DFPT & linear response

Density-functional perturbation theory computes the response of the Kohn–Sham system to a static perturbation analytically, without finite differences and — crucially — without supercells. For lattice dynamics it delivers the dynamical matrix at arbitrary $\mathbf{q}$ from a single primitive-cell calculation, the complementary route to the finite-displacement supercell method.

The Sternheimer equation

Expand everything in a perturbation parameter $\lambda$ (an atomic displacement, an electric field, a strain). The first-order density requires the first-order orbitals $|{\psi }_n^{(1)}⟩$, which satisfy the Sternheimer (linear-response) equation

$$(H^{(0)}-{\epsilon }_n^{(0)})|{\psi }_n^{(1)}⟩=-{\hat{P}}_c{V}^{(1)}|{\psi }_n^{(0)}⟩,{\hat{P}}_c=1-\sum _{m\in \mathrm{occ}}|{\psi }_m^{(0)}⟩⟨{\psi }_m^{(0)}|,$$

where ${\hat{P}}_c$ projects onto the unoccupied manifold, so the response $|{\psi }_n^{(1)}⟩$ is itself a conduction-space object and the empty Kohn–Sham eigenstates never appear explicitly. This sidesteps the slowly convergent sum-over-empty-states of ordinary second-order perturbation theory: only occupied orbitals and a linear solve are needed. The perturbing potential is self-consistent,

$$V^{(1)}=V_{\mathrm{ext}}^{(1)}+\int \frac{n^{(1)}({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d{\mathbf{r}}^{\prime }+\frac{\delta V_{\mathrm{XC}}}{\delta n}{n}^{(1)},n^{(1)}(\mathbf{r})=2\mathrm{Re}\sum _n{\psi }_n^{(0)\ast }(\mathbf{r}){\psi }_n^{(1)}(\mathbf{r}),$$

so the Sternheimer equation and the density update are iterated to self-consistency exactly as the ground-state SCF is.

Monochromatic perturbations at finite $\mathbf{q}$

A phonon of wavevector $\mathbf{q}$ is a perturbation $\propto e^{i\mathbf{q}\cdot \mathbf{R}}$. The key structural fact is that, although such a perturbation breaks the translational periodicity of the crystal, its response couples a Bloch state $|{\psi }_{n\mathbf{k}}⟩$ only to $|{\psi }_{n,\mathbf{k}+\mathbf{q}}^{(1)}⟩$ — a monochromatic response. One therefore works at a single $\mathbf{q}$ in the primitive cell, with no commensurate supercell required, for any $\mathbf{q}$ including incommensurate. The second derivative of the energy gives the interatomic force constants in reciprocal space directly, hence $D_{\alpha \beta }(\mathbf{q})$ at that $\mathbf{q}$; a Fourier interpolation over a coarse $\mathbf{q}$-grid then yields dispersions. This is the decisive advantage over frozen phonons, where a zone-boundary mode needs a doubled cell and a small incommensurate $\mathbf{q}$ is inaccessible.

The 2n+1 theorem

The variational structure means orbitals known to order $n$ in $\lambda$ determine the energy to order $2n+1$. Concretely, the first-order wavefunctions suffice for the energy through third order, so anharmonic cubic force constants — the inputs to phonon–phonon scattering and lattice thermal conductivity — are accessible from first-order DFPT response alone, without ever computing $|{\psi }^{(2)}⟩$. The $2n+1$ theorem is what makes third-order DFPT tractable from first-order response alone. (Gonze & Vigneron, PRB 39, 13120 (1989); Baroni, de Gironcoli, Dal Corso & Giannozzi, Rev. Mod. Phys. 73, 515 (2001); Born charges and LO–TO: Gonze & Lee, PRB 55, 10355 (1997).)

Born charges and ${\epsilon }_{\infty }$ — the LO–TO inputs

Choosing the perturbation to be a homogeneous electric field $\mathcal{E}$ rather than a displacement gives, as DFPT byproducts in the same linear-response framework, exactly the two tensors the polar lattice-dynamics correction needs:

• the high-frequency (clamped-ion, electronic) dielectric tensor ${\epsilon }_{\alpha \beta }^{\infty }={\delta }_{\alpha \beta }+4\pi \frac{\partial P_{\alpha }}{\partial {\mathcal{E}}_{\beta }}$, the response of the polarization at fixed nuclei;
• the Born effective charge of atom $\kappa$, $$Z_{\kappa ,\alpha \beta }^{\ast }=\Omega \frac{\partial P_{\alpha }}{\partial u_{\kappa \beta }}=\frac{\partial F_{\kappa \beta }}{\partial {\mathcal{E}}_{\alpha }},$$ the mixed second derivative linking macroscopic polarization to sublattice displacement (equivalently the force induced by a field), and the dynamical charge that sources the long-range dipole field.

These feed the non-analytic term that splits LO from TO modes at $\Gamma$ and removes the spurious degeneracy there,

$$D_{\kappa \alpha ,{\kappa }^{\prime }\beta }^{\mathrm{NA}}(\mathbf{q}\to 0)=\frac{4\pi }{\Omega }\frac{(\mathbf{q}\cdot {\mathbf{Z}}_{\kappa }^{\ast }{)}_{\alpha }(\mathbf{q}\cdot {\mathbf{Z}}_{{\kappa }^{\prime }}^{\ast }{)}_{\beta }}{\mathbf{q}\cdot {\mathbf{\epsilon }}^{\infty }\cdot \mathbf{q}}\frac{1}{\sqrt{M_{\kappa }{M}_{{\kappa }^{\prime }}}},$$

which is precisely the correction the dynamical matrix & ASR note invokes for polar crystals. DFPT is the natural source of $Z^{\ast }$ and ${\epsilon }^{\infty }$ — though note the modern-theory-of-polarization (Berry-phase) route reaches $Z^{\ast }$ from the other side, as the polarization derivative.

Sum rules on $Z^{\ast }$

Two exact constraints must be re-imposed numerically, the charge analogues of the acoustic sum rule on the force constants:

• Charge neutrality (acoustic sum rule on $Z^{\ast }$): ${\sum }_{\kappa }{Z}_{\kappa ,\alpha \beta }^{\ast }=0$. A rigid translation of the whole crystal produces no macroscopic polarization, so the Born charges must sum to zero; discretization breaks this at the ~0.01–0.1$e$ level and it is enforced by symmetric redistribution.
• Symmetry: the site symmetry of $\kappa$ constrains the form of $Z_{\kappa }^{\ast }$ (it need not be diagonal or symmetric in general); the full-crystal $Z^{\ast }$ tensors obey the acoustic sum rule above and the lattice point group.

Skipping neutrality leaves a residual net charge that corrupts the $\mathbf{q}\to 0$ limit of the non-analytic term and shifts LO frequencies.

Prerequisites

Kohn–Sham DFT · Hellmann–Feynman forces · Rayleigh–Schrödinger perturbation theory · the dynamical matrix & ASR

Builds toward: finite-displacement phonons (sibling) · electron–phonon coupling

Key references

• DFPT foundations — S. Baroni, P. Giannozzi & A. Testa, PRL 58, 1861 (1987); review: S. Baroni, S. de Gironcoli, A. Dal Corso & P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001).
• Variational structure & the 2n+1 theorem — X. Gonze & J.-P. Vigneron, PRB 39, 13120 (1989); X. Gonze, PRB 55, 10337 (1997).
• Born charges, ε∞, LO–TO splitting — X. Gonze & C. Lee, PRB 55, 10355 (1997); R. M. Pick, M. H. Cohen & R. M. Martin, PRB 1, 910 (1970).
• Polarization (Berry-phase) route to Z* — R. D. King-Smith & D. Vanderbilt, PRB 47, 1651 (1993); R. Resta, Rev. Mod. Phys. 66, 899 (1994).