Band Structure Workflows

A band calculation is an inference workflow, not a decorative path plot. It begins with a declared periodic structure and represented electronic problem, passes through a converged self-consistent density, and ends only after the full Brillouin zone and the requested downstream object have been validated. The final product is a reproducible dossier and a bounded claim.

This page owns that material-facing execution chain. It does not derive density-functional theory, rank exchange-correlation approximations in the abstract, construct pseudopotentials, teach reusable eigensolvers, perform Wannierization, or turn Kohn–Sham eigenvalues directly into spectra, transport, topology, or measured intensity.

Required background. Crystals and Lattices supplies primitive and magnetic cells, species, fractional coordinates, boundaries, and structure provenance. Band Theory Overview supplies Bloch bands, filling, band-object semantics, and the distinction between direct and global gaps.

Helpful background. The Computational Quantum Matter gateway supplies the common claim ledger. Electronic Structure Methods Map routes method families, while From Quantum Mechanics to Materials helps separate an effective one-electron representation from a material claim.

Declare the Band-Structure Claim

Start with one sentence:

For this crystal, magnetic state, charge state, boundary condition, and represented Hamiltonian, compute this band-like object to this tolerance in order to test this bounded physical claim.

The sentence must name both the computed object and the proposed conclusion. Examples of computed objects include a Kohn–Sham global gap, a fixed-density Fermi surface, a symmetry-labelled connected subspace, or the energy ordering of specified magnetic fixed points. These do not automatically equal a fundamental gap, a quasiparticle pole, an optical threshold, a topological index, an ARPES ridge, or a thermodynamic phase.

A complete workflow has five stages:

1. freeze the physical and representation choices that define the problem;
2. converge the self-consistent density and search relevant fixed points;
3. evaluate dense, fixed-density full-zone outputs;
4. validate numerical, model, and forward-model uncertainty separately;
5. archive the record and state the strongest conclusion it licenses.

The stopping rule is part of the claim. If a result changes sign under a physically reasonable structure, functional, core treatment, magnetic seed, or mesh refinement, the workflow has discovered sensitivity, not failure.

Write the Ten-Field Workflow Record

Use the same ten labels as the chapter gateway. Fill every field before a production run; write unresolved rather than hiding a missing input.

1. Physical problem. Give species, isotope or stoichiometry if relevant, primitive or magnetic cell, fractional coordinates, defects, surfaces, strain, and structure provenance.
2. State and limit. Declare charge, electron count, magnetic preparation, ensemble, physical temperature, equilibrium status, dimensional boundary conditions, and the bulk, slab, or finite-size limit.
3. Claim and accuracy. Name the band-like object, requested units and tolerance, and the physical conclusion the object is meant to test.
4. Representation and provenance. Record functional, pseudopotential or all-electron treatment, valence partition, basis, relativistic level, Hubbard or hybrid corrections, code and version, structure transformations, and hashes of all external inputs.
5. Method and controlled domain. State the independent-particle or Kohn–Sham approximation, occupation rule, symmetry constraints, magnetic ansatz, and why the represented problem is adequate for the bounded claim.
6. Finite numerical problem. Give basis cutoffs, real- and reciprocal-space grids, mesh offsets and weights, smearing, eigensolver tolerances, band count, SCF mixing, symmetry tolerance, and every competing-state seed.
7. Estimator and forward model. Define the global gap, density of states, Fermi-surface sheet, projector weight, unfolded weight, energy difference, or probe-facing quantity actually extracted from raw outputs.
8. Convergence and uncertainty. Vary all relevant numerical axes, separate numerical from model and structure spread, retain covariance, and set a prospective acceptance tolerance.
9. Verification, validation, and provenance. Check electron count, symmetry, sum rules, cell equivalence, known limits, independent implementations where possible, raw histories, failed seeds, and a machine-readable environment record.
10. Licensed claim and stopping rule. State the strongest supported claim, alternatives that remain, a falsifier, and the owner or more complete method needed to proceed.

The record should be readable without the plotting script. A filename such as bands-final.dat is not provenance; a plot whose density, mesh, Hamiltonian, and path cannot be recovered is not a result.

Fix Crystal, Magnetic, Charge, Unit, and Boundary Conventions

Record the input structure in both its submitted and canonicalized forms. Give lattice vectors, origin, handedness, species, fractional coordinates, occupancies, cell transformations, and the tolerance used to infer symmetry. A conventional cell, primitive cell, magnetic cell, and unfolded reference cell serve different purposes and must not be silently exchanged.

For a retained nonorthogonal basis, solve

$$H_{\mathbf k}c_{n\mathbf k} = \epsilon_{n\mathbf k}S_{\mathbf k}c_{n\mathbf k}, \qquad c_{m\mathbf k}^{\dagger}S_{\mathbf k}c_{n\mathbf k} = \delta_{mn}.$$

The retained overlap matrix $S_{\mathbf k}$ must be positive definite after linear dependencies are pruned. Eigenvectors normalized with the Euclidean norm instead of the $S$ metric give wrong populations, overlaps, and matrix elements.

Normalize reciprocal weights by

$$\sum_{\mathbf k}w_{\mathbf k}=1,$$

and count electrons as

$$N_e = g_s\sum_{n\mathbf k} w_{\mathbf k} f_\sigma\!\left(\epsilon_{n\mathbf k}-\mu\right).$$

Use $g_s=2$ only when spin degeneracy is implicit. Use $g_s=1$ when separate spin channels or spinor states are explicitly enumerated. In an all-electron or norm-conserving representation, the same convention appears in the density,

$$\rho_{\mathrm{out}}(\mathbf r) = g_s\sum_{n\mathbf k} w_{\mathbf k} f_\sigma\!\left(\epsilon_{n\mathbf k}-\mu\right) \left|\psi_{n\mathbf k}(\mathbf r)\right|^2.$$

PAW and ultrasoft representations add their declared augmentation terms; the simple pseudo-wavefunction sum is not the complete reconstructed density in those methods.

Declare a dimensionless SCF residual, for example

$$R_\rho = \frac{ \int_{\mathrm{cell}} \left|\rho_{\mathrm{out}}-\rho_{\mathrm{in}}\right|d^3r }{N_e},$$

and supplement it with total-energy, force, moment, and occupancy stability when those quantities enter the claim. A small code-specific residual alone does not prove a unique or lowest-energy fixed point.

State whether energies are per cell, formula unit, or atom; whether reciprocal coordinates use $2\pi$; whether moments are total-cell, atom-projected, or interstitial-inclusive; and whether a slab contains dipole corrections or a charged-cell compensating background.

Choose Functional, Core Treatment, Basis, and Cutoff

Separate model choices from numerical choices.

• The exchange-correlation functional, relativistic level, pseudopotential or all-electron partition, Hubbard correction, and magnetic ansatz change the represented Hamiltonian.
• Plane-wave cutoff, localized-basis size, augmentation grid, integration mesh, eigensolver threshold, and SCF tolerance approximate that represented Hamiltonian.
• Structure, stoichiometry, strain, defects, and magnetic cell define the physical input rather than an error bar on a fixed calculation.

Choose core and valence states according to the requested object. Semicore states can affect bonding, spin–orbit splittings, magnetic moments, and pressure response. A higher plane-wave cutoff cannot repair a pseudopotential with the wrong valence partition or poor transferability.

For each core treatment, archive the generator or library release, element and valence configuration, scalar- or fully-relativistic status, nonlinear-core correction, recommended cutoff, and file hash. When comparing codes, first match the represented Hamiltonian: using different core partitions or relativity is a model comparison, not an implementation check.

Converge basis errors against the quantity in the claim. Total energies may look stable while stress, force, band curvature, or a small gap is not. Pulay force and stress errors require explicit basis tests, especially when cells or atomic positions are relaxed.

Separate SCF, NSCF, Full-Zone, and Display-Path Calculations

The four calculations answer different questions.

1. SCF integration. A weighted reciprocal mesh updates the density until the declared residuals and state variables converge.
2. Dense fixed-density NSCF integration. A finer uniform or adaptively refined mesh evaluates global extrema, occupations, DOS, Fermi surfaces, projectors, velocities, and source data without changing the density.
3. Display path. Zero-weight points along a conventional path show selected dispersions. They neither integrate the Brillouin zone nor prove a global gap.
4. Downstream export. Raw eigenvalues, eigenvectors or projectors, symmetry data, mesh geometry, and provenance are packaged for a canonical localized, response, topology, transport, or probe owner.

A denser path does not repair an unconverged density. Conversely, recomputing a density for every visualization path makes comparisons less controlled. Record which density file each fixed-density output used and verify that symmetry, electron count, chemical potential, and represented Hamiltonian are unchanged.

For an isolated nondegenerate band in a differentiable nonorthogonal representation, the velocity satisfies

$$\hbar v_{n\alpha}(\mathbf k) = c_{n\mathbf k}^{\dagger} \left[ \partial_{k_\alpha}H_{\mathbf k} - \epsilon_{n\mathbf k}\partial_{k_\alpha}S_{\mathbf k} \right] c_{n\mathbf k}.$$

At a degeneracy, evaluate the operator within the retained subspace rather than differentiating energy-sorted labels. This is one reason the workflow must preserve eigenvectors or projectors, not only plotted eigenvalues.

Control k-Point Quadrature, Occupations, and Smearing

Mesh convergence depends on dimensionality, symmetry reduction, offsets, Fermi-surface geometry, and the target estimator. Reporting only an integer mesh such as $12^3$ is incomplete: retain reciprocal basis, shift, irreducible weights, symmetry tolerance, and the mapping to the full zone.

Use occupations and smearing deliberately. A numerical smearing parameter may stabilize metallic integration, but it is not automatically a physical temperature, lifetime, disorder width, or instrumental resolution. If a code optimizes a smeared free-energy functional, compare the appropriate zero-smearing extrapolate or corrected energy under one declared convention.

Advanced Methfessel–Paxton and cold-smearing occupations are nonmonotonic and can admit more than one chemical potential consistent with the requested electron count. Archive the selected root and its search interval. For a small-gap insulator, prefer fixed occupations or demonstrate that the density, occupied projector, chemical-potential branch, and gap are stable as the width goes to zero.

For a tested even smearing expansion, one may fit

$$\Delta E(\sigma) = \Delta E_0+a\sigma^2+O(\sigma^4),$$

but only after defining the energy or free-energy object and demonstrating that the selected occupation rule has this asymptotic form over the fitted range. For Fermi–Dirac occupations, for example, a Sommerfeld expansion of an internal energy can have a leading $\sigma^2$ term when the relevant density of states is smooth. Extrapolation is a numerical procedure, not a claim that the material was measured at temperature $\sigma/k_{\mathrm B}$.

For insulators, verify that the electron count and occupied subspace remain fixed as the mesh is refined. For metals and small-gap systems, inspect local refinement around candidate pockets or extrema. A uniform mesh can converge a total energy while missing a small Fermi pocket that decides the claim.

Tetrahedron and smearing methods have different error structures. Compare them only after matching the represented bands and defining how degeneracies, partial occupations, and the chemical potential are handled.

Treat SOC, Magnetism, Symmetry, and Competing States

Spin–orbit coupling changes both the Hilbert-space convention and the symmetry classification. Record whether spin degeneracy is implicit, whether spinors are explicit, which double group or magnetic group is enforced, and whether the magnetization direction is constrained or relaxed. Do not multiply an explicit spinor calculation by an extra factor of two.

Audit the electron filling against the double space group before licensing a gap. Nonsymmorphic connectivity can enforce band multiplets larger than a Kramers pair. A positive numerical gap at a forbidden filling signals a wrong electron count, hidden symmetry breaking, a truncated band manifold, or another workflow error—not a symmetry-preserving band insulator.

Broken-symmetry searches require a candidate set rather than one initialization. At minimum, record nonmagnetic, ferromagnetic, relevant antiferromagnetic or noncollinear patterns, multiple moment magnitudes, cell sizes, and constraints. A converged SCF fixed point is not necessarily the lowest-energy state, and the lowest state in a narrow candidate list is not necessarily the material ground state.

Compare magnetic energies using

$$\Delta e_{A-B} = \frac{E_A}{N_A} - \frac{E_B}{N_B},$$

with the sign and normalization stated explicitly. Candidate states must share composition, geometry protocol, core treatment, cutoff, reciprocal-point density, occupation rule, and smearing extrapolation. If cells differ, confirm that $N_A$ and $N_B$ refer to the same atom or formula-unit normalization.

Total magnetization is an integrated observable of the represented cell. Atom-projected moments depend on sphere radius, projector, basis, and partition of the interstitial region. Use projected moments to diagnose a calculation, not as convention-free atomic observables.

Symmetry of Bloch States owns irreducible representations, sewing matrices, compatibility, and connectivity. This workflow records the code, convention, tolerance, and raw symmetry data used to apply that theory.

Interpret Bands, Projectors, DOS, Fermi Surfaces, and Unfolding

Energy sorting is unsafe at crossings and near degeneracies. Track a retained subspace using metric-aware overlaps, projectors, symmetry labels, and continuity. For two neighboring meshes, the singular values of the subspace overlap are more stable than a one-to-one label chosen by energy order. Archive the transformation if a band index is permuted.

When electron count defines valence and conduction manifolds, the minimum direct and global independent-particle gaps are

$$E_g^{\mathrm{dir}} = \min_{\mathbf k} \left[ \epsilon_c(\mathbf k)-\epsilon_v(\mathbf k) \right],$$ $$E_g^{\mathrm{global}} = \min_{\mathbf k}\epsilon_c(\mathbf k) - \max_{\mathbf k}\epsilon_v(\mathbf k).$$

A positive direct gap can coexist with a negative global gap. A display path can miss either extremum. Search the full zone, locally refine candidate extrema, and retain the momentum, band/subspace identity, and uncertainty of each extremum.

Density of States owns DOS normalization and singularity structure. Fermi Surface owns sheet geometry and low-energy kinematics. A plotted Gaussian broadening is not a physical lifetime, and a visually smooth isosurface is not a convergence test.

Orbital weights and projected DOS depend on the projector, local axes, sphere or augmentation region, and basis. Unfolded weights additionally depend on the chosen primitive reference and embedding. They are representation diagnostics, not a many-body spectral function.

The source package for Wannier Functions may contain the requested energy coverage, raw states or projectors, symmetry data, mesh provenance, and the downstream accuracy target. Window selection, disentanglement, localization, interpolation residuals, and operator interpolation belong to Wannierization Workflows, not this page.

Validate Global Gaps, Subspaces, Convergence, and Uncertainty

Convergence is claim-specific. A useful acceptance record separates at least four uncertainty classes:

• numerical: basis, mesh, band count, eigensolver, SCF, symmetry tolerance, smearing, and any local fit or refinement used only to search for an extremum;
• representation: functional, core/valence partition, relativistic level, Hubbard or hybrid correction, magnetic ansatz, and correlated subspace;
• structure and state: atomic positions, strain, defects, charge, temperature model, and competing phases;
• forward model: matrix elements, surface sensitivity, many-body self-energy, lifetime, resolution, and experimental geometry.

Do not combine these into one root-sum-square number unless their statistical meaning and covariance justify it. Model spread is often the dominant result.

Verification asks whether the represented equations were solved correctly. Check electron count, Hermiticity, overlap positivity, symmetry degeneracies, cell equivalence, sum rules, density and energy residuals, and known analytic or atomic limits. Cross-code agreement is meaningful only after the Hamiltonian and numerical conventions are matched; two codes using the same pseudopotential library are not fully independent validation.

Validation asks whether the represented problem supports the physical claim. A converged Kohn–Sham path does not by itself validate a quasiparticle gap, optical onset, transport coefficient, topological index, or ARPES intensity. Those questions require their canonical forward models and state-matched data.

Archive the original and canonicalized structures, coordinate transformation, pseudopotential hashes, code and environment versions, input and output files, mesh points and weights, symmetry reports, SCF histories, failed seeds, raw eigenvalues and projectors, analysis scripts, random seeds, and checksums. A claim should be reproducible without reverse-engineering a figure.

Worked Audit: A Small-Gap SOC Crystal

This is a synthetic teaching record, not a prediction for an existing compound. It tests whether a small positive auxiliary gap survives the declared PBE+SOC numerical workflow and whether the conclusion stops at the proper boundary.

1. Physical problem. The hypothetical bulk crystal is tetragonal $\mathrm{Bi_2Te_2}$ in the symmorphic group $P4/mmm$ (No. 123), with $a=4.10\,\text{Å}$ and $c=6.20\,\text{Å}$. Bi occupies fractional positions $(0,0,0.22)$ and $(0,0,0.78)$; Te occupies $(1/2,1/2,0.28)$ and $(1/2,1/2,0.72)$. The shortest like-species separation is $0.44c=2.728\,\text{Å}$. The archived input fixes the origin, conventional-to-primitive transform, right-handed cell, species order, and coordinates.

2. State and limit. The target is a neutral, nonmagnetic, inversion- and time-reversal-symmetric infinite periodic bulk auxiliary Kohn–Sham problem in fixed-electron-number equilibrium at zero physical temperature, zero field, zero strain, zero drive, and fixed structure. It has 62 explicitly enumerated spinor electrons per primitive cell and $g_s=1$. All SCF and fixed-density runs occupy the lowest 62 spinor states with zero-width fixed occupations; no fictitious-temperature or advanced smearing root is used in this small-gap record.

3. Claim and accuracy. Determine the sign of the global PBE+SOC auxiliary gap to $10\,\text{meV}$. The claim is not that the material is a semiconductor, nor that the result is a fundamental, optical, or topological gap.

4. Representation and provenance. The central branch uses PBE, fully relativistic norm-conserving 15-valence-electron Bi and 16-valence-electron Te teaching pseudopotentials, a plane-wave basis, and no $+U$, hybrid correction, or structural relaxation. A PBEsol branch uses the same valence and relativistic conventions. The synthetic dossier assigns input IDs, hashes, code version, and environment fields; a real calculation must replace these teaching identifiers with retrievable artifacts.

   The explicit valence partitions are Bi $5d^{10}6s^26p^3$ and Te $4d^{10}5s^25p^4$, giving $2(15)+2(16)=62$ electrons.

5. Method and controlled domain. The calculation represents an independent-particle Kohn–Sham spectrum for one fixed crystal. PBE versus PBEsol tests one approximation axis. Missing electron self-energy, electron–hole attraction, phonons, temperature-dependent structure, and surface effects remain outside the controlled domain.

6. Finite numerical problem. Plane-wave cutoffs are 60, 80, and 100 Ry. SCF meshes are $8^3$, $12^3$, and $16^3$; full-zone fixed-density meshes are $24^3$, $32^3$, and $40^3$. All are unshifted $\Gamma$-centred meshes whose symmetry-reduced weights sum to one. Each run fixes the 62 occupied states among 96 explicit spinor bands. The fractional-coordinate symmetry tolerance is $10^{-7}$, and the generalized-eigenpair backward error, using operator and vector 2-norms, is

   $$\frac{ \lVert Hc-\epsilon Sc\rVert_2 }{ \left( \lVert H\rVert_2+|\epsilon|\lVert S\rVert_2 \right) \lVert c\rVert_2 } <10^{-10}.$$

   Pulay mixing uses coefficient 0.30 with eight stored residuals. The display path has zero integration weight.

7. Estimator and forward model. The primary estimator is $E_g^{\mathrm{global}}$ from the dense full zone with local refinement of both extrema. A conventional path supplies a minimum direct separation only for visualization. No optical matrix element, self-energy, surface, or experimental-resolution model is applied.

8. Convergence and uncertainty. At fixed 80 Ry, each SCF mesh is followed by a $32^3$ fixed-density search; the $8^3$, $12^3$, and $16^3$ densities give global gaps 16.5, 18.0, and 18.0 meV. Holding the accepted $16^3$, 80 Ry density fixed, the $24^3$, $32^3$, and $40^3$ NSCF meshes give 14, 18, and 19 meV. For the cutoff sweep, the density is recomputed on $16^3$ at each of 60, 80, and 100 Ry and evaluated on a fixed $32^3$ NSCF mesh; the gaps are 17, 18, and 18.5 meV. The path minimum direct separation is 52 meV. The final PBE+SOC numerical spread is below 10 meV, whereas the matched PBEsol+SOC branch gives 74 meV and dominates the error ledger.

9. Verification, validation, and provenance. The accepted runs reproduce 62 electrons, the allowed even filling of the symmorphic spinful group, inversion and Kramers degeneracy, a dimensionless density residual below $10^{-9}$, and forces below $10^{-3}\,\text{eV}\,\text{Å}^{-1}$. Mesh points, weights, symmetry maps, extremum coordinates, raw states, failed runs, scripts, hashes, and environment data are retained. An eight-band source subspace may be exported, but no Wannier window or interpolation claim is made here.

10. Licensed claim and stopping rule. The declared fixed synthetic crystal has a positive PBE+SOC auxiliary global gap that is numerically stable at the requested $10\,\text{meV}$ scale. The PBEsol spread prevents a sharper model-independent gap claim. Stop if the sign changes under structure, pseudopotential, functional, or local mesh refinement. Escalate a quasiparticle, optical, topological, or Wannier-interpolation claim to its canonical owner.

The key lesson is the 52 meV path separation versus the 18–19 meV full-zone gap. A path plot would have overstated the relevant scale by about a factor of three even though both numbers were individually converged on their own sampling domains.

Worked Audit: A Magnetic Metal with Competing States

This second record is also synthetic. It asks whether a high-spin ferromagnetic fixed point is lowest within a declared PBE candidate set, while testing whether functional sensitivity blocks a material ground-state claim.

1. Physical problem. The teaching crystal is neutral bcc $M$ with $a=2.86\,\text{Å}$ and eight explicit-spin valence electrons per atom. Every candidate is represented in the same $2\times1\times1$ primitive supercell, containing 16 valence electrons.

2. State and limit. The geometry is fixed. Candidate states are nonmagnetic, high- and low-spin ferromagnetic, and a commensurate antiferromagnetic pattern in fixed-electron-number equilibrium and the infinite periodic limit at zero physical temperature, zero external field, and zero drive. The calculation is collinear and scalar relativistic. Numerical smearing is a quadrature control; the compared object is the tested zero-smearing extrapolate of the internal total energy per atom, not a finite-temperature free energy or magnetic ensemble.

3. Claim and accuracy. Rank the specified fixed points to $\pm2\,\text{meV}$ per atom and decide whether the result licenses a material magnetic ground state. The band-level secondary claim is only whether the represented Kohn–Sham bands cross the chemical potential.

4. Representation and provenance. The central model is fixed-geometry collinear PBE-PAW with no SOC, $+U$, phonons, or noncollinear order. The PAW dataset, 1.25 Å projector-sphere convention, code version, input hashes, structure transform, and environment are archived. A matched PBEsol branch changes only the functional.

5. Method and controlled domain. Initial uniform moments are 0, 0.5, 2, and $4\,\mu_{\mathrm B}$; staggered seeds are $\pm2$ and $\pm4\,\mu_{\mathrm B}$. The search tests convergence to multiple collinear fixed points, not all possible magnetic cells, spin spirals, structural distortions, or dynamical correlations.

6. Finite numerical problem. Supercell meshes are $8\times16\times16$, $10\times20\times20$, and $12\times24\times24$, corresponding to matched primitive reciprocal densities of $16^3$, $20^3$, and $24^3$. Cutoffs are 400, 500, and 600 eV. All meshes are unshifted $\Gamma$-centred grids; symmetry-reduced weights sum to one. Fermi–Dirac electronic widths are 0.20, 0.10, and 0.05 eV. All candidates use the same composition, cell, mesh family, cutoff, occupation rule, and energy normalization.

7. Estimator and forward model. For width $\sigma=k_{\mathrm B}T_e$, define the internal electronic energy as $E_{\mathrm{int}}=F+\sigma S$, where $S$ is the dimensionless electronic entropy used by the declared Fermi–Dirac functional. The reported estimator is the $\sigma\to0$ extrapolate

   $$\Delta e_{A-\mathrm{FM}}(\sigma) = \frac{E_{\mathrm{int},A}(\sigma)}{N_A} - \frac{E_{\mathrm{int},\mathrm{FM}}(\sigma)}{N_{\mathrm{FM}}},$$

   so a positive value places state $A$ above the high-spin FM reference. The workflow also records total-cell and projector-sphere moments and searches the full zone for chemical-potential crossings. No thermodynamic magnetic, photoemission, or transport forward model is applied.

8. Convergence and uncertainty. Every point is a fresh SCF calculation, not a fixed-density energy. At 600 eV and the final $12\times24\times24$ mesh, widths 0.20, 0.10, and 0.05 eV give AFM–FM internal-energy differences 8.00, 11.00, and 11.75 meV per atom; the tested $\sigma^2$ fit gives $12.0\,\text{meV/atom}$. After applying that three-width extrapolation at every control, the mesh sequence is 8.5, 11.4, and 12.0 meV per atom at 600 eV, while the cutoff sequence is 10.4, 11.8, and 12.0 meV per atom on the final mesh. The acceptance order is candidate identification, fresh SCF at all three widths for every control point, zero-width extrapolation, the mesh sweep at 600 eV, and the cutoff sweep on the final mesh. A reverse-order cross-check changes the result by less than 0.4 meV per atom.

   The matched PBEsol width sequence is $-8.00$, $-6.50$, and $-6.125\,\text{meV/atom}$, giving a $-6.0\,\text{meV/atom}$ intercept. Repeated mesh and cutoff controls give a $\pm2\,\text{meV/atom}$ allowance for both functional branches. The sign reversal is therefore a model-axis effect larger than the numerical uncertainty.

9. Verification, validation, and provenance. Final zero-width PBE extrapolates place high-spin FM at 0, low-spin FM at +7, AFM at +12, and NM at +86 meV per atom. The FM total moment is $2.10\,\mu_{\mathrm B}$ per atom, while the declared sphere gives $1.86\,\mu_{\mathrm B}$. The AFM cell has zero total moment and local projected moments $\pm1.92\,\mu_{\mathrm B}$. Primitive- and doubled-cell FM energies agree within 0.5 meV per atom. Raw SCF histories, every seed, symmetry state, weights, occupations, energies, moments, and environment hashes are archived.

10. Licensed claim and stopping rule. High-spin FM is lowest among the tested fixed-geometry PBE candidates, and the represented PBE bands cross the chemical potential. The PBEsol reversal blocks a material ground-state claim. Stop before calling the system experimentally ferromagnetic or metallic; expand the structural, magnetic, correlation, and probe models first. Local moments remain projector dependent.

The model-axis reversal is the central result. Reporting only the well-converged PBE ordering would confuse numerical precision with physical robustness.

Exit Checkpoint, Failure Modes, and Canonical Handoffs

Before exporting a result, ask:

• Did every plotted band use a named converged density?
• Were electron count, spin multiplicity, mesh weights, and units audited?
• Was the requested object evaluated over the full zone rather than a path?
• Were crossings tracked by subspace, symmetry, or overlaps rather than band index alone?
• Were relevant magnetic cells and seeds attempted and failed runs retained?
• Were numerical, structure, representation, and forward-model errors kept separate?
• Does the conclusion say Kohn–Sham or independent-particle where required?
• Can another researcher recover the structure, Hamiltonian, mesh, states, raw output, scripts, and environment?

Common failures include using a display path as an integration grid, comparing energies from unmatched cells or smearings, double-counting spin degeneracy, calling a broadened DOS a lifetime, treating projector weights as absolute orbital occupations, discarding metastable seeds, and interpreting agreement between two similarly parameterized codes as independent material validation.

Canonical handoffs keep the workflow bounded:

• Band Theory and Electronic Structure routes band, functional, and output-semantics questions.
• Electronic Structure Methods Map compares method families; this page records and tests a chosen solids workflow rather than ranking every approximation.
• Symmetry of Bloch States, Density of States, and Fermi Surface own the corresponding physical objects.
• Wannier Functions owns existence, localization, gauge, and obstruction theory. Numerical windows, disentanglement, spread minimization, and interpolation validation belong to the live numerical Wannierization workflow.
• Spectral Functions owns quasiparticle and incoherent spectral weight, while ARPES owns measured intensity, matrix elements, surfaces, backgrounds, and resolution.
• The Computational Quantum Matter gateway routes topology, transport, response, disorder, and interacting calculations. Reusable numerical assembly, diagonalization, parallelization, and package-level implementation belong to Computational QM rather than this material workflow.

Exercises

Exercise 1: SCF or fixed-density work?

One path plot uses an unconverged $8^3$ SCF density. A second uses a converged $12^3$ SCF density, but the plotted path was evaluated with a new functional. A dense uniform NSCF mesh then places a candidate band extremum between grid points. Classify the SCF, dense-NSCF, local-refinement, and display-path tasks and design a controlled repair.

Solution

The first path is a fixed-density output of an unconverged represented state; increasing path density cannot repair it. Recompute the density on a converged SCF mesh, then evaluate the path at zero integration weight.

Changing the functional changes $H_{\mathrm{KS}}[\rho]$ and therefore requires a new self-consistent density. The second path cannot be labelled a controlled functional comparison if it reuses the old density. Run matched SCF calculations for both functionals, then evaluate identical full-zone and path outputs. Keep model spread separate from mesh error.

The dense fixed-density NSCF mesh is the global estimator. Once it identifies a candidate extremum between points, refine a local three-dimensional region at the same density and represented Hamiltonian until the extremum is stable. A denser display path samples only its prescribed line and cannot substitute for that local full-zone refinement.

Exercise 2: Electron counting and spin convention

An irreducible mesh has weights $1/8$, $3/8$, and $4/8$. Two bands are occupied at every point when spin degeneracy is implicit. Compute $N_e$. Then repeat for four explicitly enumerated occupied spinor bands.

Solution

The weights sum to one. With two occupied bands and implicit spin degeneracy,

$$N_e = 2\times2\times \left(\frac18+\frac38+\frac48\right) =4.$$

With four explicit spinor bands, use $g_s=1$:

$$N_e = 1\times4\times1 =4.$$

Multiplying the spinor result by two would double-count spin.

Exercise 3: Direct gap versus global overlap

A dense search finds $\max_{\mathbf k}\epsilon_v=0.10\,\text{eV}$ and $\min_{\mathbf k}\epsilon_c=0.06\,\text{eV}$ at a different momentum. The minimum direct separation is $0.22\,\text{eV}$. Classify the represented band state.

Solution

The global gap is

$$E_g^{\mathrm{global}} = 0.06\,\text{eV}-0.10\,\text{eV} = -0.04\,\text{eV}.$$

The bands overlap globally even though they are directly separated at every tested momentum. The represented state is an indirect-overlap semimetal, subject to convergence of both extrema. Carrier compensation would require a separate Fermi-volume count. A path-only direct-gap claim would miss the overlap.

Exercise 4: Numerical-smearing extrapolation

At Fermi–Dirac widths 0.20, 0.10, and 0.05 eV, an internal-energy difference is 8.00, 11.00, and 11.75 meV per atom. Fit the tested form $\Delta E(\sigma)=\Delta E_0+a\sigma^2$.

Solution

The three points obey

$$\Delta E(\sigma) = 12.0\,\frac{\text{meV}}{\text{atom}} - 100\,\frac{\text{meV}}{\text{atom}\,\text{eV}^2}\sigma^2.$$

Thus $\Delta E_0=12.0\,\text{meV}$ per atom. This is a numerical extrapolation of the declared Fermi–Dirac internal-energy object, not a universal smearing law or a thermodynamic fit versus physical temperature. One must still vary mesh, cutoff, state initialization, and model choices.

Exercise 5: A generalized eigenproblem

Solve $Hc=\epsilon Sc$ for

$$H= \begin{pmatrix} 1&0\\ 0&4 \end{pmatrix}\text{eV}, \qquad S= \begin{pmatrix} 1&0\\ 0&2 \end{pmatrix},$$

and give $S$-normalized eigenvectors.

Solution

The generalized eigenvalues satisfy $1-\epsilon=0$ and $4-2\epsilon=0$, so they are 1 and 2 eV. An $S$-normalized choice is

$$c_1= \begin{pmatrix}1\\0\end{pmatrix}, \qquad c_2= \begin{pmatrix}0\\1/\sqrt2\end{pmatrix}.$$

Indeed, $c_i^\dagger S c_j=\delta_{ij}$. The Euclidean vector $(0,1)^T$ is not normalized in the physical overlap metric.

Exercise 6: Band labels through a crossing

Between adjacent mesh points, the squared overlap matrix for two energy-sorted states is

$$\left|M\right|^2 = \begin{pmatrix} 0.02&0.98\\ 0.98&0.02 \end{pmatrix}.$$

What happened to the labels, and which object should be retained?

Solution

Each state overlaps almost entirely with the oppositely indexed state at the next point. Energy sorting has swapped the labels. Relabel by maximum overlap only if the individual states are isolated and the gauge convention is controlled.

Near an exact or unresolved degeneracy, retain the two-dimensional projector instead. The displayed elementwise magnitudes diagnose the label swap but do not determine singular values; those require the archived complex overlap matrix $M$. The subspace can remain stable even when an arbitrary unitary rotation makes individual eigenvectors discontinuous.

Exercise 7: A bounded magnetic-state claim

The PBE fixed points lie at high-spin FM 0, low-spin FM +7, AFM +12, and NM +86 meV per atom. PBEsol gives AFM–FM $=-6\,\text{meV}$ per atom. Assign a $\pm2\,\text{meV}$ numerical allowance to every value in both branches and state the strongest conclusion.

Solution

Within the tested fixed-geometry PBE candidate set, high-spin FM is lowest. Its separation from low-spin FM is only modestly larger than the numerical envelope, whereas AFM and NM remain higher by wider margins within that model.

Treating the quoted $\pm2\,\text{meV/atom}$ as an allowance on each relative energy gives low-spin FM $[5,9]$, AFM $[10,14]$, and NM $[84,88]$ meV per atom above high-spin FM. The matched PBEsol AFM–FM interval is $[-8,-4]$ meV per atom, so it remains negative throughout its allowance. If the $\pm2$ values instead described individual correlated total energies, the relative-energy covariance would have to be propagated before using these intervals.

PBEsol reverses the FM/AFM ordering by more than the numerical uncertainty. Therefore no functional-robust material ground-state claim is licensed. The record instead establishes model sensitivity and motivates additional functionals, structural relaxation, magnetic cells, correlation treatments, and experimental validation.

Exercise 8: A computed gap and an ARPES-like ridge

A new crystal calculation reports a $18\pm3\,\text{meV}$ global Kohn–Sham gap. An ARPES-like ridge separation is $25\pm8\,\text{meV}$. For the numerical exercise, treat both quoted errors as one-standard uncertainties and initially take them as independent. Complete a compact ten-field audit and state the handoffs.

Solution

1. Physical problem: record bulk and cleaved-surface structures, stoichiometry, domain population, and sample provenance.

2. State and limit: match temperature, doping, chemical potential, magnetic state, bulk versus surface geometry, equilibrium assumptions, and the occupation or Fermi-cutoff convention on both sides.

3. Claim and accuracy: ask whether a represented global auxiliary gap and a measured ridge separation are mutually compatible; do not declare equality.

4. Representation and provenance: archive structure, functional, core treatment, SOC, mesh, raw bands, and code environment.

5. Method and controlled domain: identify the Kohn–Sham approximation and the missing self-energy, surface, and matrix-element physics.

6. Finite numerical problem: provide SCF and dense meshes, cutoffs, extremum refinement, electron count, residuals, and convergence histories.

7. Estimator and forward model: define the global band estimator separately from the ARPES peak/ridge extraction, photon polarization, geometry, background, Fermi factor, and resolution convolution. Ordinary ARPES samples occupied removal weight; it does not automatically observe the unoccupied conduction extremum that enters a global gap.

8. Convergence and uncertainty: separate the 3 meV numerical standard uncertainty, functional and structure spread, and the 8 meV experimental standard uncertainty. For a difference $D=\Delta_{\mathrm{ARPES}}- \Delta_{\mathrm{KS}}$,

   $$\sigma_D^2 = \sigma_{\mathrm{ARPES}}^2 + \sigma_{\mathrm{KS}}^2 - 2\,\operatorname{Cov} \left( \Delta_{\mathrm{ARPES}},\Delta_{\mathrm{KS}} \right).$$

   Under the exercise’s independence assumption, $D=7\,\text{meV}$ and $\sigma_D=\sqrt{8^2+3^2}\,\text{meV}\simeq8.5\,\text{meV}$. Shared energy calibration or fitted chemical potential would require the covariance term.

9. Verification, validation, and provenance: check symmetry, full-zone extrema, detector calibration, surface stability, repeated zones, and raw data plus analysis scripts.

10. Licensed claim and stopping rule: the two scales differ by less than one combined standard uncertainty under the stated independence model, but a Kohn–Sham gap is not thereby measured. A falsifier is a state-matched surface/bulk or photon-energy dependence inconsistent with the proposed band assignment.

Band execution stays here. Quasiparticle poles go to Spectral Functions, and intensity, matrix elements, surface sensitivity, background, and resolution go to ARPES.

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