Optical excitations in nanographenes from the Bethe-Salpeter equation and time-dependent density functional theory: absorption spectra and spatial descriptors

TL;DR

This work addresses the accurate prediction of optical excitations and their spatial extent in nanographenes, where bound excitons challenge standard TDDFT. The authors implement $GW$-$BSE$ in CP2K, validate it against Thiel's organic set, and apply it to nanographenes to obtain both absorption spectra and spatial descriptors of excitations. They introduce and compute spatial descriptors (d_exc, sigma_e, sigma_h, R_eh) from the BSE eigenvectors and compare to TDDFT across various functionals. The key finding is that $GW$-$BSE$ reproduces experimental spectra and exciton sizes (with an extrapolated size of about $7.6$ Å), while TDDFT cannot simultaneously capture the spectral and spatial properties, underscoring the necessity of many-body methods for these nanostructures.

Abstract

The GW plus Bethe-Salpeter equation (GW-BSE) formalism is a well-established approach for calculating excitation energies and optical spectra of molecules, nanostructures, and crystalline materials. We implement GW-BSE in the CP2K code and validate the implementation for a standard organic molecular test set, obtaining excellent agreement with reference data, with a mean absolute error in excitation energies below 3 meV. We then study optical spectra of nanographenes of increasing length, showing excellent agreement with experiment. We further compute the size of the excitation of the lowest optically active excitation which converges to about 7.6 $Å$ with increasing length. Comparison with time-dependent density functional theory using functionals of varying exact-exchange fraction shows that none reproduce both the size of the excitation and optical spectra of GW-BSE, underscoring the need for many-body methods for accurate description of electronic excitations in nanostructures.

Figures (13)

• Figure 1: Workflow of a GW-BSE calculation within this work, where we employ BSE@ev$GW_0$@PBE. A KS–DFT calculation provides the KS energies $\varepsilon_p^{\mathrm{KS}}$ and orbitals $\psi_p^{\mathrm{KS}}$ as input. In the ev$GW_0$ scheme, the quasiparticle energies are iterated (bold arrows) in a self-consistent loop, starting from $G=G_0$ and using $W_0$ from the DFT starting point. This yields the ev$GW_0$ quasiparticle energies $\varepsilon_p^{\mathrm{ev}GW_0}$ [Eq. \ref{['eq-evGW0_energies']}]. BSE matrices $\mathds{A},\mathds{B}$ [Eq. \ref{['eq-BSE_ingredients']}] are constructed from $\varepsilon_p^{\mathrm{ev}GW_0}$, $\psi_p^{\mathrm{KS}}$, and the statically screened Coulomb interaction $W_0(\omega=0)$. Diagonalizing Eq. \ref{['eq-BSE_ABBA']} gives the excitation energies $\Omega^{(n)}$, the absorption spectrum $\alpha_{\mu\mu'}(\omega)$, and the excitation wavefunction $\Psi^{(n)}_{\mathrm{exc}}(\mathbf{r}_e,\mathbf{r}_h)$.
• Figure 2: Absolute error $|\Omega^{(n)}_\mathrm{CP2K}-\Omega^{(n)}_\mathrm{aims}|$ of BSE excitation energies $\Omega^{(n)}$ computed from CP2K and FHI-aims by solving Eq. \ref{['eq-BSE_ABBA']} with BSE@ev$GW_0$@PBE. The mean absolute error \ref{['eq-MAE_CP2K_aims']} over the ten lowest excitation energies across all molecules is only 2.7 meV. For a benchmark on the impact of parameters of $GW$ calculations on BSE excitation energies, we refer to App. \ref{['app-thiels_set']}.
• Figure 3: Optical absorption spectrum from the imaginary part of the dynamical dipole polarizability $\alpha_{\mu\mu'}(\omega)$\ref{['eq-BSE_polarizatbility']} ($\eta=0.05$ eV) for finite nanographene flakes with increasing number of repeating units $L$ from BSE@ev$GW_0$@PBE. (a) Geometry of a nanographene with $L=13$ anthracene units in $x$-direction. (b-d) Components of the dynamical dipole polarizability and the sum, $\alpha_\mathrm{sum} = \alpha_{xx}+\alpha_{yy}+\alpha_{zz}$ as function of the light frequency. The out-of-plane component $\alpha_{zz}$ is not shown as its absolute value is below $\text{Im}\ \alpha_{zz} = 156~(e^2 a_B^2)/\mathrm{Ha}$. Computational details are described in App. \ref{['app-computational_details']}.
• Figure 4: Excitation frequencies $\Omega^{(p=1,2)}$ of the first two dominant peaks ($p=1,2$) in $\mathrm{Im\ }{\alpha_{xx}}^{(L)}(\omega)$ for $L\in[4,13]$ (see example spectra in Fig. \ref{['fig-nanographene_geo_and_spectra']}). We fit the the obtained peak frequencies by $\Omega^{(p)}(L) = a^{(p)} + b^{(p)}/L + c^{(p)}/L^2$ (dashed lines), from which we obtain $\underset{L \rightarrow \infty}{\lim}\Omega^{(p=1)} = 2.08$ eV and $\underset{L \rightarrow \infty}{\lim} \Omega^{(p=2)} = 2.19$ eV. The obtained parameters of the fits for $p=1,2$ are reported in Table \ref{['tab-fit_parameters_peak_frequencies']}. The vertical dashed line in green indicates the average length of the experimental nanographene $\mathcal{L}^\text{(exp.)}\approx20$ nm Denk2014. The fit yields $\Omega^{(p=1)}(\mathcal{L}^\text{(exp.)})=2.09$ eV and $\Omega^{(p=2)}(\mathcal{L}^\text{(exp.)})=2.26$ eV .
• Figure 5: Optical absorption spectrum (a) reproduced from experimental data Denk2014 via $\text{Im}\ \epsilon_x(\omega)$ and (b) calculated using $GW$-BSE (BSE@ev$GW_0$) for $L=13$ via $\text{Im}\ \alpha_{xx}(\omega)$, which are related by a frequency-independent factor $C$ [Eq. \ref{['eq-relation_prop_epsilon_alpha']}]. For the $GW$-BSE calculation, we employ a broadening of $\eta=0.3$ eV. The individual peak heights at the resonance energies $\Omega^{(n)}$ are computed from Eq. \ref{['eq-absorption_peaks_from_exp']}/\ref{['eq-absorption_peaks_from_theory_as_epsilon']} in App. \ref{['app-diel_func_from_pol']}. Green vertical dashed lines denote the extrapolated peak frequencies $\Omega^{(p=1)}(\mathcal{L}^\text{(exp.)})=2.09$ eV and $\Omega^{(p=2)}(\mathcal{L}^\text{(exp.)})=2.26$ eV from Fig. \ref{['fig-fit_convergence_peak_frequency']}.