Harmonic Phonons

Harmonic phonons describe the local curvature of the Born—Oppenheimer potential-energy surface around one declared reference structure. They answer how small collective displacements vibrate in that model; they do not by themselves establish finite-temperature stability, a phase transition, thermal conductivity, electron—phonon coupling, or an experimentally observed spectrum.

Build the dispersion and watch the modes

Start from an accepted reference state whose forces, stress, electronic convergence, k sampling, and occupations are trustworthy for force response. Choose DFPT when the code provides the required perturbations, or finite displacement when a supercell force route is more appropriate; neither route is intrinsically the universal default. A Gamma calculation yields Gamma modes only. A dispersion requires a converged q mesh or supercell force-constant range, interpolation, and checks away from Gamma.

Open the dispersion and phonon DOS, select important or imaginary points, and animate their eigenvectors in a phonon viewer. Check which atoms move, the direction and phase of the displacement, whether an apparent acoustic problem is translational, and whether a soft mode suggests a plausible lower-symmetry distortion. Visual inspection can expose a wrong cell or mode interpretation, but it does not replace q/supercell, force, electronic, acoustic-sum, or non-analytic-correction tests. Converge the dispersion, DOS, or downstream quantity actually used. The Silicon guide is a real one-Gamma frequency ledger with no dispersion or mode animation. Major DFPT, finite-displacement, and viewer routes are collected under lattice dynamics.

One documented Phonopy 4.4.0 route is to set ANIME to the selected q point and run Phonopy to produce anime.ascii.

Open that file for manual mode playback:

v_sim anime.ascii

This opens anime.ascii in the v_sim executable for manual mode playback. In the v_sim GUI, open the Phonons tab, select the mode, and press play. For a q point away from $\Gamma$, expand the displayed nodes under Box and symmetry so that the longer-period modulation is visible. Record the q point, branch or frequency, expansion, and what moves. This visual inspection complements numerical diagnosis; it does not prove that an imaginary mode is a physical instability.

For a full Quantum ESPRESSO dispersion route, keep scf.in/out/err, ph.in/out/err, every required dynamical matrix, and the q2r.x and matdyn.x inputs and outputs as one parent chain; do not run q2r.x over a partial q set. A Phonopy route instead needs the generated displacement cells, a one-to-one force-output mapping with unchanged atom order, the resulting force constants, and the band or mesh output. The Software Bridge phonon route shows the corresponding QE, VASP, ABINIT, and CP2K objects. When a negative mode appears, preserve its raw frequency and eigenvector and follow the imaginary-frequency troubleshooting route before changing the structure or enforcing a sum rule.

From force constants to normal modes

For atoms $\kappa,\kappa'$, Cartesian directions $\alpha,\beta$, and lattice translations $\mathbf R$, the harmonic force constants are

$$\Phi_{\kappa\alpha,\kappa'\beta}(\mathbf R) = \frac{\partial^2 E} {\partial u_{\kappa\alpha}(\mathbf 0)\, \partial u_{\kappa'\beta}(\mathbf R)}.$$

$E$ is the total energy of the declared electronic state and $u$ is a displacement. Fourier transforming mass-weighted force constants gives the dynamical matrix $D(\mathbf q)$ at wavevector $\mathbf q$; its eigenvalues are $\omega^2(\mathbf q\nu)$ and its eigenvectors label branch $\nu$. A phonon dispersion is therefore not raw force output: it is an interpolation or DFPT result whose meaning includes structure, masses, cell, force-constant convention, reciprocal mesh, non-analytic terms, and branch labels.

There are $3N$ branches for a primitive cell with $N$ atoms. Three acoustic branches approach zero frequency at Γ in a translationally invariant three-dimensional crystal. Optical branches are not “more stable” merely because they are finite at Γ; stability is assessed over the relevant reciprocal space, not from one labelled branch or one Γ-point calculation.

DFPT and finite displacements are alternative constructions

DFPT differentiates the self-consistent electronic state with respect to an infinitesimal displacement and builds the response directly. A finite-displacement calculation estimates derivatives from a set of signed displaced supercells and forces. Either route needs a compatible reference state and observable-specific convergence. Finite displacements require a supercell large enough for the represented force constants, symmetry-aware displacement construction, force accuracy, and a displacement range tested for linearity. DFPT requires convergence of the reference electronic state, reciprocal sampling, response solver, and q mesh.

Do not combine a force-constant set from one Hamiltonian, magnetic state, charge, cell, or pseudopotential with a dynamical-matrix correction from another. A successful SCF or response calculation is evidence that a numerical procedure ended, not that every phonon frequency or downstream observable has converged.

Long-range electrostatics are part of a polar phonon model

In a polar insulator, the $\mathbf q\to0$ dynamical matrix contains a direction-dependent non-analytic long-range contribution. It uses Born effective charges and an electronic dielectric tensor under compatible conventions, and produces LO—TO splitting. A Γ-point analytic dynamical matrix without this contribution does not contain the corresponding LO—TO splitting. The correction must not be borrowed from a different structural or electronic state, nor should it be applied to a metal as if the same macroscopic-field model held.

The acoustic sum rule, schematically $\sum_{\kappa'\mathbf R}\Phi_{\kappa\alpha,\kappa'\beta}(\mathbf R)=0$, checks translational invariance of the force constants. Enforcing it can remove a small numerical drift, but it cannot repair inadequate supercells, an inconsistent reference, a broken symmetry, or a genuine unstable branch. Preserve whether and how it was imposed.

Imaginary frequencies need diagnosis, not a one-word verdict

Many plots display an imaginary harmonic mode as a negative real frequency. It means a negative curvature in the harmonic model at that sampled $\mathbf q$ and reference structure. A robust imaginary branch may identify an athermal local instability and motivate following the eigenvector to a lower-symmetry candidate. It is not automatically a synthesis prediction, a finite-temperature phase, or a reason to replace the structure without checking the path and the numerical model.

Conversely, a small isolated imaginary acoustic value near Γ can arise from incomplete acoustic-sum closure, interpolation, finite-size effects, or convergence error. Diagnose its q dependence, magnitude under systematic numerical refinements, eigenvector character, symmetry, reference forces/stress, and the convergence of the target downstream observable. Do not silently take absolute values of imaginary modes in a free-energy calculation and then call the parent structure dynamically stable.

What must converge and what the result can support

Converge the dispersion, phonon DOS, mode frequencies, or thermodynamic quantity actually needed. Relevant variables include the electronic representation, reciprocal mesh, q mesh or supercell range, force or response accuracy, interpolation, polar correction, smearing where physically applicable, structural state, and all constraints. No single cutoff, mesh, displacement, force threshold, or imaginary-frequency tolerance is a universal prescription.

An adequately converged harmonic spectrum supports a conditional local-curvature statement for the declared athermal model. It does not establish anharmonic renormalization, thermal expansion, finite-temperature stabilization, diffusion, thermal conductivity, superconductivity, linewidths, electron—phonon coupling, or experimental agreement. Retain the reference lineage, force-constant or DFPT inputs, displacements and forces where used, q and k meshes, non-analytic data, sum-rule treatment, eigenvectors, convergence traces, and every transformation used to make a dispersion or DOS.

Sources and methods

• Togo and Tanaka, first-principles phonon calculation review
• Baroni et al., DFPT review
• Quantum ESPRESSO ph.x input description
• Quantum ESPRESSO q2r.x input description
• Quantum ESPRESSO matdyn.x input description
• Quantum ESPRESSO dynmat.x input description
• Quantum ESPRESSO single-q phonon guide
• Phonopy command and force-constant workflow
• Phonopy setting tags and acoustic sum-rule note