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Chapter III · page 8 of 12QHD, continued
QHD · the method

How we test it.

Five steps, three working equations, and the details for scientists.

For scientists: the method

In practice QHD is a soft-mode search. We start from a deliberately high-symmetry, often dynamically unstable parent; compute its Γ-point phonons; displace along the imaginary (soft) modes and along phase-mixed combinations of near-degenerate modes, Q(θ) = QT1 cos θ + QT2 sin θ; and relax with all symmetry switched off. A result is accepted only when it is force-converged, has no imaginary phonons, and is placed on the convex hull against references computed with identical settings. Every expected outcome is time-stamped before the run and scored afterwards, misses included.

The 3 T's set what we measure along a pathway: Torsion, the rotational and chiral content of a mode; Tensor, the full stress–strain response of the cell (variable-cell relaxation, anisotropic strain); Time, the order and relative phase in which coupled modes act. The deeper theory remains in the author's journals and is not claimed beyond what the calculations show.

Quantum ESPRESSO 7.6 (PBE/PBEsol, PAW) · DFPT phonons · ab initio MD · spglib · data on Zenodo

“Do not force the evidence to fit QHD. Force QHD to face the evidence.”
  1. 01DocumentRecord the idea and its origin, unedited.
  2. 02FormalizeTurn it into geometry and equations.
  3. 03PredictWrite the expected result down first.
  4. 04CalculateRun density functional theory (Quantum ESPRESSO).
  5. 05FalsifyScore hit or miss, and keep both.
E(Q) = E₀ + aQ² + bQ⁴Soft-mode energy along a distortion Q: a < 0 means the structure wants to bend.
det | D(q) − ω² I | = 0Phonon frequencies: every ω real means the structure is dynamically stable.
Q(θ) = QT1cos θ + QT2sin θPhase-mixing two near-equal modes to open paths neither reaches alone.