Three-dimensional dynamical matrices

Expand the potential energy to second order in displacements about an equilibrium crystal. Define $\Phi_{s\alpha,t\beta}(\mathbf R)$ as the mixed second derivative with respect to displacement of atom s in the reference cell and atom t in the cell at R. Its units are force per length. The equation of motion has minus the force-constant matrix acting on displacements; a stable quadratic energy therefore gives nonnegative frequency squares. A first derivative would describe an unbalanced force and should vanish at the reference equilibrium.

Choose the mode convention $u_{\ell s\alpha}=e_{s\alpha}e^{i(\mathbf q\cdot\mathbf R_\ell-\omega t)}/\sqrt{M_s}$. Basis offsets are not included in this Fourier phase; putting them there instead produces a phase-related matrix representation with the same frequencies. Substitution gives the mass-weighted dynamical matrix below. The symmetry of mixed derivatives makes it Hermitian at real wavevector. Mass weighting turns the original generalized eigenproblem into an ordinary Hermitian one and makes orthogonality and normalization straightforward.

A uniform shift of every atom cannot change the energy of a free unpinned crystal. Consequently the force constants sum to zero over all atoms and cells for each reference coordinate and each translation direction. At zero wavevector this gives three null eigenvectors in a three-dimensional stable crystal. Their mass-weighted components are proportional to square root of the atomic mass times a common displacement direction. Dividing by square root of mass recovers equal physical displacements, as required for rigid translation.

A numerical force-constant set that violates the sum rule may create spurious acoustic gaps or small imaginary frequencies near the zone centre. Enforcing translational symmetry can repair numerical inconsistency, but should not hide a genuine unstable structure. A negative eigenvalue of D corresponds to an imaginary harmonic frequency and a downhill displacement direction of the reference energy. It is evidence that the chosen equilibrium is not a local harmonic minimum, unless the sign is a sufficiently small and well-understood numerical error.

For each wavevector, the eigenvectors describe which atoms move, in which directions, and with which relative phases. Their eigenvalues determine frequencies, but not every mode is purely longitudinal or transverse in an anisotropic crystal. Degenerate modes can be mixed by changing the orthonormal basis within their degenerate subspace without changing the physical spectrum. Comparing eigenvectors between neighboring wavevectors therefore requires care with phase and degeneracy conventions, particularly when following branches through crossings.

Force constants may come from a fitted interatomic model, a finite-displacement calculation, or electronic linear-response theory. Each route adds approximations beyond diagonalizing D. Long-range electrostatic forces in polar insulators produce a special direction-dependent long-wavelength contribution, important for optical splitting. A finite-range spring model that omits that term can remain internally consistent while missing real polar behavior. Always distinguish an exact eigenvalue calculation for an assumed matrix from the accuracy of the matrix as a description of the actual material.