6.1 Finite-displacement phonons with Phonopy

These are unexecuted teaching templates, with no ABACUS results. Baseline: ABACUS 3.9.0. Verify executable and external-tool versions.

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6.1.1 Scientific question

Can a relaxed crystal resist a small atomic displacement? Phonons connect the curvature of the potential-energy surface to collective vibrations. A stable structure has restoring forces near its local minimum. An imaginary harmonic frequency can indicate a structural instability, but numerical force noise, inadequate supercell size or a poorly relaxed reference can produce a similar symptom. This lesson teaches how to distinguish these possibilities before interpreting a dispersion plot. It uses bulk silicon and the matched data family from the first LCAO case. The example supercell is a proposed starting point, not a converged physical result.

6.1.2 Model and prerequisites

Finish the ionic and cell relaxation lessons first. Freeze the resulting equilibrium cell and atomic positions as the undisplaced reference. Measure its residual forces with tighter SCF and basis settings than the intended displacement signal. Use the same pseudopotential, orbital family, functional, electron number and spin treatment for every displaced configuration. Save ABACUS and Phonopy versions. ABACUS documentation identifies the external interface as available from Phonopy 2.19.1; test the installed parser on a small output before organizing many directories. A force calculation uses ABACUS; symmetry reduction, fitting force constants and constructing the dynamical matrix belong to Phonopy.

6.1.3 Worked template and interpretation

The first command assumes a correctly named parent STRU in the current directory. It creates displaced supercells and their displacement inventory. Record the actual atom count and lattice matrix. Put each displaced STRU into its own directory and replace the listed INPUT keys once, preserving all converged electronic controls. Never append duplicate keys to an existing file. Adapt KPT to the reciprocal dimensions of the supercell; identical mesh integers in primitive and enlarged cells do not mean identical sampling density. Do not relax a displaced structure: relaxation removes the perturbation needed to measure the response. The wildcard force-collection command is useful only after confirming its expansion covers exactly the ordered displacement list.

# External Phonopy command, not an ABACUS INPUT keyword:
phonopy -d --dim="2 2 2" --abacus

# Replace keys in each separate displaced-structure INPUT:
calculation scf
suffix phonon_001
stru_file STRU-001
cal_force 1
symmetry 0
scf_thr 1e-10

# After all separately inspected force calculations:
phonopy -f disp-*/OUT.*/running_scf.log

6.1.4 Physics and units

Force constants have dimensions of energy per length squared. With forces in eV/Å and displacements in Å, the fitted constants are in eV/Å². Mass weighting is essential: the curvature alone is not a frequency. Document the external tool's conversion to its reported THz or inverse-centimeter frequency units. Frequencies are eigenvalues of a mass-weighted response problem, not differences between SCF total energies. Acoustic translation should approach zero frequency at Γ. A small offset is a numerical diagnostic; enforcing an acoustic sum rule can improve a consistent calculation but cannot establish convergence by itself.

\[ \Phi_{i\alpha,j\beta}=-\frac{\partial F_{i\alpha}}{\partial u_{j\beta}},\qquad D_{i\alpha,j\beta}(\mathbf q)=\frac{\Phi_{i\alpha,j\beta}(\mathbf q)}{\sqrt{m_i m_j}} \]

6.1.5 Outputs and analysis

Inspect every running_scf.log for successful electronic convergence and complete force tables. Compare species ordering with the displacement inventory, retain signs and Cartesian components, and reject an incomplete log. Save FORCE_SETS alongside the parent STRU and displacement YAML. Set a band path in a separate, explicitly reviewed Phonopy configuration; use labels appropriate to the chosen primitive convention. Plot frequency against cumulative reciprocal path length, marking Γ and segment boundaries. This tutorial provides no calculated dispersion. If a mode is imaginary, inspect its displacement eigenvector and the reference geometry before declaring a phase transition.

6.1.6 Convergence and acceptance

Repeat a representative displacement with a smaller and larger amplitude. Too small an amplitude amplifies force noise; too large an amplitude includes anharmonic response. Cross this amplitude test with stricter SCF and richer orbitals or a higher PW cutoff. Increase the supercell and compare the modes central to the scientific question at equivalent q points. Keep the physical k spacing controlled throughout. Record acoustic offsets before and after any sum-rule treatment. A meaningful acceptance criterion is a bounded change in selected frequencies and force constants, rather than the absence of a fatal parser error. For polar materials, separately verify nonanalytic electrostatic corrections and dielectric inputs; silicon avoids that additional branch.

6.1.7 Pitfalls and exercises

Do not mix primitive-cell force files with supercell displacement labels. Do not silently average different species orderings. One visually smooth dispersion can still reflect a force-unit mistake. A reconstruction instability in a slab and a bulk instability are different questions. Exercises: (1) derive the central-difference estimator from opposite displacements and explain why its leading truncation error differs from a one-sided estimator; use a harmonic spring plus a quartic correction as guidance. (2) Design a two-amplitude, two-SCF-tolerance test and identify the signature of force noise. (3) Explain why changing all isotope masses rescales frequencies without changing the Born–Oppenheimer force constants.