Calculation and analysis of exciton couplings via a subsystem formulation of the $GW$-Bethe-Salpeter Equation

TL;DR

The paper addresses the challenge of characterizing charge-transfer (CT) excitations in large molecular assemblies by introducing a fragment-based, orthonormal fragment-localized orbital framework within the $GW$-$BSE$ linear-response formalism. It develops a top-down localization procedure that yields quasi-diabatic exciton states and explicitly incorporates CT couplings in the reduced basis, enabling straightforward analysis of local and CT contributions to exciton interactions. Benchmarking on ethylene and pyrene dimers, and a biologically relevant chlorophyll dimer, reveals that CT states substantially influence exciton energies and Davydov splittings at short interfragment separations, while the truncated quasi-diabatic basis can reproduce canonical MO results within controllable errors when CT space is sufficiently expanded. The framework provides a tractable route to interpret exciton behavior in complex systems and lays the groundwork for fragment-based reconstruction of full exciton-coupling matrices in large assemblies, with distance-dependent couplings showing expected $R^{-3}$ (LE-LE) and $R^{-1}$ (LE-CT) trends.

Abstract

We present a fragment-based framework for analyzing exciton couplings within the $GW$-Bethe-Salpeter Equation formalism using localized molecular orbitals, and assess how excitonic states in molecular dimers can be decomposed into local and charge-transfer (CT) sectors. Our localization procedure preserves orbital orthonormality via a block-diagonal unitary transformation, enabling a simple and interpretable analysis of excitonic interactions. Using ethylene and pyrene dimers as model systems, we identify key effects of excitonic basis truncation and coupling approximations on excitation energies. We then extend the method to chlorophyll dimers, where weak CT asymmetries emerge due to geometric distortions. This framework offers a tractable route to analyze excitonic behavior in complex systems and paves the way for future fragment-based reconstruction of full exciton coupling matrices in large molecular assemblies.

Figures (4)

• Figure 1: Upper panel: Excitation energies computed using $G_0W_0$ within the QD basis, obtained by diagonalising the local (shades of blue) and charge-transfer (CT) excitation blocks (shades of red) separately. Lower panel: Dimer bonding energies (ground state) plus excitation energies computed using $G_0W_0$-BSE. Discrete blue/red markers correspond to the coupled states in the QD basis, black squares represent singlet excited states calculated in the CMO basis and the continuous black line is the interpolation of the black squares. Symmetry labels on the left denote the supramolecular (dimer) symmetry of the CMO states, whereas labels on the right indicate the underlying monomer fragment symmetries.
• Figure 2: Upper panel: Excitation energies computed using $G_0W_0$ within the QD basis, obtained by diagonalising the local and charge-transfer (CT) excitation blocks separately. Lower panel: Lower panel: Dimer bonding energies (ground state) plus excitation energies computed using $G_0W_0$-BSE. Discrete blue/red/black (black indicating purely CT state, red indicating mixture and blue indicating purely a local excited state) markers correspond to the coupled states in the QD basis, black squares represent singlet excited states added to the bonding energies calculated in the CMO basis and the continuous black line is the interpolation of the black squares. Symmetry labels on the left denote the supramolecular (dimer) symmetry of the CMO states, whereas labels on the right indicate the underlying monomer fragment symmetries.
• Figure 3: Upper panel : Largest couplings between QD states. Lines with shades of red show couplings of the first LE state, lines with shades of blue show couplings of the second LE state. Lower panel : Couplings between the first two LE states of the pyrene dimer. The blue and red square scatter points show the coupling in the CMO basis, computed as half the splitting of $B_{1g}$ and $B_{2g}$ states. The black triangular points represent couplings calculated by including all CT states via Eq. \ref{['comp_coupling']}. The lines (without markers) indicate couplings obtained by limiting the number of CT states included in the QD basis.
• Figure 4: Comparison of canonical and localized molecular orbitals involved in excitation processes. Left: Canonical occupied $|\phi_i\rangle\;|\phi_j\rangle$ and virtual $|\phi_a\rangle\;|\phi_b\rangle$ orbitals of the full dimer system. Right: Corresponding localized orbitals on fragments A and B, showing occupied $|\phi_i^A\rangle$, $|\phi_j^B\rangle$ and virtual $|\phi_a^A\rangle$, $|\phi_b^B\rangle$ orbitals. The localized representation captures spatial separation, enabling fragment-based excitation analysis.