Density-functional theory recasts an interacting many-electron ground-state problem in terms of the electron density $n(\mathbf r)$ rather than the full many-body wavefunction. Its exact foundation says that the ground-state density determines the external potential, and that the correct density minimizes an energy functional. Practical Kohn–Sham DFT introduces noninteracting orbitals that reproduce that density, leaving exchange and correlation in an unknown functional that must be approximated. A credible calculation therefore joins theorem, approximation, basis, boundary conditions, pseudopotential or all-electron treatment, self-consistency, structural optimization, convergence, and comparison to an observable.

<svg viewBox="0 0 760 470" xmlns="http://www.w3.org/2000/svg" font-family="Arial,sans-serif"><rect x="0" y="0" width="760" height="470" rx="18" fill="#0d1117"/><text x="380" y="34" fill="#e6edf3" font-size="21" font-weight="700" text-anchor="middle">DFT replaces a many-electron wavefunction with a density</text><text x="380" y="60" fill="#8b949e" font-size="12" text-anchor="middle">Kohn–Sham orbitals are an auxiliary route to the interacting ground-state density</text><rect x="35" y="110" width="210" height="245" rx="14" fill="#161b22" stroke="#f85149" stroke-width="2"/><text x="140" y="145" fill="#ff7b72" font-size="14" font-weight="700" text-anchor="middle">Many-body problem</text><text x="140" y="205" fill="#e6edf3" font-size="13" text-anchor="middle">Ψ(r₁,…,rₙ)</text><text x="140" y="250" fill="#c9d1d9" font-size="11" text-anchor="middle">3N spatial variables</text><path d="M245 230H295" stroke="#58a6ff" stroke-width="4"/><polygon points="295,230 282,222 282,238" fill="#58a6ff"/><rect x="295" y="90" width="170" height="285" rx="14" fill="#161b22" stroke="#58a6ff" stroke-width="2"/><text x="380" y="130" fill="#79c0ff" font-size="14" font-weight="700" text-anchor="middle">Density</text><path d="M325 290C350 150 410 150 435 290" fill="none" stroke="#58a6ff" stroke-width="4"/><text x="380" y="325" fill="#e6edf3" font-size="13" text-anchor="middle">n(r)</text><text x="380" y="350" fill="#c9d1d9" font-size="11" text-anchor="middle">3 spatial variables</text><path d="M465 230H515" stroke="#3fb950" stroke-width="4"/><polygon points="515,230 502,222 502,238" fill="#3fb950"/><rect x="515" y="110" width="210" height="245" rx="14" fill="#161b22" stroke="#3fb950" stroke-width="2"/><text x="620" y="145" fill="#7ee787" font-size="14" font-weight="700" text-anchor="middle">Kohn–Sham system</text><text x="620" y="205" fill="#e6edf3" font-size="12" text-anchor="middle">{φᵢ(r)} → n(r)</text><text x="620" y="250" fill="#c9d1d9" font-size="11" text-anchor="middle">self-consistent auxiliary orbitals</text><text x="380" y="420" fill="#e6edf3" font-size="12" text-anchor="middle">The reduction is exact in principle; practical accuracy is controlled by Eₓ꜀[n] and numerical choices.</text></svg>

The first Hohenberg–Kohn theorem establishes density as sufficient ground-state information. For interacting electrons in a fixed particle number under suitable conditions, the ground-state density determines the external scalar potential up to an additive constant and therefore determines the Hamiltonian and ground-state observables. This is an existence-and-uniqueness result, not a recipe for writing the functional. Degenerate ground states require careful ensemble formulations, and magnetic vector potentials demand current- or spin-density extensions rather than an unqualified scalar-density statement.

The second Hohenberg–Kohn theorem is a variational principle over densities. The universal functional $F[n]=T[n]+V_{ee}[n]$ combined with $\int v_{ext}(\mathbf r)n(\mathbf r)d\mathbf r$ gives the ground-state energy when minimized over physically admissible densities with the correct particle number. Any trial density gives an energy no lower than the true ground-state energy when the exact functional is used. With approximate functionals, comparing energies retains practical value but loses a universal rigorous upper-bound guarantee.

Universality means independence from the external potential, not independence from the particle interaction. $F[n]$ is common to atoms, molecules, solids, and surfaces sharing the same electron–electron interaction. Nuclear positions and species enter through $v_{ext}$. Changing dimensionality, screened interaction, relativistic Hamiltonian, or model interaction changes the universal functional class. This distinction enables transfer of approximate exchange–correlation forms while explaining why a functional calibrated for one interaction cannot be assumed exact for another.

The density reduction does not make the exact functional simple. The many-body complexity is compressed into the functional dependence of kinetic and interaction energy on $n$. Thomas–Fermi theory supplies a direct local kinetic approximation but misses shell structure and much bonding. Kohn–Sham construction treats most kinetic energy exactly for an auxiliary noninteracting system and isolates the remaining difficulty in exchange–correlation energy. Computational tractability comes from this construction plus approximation, not from the theorem alone.

Kohn–Sham orbitals reproduce density but are not the interacting many-electron wavefunction. For a spin-unpolarized closed-shell system, $n(\mathbf r)=2\sum_i^{occ}|\phi_i(\mathbf r)|^2$ under the appropriate occupation convention. The Slater determinant of Kohn–Sham orbitals belongs to a fictitious noninteracting reference. It delivers the exact density if the exact functional is known, but its orbitals and most eigenvalues are auxiliary. Treating every orbital plot as an observable overstates the theorem.

The Kohn–Sham energy partitions known and unknown terms. A common form is $E[n]=T_s[n]+\int v_{ext}n+E_H[n]+E_{xc}[n]+E_{NN}$ under Born–Oppenheimer nuclei. $T_s$ is the noninteracting orbital kinetic energy, $E_H$ is classical Coulomb self-energy, $E_{xc}$ contains nonclassical exchange, correlation, and the difference between interacting and noninteracting kinetic energy, and $E_{NN}$ is nuclear repulsion. Alternate bookkeeping is acceptable only when double counting is handled consistently.

Functional differentiation yields the effective one-electron equations. Variation under orbital orthonormality gives $[-\hbar^2\nabla^2/(2m)+v_{ext}+v_H+v_{xc}]\phi_i=\epsilon_i\phi_i$, where $v_{xc}=\delta E_{xc}/\delta n$. The Hartree potential solves a Poisson equation for electron density with sign and units consistent with nuclear potentials. Because $v_H$ and $v_{xc}$ depend on the orbitals through density, these eigenproblems are nonlinear and must be solved self-consistently.

The total energy is not the sum of occupied Kohn–Sham eigenvalues. That sum counts effective Hartree and exchange–correlation potentials in a way that requires subtraction and correction. Codes evaluate a total-energy expression with ion–ion, Hartree, exchange–correlation, and pseudopotential contributions under their conventions. Comparing a raw eigenvalue sum between structures can give wrong energetics even when orbitals are converged. Forces likewise require derivatives of the total energy, not sums of eigenvalue derivatives alone.

<svg viewBox="0 0 760 470" xmlns="http://www.w3.org/2000/svg" font-family="Arial,sans-serif"><rect x="0" y="0" width="760" height="470" rx="18" fill="#0d1117"/><text x="380" y="34" fill="#e6edf3" font-size="21" font-weight="700" text-anchor="middle">Self-consistency closes density and effective potential</text><text x="380" y="60" fill="#8b949e" font-size="12" text-anchor="middle">A small eigenproblem residual is not the same as a converged density</text><rect x="70" y="120" width="170" height="160" rx="14" fill="#161b22" stroke="#58a6ff" stroke-width="2"/><text x="155" y="160" fill="#79c0ff" font-size="14" font-weight="700" text-anchor="middle">Input density</text><text x="155" y="220" fill="#e6edf3" font-size="13" text-anchor="middle">n⁽ᵏ⁾(r)</text><path d="M240 200H300" stroke="#3fb950" stroke-width="4"/><polygon points="300,200 287,192 287,208" fill="#3fb950"/><rect x="300" y="95" width="160" height="210" rx="14" fill="#161b22" stroke="#3fb950" stroke-width="2"/><text x="380" y="135" fill="#7ee787" font-size="14" font-weight="700" text-anchor="middle">Build and solve</text><text x="380" y="190" fill="#e6edf3" font-size="12" text-anchor="middle">vₑ𝒻𝒻[n⁽ᵏ⁾]</text><text x="380" y="235" fill="#e6edf3" font-size="12" text-anchor="middle">Hₖₛφᵢ = εᵢφᵢ</text><path d="M460 200H520" stroke="#d29922" stroke-width="4"/><polygon points="520,200 507,192 507,208" fill="#d29922"/><rect x="520" y="120" width="170" height="160" rx="14" fill="#161b22" stroke="#d29922" stroke-width="2"/><text x="605" y="160" fill="#e3b341" font-size="14" font-weight="700" text-anchor="middle">Output density</text><text x="605" y="220" fill="#e6edf3" font-size="13" text-anchor="middle">nₒᵤₜ⁽ᵏ⁾(r)</text><path d="M605 280V360H155V280" fill="none" stroke="#a371f7" stroke-width="4"/><polygon points="155,280 147,293 163,293" fill="#a371f7"/><text x="380" y="390" fill="#d2a8ff" font-size="12" text-anchor="middle">mix residual nₒᵤₜ − nᵢₙ; update until energy, density, and forces agree</text><text x="380" y="435" fill="#c9d1d9" font-size="11" text-anchor="middle">Charge sloshing demands dielectric-aware preconditioning rather than blind mixing.</text></svg>

Self-consistent field iteration is a nonlinear fixed-point problem. Begin with a density, construct effective potential, solve Kohn–Sham eigenstates, occupy them, form a new density, and mix toward the next input. Convergence can be linearized in terms of dielectric response. Simple mixing works for benign molecules and insulators but fails for metals, large cells, slabs, and heterogeneous systems through long-wavelength charge sloshing. Pulay or Broyden mixing and Kerker-like preconditioning approximate inverse response.

SCF convergence requires multiple independent criteria. Monitor density or potential residual, total-energy change, eigenproblem residual, electron count, spin moment, and forces. A flat total energy can coexist with noisy forces or an unconverged density because energy is variational to second order near a stationary point. Loose inner diagonalization can corrupt the outer residual. Report tolerances in physical units and demonstrate that tightening them does not change the requested energy difference or observable.

Occupation and smearing are part of the numerical and physical model. Insulators at zero temperature have integer occupations separated by a gap. Metals need Brillouin-zone integration across a Fermi surface; Fermi–Dirac or auxiliary smearings stabilize sampling. Finite smearing introduces entropy or extrapolation terms whose code-specific energy must be interpreted correctly. A broad smearing can alter magnetism, phase stability, and forces. Converge both $k$ mesh and smearing, preferably along more than one path.

The local-density approximation imports uniform-electron-gas physics pointwise. $E_{xc}^{LDA}=\int n(\mathbf r)\epsilon_{xc}^{unif}(n)d\mathbf r$ is exact for a uniform density and often surprisingly useful for slowly varying solids. It tends to overbind many systems and misses long-range dispersion, but its errors are not universal slogans. Spin-LDA uses spin densities. Modern parameterizations rely on accurate uniform-gas data, and different parameterizations should be identified rather than labeled only “LDA.”

Generalized-gradient approximations add local density-gradient information. GGAs such as PBE write exchange–correlation using $n$ and $\nabla n$ subject to selected exact constraints. They often improve atomization energies and structures relative to LDA but can overestimate lattice constants and still miss nonlocal correlation. PBEsol restores behavior aimed at densely packed solids at a tradeoff for molecular energetics. Functional names encode different design goals; “GGA” is not a reproducible method specification.

Meta-GGAs add kinetic-energy density or higher local ingredients. Functionals such as SCAN use orbital kinetic-energy density to recognize bonding environments while satisfying more constraints. They are still semilocal and can be numerically sensitive to integration grids or pseudopotential consistency. Regularized variants improve stability. Increased formal rung does not guarantee monotonic accuracy for every property. Benchmark the exact property and chemistry rather than treating Jacob's ladder as a universal ranking.

Hybrid functionals mix nonlocal exact exchange with semilocal terms. Global hybrids such as PBE0 use a fixed fraction; range-separated hybrids partition Coulomb interaction so short- and long-range exchange receive different treatment; screened hybrids such as HSE reduce solid-state cost and long-range exchange. They often improve gaps, localization, and reaction energetics, but cost rises sharply and optimal mixing can depend on screening. The Kohn–Sham operator becomes nonlocal, changing algorithms and $k$-point convergence.

Exact exchange in a Kohn–Sham framework does not eliminate correlation error. Hartree–Fock exchange cancels one-electron self-interaction in the exchange term but omits dynamical correlation. A hybrid retains an approximate correlation functional and only part or range of exchange. Orbital-dependent functionals may require generalized Kohn–Sham nonlocal operators or optimized effective potentials. Calling hybrid eigenvalues “many-body quasiparticles” remains an approximation even when gaps improve.

Dispersion corrections address long-range correlation absent from semilocal functionals. Pairwise DFT-D schemes add damped atom-pair terms with functional-dependent parameters. Nonlocal van der Waals density functionals incorporate spatial density kernels. Many-body dispersion accounts for collective polarizability. Damping prevents double counting at short range, so mixing a correction with an unparameterized base functional is unsafe. Layer binding, adsorption, molecular crystals, and conformers can be qualitatively controlled by the selected dispersion treatment.

Self-interaction and delocalization error distort fractional charge. The exact energy is piecewise linear between integer electron numbers, whereas many semilocal functionals are convex and favor overly delocalized charge. Consequences include underestimated gaps, incorrect dissociation, shallow defects, and excessive charge transfer. Hartree self-repulsion is not perfectly canceled by approximate exchange–correlation. Hybrids, DFT+$U$, self-interaction corrections, and tuned range separation can reduce selected symptoms but introduce choices that require validation.

Static-correlation error appears when one determinant cannot represent competing configurations. Stretched bonds, transition-metal spin states, Mott insulators, and near-degenerate orbitals can defeat standard Kohn–Sham approximations. Broken-symmetry solutions sometimes recover energies but contaminate spin and hide multireference character. Hybrid exchange alone is not a universal fix. Diagnostics, multiple initial occupations, higher-level wavefunction methods, DFT+DMFT, or quantum Monte Carlo may be needed.

DFT plus Hubbard $U$ adds a localized-subspace correction. DFT+$U$ penalizes fractional occupancy in selected localized orbitals and subtracts a double-counted interaction already approximated by the base functional. Results depend on projector definition, $U$ and $J$, double-counting form, oxidation state, structure, and magnetic order. Linear-response or constrained calculations can estimate parameters, but transfer across environments is not automatic. Report all subspace and parameter details.

The exchange–correlation functional is the dominant model choice, not a cosmetic dropdown. Bond lengths, cohesive energies, barriers, surface energies, magnetic moments, gaps, dielectric response, adsorption, and defect localization respond differently to functional error. Select using exact constraints, known failure modes, and benchmarks closest to the target. Comparing several correlated functionals is not a statistical uncertainty estimate, but it reveals sensitivity. Experimental agreement after structural or parameter fitting should not be called first-principles prediction without qualification.

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A basis set defines the variational space of Kohn–Sham orbitals. Plane waves, localized atomic orbitals, real-space grids, finite elements, wavelets, and augmented methods trade systematic convergence, locality, boundary flexibility, and all-electron resolution. Basis incompleteness affects energy, forces, stress, response, and basis-set superposition error differently. Comparing two codes without converging their representations to a common physical result conflates implementation with theory.

Plane waves provide a systematic kinetic-energy cutoff for periodic systems. Include reciprocal vectors satisfying $\hbar^2|\mathbf k+\mathbf G|^2/(2m)

Localized basis sets trade compactness for completeness diagnostics. Numerical atomic orbitals or Gaussian functions can describe molecules and large sparse systems efficiently. Basis shape, confinement radius, polarization, diffuse functions, and multiple-zeta levels matter. Basis-set superposition error artificially stabilizes fragments because each borrows the other's functions; counterpoise analysis or systematic enlargement diagnoses it. Forces include Pulay terms when basis functions move with nuclei.

Pseudopotentials remove chemically inert core oscillations under a transferability assumption. Norm-conserving, ultrasoft, and projector-augmented-wave approaches replace or transform core behavior so valence orbitals need less basis resolution. Each dataset fixes valence configuration, core radius, relativistic treatment, and reference functional. Semicore states may be essential for bonding, pressure, polarization, or spectroscopy. A pseudopotential generated with one functional and used with another introduces inconsistency that can exceed target energy differences.

PAW is a transformation between smooth auxiliary and all-electron-like quantities. It reconstructs nodal behavior within augmentation spheres using partial waves and projectors while retaining plane-wave efficiency. Frozen-core and finite-projector approximations remain. Local quantities near nuclei, hyperfine fields, and high-pressure overlap demand dataset checks. Calling every PAW result “all electron” without its frozen-core qualification is misleading.

All-electron methods resolve core and valence without a pseudopotential approximation. LAPW, augmented-plane-wave, numerical atomic orbital, and finite-element schemes treat rapid nuclear-region variation explicitly with specialized bases. They are valuable references for pseudopotential transferability and core-sensitive properties. They still have basis, linearization, relativistic, and functional approximations. Agreement between two all-electron codes requires convergence of sphere radii, angular cutoffs, local orbitals, and interstitial representation.

Relativistic effects separate into scalar and spin–orbit contributions. Scalar relativity contracts and stabilizes some orbitals and alters heavy-element bonding; spin–orbit coupling splits bands, changes anisotropy, topology, and optical selection. Fully relativistic pseudopotentials or four-component approaches encode different levels. Adding spin–orbit non-self-consistently may be adequate for small perturbations but fails when it changes occupations or density. Compare energy scales with structural, magnetic, and thermal targets.

Periodic boundary conditions create an infinite crystal from one cell. A primitive cell exploits translational symmetry; a supercell represents defects, disorder, surfaces, molecules, or long-period order through periodic images. Vacuum does not remove periodicity; electrostatic, elastic, and dispersion interactions can persist. Charged cells require a compensating convention and finite-size correction. Dipole corrections, Coulomb truncation, and larger cells address specific interactions but must match geometry.

Brillouin-zone integration replaces an infinite set of crystal states by weighted samples. Monkhorst–Pack meshes, Gamma-centered grids, symmetry reduction, tetrahedra, and smearing approximate $k$ integrals. Metals need dense sampling near Fermi surfaces; large supercells need fewer points; anisotropic cells need anisotropic meshes. Convergence by equal grid dimensions can be misleading across changing cells, so reciprocal spacing or density is more transferable. Band paths used for plotting do not replace integration meshes.

Symmetry reduces cost but can suppress physically relevant broken states. Space-group operations reduce irreducible $k$ points and constrain density, forces, and stress. Defects, magnetism, ferroelectricity, Jahn–Teller distortions, charge order, and soft modes may lower symmetry. Starting from a high-symmetry density and enforcing symmetry can trap a saddle point. Disable or deliberately reduce symmetry when exploring broken-symmetry states, then classify the converged solution afterward.

Spin-polarized DFT promotes density to spin components or magnetization density. Collinear calculations use $n_\uparrow$ and $n_\downarrow$; noncollinear calculations use spinors and vector magnetization, commonly with spin–orbit coupling. Multiple magnetic arrangements and initial moments should be explored because SCF can converge to metastable states. Energy differences may be tiny relative to total energy and sensitive to functional, $U$, cell, and $k$ mesh. A local spin density is not identical to a unique atomic oxidation state.

Born–Oppenheimer DFT treats nuclei as parameters on an electronic energy surface. For each nuclear geometry, solve the electronic ground state and add nuclear repulsion. This neglects nonadiabatic electron–nuclear coupling and usually treats nuclei classically in geometry optimization or molecular dynamics. Zero-point motion, isotope effects, tunneling nuclei, and finite-temperature anharmonicity need phonon, path-integral, or beyond-adiabatic treatment. “DFT temperature” from electronic smearing is not the ionic temperature.

Hellmann–Feynman forces require a stationary and complete-enough electronic solution. At self-consistency, derivatives of the Hamiltonian expectation yield forces, but basis dependence introduces Pulay terms and pseudopotential/PAW terms require consistent implementation. Loose SCF, finite grids, and incomplete bases create force noise. Validate forces by finite energy differences for representative displacements. Geometry convergence should state maximum force, energy change, stress, and displacement tolerances.

Stress and cell optimization amplify basis and sampling errors. Pulay stress arises when a fixed plane-wave cutoff changes the effective basis as cell vectors vary. $k$-point density also changes unless controlled. Optimize internal positions and lattice under a declared external stress and symmetry choice. Energy–volume curves fitted to an equation of state offer a more transparent bulk-modulus check than one optimizer result. Dispersion and functional choice often dominate equilibrium volume after numerical convergence.

Ab initio molecular dynamics propagates nuclei on repeatedly solved DFT forces. Born–Oppenheimer MD converges SCF each step; Car–Parrinello-like methods introduce auxiliary orbital dynamics. Time step must resolve fastest ionic motion and electronic convergence must keep force drift below integration error. Thermostats sample chosen ensembles only when parameters and equilibration are appropriate. Finite system size, trajectory length, rare events, and functional error usually dominate statistical uncertainty.

<svg viewBox="0 0 760 470" xmlns="http://www.w3.org/2000/svg" font-family="Arial,sans-serif"><rect x="0" y="0" width="760" height="470" rx="18" fill="#0d1117"/><text x="380" y="34" fill="#e6edf3" font-size="21" font-weight="700" text-anchor="middle">A converged DFT workflow separates numerical and model choices</text><text x="380" y="60" fill="#8b949e" font-size="12" text-anchor="middle">Each arrow needs its own stopping and validation evidence</text><rect x="40" y="120" width="130" height="170" rx="12" fill="#161b22" stroke="#58a6ff"/><text x="105" y="155" fill="#79c0ff" font-size="13" font-weight="700" text-anchor="middle">Model</text><text x="105" y="205" fill="#c9d1d9" font-size="10" text-anchor="middle">functional</text><text x="105" y="230" fill="#c9d1d9" font-size="10" text-anchor="middle">spin · charge</text><text x="105" y="255" fill="#c9d1d9" font-size="10" text-anchor="middle">boundary</text><path d="M170 205H205" stroke="#3fb950" stroke-width="4"/><polygon points="205,205 193,198 193,212" fill="#3fb950"/><rect x="205" y="100" width="140" height="210" rx="12" fill="#161b22" stroke="#3fb950"/><text x="275" y="135" fill="#7ee787" font-size="13" font-weight="700" text-anchor="middle">Numerics</text><text x="275" y="190" fill="#c9d1d9" font-size="10" text-anchor="middle">basis cutoff</text><text x="275" y="220" fill="#c9d1d9" font-size="10" text-anchor="middle">k mesh · cell</text><text x="275" y="250" fill="#c9d1d9" font-size="10" text-anchor="middle">SCF tolerance</text><path d="M345 205H380" stroke="#d29922" stroke-width="4"/><polygon points="380,205 368,198 368,212" fill="#d29922"/><rect x="380" y="100" width="140" height="210" rx="12" fill="#161b22" stroke="#d29922"/><text x="450" y="135" fill="#e3b341" font-size="13" font-weight="700" text-anchor="middle">State</text><text x="450" y="190" fill="#c9d1d9" font-size="10" text-anchor="middle">SCF density</text><text x="450" y="220" fill="#c9d1d9" font-size="10" text-anchor="middle">geometry</text><text x="450" y="250" fill="#c9d1d9" font-size="10" text-anchor="middle">magnetic branch</text><path d="M520 205H555" stroke="#a371f7" stroke-width="4"/><polygon points="555,205 543,198 543,212" fill="#a371f7"/><rect x="555" y="120" width="165" height="170" rx="12" fill="#161b22" stroke="#a371f7"/><text x="637" y="155" fill="#d2a8ff" font-size="13" font-weight="700" text-anchor="middle">Observable</text><text x="637" y="205" fill="#c9d1d9" font-size="10" text-anchor="middle">energy · force · gap</text><text x="637" y="235" fill="#c9d1d9" font-size="10" text-anchor="middle">response · spectrum</text><text x="380" y="375" fill="#e6edf3" font-size="12" text-anchor="middle">Converge the requested difference, not merely the absolute total energy.</text><text x="380" y="425" fill="#c9d1d9" font-size="11" text-anchor="middle">Validate functional error separately from basis, sampling, cell, and solver error.</text></svg>

Energy differences are meaningful only between consistently defined calculations. Use the same functional, pseudopotential families, valence states, relativistic level, cutoff, sampling quality, smearing convention, and finite-size treatment. Atom, molecule, bulk, slab, and charged-cell references may require different boxes but compatible numerical limits. A cancellation of large total energies can be excellent when errors are correlated and disastrous when reference states use inconsistent approximations.

Formation energies require explicit reservoirs and stoichiometric bookkeeping. A defect formation energy combines defective and pristine supercell energies, chemical potentials, charge terms, and corrections. Chemical potentials are constrained by phase stability, not arbitrary elemental constants. Compound competing phases define allowable growth conditions. Comparing values without reservoir conventions can reverse conclusions. Functional errors in elemental molecules or metals may need validated corrections rather than silent empirical shifts.

Charged defects require electrostatic and band-edge finite-size corrections. Periodic codes usually neutralize a charged cell with a background, creating spurious image interactions and potential offsets. Corrections use dielectric screening, cell geometry, localized charge assumptions, and potential alignment. Defect transition levels also depend on band-edge placement and gap error. Convergence with supercell size and correction scheme should be shown; a single corrected small cell is not definitive.

Defect localization must be tested against initial state and functional bias. Semilocal functionals can delocalize a nominal defect carrier across the host band. Seed different occupations, local distortions, magnetic moments, and charge localization; compare hybrid or DFT+$U$ where justified. A symmetry-constrained pristine geometry can suppress polaron formation. Charge-density differences should be referenced to aligned calculations and integrated, not interpreted from a plotting isovalue alone.

Surface energies depend on slab construction and chemical termination. Symmetric slabs avoid dipoles but may double reconstruction constraints; asymmetric slabs need dipole treatment and separate surface accounting. Converge slab thickness, vacuum, $k$ sampling, relaxation depth, and electrostatic correction. Polar surfaces may require reconstruction, adsorption, charge transfer, or nonstoichiometric thermodynamics rather than a naive bulk truncation. Cleavage energy and relaxed surface free energy are distinct.

Adsorption energies combine surface, molecule, and coverage conventions. State adsorption site, coverage, cell, molecular reference, spin, zero-point correction, and whether fragments remain bound. Dispersion, basis superposition, slab dipoles, and finite coverage can dominate weak adsorption. Gas-phase chemical potentials add temperature and pressure through statistical mechanics. A zero-K electronic adsorption energy is not directly a catalytic free energy.

Reaction barriers require a path search rather than interpolation of endpoint energies. Nudged elastic band and related chain-of-states methods optimize images toward a minimum-energy path; dimer methods seek a saddle from a local region. Converge images, spring/path forces, cell, spins, and electronic states. The highest unrefined image is not necessarily the transition state. Free-energy barriers require vibrational, entropic, solvent, and dynamical corrections beyond the electronic saddle energy.

Phonons are second derivatives of the DFT energy surface. Finite-displacement supercells or density-functional perturbation theory produce force constants and dynamical matrices. Acoustic sum rules encode translational invariance. Imaginary frequencies can indicate structural instability, insufficient convergence, interpolation artifacts, or a saddle-point phase. Converge supercell or $q$ mesh, forces, $k$ sampling, and long-range nonanalytic corrections in polar materials.

Density-functional perturbation theory computes linear response self-consistently. Small atomic displacements, electric fields, strain, or other perturbations induce first-order orbitals and density. The $2n+1$ theorem relates lower-order wavefunction response to higher energy derivatives. DFPT yields phonons, dielectric tensors, Born effective charges, electron–phonon coupling, and elastic response under method-specific conditions. Metals require occupation derivatives and careful Fermi-surface sampling.

Berry-phase polarization is a bulk geometric quantity defined modulo a quantum. In a periodic insulator, absolute position times charge is ill-defined; the modern theory uses occupied-band Berry phases. Physical polarization changes follow a continuous insulating path, with branch choice tracked. Born effective charges are derivatives of polarization, not static ionic charges. Metallic states lack the same bulk polarization definition. Ferroelectric switching comparisons must preserve band insulation and branch continuity.

Band structures plot Kohn–Sham eigenvalues along a chosen reciprocal-space path. The path is a visualization convention, while the self-consistent density came from a full integration mesh. High-symmetry labels depend on lattice convention and standardization. Band crossings require character or symmetry analysis. A path can miss an indirect band extremum away from its lines; dense searches or interpolation are required for effective masses and transport valleys.

The fundamental gap is not generally the Kohn–Sham eigenvalue gap. Exact DFT's fundamental gap equals the Kohn–Sham gap plus the exchange–correlation derivative discontinuity. Semilocal approximations largely miss this discontinuity and often underestimate gaps. The highest occupied exact Kohn–Sham eigenvalue has a special ionization-potential relation under conditions, but unoccupied eigenvalues lack a general quasiparticle theorem. Hybrid, meta-GGA, $GW$, or tuned approaches can improve gaps without making every band an exact excitation.

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Quasiparticle $GW$ corrections address charged excitation energies beyond ordinary DFT. The self-energy $\Sigma=iGW$ replaces a static local or generalized Kohn–Sham exchange–correlation potential with an energy-dependent nonlocal object. One-shot $G_0W_0$ depends on starting functional; partial or full self-consistency changes screening and cost. Converge empty states, dielectric cutoff, frequency treatment, $k$ mesh, and finite-size effects. $GW$ is not an exchange–correlation functional for ground-state geometry in its common use.

Optical excitations require electron–hole interaction or a response functional. The Bethe–Salpeter equation built on quasiparticle states captures excitons and redistribution of oscillator strength. Time-dependent DFT uses a time-dependent exchange–correlation kernel; adiabatic semilocal kernels can miss long-range excitons and charge-transfer states. Independent-particle Kohn–Sham transitions omit both quasiparticle and excitonic corrections. Compare spectra only after broadening, polarization, temperature, and experimental geometry are declared.

Time-dependent DFT extends density ideas to driven dynamics. The Runge–Gross foundation establishes a time-dependent density–potential mapping under assumptions, and time-dependent Kohn–Sham equations propagate auxiliary orbitals. Practical accuracy depends on temporal and memory dependence of the exchange–correlation potential. Real-time propagation yields spectra and nonlinear dynamics; linear-response TDDFT yields excitation equations. Strong ionization, double excitations, charge transfer, and memory challenge standard adiabatic approximations.

Finite-temperature DFT minimizes a free-energy functional rather than only energy. Mermin's extension treats equilibrium density matrices and electron density at nonzero temperature. Electronic entropy matters for warm dense matter and metals; common smearing schemes used for integration are not all physical Fermi–Dirac temperatures. Ionic free energy additionally requires vibrations, configurational disorder, and anharmonicity. Separating electronic and ionic temperatures is essential in ultrafast or two-temperature conditions.

Solvation and environmental models add a second approximation layer. Explicit solvent captures local structure but demands sampling; continuum dielectric models average polarization and define a cavity through density or atoms; hybrid embedding combines regions. Electrode potentials, ions, and constant-charge versus constant-potential ensembles require careful thermodynamics. A vacuum DFT energy plus an empirical solvent number may miss geometry-dependent polarization and entropic contributions.

Constrained DFT defines states by imposed charge or occupation conditions. Lagrange multipliers enforce fragment charge, spin, or subspace occupation, enabling charge-transfer energies, diabatic states, and interaction parameters. Results depend on weight functions and subspaces. The constraint work must be included consistently. A converged constrained state is not the unconstrained ground state; it represents a deliberately selected state whose physical preparation must be justified.

Orbital projections are analysis choices rather than unique observables. Projected density of states, atomic charges, bond orders, Wannier functions, and orbital populations depend on basis, projector radius, partition method, and gauge within occupied subspaces. Bader density basins, Mulliken populations, Löwdin populations, and PAW projections answer different questions. Trends can be robust, but integer oxidation states should not be inferred from one arbitrary projection threshold.

Wannier functions transform Bloch states into localized orbitals within a chosen gauge. Maximally localized Wannier functions support interpolation of bands, velocities, Berry curvature, electron–phonon coupling, and tight-binding models. Entangled bands require disentanglement windows and initial projections. Localization can settle in different minima. Interpolated quantities must reproduce direct calculations over the active energy range, particularly around crossings and topology.

Topological invariants require wavefunction geometry beyond charge density plots. Berry phases, Chern numbers, $Z_2$ indices, Wilson loops, and symmetry indicators use occupied Bloch subspaces. Spin–orbit coupling, band ordering, and gap opening must be converged. A semilocal functional's wrong ordering can give a wrong topological classification; hybrid or $GW$ checks may be needed. Surface states in finite slabs additionally depend on termination and thickness.

Machine-learned potentials inherit the DFT reference definition. Training energies and forces from one functional, pseudopotential, spin state, cutoff, and convergence policy defines a particular potential-energy surface. Active learning expands configuration coverage but cannot exceed systematic reference accuracy without extra data. Energy offsets across inconsistent datasets produce artifacts. Validate forces, stresses, phase energies, defects, and extrapolation indicators, not only aggregate test RMSE.

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Verification begins with reproducible numerical convergence, not agreement with experiment. Vary cutoff, basis, $k$ mesh, smearing, cell size, vacuum, SCF tolerance, force threshold, and response grids while holding the physical model fixed. Converge energy differences, forces, stress, gaps, polarization, phonons, or barriers to tolerances tighter than the scientific conclusion. One-at-a-time scans can miss coupled errors, so test representative combinations near the chosen point.

Cross-code comparison is strongest when inputs define equivalent Hamiltonians. Match functional version, relativistic level, valence electrons, geometry, occupations, smearing, $k$ points, and convergence. Pseudopotential and all-electron calculations need absolute-energy-independent comparisons such as structures, energy differences, or eigenvalue alignments. Agreement between independently implemented basis families is powerful evidence against numerical defects, but shared functional error remains.

Analytical and symmetry limits catch implementation errors cheaply. An isolated hydrogen atom tests one-electron self-interaction behavior and spin; uniform electron gas recovers LDA reference; separated fragments test size consistency and fractional charge; translating or rotating an isolated system should not change energy beyond grids; crystal symmetry constrains forces, stress, degeneracies, and tensors. Acoustic phonons vanish at Gamma under translation invariance.

Energy–force consistency tests differentiation and self-consistency together. Displace one atom by positive and negative small steps and compare the central energy derivative with analytic force over a shrinking-step range. Too large a step measures anharmonicity; too small reveals SCF and floating-point noise. Repeat for strain and stress. This detects Pulay, pseudopotential, grid, and incomplete-SCF problems that a stable optimizer can conceal.

Validation maps computed quantities through the experiment's thermodynamic and instrumental conditions. Diffraction sees finite-temperature average structure, photoemission sees spectral removal energies and matrix elements, optical absorption sees electron–hole excitations, calorimetry sees free energies, and transport sees scattering absent from static bands. Broaden spectra, account for temperature and pressure, and compare the matching observable. Agreement of a Kohn–Sham eigenvalue with a peak may be useful but does not retroactively make it exact.

Uncertainty should separate numerical, functional, parameter, structural, and experimental components. Numerical convergence can be bounded directly. Functional sensitivity can be sampled across defensible approximations but is correlated and not a calibrated probability by default. $U$, exact-exchange fraction, dispersion parameters, defect chemical potentials, and finite-temperature corrections contribute parameter uncertainty. Unknown polymorph, disorder, stoichiometry, and surface termination contribute structural uncertainty. State each layer rather than one undifferentiated error bar.

Data provenance is part of first-principles reproducibility. Archive code and version, functional identifiers, pseudopotential files and hashes, input/output, cell and coordinates, $k$ meshes and paths, cutoff, smearing, occupations, spin seeds, SCF and ionic tolerances, corrections, scripts, and postprocessing definitions. Database labels like “PBE PAW” are insufficient because datasets and defaults change. Preserve failed or metastable branches when they inform state selection.

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Define composition, charge, spin, periodicity, thermodynamic state, and target observable
  -> Select ground-state DFT or an extension appropriate to excitation, temperature, or strong correlation
  -> Choose exchange–correlation functional, dispersion, +U, relativistic level, and core treatment
  -> Converge basis, k mesh, smearing, supercell, vacuum, SCF, force, stress, and response parameters
  -> Explore symmetry, magnetic, occupation, charge-localization, and structural starting states
  -> Optimize the relevant geometry or sample the declared finite-temperature ensemble
  -> Verify density, electron count, energy, forces, stress, symmetry, and global electrostatics
  -> Apply defect, dipole, charged-cell, free-energy, quasiparticle, excitonic, or reservoir corrections as needed
  -> Repeat convergence for the final energy difference, derivative, band, response, or spectrum
  -> Validate against independent observables at matching temperature, pressure, frequency, and environment
  -> Archive exact inputs, datasets, hashes, branches, corrections, and uncertainty ledger

A useful diagnostic divides failures into theorem misuse, functional error, state selection, representation, periodic finite size, self-consistency, geometry, postprocessing, and experimental mapping. Calling an auxiliary eigenvalue an excitation is theorem misuse; wrong adsorption from missing dispersion is functional error; converging to low spin after a high-spin start was never tried is state selection; cutoff-sensitive stress is representation; charged-defect drift is finite size. Changing the mixing parameter cannot cure the other layers.

The hydrogen atom is a revealing exact-condition benchmark. Its Hartree self-interaction should be canceled by exact exchange–correlation, leaving the exact one-electron energy and density. Semilocal functionals do not cancel perfectly and often give a too-shallow asymptotic potential. This single-electron case isolates self-interaction without many-electron correlation, while hydrogen molecule dissociation exposes static correlation and fractional-spin error.

Separated fragments test size consistency and charge localization. At infinite separation, total energy should equal fragment energies under compatible spin and charge states. Approximate convex energy versus electron number can spuriously transfer fractional charge between fragments until chemical potentials equalize. A correct total integer electron count does not prevent this internal error. Range-separated or constrained approaches can diagnose and reduce it.

The uniform coordinate-scaling relations constrain exchange and correlation. Scaling density as $n_\gamma(\mathbf r)=\gamma^3n(\gamma\mathbf r)$ gives exact behavior for kinetic, Hartree, exchange, and limiting correlation contributions. Functionals designed around exact constraints use such relations to improve transferability. Passing constraints does not guarantee accuracy, but violating them predicts failures in density or coupling-strength limits.

The virial theorem supplies another global consistency relation for bound Coulomb systems. Exact stationary densities link kinetic, interaction, and external-potential terms under coordinate scaling. Approximate functionals have corresponding virial relations when solved self-consistently. Evaluating these can distinguish incomplete SCF or basis errors from functional behavior. Pseudopotentials and periodic systems modify the direct form and require the implemented Hamiltonian's consistent relation.

Molecules require diffuse space, correct spin, and counterpoise awareness. Anions and Rydberg states need diffuse basis support and an exchange–correlation potential with suitable asymptotic behavior. Open-shell atoms require correct multiplet interpretation beyond one determinant. Atomization energies need spin-polarized atomic references and zero-point corrections for experiment. A molecular box must suppress image interaction without making numerical grids unnecessarily expensive.

Metals require Fermi-surface and magnetic care. Dense $k$ sampling, controlled smearing, and accurate relative phase energies are essential. Small density-of-states changes can trigger Stoner magnetism; volume and functional shift the balance. Surface and defect supercells can create quantum-size oscillations. Metallic screening also makes hybrid exchange convergence expensive and changes the physical justification for long-range exact exchange.

Two-dimensional materials require truncated electrostatics or large-vacuum analysis. Periodic image screening contaminates charged excitations, dielectric constants, polar phonons, defects, and dipoles. Report polarizability per area rather than a vacuum-dependent three-dimensional dielectric constant. $k$ sampling, spin–orbit coupling, substrate screening, and van der Waals stacking affect gaps and topology. A monolayer in vacuum is not automatically the experimental supported layer.

Amorphous materials require ensembles rather than one convenient cell. Generate structures through melt–quench, deposition-like, or data-driven protocols; validate density, coordination, rings, pair distributions, and electronic tails. Finite cells discretize disorder and may miss rare defects. Average observables across independent structures and separate quench-rate bias from functional error. Relaxing one random network to a local minimum is not a thermodynamic amorphous prediction.

Alloys and configurational disorder require sampling or effective Hamiltonians. Ordered small cells can exaggerate periodic correlations; special quasirandom structures match selected correlation functions; cluster expansions map DFT energies to configurational thermodynamics; coherent potential approaches average scattering differently. Converge cell and configuration ensemble. Chemical short-range order, strain, magnetism, and charge transfer can couple, so independent random substitutions may miss the relevant state.

Free energies add vibrational, electronic, configurational, rotational, translational, and environmental terms according to the system. Harmonic phonons work near stable minima; quasiharmonic volumes approximate thermal expansion; thermodynamic integration treats anharmonicity; gas molecules need standard-state translation and rotation; surfaces use chemical potentials per area. Mixing standard states or omitting symmetry numbers produces errors larger than many electronic energy differences.

Computational scaling shapes feasible accuracy. Semilocal Kohn–Sham diagonalization often scales roughly cubically with electron count in conventional implementations, while exact exchange and response add larger prefactors and communication. Linear-scaling methods exploit density-matrix locality in gapped systems. GPU acceleration changes kernels but not convergence obligations. Report time to solution including SCF steps, exact exchange, $k$ points, forces, and postprocessing at equal accuracy.

Database-scale DFT demands standardized workflows without hiding exceptions. Automated symmetry, magnetic seeds, convergence, error recovery, and provenance enable materials screening, but transition metals, f electrons, molecules, charged systems, and metastable phases need specialized handling. A uniform parameter set produces uniform data, not uniformly accurate data. Quality flags should record state ambiguity, corrections, and failed convergence alongside successful values.

The exact Hohenberg–Kohn statements, approximate Kohn–Sham machinery, and property-specific extensions should remain conceptually separate. The theorem licenses density as a ground-state variable. Kohn–Sham orbitals make the kinetic part tractable. The exchange–correlation choice controls model error. $GW$, BSE, TDDFT, phonons, thermodynamics, and transport map the ground state toward other observables under added approximations. Collapsing these layers into “DFT predicts” prevents honest diagnosis.

Read density-functional theory through a density-variational-functional-and-evidence lens rather than a black-box-band-structure lens.