Methods

Practical DFT and beyond — the machinery behind the physics. What I run, why, and where it lies to you. Depth is in each level/* tag; The Atlas has the full ladder.

Foundations

• Kohn–Sham DFT L0 🌳 — HK theorems, the non-interacting mapping, $F=T_s+E_H+E_{xc}$; $E_{xc}$ small but decisive
• Hartree–Fock theory L1 🌳 — single determinant, Fock operator, exact-exchange SIE cancellation, Koopmans, Roothaan–Hall

Functionals & corrections

• XC functionals & Jacob’s ladder L1 🌳 — LSDA/GGA/meta-GGA, exact constraints; the semilocal axis is orthogonal to the SIE axis
• DFT+U L1 🌳 — Dudarev $U_{\mathrm{eff}}$ vs. Liechtenstein $U,J$; occupation matrix & projector ambiguity; FLL vs. AMF double counting; linear-response / cRPA $U$
• Hybrid functionals L1 🌳 — adiabatic connection; PBE0 ($\frac{1}{4}$ EXX) vs. HSE range separation ($\omega$); gaps as partial SIE cancellation; Fock-exchange / $k$-mesh cost
• Density-driven error (DC-DFT) L2 🌱 — separating functional error from density error

Beyond-DFT & correlation

• Self-interaction & delocalization error L1 🌳 — one- vs many-electron SIE; convexity / flat-plane condition
• The beyond-DFT ladder L1 🌳 — which error each rung fixes; +U vs hybrids vs GW vs DMFT vs DC-DFT
• GW & quasiparticles L2 🌳 — $\Sigma =iGW$, $G_0{W}_0$ vs QSGW, quasiparticle gaps & the derivative discontinuity

Lattice dynamics workflows

• Hellmann–Feynman forces & the Pulay term L1 🌳 — forces as the PES gradient; electrostatic force theorem; the Pulay caveat for atom-centred bases
• Phonons by finite displacement L1 🌳 — phonopy + CRYSTAL23 supercell route, symmetry-reduced displacements, the supercell-vs-range tradeoff, ASR/FC_SYMMETRY; vs DFPT
• Imaginary modes: artifact vs. instability L2 🌳 — artifact vs. true instability: ASR/basis artifacts, plus the Born–Huang rotational & Huang vanishing-stress conditions for the 2D flexural (ZA) branch
• DFPT & linear response L2 🌳 — Sternheimer, 2n+1, $Z^{\ast }$ & ${\epsilon }_{\infty }$ for LO–TO; vs finite displacement

Machine-learned potentials

• Machine-learned interatomic potentials L2 🌳 — MACE/NequIP vs GAP/ACE, force/stress training, active learning, extrapolation failure
• Phonons & anharmonicity from MLIPs L2 🌿 — force constants + anharmonic sampling from MLIPs; the DC-DFT caveat