Downfolding approaches to electron-ion coupling: Constrained density-functional perturbation theory for molecules

Constrained electronic-structure theories enable the construction of effective low-energy models consisting of partially dressed particles. However, the interpretation and physical content of these theories is not straightforward. Here, we carefully explore the properties of downfolding theories for electron-ion problems, in particular constrained density-functional perturbation theory (cDFPT). We show that the dipole selection rules determine whether the partially dressed phonons satisfy Goldstone's theorem, and we prove that electronic screening always lowers the phonon frequencies. We illustrate the theory with cDFPT calculations for minimal example systems: the nitrogen and benzene molecule as well as graphene.

Figure 1

Relation between dressed and bare electron-phonon coupling.

Figure 2

Electronic and phononic spectra of an ${\mathrm{N}}_2$ molecule. Note that the 1s electrons are incorporated into the pseudopotential in our calculations, so they are not shown here.

Figure 3

Bare susceptibility ${\chi }^0$ of ${\mathrm{N}}_2$. The susceptibility is nonzero only for transitions across the Fermi level.

Figure 4

Displacement potential $d^2$ in ${\mathrm{N}}_2$. Every panel stands for the coupling to a single phonon mode; there is no mode-mode coupling in ${\mathrm{N}}_2$. The colored matrix shows which electronic states couple to the phonon. Degenerate electronic states have been summed over.

Figure 5

Selected phonons in benzene. The dressed (DFPT) dynamical matrix in the basis of bare (cDFPT) eigenmodes. The target subspace spans all bands up to ${\pi }_6^{*}$. The dynamical matrix is block diagonal and only a $6\times 6$ block is shown here (out-of-plane displacements, even under 180-degree rotation symmetry). The bare eigenmodes are shown at the top, with their corresponding energies at the left side of the matrix.

Figure 6

DFPT and cDFPT phonons in benzene. $x,y$ ($z$) labels in-plane (out-of-plane) modes. The cDFPT phonons exclude screening from electronic transitions between the six ${\mathrm{p}}_z$ states. The out-of-plane modes are not affected by this constraint due to the mirror-symmetry selection rule.

Figure 7

(Left) An in-plane phonon (DFPT eigenmode) with substantial coupling to the ${\mathrm{p}}_z$ electrons. (Middle) $d^2$ for this mode. (Right) Bare electronic susceptibility ${\chi }^0$. $d^2$ and ${\chi }^0$ are both visualized in terms of the eigenstates of the electronic ${\mathrm{p}}_z$ space. The contribution to the phonon self-energy is obtained as the pointwise product of these two squares.

Figure 8

DFPT and cDFPT phonons of graphene. L/T (Z) labels longitudinal/transverse in-plane (out-of-plane) modes from DFPT. The cDFPT phonons exclude electronic screening from within the two ${\mathrm{p}}_z$ bands. Please note that for the out-of-plane modes, we Fourier interpolated ${\widehat{\omega }}_{\mathbf{q}}$ instead of ${\widehat{\omega }}_{\mathbf{q}}^2$.