Finite-Field Approach to Solving the Bethe-Salpeter Equation

We present a method to compute optical spectra and exciton binding energies of molecules and solids based on the solution of the Bethe-Salpeter equation and the calculation of the screened Coulomb interaction in a finite field. The method does not require either the explicit evaluation of dielectric matrices or of virtual electronic states, and can be easily applied without resorting to the random phase approximation. In addition, it utilizes localized orbitals obtained from Bloch states using bisection techniques, thus greatly reducing the complexity of the calculation and enabling the efficient use of hybrid functionals to obtain single particle wave functions. We report exciton binding energies of several molecules and absorption spectra of condensed systems of unprecedented size, including water and ice samples with hundreds of atoms.

Figure 1

The lowest singlet excitation energies of the 28 molecules of the Thiel’s set computed by solving the Bethe Salpeter equation in finite field (FF-BSE) with (green) and without (blue) the random phase approximation (RPA), using the PBE and the PBE0 hybrid functional (red). Results are compared ($\mathrm{\Delta }E$) with the best theory estimates obtained using quantum chemistry methods [65, 66]. The horizontal lines denote the maximum, mean, and minimum of the distribution of results, compared with quantum chemistry methods. $\chi$ denotes the response function computed with and without the RPA. The numerical values are reported in the Supplemental Material [37].

Figure 2

Optical absorption spectra of ${\mathrm{C}}_{60}$ in the gas phase computed by solving the BSE with several thresholds $\xi$ for the screened exchange integrals. The resulting number of integrals is indicated. The inset shows the same spectra plotted as a function of $\omega -E_{\mathrm{g}}$, and compared with experiment [75]. $E_{\mathrm{g}}$ is the electronic gap. Note that an accurate spectrum is obtained when using 4284 integrals instead of the total number, which is more than three times larger (14 400).

Figure 3

Imaginary part of the macroscopic dielectric constant (${\epsilon }_M$) as a function of the photon frequency ($\omega$) for a proton-disorder hexagonal ice model (left panel) and liquid water (right panel) computed as an average over nine samples extracted from path-integral molecular dynamics (PIMD) trajectories [79] generated with the MB-Pol potential [89]. Experimental results (from Refs. [76, 77] for water and ice, respectively) are shown by the blue solid lines. The black and red arrows indicate the positions of the first excitonic peak and the onset of the spectra, respectively.