic x=0.12 La2−xSrxCuO4 solution with upper and lower Hubbard bands and a low-energy incoherent peak at EF.The three-peak structure is characteristic of strongly correlated metals near a metal–insulator transition.
  • Limitations: Single-site DMFT omits longer-range spin or nematic fluctuations, so it cannot explain Fermi arcs in weakly hole-doped cuprates or momentum-dependent electron-pocket suppression in FeSe.These limitations arise because the relevant fluctuations and diagrams are nonlocal rather than purely local.

6. Susceptibilities

Questaal develops linear-response methods for charge and spin susceptibilities, including DMFT-based calculations of magnetic, optical, and superconducting responses. Applications show how approximations and many-body corrections affect predicted spin waves, dielectric spectra, and gap structures.

  • General framework: Susceptibilities describe material responses to perturbations and can be formulated in momentum, position, and frequency space for periodic systems.The framework focuses on transverse spin susceptibility while also relating it to charge response.
  • General framework: The transverse spin response decouples from longitudinal and charge channels in collinear systems, whereas the full response can form a 4×4 spin-charge matrix.The transverse components χ+− and χ−+ remain uncoupled from charge in the stated setting.
  • General framework: The noninteracting susceptibility is obtained by linearising Dyson’s equation, while interaction effects enter through a magnetic kernel relating the bare and full responses.The kernel may depend on position, frequency, and the chosen approximation.
  • Spin response: Lichtenstein’s spin-interaction formula is reliable in the long-wave limit but underestimates large-q exchange, producing discrepancies from full spin-wave calculations.For Zb-MnAs, QSGW predicts positive spin-wave frequencies and a Curie temperature of order 600 K, contrasting with LDA’s negative frequencies.
  • Spin response: QSGW and LDA yield different exchange interactions because LDA underestimates 3d exchange splitting by approximately 1 eV, shifting minority states toward the Fermi level.This generates longer-range antiferromagnetic interactions in LDA, whereas QSGW finds that nearest-neighbour exchange dominates and is positive.
  • Optical and correlated response: Bethe–Salpeter corrections improve optical spectra in correlated systems, but the description can break down when QSGW poorly captures strong spin fluctuations.Adding ladder diagrams to W within QSGW was reported to improve the dielectric response of La2CuO4.
  • Implementation limits: Second-harmonic coefficients are available in the ASA code but are not yet updated for the full-potential lmf implementation or QSGW band structures.An update is intended, but the cited implementation remains limited to the ASA context.
  • DMFT response: QSGW+DMFT calculations recover dominant Néel fluctuations, a temperature-dependent spin gap, and a dx2−y2 superconducting gap in cuprate systems.The spin response peaks at q=(π, π), while the x=0.12 system becomes spin gapped below a certain temperature.

7. Towards a High-Fidelity Solution of the Many-Body Problem

Questaal proposes a hierarchical strategy for high-fidelity many-body calculations, combining QSGW, perturbative diagrammatic corrections, and nonperturbative local treatments. The strategy assigns different approximations to charge and spin fluctuations according to their characteristic scales and locality.

  • Strategy: The strategy begins from the premise that high-fidelity many-electron calculations require multiple levels of theory rather than a single universally adequate approximation.QSGW is presented as the lowest level of the hierarchy.
  • Strategy: QSGW is chosen as the preferred noninteracting starting point because it reduces dependence on conventional DFT starting potentials.The paper frames QSGW as central to the hierarchy, while acknowledging that it is not adequate for every problem.
  • Fluctuation channels: Long-range charge fluctuations are treated with low-order perturbation theory, whereas strong short-range spin fluctuations require a single-site effective Hamiltonian or action.The distinction follows the assumed locality and strength of the respective interactions.
  • Fluctuation channels: Charge and spin fluctuations occupy separated energy scales, with plasmons typically near 5 eV and magnons near 100 meV or less.The paper uses this separation to motivate approximately independent spin and charge contributions to the self-energy.
  • Routes beyond QSGW: The perturbative route adds diagrams such as charge-channel ladders to W, while spin-fluctuation diagrams such as the T matrix were not yet implemented in Questaal.The framework also identifies nonperturbative routes for treating strongly correlated local physics.
  • Future direction: A unified Diagrammatic Monte Carlo treatment is considered promising but remains impractical for realistic Hamiltonians because of enormous time and memory costs.The paper suggests the JPO basis as a possible framework for future realistic implementations.

8. Software Aspects

Questaal is a modular, open-source software suite with documented interfaces, testing, parallel execution, flexible input handling, and a broad record of electronic-structure and response-function capabilities. Its implementation spans established methods and ongoing or scope-limited features.

  • Software design: Questaal’s modular components interoperate through shared interfaces and file formats, supporting a long-evolving suite of electronic-structure programs.The package combines legacy Fortran with modern Fortran features and performance libraries.
  • Software design: Continuous integration includes coding standards, regression testing, coverage checks, and performance testing for ongoing development.The documented requirements include approximately 350 tests and at least 90% code coverage.
  • Performance: The suite supports MPI multiprocessing, multithreading, and selected GPU acceleration, with parallelisation strategies differing across full-potential, tight-binding, and GW programs.The GW code is reported to perform well on many-core vector architectures for systems of at least 15–20 atoms.
  • User interface: Command-line preprocessing allows users to inspect defaults and override nearly any input value, while documentation, tutorials, and examples support use of the executables.The input system is illustrated by full-potential GGA calculations with configurable k-point sampling and automatic LMTO basis setup.
  • Capabilities: The package encompasses DFT, QSGW, transport, magnetic, optical, spin-dynamics, alloy, and susceptibility-related developments documented across the cited application history.Examples include nonequilibrium transport, spin-wave theory, dielectric ladders, QSGW+DMFT, and Bethe–Salpeter susceptibilities.
  • Distribution: Questaal is distributed under the GNU General Public License version 3, with contributions accepted under compatible licensing terms.The distribution has historically included source tarballs and an online repository.

9. Conclusions

The paper unifies Questaal’s historical methods and newer developments to clarify how its components fit together. It presents the suite as a promising high-fidelity path spanning many properties, materials, and approximation levels while acknowledging both strengths and limits.

  • Scope and contribution: The paper consolidates classic expressions and recent unpublished developments into a unified account of Questaal’s evolution and connections.Questaal descends from early all-electron methods developed in Stuttgart.
  • Conclusion: The authors present Questaal as a promising route to efficient, high-fidelity electronic-structure calculations across diverse properties, materials, and approximation levels.The conclusion explicitly frames broad scope as a distinctive potential of the suite.

Appendices

The appendices define notation for augmentation radii and angular-momentum indices used throughout the formalism.

  • The augmentation radius is denoted by s, while s_R denotes a site-dependent radius at site R.
  • A subscript R on a function indicates that its coordinate r is measured relative to site R.
  • An uppercase angular-momentum label L combines the angular quantum number ℓ and magnetic quantum number m.

A. Definition of Real and Spherical Harmonics

Questaal uses specified spherical-harmonic conventions and predominantly works with real harmonics, whose products and coupling coefficients preserve polynomial structure.

  • Questaal uses the spherical-harmonic definitions of Jackson and predominantly employs real harmonics related to spherical harmonics.The real harmonics are real polynomials in x, y, and z.
  • The functions with −m and +m are related by the stated spherical-harmonic identities.
  • Products of spherical-harmonic polynomials expand as linear combinations of the same function set through Gaunt coefficients.
  • Gaunt coefficients C_KLM are nonzero only when k + ℓ − m is even, so the resulting expression remains a polynomial in x, y, and z.
  • The appendix derives additional relations for imaginary α and for p = 1, 2, 3, including zero-energy Hankel and Bessel functions and definitions of C^(±).
  • C^(−) is the transpose of C^(+), while the two are kept distinct for couplings to ℓ + 1 and ℓ − 1 channels.

B. Definition of Hankel and Bessel Functions

The appendix establishes notation and properties for spherical Hankel and Bessel functions, including their real-energy behavior, differential construction, and alternative historical conventions.

  • Spherical radial parts use subscript ℓ, while solid functions use uppercase L for combined angular and magnetic indices.
  • Questaal generally follows Methfessel’s definitions for Hankel and Bessel functions.
  • H_L and J_L are real at real energy, and the structure constants satisfy S_KL(R) = S*_LK(−R).
  • Higher-order H_L functions can be generated from ℓ = 0 functions using the operator Y_L(−∇), which also supports gradient expansions.
  • For E ≤ 0 and κ^2 = −E, the appendix distinguishes Andersen’s historically motivated definitions from Questaal’s standard definitions.The alternative definitions are convenient for the atomic-sphere approximation at E = 0.
  • The scale w is an arbitrary length, usually chosen as an average of muffin-tin radii.

C. Calculations in Real and Reciprocal Space

Questaal supports reciprocal- and real-space treatments, using uniform k meshes and distinct constructions for short-ranged and standard muffin-tin orbitals.

  • Questaal calculations generally use reciprocal space, except lmpg, which uses real space along the principal-layer direction.
  • The k mesh is uniform, with the user choosing either a point at k = 0 or points symmetrically straddling it along each axis.
  • For screened MTOs, Bloch sums are formed directly from short-ranged orbitals, whereas the standard lmf basis uses an Ewald technique.
  • The lattice-vector sum is periodic and can therefore be represented as a Fourier series in reciprocal lattice vectors G.
  • The Bloch-summed H_L is obtained from this reciprocal-lattice representation, with V denoting the unit-cell volume.

D. Matrix elements of Smooth Hankel Envelope Functions

This section derives analytic matrix elements for overlap, Laplacian, gradient, and position operators between smooth Hankel functions by evaluating products in reciprocal space. Products reduce to linear combinations of the same function family, with special treatment for equal energies and generalized Gaussians.

  • Analytic strategy: Parseval’s identity enables analytic evaluation of overlap, Laplacian, gradient, and position matrix elements in reciprocal space.The derivation applies reciprocal-space representations of smooth Hankel functions and their products.
  • Overlap and Laplacian: Overlap and Laplacian matrix elements belong to the same function class because ∇^2HkL = Hk+1L.Products of Fourier-transformed functions are expanded using angular and radial decompositions.
  • Product reduction: Products of two smooth Hankel functions reduce to linear combinations of HkM functions centered at R1 − R2 with smoothing radius γ1 + γ2.The resulting indices satisfy k′ = k1 + k2 + (ℓ1 + ℓ2 − m)/2 and γ = γ1 + γ2.
  • Special cases: Equal energies ε1 = ε2 require a new radial function because the limiting radial expression is not contained in the HkL family.The derivation introduces this additional function to handle the special limit.
  • Derived functions: Higher-order WkL functions can be generated from known real-space forms of H0L and its energy derivative.The construction uses the close relationship between WkL and the energy derivative of HkL.
  • Generalized Gaussians: The resulting formulas also encompass matrix elements between generalized Gaussians through the relation Hk+1L(ε=0) = −4πGkL.This extends the matrix-element framework beyond ordinary smooth Hankel functions.

D.1. The momentum and position operators acting on a smooth Hankel function

This section develops reciprocal-space rules for the gradient and position operators acting on smooth Hankel functions. The position operator is more involved, so its derivation is restricted to the k=0 case.

  • Reciprocal-space operators: The gradient and position operators are represented in reciprocal space using standard Fourier-transform rules.The construction combines these rules with the defining Fourier representations of the smooth Hankel functions.
  • Operator structure: For fixed L and p, ML;p has two nonzero elements, while the position operator requires restricting the derivation to k=0.The position operator has a more complicated structure than the gradient operator.
  • Operator structure: The second contribution to the operator expression follows directly from the angular Fourier-transform identity in Eq. (A.10).The derivation uses Eq. (A.11) for the reciprocal-space operator action and Eq. (A.10) for the second term.

D.2. Matrix elements of position and gradient operators

This section shows that gradient matrix elements remain within the smooth Hankel family, while position matrix elements additionally involve overlaps between HkL and its energy derivative. These quantities are evaluated in reciprocal space.

  • Gradient matrix elements: Gradient matrix elements are linear combinations of smooth Hankel functions evaluated at the vector connecting the two atom centres.The gradient of HkL generates functions from the same HkL family.
  • Position matrix elements: Position matrix elements involve overlaps between HkL and ˙HkL rather than only the original smooth Hankel functions.Their evaluation proceeds in reciprocal space using Parseval’s identity.
  • Radial expansions: The radial product involving ˙H2 is expanded into terms containing bw0 and energy-dependent exponential factors.This expansion supplies the radial component needed for the position-operator matrix elements.
  • Final expressions: The resulting Wk′M and Hk′M functions share the arguments k′, γ, and R used in the earlier product reduction.Combining the cited equations completes the gradient and position matrix-element expressions.
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