Real-Space Quantification of Exciton Localization in Acene Crystals Using Wannier Function Decomposition

TL;DR

This work introduces the Wannier function decomposition of excitons (WFDX), a real-space framework that maps exciton states from the Bloch electron–hole basis used in GW-BSE onto products of maximally localized Wannier functions. By defining real-space amplitudes $A_{a i riangle ar{ extbf{R}}}^{S extbf{Q}}$ and the electron–hole center separation $ riangle extbf{R}$, WFDX yields simultaneous orbital- and site-resolved measures of Frenkel and charge-transfer character, while enabling efficient interpolation and gauge-invariant evaluation of position-operator observables. Applied to acene crystals (2–5 rings), WFDX reveals clear trends in exciton localization with molecular length, spin multiplicity, and center-of-mass momentum, and uncovers nonsymmorphic symmetry effects in real space that are otherwise obscured in reciprocal-space analyses. The approach offers a practical, general tool for analyzing and computing excitonic properties in solids, complementing NTOs and exciton-correlation methods and enabling new insights into exciton dynamics and related optoelectronic phenomena.

Abstract

We introduce the Wannier function decomposition of excitons (WFDX) method to quantify exciton localization in solids within the ab initio Bethe-Salpeter equation framework. By decomposing each Bloch exciton wavefunction into products of single-particle electron and hole maximally localized Wannier functions, this real-space approach provides well-defined orbital- and spatial- resolved measures of both Frenkel and charge-transfer excitons at low computational cost. We apply WFDX to excitons in acene crystals, quantifying how the number of rings, the exciton spin state, and the center-of-mass momntum affect spatial localization. Additionally, we show how this real-space representation reflects structural nonsymmorphic symmetries that are hidden in standard reciprocal-space descriptions. We demonstrate how the WFDX framework can be used to efficiently interpolate exciton expansion coefficients in reciprocal-space and outline how it may facilitate evaluation of observables involving position operators, highlighting its potential as a general tool for both analyzing and computing excitonic properties in solids.

Figures (24)

• Figure 1: Reciprocal-space decomposition of the lowest-energy zero-COM-momentum singlet exciton in crystalline naphthalene. The size of the data point at each $\mathbf{k}$ is a reflection of the state's contribution: for a valence state $\left|v\mathbf{k} \right \rangle$, size $\propto \sum_{c}|A_{cv\mathbf{k}}^{S\mathbf{Q}}|^2$; for a conduction state $\left|c\mathbf{k} \right \rangle$, size $\propto \sum_{v}|A_{cv\mathbf{k}}^{S\mathbf{Q}}|^2$. The exciton expansion coefficients are computed on a uniform k-gird and interpolated onto the selected high-symmetry k-path via Eq. \ref{['eq:iFFT']}, see Sec.\ref{['sec:dis']} for details. The underlying electronic band structure shows $GW$ quasiparticle energies referenced to the valence band maximum (VBM = $0$ eV). For clarity, each pair of bands is displayed in a vertically zoomed window with its own energy scale; the y-axis is broken between windows.
• Figure 2: Workflow for WFDX in the $G_{0}W_{0}$–BSE approach. Left: wannierization of valence and conduction subspaces to obtain $U(\mathbf{k})$. Right: BSE solution for $A_{cv\mathbf{k}}^{S\mathbf{Q}}$. Software used in this work is indicated, but the workflow is independent of codes.
• Figure 3: Left: $GW$ quasiparticle band structure of crystalline naphthalene, referenced to the valence band maximum (VBM energy set to 0 eV). Curves are colored by MLWF sublattice character (dark: site A; light: site B). The near-midpoint colors across the bands indicate that each Bloch state is approximately an equal-weight superposition of MLWFs on the A and B sublattices. Right: Isosurfaces of the MLWFs for HOMO–2, HOMO–1, HOMO, LUMO, LUMO+1 and LUMO+2, columns from left to right show A site, top, side and B site views.
• Figure 4: Trends in lowest-lying singlet and triplet excitons in acene solids. Frenkel ($\mathcal{P}_\mathrm{F}$, green bars) versus charge-transfer ($\mathcal{P}_\mathrm{CT}$, orange bars) character (left black axis) and root-mean-square electron–hole separation (red points and right red axis) for the lowest-energy $\mathbf{Q}=\mathbf{0}$ (a) singlet and (b) triplet excitons, plotted against the number of rings in the acene molecule making up the crystal.
• Figure 5: (a) Exciton dispersion for singlet in crystal anthracene. Connectivity of the bands are chosen to maximize the overlap of eigenstates at adjacent Q-points. (b) The spatial character change for the lowest four exciton along the Q-path.